Université Paris VII - Denis Diderot Office National d’Études et de Recherches Aérospatiales Diplôme d’Habilitation à Diriger des Recherches Discipline : PHYSIQUE

présenté et soutenu publiquement par Thierry FUSCO

le [...]

Titre

Optique adaptative et traitement d’images pour l’astronomie : de nouveaux enjeux et de nouvelles solutions JURY M. Jean-Gabriel Cuby MME. Anne-Marie Lagrange M. Gérard Rousset M. Jean-Luc Beuzit MME. Sylvie Roques M. Farokh Vakili

(Rapporteur) (Rapporteur) (Rapporteur) (Examinateur) (Examinatrice) (Examinateur)

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Table des matières I Synthèse des travaux de recherches et perspectives

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1 Introduction

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2 De l’OA classique à l’OA extrême

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2.1 2.2

Définition de système d’OA . . . . . . . . . . . . . . . . . . . . . Analyse de front d’onde optimisée pour la XAO . . . . . . . . . . .

15 16

2.2.1

Effets de repliement . . . . . . . . . . . . . . . . . . . . .

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2.2.2

Optimisation de la mesure . . . . . . . . . . . . . . . . . .

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Vibrations et filtrage de Kalman . . . . . . . . . . . . . . . . . . . Pré-compensation des aberrations non vues . . . . . . . . . . . . .

18 19

3 Optique Adaptative à grand champ 3.1 Principe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3 2.4

3.2

Reconstruction optimale pour l’optique adaptative à grand champ .

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3.3

Concept d’analyse pour l’optique adaptative à grand champ . . . . .

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3.4 3.5

Couverture de ciel en MCAO . . . . . . . . . . . . . . . . . . . . . De nouveaux besoins et de nouvelles solutions . . . . . . . . . . . .

25 25

3.5.1 3.5.2

OA tomographique pour étoiles laser (LTAO [Laser Tomographic AO]) . . . . . . . . . . . . . . . . . . . . . . . . . L’OA multi-objet (MOAO [Multi-Object AO]) . . . . . . .

3.5.3

L’OA corrigeant la couche au sol (GLAO [Ground Layer AO]) 27

4 Traitements a posteriori

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4.1

Reconstruction de FEP et déconvolution . . . . . . . . . . . . . . .

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4.2

Déconvolution myope - l’algorithme MISTRAL . . . . . . . . . . .

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4.3 4.4

“Déconvolution” anisoplanétique . . . . . . . . . . . . . . . . . . . traitement d’image et haute dynamique . . . . . . . . . . . . . . . .

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4.4.1

Description du problème direct . . . . . . . . . . . . . . . .

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4.4.2

Imagerie différentielle spectrale et angulaire . . . . . . . . .

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4.4.3

Imagerie coronographique . . . . . . . . . . . . . . . . . .

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TABLE DES MATIÈRES

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Perspectives

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Bibliographie

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II Éléments de Curriculum Vitae

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III Activités d’encadrement

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IV

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Liste des publications

A On- and off-axis statistical behavior of adaptive-optics-corrected shortexposure Strehl ratio 79 B NAOS on-line characterization of turbulence parameters and adaptive optics performance 93 C High order adaptive optics feasibility and requirements for direct detection of extra-solar planets 107 D Closed loop experimental validation of the spatially filtered Shack-Hartmann concept 129 E Improvement of Shack-Hartmann wave-front sensor measurement for extreme adaptive optics 133 F Comparison of centoid computation algorithms in a Shack-Hartmann Sensor 137 G First laboratory validation of vibration filtering with LQG control law for Adaptive Optics 171 H Calibration of NAOS and CONICA static aberrations

181

I

Calibration and precompasenation of non-common path aberrations for extreme adaptive optics 205

J

Optimal wave-front reconstruction strategies for multiconjugate adaptive optics 219

K Optimization of star-oriented and layer-oriented wavefront sensing concepts for ground layer adaptive optics 233

TABLE DES MATIÈRES

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L Experimental comparaison of optimized star oriented and numerical layer oriented using the MAD test bench 247 M Sky coverage estimation for multiconjugate adaptive optics systems : strategies and results 271 N MISTRAL : a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images299 O Post processing fo differential image for direct extrasolar planet detection from the ground 315

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TABLE DES MATIÈRES

Resumé Mon activité de recherche s’insère dans l’équipe «Haute Résolution Angulaire» du «Département d’Optique Théorique et Appliquée» de l’ONERA. Elle porte sur les techniques d’imagerie à haute résolution angulaire pour l’astronomie en se concentrant plus particulièrement sur l’Optique Adaptative (OA), l’analyse de front d’onde et le traitement des images obtenues au foyer d’un télescope après correction par OA. Les activités que j’ai menées englobent à la fois des aspects théoriques, des simulations numériques poussées, des validations expérimentales, pour certains cas des intégrations de systèmes sur le ciel et enfin une exploitation des données scientifiques en nouant des collaborations actives avec de nombreux astronomes. Pour ce faire, il est essentiel de maîtriser l’ensemble de la problématique physique (turbulence, OA, analyse de front d’onde, commande, formation d’image, problèmes inverses ...) et de mettre en œuvre des outils provenant de diverses disciplines (optique, automatique, traitement du signal ...). Je propose dans ce manuscrit une description non exhaustive de mes travaux de recherche en les recentrant sur les thématiques liées à l’astronomie (je ne parlerai pas de modélisation de profil de turbulence, d’observations endo-atmosphérique et de focalisation de faisceaux lasers, thèmes qui ont pourtant constitué une partie importante des travaux que j’ai menés à l’ONERA). Dans un premier temps, je détaillerai les activités liées à l’optimisation de systèmes d’OA, en mettant l’accent sur l’étude de nouveaux concepts (que ce soit l’OA à haute dynamique ou les OA à grand champ) et l’optimisation des composants clés de l’OA que sont l’analyseur de front d’onde et la commande. Puis je m’intéresserai à la problématique du traitement d’images corrigées par OA. Pour les deux grandes thématiques (OA et traitement d’images) les différents axes de recherche seront brièvement résumés et illustrés dans chaque cas par un article de revue (présenté en annexe).

TABLE DES MATIÈRES

7 Table des acronymes

BOA DOTA EAGLE ELT E-ELT EPICS ESO FEP GLAO GPI GRAAL HOT LQG LTAO LO MAD MASSIVE MISTRAL NAOS NCPA OA OAMC ONERA PDR PFI SAXO SH SO SPARTA SPHERE TMT VLT WCOG

Banc d’Optique Adaptative de l’ONERA Département d’Optique Théorique et Appliquée ELT Adaptive Optics for GaLaxies Evolution Extremely Large Telescope European ELT Exo-Planets Imaging Camera and Spectrograph European Southern Observatory Fonction d’Etalement de Point Ground Layer AO Gemini Planet Imager GRound-layer Ao Assisted by Lasers High Order Testbench Linear Quadratic Gaussian Laser Tomographic AO Layer Oriented Multiconjugate AO Demonstrator Multiconjugate AO SyStem for Infrared and VisiblE wavelenghts Myopic Iterative STep-preserving Restoration ALgorithm Nasmyth Adaptive Optics System Non-Common Path Aberrations Optique Adaptative Optique Adaptative MultiConjuguée Office National d’Etudes et de Recherches Aéorspatiales Preliminary Design Review Planet Formation Imager Sphere Ao for eXoplanet Observation Shack-Hartmann Star-Oriented Standard Platform for Adaptive optics Real Time Applications Spectro-Polarimatry High-contrast Exoplanet REsearch Thirty Meters Telescope Very Large Telescope Weighted Center Of Gravity

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TABLE DES MATIÈRES

Première partie Synthèse des travaux de recherches et perspectives

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Chapitre 1 Introduction

Depuis une centaine d’années, la taille des télescopes n’a cessé d’augmenter pour atteindre des diamètres de l’ordre de la dizaine de mètres, la nouvelle génération, actuellement à l’étude, devrait conduire à des diamètres de plus de 30 mètres. Cette croissance importante a deux principaux buts, augmenter le flux total collecté, réduisant par là même le bruit de photons qui représente la limite fondamentale de toute observation et améliorer la résolution angulaire sur l’objet observé. Si le premier objectif est atteint (le nombre de photons collectés augmente avec le carré du diamètre du télescope), il n’en est, hélas, pas de même pour la résolution angulaire [Roddier-81a]. En effet, la présence de l’atmosphère terrestre limite de manière importante cette résolution. Cette dernière ne dépasse jamais la résolution théorique d’un télescope de quelques dizaines de centimètres aux longueurs d’onde optique et ce, quelque soit le diamètre considéré [Fried-a-66]. En effet, les fronts d’onde, issus d’un objet, sont perturbés par les fluctuations d’indice de réfraction de l’air dans l’atmosphère. Ces perturbations entraînent un élargissement de la tache image au foyer du télescope ce qui introduit, in fine, une perte sensible de performance. La résolution obtenue, dépendant de la longueur d’onde d’observation, peut être plusieurs dizaines de fois inférieures à la résolution théorique attendue. En 1953, Babcock [Babcock-53] propose une technique, appelée Optique Adaptative (OA), pour compenser partiellement cet effet : un miroir est déformé par des moteurs pilotés en temps réel pour compenser les avances ou retards de phase introduits par la turbulence le long du trajet optique. Il faut néanmoins attendre plus de 35 ans pour que cette idée soit mise en pratique en astronomie sur le télescope de 3,6 m de l’European Southern Observatory (ESO) à la Silla (Chili) [Rousset-a-90]. Aujourd’hui, la quasi-totalité des grands télescopes sont équipés d’OA ( [Roussetp-02a, Wizinowitch-p-00, Eisenhauer-p-03]). Dans ces systèmes, qui fonctionnent en boucle fermée sur une étoile appelée étoile guide, l’onde réfléchie sur le miroir déformable est envoyée sur un analyseur de front d’onde. Les aberrations résiduelles mesurées par l’analyseur sont utilisées pour contrôler le miroir déformable placé en général dans un plan pupillaire. L’OA constitue, sans nul doute, la solution d’avenir pour l’observation astronomique depuis le sol. Tous les nouveaux projets de télescope incluent à présent 11

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CHAPITRE 1. INTRODUCTION

l’OA dès les toutes premières étapes de leur conception, certain comme l’“EuropeanExtremely Large Telescope” (E-ELT) de l’ESO, intègre l’OA dans le télescope luimême, en amont des instruments scientifiques. Les vingt dernières années ont fait passer l’OA du stade de démonstration à celui de technique à la fois éprouvée et opérationnelle [Lacombe-p-02], mais surtout foisonnant de nouvelles idées et de nouveaux concepts pour améliorer les performances, répondre aux besoins (de plus en plus exigeants) des astronomes et tenter de s’affranchir des restrictions liées au concept original. En effet, certaines limitations technologiques (temps de réponse des composants du système, bruit de lecture des détecteurs, ...) ou fondamentales (bruit de photons) dégradent la qualité de correction des systèmes d’OA lors d’observation sur l’axe (c’est-à-dire dans la direction d’analyse). Le front d’onde n’est alors que partiellement corrigé et des résidus de turbulence viennent perturber l’image finale. Une autre limitation des systèmes d’OA provient du domaine angulaire restreint (domaine isoplanétique [Fried-a-82]) où l’on peut considérer que les dégradations sur le front d’onde sont identiques et donc où la correction du système sera la plus efficace. En dehors de ce domaine la qualité de correction (et donc la performance de l’OA) va chuter en fonction de l’angle. Ce domaine isoplanétique, qui fixe le champ de vue utile d’une OA, n’est que de quelques secondes d’arc aux longueurs d’onde visibles et de quelques dizaines dans l’infra-rouge. Un système d’optique adaptative ne fournira donc que des images partiellement corrigées (comportant donc un flou résiduel) sur un champ de vue restreint. Les activités de recherche menées lors de ma thèse et poursuivies durant les sept dernières années ont pour objectif de tenter de s’affranchir de ces limitations et d’améliorer les performances finales des systèmes d’OA (existants ou futurs). Cette amélioration passe par deux grands axes de recherche : • Le premier axe consiste à agir sur le système lui-même en cherchant les meilleurs compromis lors de sa définition tout en imaginant de nouvelles solutions et de nouveaux concepts pour chacun de ses composants clés. Il faut pour cela : – mener des études amonts sur les concepts d’OA eux même (optique adaptative multiconjuguée par exemple) et sur les éléments clés de ces systèmes : l’analyse de surface d’onde et la commande (pour ce qui me concerne). Il faut pour cela proposer des solutions innovantes, optimiser des concepts existants, valider expérimentalement de nouvelles idées ; – avoir une vue globale du système et maîtriser l’ensemble de la problématique (modèle de turbulence, propagation, analyse de front d’onde, commande, miroirs déformables, formation d’images et problèmes inverses) afin de pouvoir analyser, simuler et quantifier les performances de l’OA, de faire les bons compromis lors de la définition et du dimensionnement de systèmes et identifier les points clés à étudier et approfondir ; – répondre aux besoins scientifiques exprimés par les astronomes. Il s’agit de proposer des solutions globales originales permettant d’y répondre et ceci en fonction du type d’objets observés, du degré de correction recherché, de la (ou des) longueur(s) d’onde d’intérêt, du champ de vue désiré, de la couverture de ciel souhaitée ...

13 • Le second axe de recherche consiste à agir sur les images fournies par l’instrument scientifique lui-même, après la correction toujours imparfaite, de l’OA. Il s’agit alors de développer des techniques de traitement a posteriori pour s’affranchir de la correction partielle (i.e. du flou résiduel dans l’image) et retrouver, au mieux, l’objet d’intérêt. En effet, malgré une atténuation qui peut être importante, toute l’information fréquentielle de l’objet jusqu’à la fréquence de coupure du télescope est présente dans l’image [Conan-t-94]. Ainsi des solutions basées sur la théorie du traitement du signal et des images permettent à partir d’une image partiellement corrigée, d’ameliorer sensiblement la qualité des images obtenues après OA. Il faut, pour ce faire, réussir à gérer deux problèmes majeurs : la propagation du bruit (de photons , de détecteur, de fond ...) dans le processus de traitement et la connaissance imparfaite de la réponse de l’instrument (variabilité spatiale et temporelle). L’intégration de ces problématiques dans le cadre rigoureux de la théorie de l’information et plus spécifiquement des problèmes inverses est essentiel. Elle se base sur une connaissance fine du problème direct (i.e. la formation d’image et les caractéristiques des objets observés) et implique : – une interaction forte avec les utilisateurs (astronomes) pour définir les contraintes (ou a priori) liées aux objets observés, – une connaissance approfondie de l’instrument (et des technologies critiques associées), Les quatre thèmes scientifiques principaux sous-tendant les recherches en optique adaptative menées ces dix dernières années sont : • Étude des objets du système solaire ; • Détection directe de planètes extra-solaires ; • Étude des étoiles et des populations stellaires (fonction de masse initiale des étoiles ...) ; • Exploration extra-galactique (étude des galaxies primordiales, cinématique des galaxies ...). Ces thèmes scientifiques se traduisent par 2 grandes classes de systèmes d’OA intégrant des solutions techniques très différentes, il s’agit de : • l’OA “classique”. Elle comporte un unique analyseur de front d’onde pour mesurer les défauts de phase dans une direction donnée et d’un système de correction (miroir déformable) conjugué optiquement de la pupille du télescope. Ce type de système est bien adapté pour l’observation d’objets peu étendus et relativement brillants. Il peut toutefois être complété par une étoile laser pour augmenter la sensibilité (une étoile naturelle est toujours nécessaire pour la mesure des tous premiers modes, en particulier le tip-tilt, de la turbulence). Pour les applications dites “à haute dynamique” (imagerie d’environnements stellaires très faibles, idéalement des planètes extra-solaires) ce concept est poussé dans ses retranchements à la fois sur le plan technologique (avec l’utilisation de miroirs à grand nombre de degrés de liberté, de détecteurs rapides et à faible bruit pour l’analyse de front d’onde et de calculateurs optimisés pour la commande) et sur le plan conceptuel (avec l’optimisation des processus de mesure de front d’onde, de commande et de calibration). On parle alors d’OA “extrême”. Il faut, de plus, intégrer dans le dimensionnement et la réalisation de

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CHAPITRE 1. INTRODUCTION l’OA les spécificités qu’imposent le couplage avec des coronographes et des systèmes d’imagerie (ou de spectroscopie) avancés. • l’OA “à grand champ”. Il s’agit là d’étendre le champ d’intérêt et/ou la couverture de ciel des OA classiques. Pour ce faire, l’idée est d’estimer le volume de turbulence (et non plus une simple phase sur la pupille du télescope) afin de pouvoir ensuite le corriger. Ainsi, l’un des points clés de tout système à grand champ (on verra au paragraphe 3 qu’il existe de nombreux concepts) reste la mesure (dite tomographique) de la turbulence. Cette problématique a été une des pierres angulaires des recherches que j’ai menées sur le sujet. A cela s’ajoute bien d’autres thématiques telle que la projection du volume de turbulence estimé sur le (ou les) miroir(s) déformable(s), la commande (et les critères de performances associés) de tels systèmes en boucle fermée, les procédures de calibrations associées, les spécificités liées à la mesure de front d’onde en utilisant des étoiles laser, pour n’en citer que quelques unes ...

Les différents aspects de mon travail de recherche sont brièvement résumés dans les chapitres suivants (en faisant référence, dans chaque cas, à des articles présentés en annexe). Le chapitre 2 traite la problématique de l’étude et de l’optimisation de l’OA extrême (du point de vue des systèmes globaux ou de sous-systèmes particuliers comme l’analyseur de front d’onde ou le calculateur temps réel). Dans le chapitre 3, je mettrai l’accent sur la problématique de l’optique adaptative à grand champ, ses défits, ses spécificités, les différents concepts qui ont vu le jour au cours des dernières années et que j’ai étudiés, améliorés et optimisés pour certains. Le chapitre 4 sera consacré aux méthodes de traitement a posteriori développées pour des systèmes existants ou envisagées pour les futurs systèmes d’OA (qu’ils soient classiques ou à grand champ). Pour finir quelques perspectives pour les années à venir seront proposées au chapitre 5.

Chapitre 2 De l’Optique Adaptative Classique à l’Optique Adaptative Extrême

Ce premier chapitre est consacré à l’optique adaptative dite “classique”, (c’est-àdire comportant un seul analyseur de front d’onde et un seul miroir déformable). Je m’intéresse dans un premier temps à la compréhension, la définition et l’intégration de systèmes complets d’optique adaptative. Puis, dans le but d’atteindre les performances ultimes du concept pour l’imagerie à haut contraste, j’ai été jusqu’à la proposition et l’étude approfondie de nouvelles approches en ce qui concerne l’analyse de surface d’onde et la commande. Pour ce faire, je me suis appuyé sur le développement d’outils de simulation les plus fins et les plus complets possible ainsi que sur l’utilisation d’un banc de laboratoire (BOA [Banc d’Optique Adaptative]) développé à l’ONERA.

2.1 Définition de systèmes d’optique adaptative et étude de performances Un partie importante de mes travaux consiste en l’étude, la définition, le dimensionnement et l’intégration de systèmes d’optique adaptative. En m’appuyant sur l’intégration et les tests (en laboratoire et sur le ciel) de la première optique adaptative du VLT [Very Large Telescope] (NAOS [Nasmyth Adaptive Optics System] [Lagrangep-02, Rousset-p-02a, Fusco-p-02d]) et en reprenant les travaux effectués par JeanMarc Conan durant sa thèse [Conan-t-94] j’ai mené une analyse quantitative et détaillée des performances d’un système d’optique adaptative. Ces études se basent dans un premier temps sur des développements analytiques complexes (voir Annexe A - T. Fusco et J.-M. Conan, JOSAA, 2004) permettant de décrire l’évolution du rapport de Strehl d’un système d’OA en fonction des paramètres turbulents (seeing, repartition de la turbulence en altitude, vitesse de vent) du degré de correction de l’OA et des paramètres observationnels (imagerie sur ou hors axe, longues ou courtes poses). 15

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CHAPITRE 2. DE L’OA CLASSIQUE À L’OA EXTRÊME

La confrontation de la théorie et des résultats obtenus sur le ciel est une étape essentielle pour affiner les modèles et prendre en compte certaines spécifitiés difficiles à appréhender analytiquement ou par simulations numériques (problématique de la calibration par exemple [Fusco-p-03]). Ceci passe par la quantification précise des performances de l’AO sur le ciel et donc par le développement d’algorithmes d’estimation de performances (voir Annexe B - T. Fusco et al., MNRAS, 2004). Un point particulièrement délicat est l’estimation précise d’un rapport de Strehl à partir d’images expérimentales obtenues en laboratoire, ou a fortiori sur le ciel. Dans le cadre des intégrations et tests de NAOS, j’ai proposé un algorithme de calcul de rapport de Strehl, basé sur une approche fréquentielle qui permet de prendre en compte les différents paramètres du système, en allant jusqu’à la réponse spatiale du détecteur et l’ensemble des imperfections qui peuvent être liées à une prise d’images expérimentales. Fort de l’expérience acquise sur NAOS, je suis, depuis maintenant 4 ans, en charge de la définition du système d’optique adaptative extrême (SAXO [Sphere AO for eXoplanet Observation]) du projet d’instrument SPHERE [Spectro-Polarimetric High-contrast Exoplanet REsearch] pour le VLT. Dans ce cadre, j’ai été amené à développer des modèles d’OA qui rendent compte de manière très précise des limitations physiques et techniques rencontrées [Fusco-p-04c, Fusco-p-07a]. Il s’agit avec ce type de systèmes d’atteindre un niveau de correction très élevé afin de pouvoir détecter une planète très peu lumineuse proche d’une étoile brillante (typiquement une différence de magnitude de 15 pour une séparation angulaire de l’ordre de quelques dixièmes de seconde d’arc) [Beuzit-p-07]. En plus de l’analyse détaillée du budget d’erreur pour un tel système (voir Annexe C - T. Fusco et al., Optics Express, 2006), je me suis intéressé à l’optimisation des composants clés de l’OA, condition sine qua none à l’obtention des performances ultimes désirées. Il s’agit de l’analyse de front d’onde, de la commande (et en particulier de la prise en compte et de la correction des vibrations du télescope) et de la calibration du système (pré-compensation des aberrations non vues par le système d’OA en particulier).

2.2 Analyse de front d’onde optimisée pour la XAO Les approches classiques d’analyse de front d’onde (telles que mises en œuvre sur NAOS par exemple) ne sont pas suffisament performantes pour répondre aux exigences de SPHERE (en terme de précision ultime et de sensibilité). J’ai donc été amené à étudier en détail plusieurs types d’analyseurs en mettant l’accent sur deux d’entre eux (les plus prometteurs vis-à-vis de l’application considérée) : le Shack Hartmann [SH] et l’analyseur Pyramide. Si la Pyramide permet en théorie d’atteindre de meilleurs performances que le SH, le manque de maturité du concept et les problèmes de stabilité et de mise en œuvre pratique rencontrés m’ont conduit à retenir le SH comme analyseur de front d’onde principal de SPHERE. Ce choix n’a été possible que grâce à une optimisation du concept initial du Shack-Hartmann afin de réduire les erreurs liées aux effets de repliement et d’améliorer la sensibilité de la mesure en modifiant l’algorithmie de calcul de centre de gravité. Les différentes améliorations proposées sont résumées dans les deux paragraphes suivants.

2.2. ANALYSE DE FRONT D’ONDE OPTIMISÉE POUR LA XAO

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2.2.1 Effets de repliement A fort flux, la principale source d’erreur sur la mesure de front d’onde est due aux effets de repliement. Pour un Shack-Hartmann et une turbulence suivant la statistique de Kolmogorov (au moins pour les hautes fréquences spatiales), on peut montrer que l’erreur liée au repliement est de l’ordre de 40 % de l’erreur de sous-modélisation elle même [Rigaut-p-01]. Cette erreur est distribuée plus ou moins uniformément sur l’ensemble des modes corrigés introduisant par là-même une des limitations principales en terme d’imagerie à haut contraste. L. Poyneer a proposé en 2004 [Poyner-a-04] une solution pour s’affranchir (au moins partiellement) de ces effets de repliement. Il s’agit de filtrer optiquement le front d’onde avant son passage dans l’analyseur. Les hautes fréquences spatiales sont bloquées avant la mesure et ne viennent plus la perturber. De nombreux points sont à étudier et à ajuster lors de de la définition d’un ShackHartmann filtré, on peut citer : • la forme et la taille du filtre optique ; • l’impact des conditions de turbulence et de la longueur d’onde d’analyse ; • l’impact de la bande spectrale de l’analyseur. L’ensemble des compromis et des optimisations dans le cadre de l’OA de SPHERE ont été obtenus en utilisant une modélisation complète du système, ils font l’objet d’un article en cours de rédaction. Parallèlement, une validation expérimentale du concept a été effectuée en utilisant le banc d’OA de l’ONERA (BOA) (voir Annexe D - T. Fusco et al., Optics Letters, 2005). Le gain amené par le filtrage spatial a clairement été démontré avec des résultats expérimentaux en bon accord avec la simulation numérique. De plus, l’expérience a mis en évidence un autre aspect intéressant du Shack-Hartmann filtré : la réduction sensible des effets du "‘mode gaufre"’. La présence du filtre spatial assure que ce mode (produit par le miroir déformable et très mal mesuré par l’analyseur) sera filtré optiquement et ne pourra donc pas se propager et être amplifié lors de l’observation. Ceci est illustré sur la figure 2.1 où deux images (une sans et une avec filtrage) sont obtenues expérimentalement sur BOA dans les mêmes conditions. Dans un cas, le mode gaufre est clairement visible et dans l’autre il est significativement atténué ce qui introduit un gain notable en terme de performance (les intensités des pics secondaires sont réduites d’un facteur 3 et on passe de 76,5 à 79 % pour le rapport de Strehl).

F IG . 2.1 – Réduction des effets du mode gaufre grâce au filtrage spatial.

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CHAPITRE 2. DE L’OA CLASSIQUE À L’OA EXTRÊME

2.2.2 Optimisation de la mesure Les programmes scientifiques liés à l’étude des planètes extra-solaires, nécessitent dans la majorité des cas, un nombre d’échantillons (i.e. d’étoiles observées et espérons le, de planètes détectées) suffisamment grand pour pouvoir effectuer des études statistiques. Ainsi la sensibilité de l’optique adaptative est un enjeu majeur. Cela conduit à des choix et à des compromis importants dans le système (sensibilité par rapport aux performances ultimes) et pose inévitablement la problématique de l’optimisation des performances de l’analyseur lui-même. Dans un analyseur de Shack-Hartmann, l’estimation optimale de la position de la tache focale en présence de bruit de détection est un problème qui a été abordé par de nombreux auteurs sans qu’une solution parfaitement satisfaisante ait pu être trouvée. La complexité du problème a deux origines. D’une part, la forme de la tache n’est pas connue a priori et évolue au cours du temps. D’autre part, le bruit de détection est la superposition d’un bruit de nature poissonnienne, lié à la statistique d’arrivée des photons et d’un bruit de nature gaussienne, intrinsèque au détecteur. Un nouvel algorithme a été proposé au cours de la thèse de M. Nicolle pour gérer au mieux le bruit dans les images de chaque sous-pupille d’un analyseur de Shack-Hartmann. Cet algorithme est particulièrement bien adapté au cas de l’OA extrême pour l’imagerie à haute dynamique. L’idée est d’affecter à chaque pixel de la tache image un coefficient de pondération représentatif de son rapport signal à bruit avant de calculer le barycentre de l’image. Cet algorithme, dit du centre de gravité pondéré (ou WCOG pour Weighted Center of Gravity) (voir Annexe E - M. Nicolle et al., Optics Letter, 2004), permet d’évaluer de façon optimale (au sens du maximum de vraisemblance) la position de la tache (supposée gaussienne) en s’appuyant sur les connaissances a priori disponibles (bruit de lecture, flux disponible, variance de la position de la tache). Des formules analytiques ont été proposées pour rendre compte du comportement de l’algorithme en présence de bruit de photons et de bruit de détecteur. Une étude approfondie des différentes sources d’erreurs intervenant dans la mesure de la position de la tache image dans une sous-pupille de Shack-Hartmann a été effectuée (dans le cadre des thèses de S. Thomas et M. Nicolle). Une comparaison des différentes approches (dont le WCOG et la corrélation) pour mesurer cette position a été menée (voir Annexe F - S. Thomas et al., MNRAS, 2006). Un gain sensible en faveur du WCOG, vis-à-vis d’algorithmes classiquement utilisés, a été démontré analytiquement, par simulation numérique et tout récemment expérimentalement sur le banc BOA (article en cours d’écriture).

2.3 Vibrations et filtrage de Kalman Une des limitations principales des systèmes d’OA réside dans les vibrations générées par le télescope et/ou le système lui-même (due à l’excitation des structures par le vent, les systèmes de refroidissement ...) [Rousset-p-02a]. Durant l’intégration et les tests de NAOS, j’ai montré que ces effets pouvaient entrainer une chute de plus de 15 % du rapport de Strehl (en bande K). Une telle perte est rédhibitoire

2.4. PRÉ-COMPENSATION DES ABERRATIONS NON VUES

19

pour l’optique adaptative extrême. Ainsi, dans le cadre du projet SPHERE, j’ai porté une attention particulière à la problématique des vibrations parasites dans un système d’OA. La stabilité extrème de l’axe optique du système est une condition sine qua none pour l’imagerie à haut contraste. Elle nécessite une correction active des vibrations par la boucle d’OA elle même et ceci à des fréquences bien au delà de la bande passante du système d’OA. Ainsi, une étroite collaboration avec Jean-Marc Conan et Cyril Petit a permis (à partir des travaux fondateurs qu’ils ont effectués dans le domaine [Petit-a-05]) la définition et la validation expérimentale d’une telle commande (voir Figure 2.2), basée sur une approche LQG (filtrage de Kalman). Cette commande a été mise en œuvre sur le banc d’optique adaptative de l’ONERA (voir annexe G C. Petit et al., Opt. Express, 2007) et sera implantée dans le calculateur temps réel du système d’OA de SPHERE. Il a été démontré qu’elle permettait de réduire l’impact des vibrations jusqu’à un niveau compatible avec les exigences d’un système d’imagerie à haute dynamique comme SPHERE.

F IG . 2.2 – Densité spectrale cumulée de la pente moyenne en x mesurée sur le banc d’OA de l’ONERA. [Tirets] Cas d’un intégrateur simple en présence d’une vibration importante à 15Hz, [Trait plein] cas d’un filtre de Kalman (la vibration est complètement éliminée), [Pointillés] Cas de référence obtenu en l’absence de vibrations.

2.4 Pré-compensation des aberrations non vues De part son principe, l’OA corrige les aberrations du front d’onde (qu’elles soient dues à la turbulence ou aux défauts internes du télescope et de l’instrument) sur l’ensemble du trajet de la lumière jusqu’à l’analyseur de front d’onde (i.e. là où elles sont mesurées). Une des limitations des systèmes d’OA réside dans les aberrations dites "‘non communes"’ (ou Non-Common Path Aberrations [NCPA] en anglais).

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CHAPITRE 2. DE L’OA CLASSIQUE À L’OA EXTRÊME

Ces aberrations statiques se situent après la lame séparatrice (en flux ou en longueur d’onde) entre la voie d’analyse du système d’OA et la voie scientifique. La correction de ces aberrations est essentielle dans le cadre de l’optique adaptative extrême. L’idée est de mesurer ces aberrations directement au niveau de la caméra d’imagerie en utilisant une technique d’analyse de front d’onde “plan focale” et de les pré-compenser en modifiant la référence (pentes de références pour un Shack-Hartmann) du système d’OA. Cette approche a été initialement proposée pour optimiser le système NAOSCONICA en utilisant la technique de la diversité de phase [Blanc-a-03b, Blanc-t-02] (voir Annexe H - A. Blanc et al. et M. Hartung et al., A& A, 2003). Les résultats obtenus ont démontré l’intérêt de la méthode (avec un gain notable en rapport de Strehl) mais ont aussi permis de mettre en évidence ses limitations : – méconnaissance de l’aberration (défocalisation) supposée connue entre les deux images de la diversité de phase, – méconnaissance du facteur d’échantillonnage de l’image, – influence d’un fond résiduel dans l’image, des défauts du détecteur, des différentes sources de bruits, – méconnaissance des paramètres de l’OA pour permettre la transformation des mesures de diversité de phase en pentes de réference afin de précompenser les aberrations mesurées. Dans le cadre de la thèse de J.-F. Sauvage une amélioration de la technique a été proposée en cherchant à la fois à optimiser l’algorithme de diversité de phase lui même (pour permettre une meilleure gestion du rapport signal à bruit dans les images) et la procédure de pré-compensation en utilisant une approche itérative (baptisée "‘Pseudo Closed Loop"’). Cette approche sera celle appliquée pour l’instrument SPHERE. Une validation expérimentale de ces différentes améliorations a été effectuée sur le banc BOA (voir Annexe I - J.-F. Sauvage et al., JOSAA, 2007). Il a ainsi été démontré que grâce à ces nouvelles optimisations le rapport de Strehl interne de BOA (mesuré sur une source en entrée du banc, à une longueur d’onde de 633 nm) passait de 78 % à plus de 96 % et que l’erreur résiduelle sur l’ensemble des 25 premiers polynômes de Zernike était inférieure 2 nm rms (soit moins de 0.4 nm rms par mode). Cette étude a démontré que le niveau de précision recherché pour l’instrument SPHERE pouvait être atteint en laboratoire, sur un vrai système d’OA. L’extrapolation des performances à un système de plus haut ordre combinée avec de nouveaux développements (changement de base pour la mesure d’un très grand nombre de modes, couplage de la diversité de phase et de la coronograhie, diversité de phase sur le ciel en présence de turbulence résiduelle) permet d’imaginer, à la fois pour SPHERE mais aussi pour les futurs systèmes d’imagerie à haute dynamique pour les ELTs, des performances bien au delà de celles attendues actuellement, et donc un gain potentiel important en terme de détectivité.

2.4. PRÉ-COMPENSATION DES ABERRATIONS NON VUES

21

Before Compensation

After Compensation

F IG . 2.3 – Aberrations internes du banc BOA (78 premiers polynômes de Zernike) avant et après pré-compensation. 28 polynômes ont été mesurés et compensés. A droite, l’image obtenue sur la caméra d’imagerie avant et après compensation.

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CHAPITRE 2. DE L’OA CLASSIQUE À L’OA EXTRÊME

Chapitre 3 Optique Adaptative à grand champ

3.1 Principe Une des limitations fondamentales de l’OA trouve son origine dans le fait que les aberrations sont créées par la traversée d’un volume de turbulence alors que leur correction est effectuée dans un seul plan, celui du miroir déformable (en général conjugué optiquement dans la pupille du télescope). Ainsi la correction est effective dans la direction de l’étoile guide mais se dégrade dans le champ de vue. Pour dépasser cette limitation, le concept d’OA multiconjuguée a été proposé dès 1975 par Dicke [Dickea-75] puis repris à la fin des années 80 par Beckers [Beckers-p-88] et au début des années 90 par de nombreux auteurs ( [Tallon-92b, Ellerbroek-a-94, Ragazzoni-a-00], pour n’en citer que quelques uns). Dans ce concept, la correction des aberrations est effectuée par plusieurs miroirs déformables conjugués le long du trajet optique dans des plans différents afin de modéliser les effets de volume. Au-delà du concept, on comprend aisément que de nombreux problèmes se posent quand on s’intéresse à de tels systèmes : combien de miroirs déformables sont nécessaires, dans quels plans doivent-ils être installés, comment commander ces miroirs, que doit-on mesurer pour les commander ? Bien que des réponses partielles aient été apportées à ces questions, aucune approche globale ne permettait de conclure quant à l’intérêt de l’OAMC. Ainsi, l’OAMC constitue un axe majeur de recherche en OA et une partie importante de mes propres travaux. J’ai développé au cours de ma thèse, et dans les années qui ont suivi, une approche cohérente pour tenter de répondre à l’ensemble de ces questions.

3.2 Reconstruction optimale pour l’optique adaptative à grand champ La reconstruction de front d’onde en OAMC constitue le coeur de la problématique. Il faut en effet être capable de corriger le volume turbulent (c’est-à-dire trouver les meilleures déformations à appliquer aux miroirs déformables) à partir de mesures 23

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CHAPITRE 3. OPTIQUE ADAPTATIVE À GRAND CHAMP

parcellaires effectuées sur des étoiles dans le champ. La mesure de front d’onde étant supposée effectuée par une technique largement éprouvée (un Shack-Hartmann par exemple), j’ai pu ainsi démontrer par modélisation numérique qu’une augmentation significative du champ de correction d’un télescope pouvait être obtenue avec un nombre restreint de miroirs déformables (typiquement 2 ou 3) et d’étoiles guides (typiquement 3 ou 4), montrant par là-même tout l’intérêt du concept d’OAMC [Fuscoa-99a]. Parallèlement, en m’appuyant sur une approche bayesienne et en exploitant les connaissances a priori sur les propriétés statistiques de la turbulence dans l’atmosphère, j’ai développé une méthode originale pour la reconstruction optimale de front d’onde appliquée à l’OAMC (voir annexe J -Fusco et al., JOSAA, 2001). Cette approche a été étendue au concept de GLAO (système de correction partielle de la couche au sol) dans le cadre de la thèse de M. Nicolle [Nicolle-t-06]. Par ailleurs, la reconstruction optimale de front d’onde a servi de base à un travail pionnier : le développement d’une commande optimale, en boucle fermée, adaptée à l’OAMC [Leroux-a-04,Petit-a-05,Petit-t-06], et un moyen de laboratoire est en cours de développement à l’Onera pour en étudier les performances.

3.3 Concept d’analyse pour l’optique adaptative à grand champ Dans le concept initialement proposé, le système d’OAMC mesure indépendamment le front d’onde dans chacune des directions matérialisées par les étoiles guides (qu’elles soient artificielles ou naturelles). A partir des mesures des perturbations du front d’onde intégrées le long du trajet, on tente d’estimer au mieux le volume de turbulence par un processus de reconstruction tomographique. Ce concept dit “Star Oriented” (SO) conduit à multiplier les senseurs (et donc les bruits qui leur sont associés) lorsque le nombre d’étoiles augmente. Un concept alternatif, dit “Layer Oriented” (LO), a été proposé par R. Ragazzoni [Ragazzoni-a-99b] dans lequel la mesure de front d’onde ne se fait plus par direction d’analyse dans le champ de vue mais par couche turbulente à corriger, chaque mesure utilisant l’ensemble des étoiles guides. Le nombre d’étoiles est généralement plus important que le nombre de miroirs déformables, le concept LO est donc naturellement moins sensible au bruit que le SO [Bello-a-03]. Toutefois, on peut montrer [Fusco-a-05c] que, dans le concept LO, une partie de l’information sur le volume turbulent est perdue lors de la coaddition du signal venant des différentes étoiles. Ainsi on obtient une meilleure propagation de bruit au prix d’une perte d’information sur le volume turbulent (et donc une perte de performances ultimes du système). Une étude comparative poussée et une optimisation des deux concepts d’analyse a été effectuée dans le cadre de la thèse de Magalie Nicolle (voir annexe K - M. Nicolle et al., JOSAA, 2006). Une validation expérimentale des algorithmes d’optimisation a été effectuée sur le démonstrateur d’optique adaptative multiconjuguée (MAD [Marchetti-p-03b]) de l’ESO. Les résultats obtenus font l’objet d’un article en cours de soumission (voir annexe L - T. Fusco et al., A&A).

3.4. COUVERTURE DE CIEL EN MCAO

25

3.4 Couverture de ciel en MCAO La notion de couverture de ciel est essentielle pour l’optique adaptative, elle quantifie la fraction de ciel observable (c’est-à-dire sur laquelle les performances seront supérieures à une certaine limite) par un système donné. En optique adaptative classique la notion de couverture de ciel est facilement quantifiable puisque directement et uniquement reliée à la magnitude de l’étoile guide du système. Elle est donc directement donnée par la statistique de répartition des étoiles (en fonction de leur magnitudes) dans le ciel. Les choses sont plus complexes dans le cas d’OA à grand champ. En effet, la performance du système va dépendre, dans ce cas, à la fois du nombre d’étoiles guides disponibles, de leur magnitude, mais aussi de leur répartition dans le champ à corriger ainsi que du concept d’analyse considéré (SO ou LO). Je me suis en particulier attaché à redéfinir la notion de champ en OA et à la séparer en quatre éléments distincts : • le champ technique, dans lequel on va rechercher les étoiles guides pour l’analyseur de front d’onde ; • le champ scientifique, dans lequel l’objectif de performance du système doit être atteint ; • le champ observable, qui est le champ dans lequel les étoiles guides permettent d’atteindre l’objectif de performance ; • le champ corrigé qui est l’intersection du champ scientifique et du champ observable et qui in fine, permet d’obtenir la couverture de ciel du système d’OA. En collaboration avec Amandine Blanc et Magalie Nicolle, une généralisation de la notion de couverture de ciel a été proposée (voir annexe M - Fusco et al., MNRAS, 2006). L’algorithme ainsi défini permet d’obtenir une estimée réaliste de la couverture de ciel d’un système d’OAMC sans la lourdeur d’une simulation statistique complète. Il a été appliqué au cas du démonstrateur d’OAMC de l’ESO (MAD).

3.5 De nouveaux besoins et de nouvelles solutions La diversité des thèmes astrophysiques et des besoins observationnels associés a conduit au fil des années à l’émergence de nouveaux concepts plus ou moins dérivés de l’idée d’OAMC originale. Comme on le verra dans les paragraphes suivants, le point commun de tous les concepts d’OA à grand champ reste la nécessité d’obtenir une information sur le volume de turbulence pour le reconstruire au mieux. Les approches présentées dans les chapitres précédents s’appliquent bien évidement à ces différents concepts avec, pour chacun, des adaptations propres et la prise en compte de spécificités propres (en termes de performances recherchées ou de contraintes techniques). Ainsi, en me basant sur un solide socle théorique, il m’a été possible d’étudier divers concepts d’OA à grand champ. Au fil des années et de l’évolution (en terme de maturité) des différents concepts, les travaux que j’ai menés sont ainsi passés d’analyses théoriques globales à des études préliminaires de systèmes, passage obligé avant la réalisation d’instruments pour les futurs télescopes géants.

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CHAPITRE 3. OPTIQUE ADAPTATIVE À GRAND CHAMP

3.5.1 OA tomographique pour étoiles laser (LTAO [Laser Tomographic AO]) Ce concept a été proposé afin de s’affranchir de l’effet de cône dû à la distance finie de focalisation d’une étoile laser et ainsi obtenir une bonne correction (typiquement la limite de diffraction dans le visible) couplée à une bonne couverture de ciel (uniquement déterminée par la nécessité d’une étoile de naturelle pour la mesure des bas ordres, tilt et défocus). On utilise ainsi plusieurs étoiles laser servant à mesurer le cylindre de turbulence correspondant à la direction de l’objet d’intérêt. Le champ de vue étant quant à lui restreint, un seul miroir conjugué dans la pupille du télescope est utilisé pour la correction. Une des spécificités de ce type de système est l’utilisation simultanée d’informations provenant d’étoiles naturelles et d’étoiles laser (mesures n’ayant pas les mêmes caracteristiques spatiales et temporelles). L’optimisation de la loi de commande, et plus généralement de l’ensemble du système dans ce cadre particulier, constituera un axe de mes recherches dans les mois et années à venir.

3.5.2 L’OA multi-objet (MOAO [Multi-Object AO]) L’idée est de généraliser le concept de LTAO (à savoir optimiser la correction dans une direction du champ à partir d’une analyse multiple) en augmentant le nombre de directions d’analyse et de zones du champ à corriger. Il s’agit donc d’optimiser la correction dans quelques dizaines de directions comprises dans un champ de vue global de l’ordre de la dizaine de minutes d’arc de diamètre [Gendron-a-05]. De nombreux défis sont à relever, en particulier le fait que la lumière arrivant sur les analyseurs de front d’onde n’est pas passé par le miroir déformable. Les deux systèmes fonctionnent donc en boucle ouverte ce qui a de nombreuses incidences sur le choix des composants et des concepts (de mesure et de correction). Les performances demandées pour ce type de système étant différentes de celles communément utilisées en imagerie, on ne parle plus de rapport de Strehl ou de variance résiduelle mais de concentration d’énergie dans des zones du champs (pour le couplage dans un spectrographe à intégrale de champ par exemple). Les tailles caractéristiques étant plus importantes que la limite de diffraction (de l’ordre de 5 à 10 fois), les compromis à effectuer seront différents de ceux envisagés pour des systèmes imageurs [Puech-a-07]. En particulier, la correction des premiers modes de la turbulence (et le tilt en premier lieu) ne sera pas aussi critique, voire dans certains cas pas nécessaire [Neichel-p-05, Neichel-p-06]. La simulation (basée sur une approche de type Fourier [Fusco-p-01b]), l’analyse fine, le dimensionnement et la démonstration (que ce soit en laboratoire ou sur le ciel) restent des thèmes de recherche ouverts en particulier dans le cadre des futurs télescopes géants (ELT). Dans le cadre du projet EAGLE (instrument spectro imageur multi-objet pour l’E-ELT), ces problématiques constituent un thème important de mes recherches.

3.5. DE NOUVEAUX BESOINS ET DE NOUVELLES SOLUTIONS

27

3.5.3 L’OA corrigeant la couche au sol (GLAO [Ground Layer AO]) Certaines applications astrophysiques nécessitent l’observation d’un très grand champ (de l’ordre de plusieurs minutes à plusieurs dizaines de minutes d’arc) sans pour autant rechercher de hautes performances en terme de correction. Les critères clés sont alors l’uniformité de la correction et la taille du champ. Pour répondre à ces objectifs un système de type GLAO va corriger uniquement la turbulence localisée près du sol. En effet, cette zone de la turbulence concentre la plus grande partie de la turbulence et le domaine isoplanétique le plus large ce qui garantira l’uniformité de la correction dans tout le champ [Rigaut-p-01, Hubin-a-05]. Si une approche simplifiée pour mesurer la couche au sol consiste à tirer parti de la décorrélation spatiale des fronts d’onde en altitude pour les différentes directions d’analyse en effectuant une simple moyenne des mesures, une approche plus optimale conduit inévitablement, à l’instar des autres concepts d’OA à grand champ, en une estimation tomographique du volume turbulent suivie d’une projection adéquate sur le miroir de correction. Comme démontré dans la thèse de Magalie Nicolle [Nicolle-t-06], cette approche optimale conduit à un gain significatif en performances. L’utilisation de lois de commandes basées sur ce principe combinée avec des techniques de traitement de données pertinentes sur les futurs systèmes en cours de réalisation (GRAAL ou GALACSI sur le VLT) ou en projets (pour l’E-ELT ou au dome C par exemple) devrait sans nulle doute rendre ce type de système encore plus attractif pour de nombreuses applications astrophysiques.

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CHAPITRE 3. OPTIQUE ADAPTATIVE À GRAND CHAMP

Chapitre 4 Traitements a posteriori

Comme décrit au chapitre 2, la correction du front d’onde effectuée par OA est l’objet de plusieurs limitations. Citons pour l’essentiel le bruit de détection dans l’analyseur, la bande passante de la boucle d’asservissement, le nombre limité de moteurs du miroir déformable et la décorrélation angulaire des fronts d’onde. Du fait de cette correction partielle, l’image finale s’éloigne (parfois sensiblement) de l’image "idéale" limitée par le seul effet de la diffraction de la pupille du télescope. La mise en œuvre de techniques de traitement a posteriori permet de s’affranchir de cette correction partielle et de retrouver tout le contenu fréquentiel présent dans l’image (jusqu’à la fréquence de coupure du télescope). Comme pour les systèmes d’OA eux-mêmes, les méthodes de traitement vont varier selon le type de systèmes d’OA envisagés, les objets d’intérêts, les paramètres astrophysiques pertinents et les conditions d’observation. Néanmoins, toutes les approches que j’ai considérées et développées l’ont été dans un même cadre général, celui de l’approche Bayesienne. La trame globale, quelque soit l’application, reste donc identique et peut se décomposer en 3 grandes étapes : • la description du problème direct. On s’intéresse là à la physique liée à l’observation. Plus la modélisation du problème direct sera fidèle, plus l’inversion (et donc la restitution des paramètres d’intérêt) sera efficace et précise. Ainsi on a intérêt à introduire dans cette description le maximum de connaissances sur : – la formation d’images elle-même. Il s’agit là de décrire le processus de acquisition (imagerie directe, coronographique, différentielle, spectrographe...) et d’intégrer les contraintes et les spécificités de l’instrument scientifique et du système d’OA. Un point clé de cette description de la formation d’image est la connaissance (ou l’estimation) de la fonction d’étalement de point (FEP) de l’instrument. – les différents bruits venant perturber les images (bruit de photons, de détecteurs, de fond, bruit spatial fixe pour les caméras infra-rouges etc ...) • La définition d’un critère à minimiser vis-à-vis des paramètres astrophysiques d’intérêt (que ce soit une image ou une description paramétrique de l’objet observé). Ce critère va, bien évidemment, s’appuyer sur le modèle direct défini précédemment. Le plus grand soin doit être apporté à la définition du 29

CHAPITRE 4. TRAITEMENTS A POSTERIORI

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critère, en effet, c’est sa pertinence qui va permettre une plus ou moins grande précision (bruit et biais) sur l’estimation des paramètres astrophysiques liés à l’objet observé. Le critère repose sur deux éléments : – un terme d’attache aux données qui va assurer la fidélité des paramètres estimés vis-à-vis des mesures et va dépendre de la nature des différents bruits venant perturber les images ; – un terme de régularisation [Titterington-a-85, Demoment-a-89] qui va éviter l’amplification du bruit lors du processus d’inversion [Tikhonov-l-77, Roota-87]. De nombreux types de régularisation (plus ou moins efficaces et pertinents) peuvent être considérés. Dans tous les cas, régulariser revient à introduire une connaissance a priori sur l’objet observé dans le critère. On maîtrise ainsi l’amplification du bruit (on réduit donc la variance de l’erreur) au prix du rajout d’un biais possible dans la solution. Toute la problématique consiste à ajuster (de manière supervisée ou non supervisé selon la complexité du critère) le poids de ce biais par rapport au terme d’attache aux données afin de minimiser l’erreur globale (c’est-à-dire maîtriser l’amplification du bruit sans pour autant trop biaiser le résultat final). • L’inversion de critère défini préalablement. Pour ce faire, on se base sur des techniques d’analyse numérique plus ou moins sophistiquées (allant d’une inversion simple de type filtre de Wiener par exemple, jusqu’aux techniques de recuit simulé [Geman-a-84] en passant par les algorithmes de descente de type gradients conjugués [Press-l-88]) selon la complexité du problème direct (critère convexe ou présentant des minima locaux ...) et les contraintes de l’utilisateur (simplicité d’utilisation, rapidité etc ...) Dans ce cadre général, je me suis intéressé à différentes problématiques reliées au traitement a posteriori d’images corrigées par AO. En me concentrant en particulier sur • la reconstruction de FEP et son utilisation dans un cadre de déconvolution dite "‘classique"’ ; • la généralisation au cas de la déconvolution myope, c’est-à-dire une déconvolution où la FEP est mal (ou pas) connue ; • La problématique de la "déconvolution1" anisoplanétique c’est-à-dire l’estimation et la prise en compte, dans le processus de traitement, d’une FEP variable dans le champ ; • La problématique spécifique de la haute dynamique, où il faut traiter des images coronographiques et/ou différentielles (c’est-à-dire obtenues à différentes longueurs d’onde, différentes positions angulaires dans le ciel, différentes polarisations ...).

4.1 Reconstruction de FEP et déconvolution Le cadre de la déconvolution classique suppose que la réponse de l’instrument (FEP) est parfaitement connue. Une telle hypothèse permet de ne chercher à estimer 1

le terme “déconvolution” est ici un abus de langage, la FEP variant dans le champ. Le terme designe ici une généralisation du concept initialement proposé pour des cas isoplanétiques

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lors du processus d’inversion que les paramètres liés à l’objet observé sans se soucier de l’instrument lui-même. Il y a deux façons d’obtenir une FEP : • la mesurer sur une étoile de référence (point source) avant ou après l’observation de l’objet d’intérêt. Cette méthode suppose que les conditions d’observation (caractéristiques de la turbulence et du flux sur l’analyseur de front d’onde) restent identiques entre l’acquisition de l’objet et de la FEP de référence. Elle a en outre l’inconvénient de réduire le temps de télescope alloué à l’observation de l’objet d’intérêt lui-même ; • la reconstruire numériquement à partir d’informations fournies par le système d’OA lors de l’observation (statistiques sur les résidus de l’analyseur de front d’onde et sur les commandes appliquées au(x) miroir(s) déformable(s)). Cette méthode proposée initialement par J.-P. Véran [Veran-a-97] permet d’avoir une estimée de la FEP dans les mêmes conditions d’observation que l’objet d’intérêt et de ne pas utiliser de temps de télescope pour l’observation d’une référence. Du fait de certaines approximations, cette méthode est actuellement limitée à des cas de bonne correction (bon rapport signal à bruit sur l’analyseur et système à grande bande passante temporelle). En collaboration avec Laurent Mugnier, Damien Gratadour et Jean-Marc Conan, j’ai participé aux études portant sur l’amélioration des approches permettant la reconstruction de FEP à partir des mesures de l’analyseur de front d’onde [Gratadour-t-05]. Cette implication dans la problématique de la reconstruction de FEP s’est poursuivie par une collaboration avec Eric Gendron et Yann Clenet portant sur le développement d’un algorithme de reconstruction applicable au système NAOS [Gendron-a06, Clenet-p-06, Clenet-p-07]. Néanmoins, en fonction des conditions d’observation (turbulence et flux disponible sur l’analyseur) ainsi que des caractéristiques du système d’OA, la connaissance de la FEP de l’instrument est souvent entachée d’incertitude et représente, in fine, la limitation principale d’une approche de déconvolution classique. Pour s’affranchir de cette limitation, il faut complexifier la procédure de déconvolution et laisser l’opportunité à l’algorithme de faire évoluer la FEP pour coller au mieux aux mesures. C’est le concept de déconvolution “aveugle” (si aucune connaissance sur la FEP n’est disponible) ou de déconvolution “myope” (qui intègre dans le critère à minimiser les connaissances, même imparfaites, disponibles sur la FEP) ;

4.2 Déconvolution myope - l’algorithme MISTRAL Pour palier à la connaissance imparfaite de la FEP, l’idée proposée par Laurent Mugnier, Jean Marc Conan et moi-même a été de modifier la déconvolution classique pour prendre en compte les variations possibles de la FEP. On ne recherche plus uniquement à estimer les paramètres de l’objet mais aussi ceux liés à la FEP. On se rapproche ici du concept de déconvolution "en aveugle" [Ayers-a-88, Holmes-a92,Lane-a-92,Schultz-a-93,Thiebaut-a-95,Sheppard-a-98], l’idée originale proposée dans nos travaux étant d’introduire une régularisation sur la FEP utilisant les informations disponibles sur cette dernière, obtenues à partir de mesures sur étoiles de

CHAPITRE 4. TRAITEMENTS A POSTERIORI

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référence [Conan-a-98] ou grâce aux données de l’analyseur de front d’onde [Fuscoa-99a]. L’ensemble des études menées sur le sujet ont été regroupées dans le cadre du développement de l’algorithme MISTRAL (Myopic Iterative STep-preserving Restoration ALgorithm). Ce dernier intègre à la fois un aspect myope mais aussi différentes régularisations sur l’objet scientifique (voir Annexe N - L. Mugnier, T. Fusco et J.-M Conan, JOSAA, 2004). Il contient en particulier une régularisation dite quadratique-linéaire [Brette-p-96, Mugnier-p-99], particulièrement bien adaptée aux objets à bords francs et aux objets ponctuels, qui permet de s’affranchir de certains artéfacts de déconvolution tels que les effets de Gibbs. Un exemple d’utilisation de MISTRAL est présenté en Figure 4.1. Il s’agit de Ganymède observée à l’OHP avec le banc d’OA de l’ONERA.

(a)

(b)

(c)

(d)

F IG . 4.1 – (a) Image de Ganymède avec le Banc d’optique Adaptative de l’ONERA sur le télescope de 1.52-m de l’Observatoire de Haute Provence (le 28/09/1997, 20 :18 UT.(b) Déconvolution avec MISTRAL. (c) Image synthétique (obtenue grace à des images de sondes spatiales de la NASA) de Ganymède vue depuis le sol au même moment (remerciement NASA/JPL/Caltech, voir http ://space.jpl.nasa.gov/).(d) Même image de synthèse convoluée par la FEP parfaite du télescope de 1.52 m, pour comparaison avec l’image déconvoluée. Durant le développement de MISTRAL, j’ai collaboré avec de nombreux astronomes pour confronter l’algorithme aux données réelles permettant ainsi de l’amélio-

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rer en prenant en compte les spécificités et contraintes liées aux observations réelles. Ces nombreuses collaborations ont permis de démontrer ses performances et de le rendre facile d’utilisation (et disponible sur demande pour n’importe quel utilisateur). Elles ont donné lieu à plus d’une vingtaine de publications dans des revues à comités de lecture portant sur de nombreux objets aux caractéristiques diverses (astéroides, satellites, planètes, environnement stellaires, étoiles multiples, galaxies à noyaux actifs ...) observés par différents systèmes d’OA (ADONIS, PUEO, NAOS, KECK). MISTRAL est à présent reconnu par l’ensemble de la communauté comme une référence en ce qui concerne la déconvolution d’images corrigées par OA.

4.3 “Déconvolution” anisoplanétique L’algorithme MISTRAL présenté plus haut se base sur une hypothèse essentielle : l’image observée est le résultat d’une convolution de l’objet d’intérêt par la FEP de l’instrument. Autrement dit, la FEP est invariante par translation dans le champ de vue. Ce dernier doit donc être plus petit que le domaine isoplanétique de l’optique adaptative, soit quelques secondes à quelques dizaines de secondes d’arc (selon les conditions atmosphériques et la longueur d’onde d’observation). Pour de plus grands champs, il faut prendre en compte la variabilité de la FEP pour effectuer un traitement efficace. J’ai développé au cours de ma thèse une expression analytique de cette évolution [Fusco-a-00a] et je l’ai intégrée dans un algorithme de déconvolution de champs stellaires. J’ai en outre proposé une solution pour gérer l’observation d’objets complexes à grand champ. Ceci permet d’obtenir des très bonnes précisions photométriques sur des champs stellaires en présence d’une FEP spatiallement variable. Ces études, menées dans le cadre de l’OA classique, doivent à présent être adaptées au cas de systèmes d’OA à grand champ afin de prendre en compte l’effet d’une analyse multi-directionnelle et d’une estimation tomographique du volume de turbulence.

4.4 Traitement d’images et haute dynamique Les succès de MISTRAL ont démontré l’importance du traitement de données après OA et la nécessité d’inclure la réflexion sur les algorithmes en parallèle de la définition du système lui-même (ajustement des compromis et des contraintes du système en fonction des besoins en terme de traitement). Ainsi, pour le projet SPHERE, la problématique du traitement de données coronographiques et différentielles (que ce soit du différentiel spectral ou angulaire) a été abordée dès le début de la phase de dimensionnement de l’instrument dans le cadre de la thèse de J.-F. Sauvage et en collaboration avec Laurent Mugnier.

4.4.1 Description du problème direct Un instrument d’imagerie à très haut contraste comme SPHERE est composé d’une OA extrême, d’un coronographe et d’un système d’imagerie multi-longueur

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d’onde. Le but ultime ici est donc d’obtenir un algorithme permettant de traiter des images d’OA longues poses coronographiques acquises simultanément à plusieurs longueurs d’onde (deux pour commencer). De plus, le champ tournant au cours de l’observation, il vient s’ajouter une information temporelle supplémentaire (que l’on peut traduire en mouvement de la planète dans le champ) à prendre en compte dans le processus global de traitement. Afin de parvenir à cette approche globale, nous avons dans un premier temps (et dans le cadre de la thèse de J.-F. Sauvage) considéré chaque problème individuellement : – l’imagerie differentielle spectrale (utilisation d’images acquises simultanément à différentes longueurs d’onde) ; – l’imagerie différentielle angulaire (prise en compte de l’évolution temporelle de la position de l’objet dans le champ) – l’imagerie coronographique (prise en compte de la réponse du coronographe et de la variabilité de la FEP dans le champ). La mise en commun des différents développements proposés dans la thèse de J.-F. Sauvage fera l’objet de la thèse d’A. Cornia qui vient de débuter.

4.4.2 Imagerie différentielle spectrale et angulaire L’imagerie différentielle est une approche qui permet de s’affranchir des défauts instrumentaux et atmosphériques résiduels dans l’image obtenue au foyer de la caméra scientifique, en tirant partie d’une évolution des paramètres de l’objet (position, structures spectrales) en fonction des conditions d’observation. 4.4.2.1 Imagerie différentielle spectrale L’idée est ici d’acquérir simultanément deux images du même objet à deux longueurs d’onde et de tirer partie de sa structure spectrale complexe [Marois-t-04], en supposant que le spectre de l’étoile suit lui l’évolution typique, et régulière, d’un corps noir. Si les longueurs d’onde sont suffisamment proches on peut, en première approximation, considérer que les speckles résiduels dans l’images sont identiques à un facteur d’échelle spatiale près, lié au rapport des longueurs d’onde. Ainsi après la mise à l’échelle d’une image par rapport à l’autre on peut les soustraire pour éliminer les speckles résiduels et ne laisser apparaître que le signal différentiel de la planète. Ce principe, en théorie très séduisant, rencontre toutefois certaines limitations : • l’approximation liée au premier ordre sur la similitude des figures de speckle à deux longueurs d’onde. Cette erreur augmente avec la différence entre les longueurs d’onde ; • l’acquisition simultanée des images. Il faut séparer en longueurs d’onde le faisceau au prix de l’introduction d’optiques et donc d’aberrations différentielles entre chaque voie. Ce second aspect est la limitation principale de la méthode. Nous avons proposé dans le cadre de la thèse de J.-F. Sauvage, une nouvelle approche pour traiter les données de l’imagerie différentielle spectrale. L’idée est d’effectuer un traitement conjoint des deux (ou plusieurs) images (au lieu d’une simple soustraction) en modélisant le problème direct pour faire intervenir la fonction de structure

4.4. TRAITEMENT D’IMAGE ET HAUTE DYNAMIQUE

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de la phase résiduelle après correction par OA ainsi que les aberrations statiques différentielles. Ainsi, à partir d’un couple (ou d’une série) d’images, de mesures préalables des aberrations différentielles pour chaque voie et des informations fournies par le système d’OA, il est possible d’estimer les paramètres d’intérêt de l’objet (voir Annexe O, Sauvage et al., SPIE Orlando 2006). Les avantages de cette approche sont multiples : • on ne passe plus par une soustraction dans le plan focal et l’on n’est plus limité par la proximité en longueur d’onde des deux images (on peut choisir les couples spectraux présentant le meilleur rapport signal à bruit) ; • on prend en compte les aberrations différentielles (que l’on aura préalablement mesurées en utilisant une méthode de type diversité de phase par exemple) dans l’algorithme de traitement ; • on peut introduire simplement un aspect "‘myope"’ dans l’algorithme pour tenir compte des incertitudes sur la mesure des aberrations différentielles et de la fonction de structure de phase résiduelle ; • enfin, on peut très simplement utiliser une paramétrisation de l’objet de type "point source" [Fusco-a-99b] qui nous permet d’obtenir une grande précision astrométrique et photométrique lors du traitement. Les premiers résultats obtenus sont très encourageants. Les études sur le sujet vont se poursuivre avec l’introduction de l’aspect "‘myope"’ dans l’algorithme (voir paragraphe 4.2), sa validation expérimentale (en utilisant par exemple des données fournies par le mode SDI de NAOS-CONICA) et son adaptation aux spécificités de SPHERE (imagerie coronographique dont nous reparlerons au paragraphe 4.4.3). 4.4.2.2 L’imagerie différentielle angulaire Le deuxième concept d’imagerie différentielle qui a été étudié est basé sur le rotation du champ de l’instrument [Labeyrie-a-82,Marois-a-06] (dans le cas de SPHERE, un système optique assure l’immobilité de la pupille laissant par là même tourner le champ). La pupille étant fixe, l’image de l’étoile (située sur l’axe optique), et en particulier les speckles fixes (liés aux aberrations résiduelles de l’instrument) restent identiques durant l’observation (on suppose ici que le temps de pose de chaque image individuelle est suffisamment grand pour moyenner les effets résiduels de la turbulence, i.e. non corrigés par le système d’OA). Un objet situé hors axe, va quant à lui évoluer au cours du temps. L’idée est donc d’utiliser cette rotation de champ pour distinguer un objet faible (idéalement une planète) des résidus liés à l’étoile elle-même. Si le concept n’est pas nouveau et a été proposé à l’origine par A. Labeyrie [Labeyriea-82] dans le cadre d’un projet coronographique pour le HST, le problème de l’identification de l’objet reste délicat. Il s’agit ici de recombiner au mieux les images obtenues à différentes positions et de proposer l’algorithme adéquat pour restituer le signal de l’objet avec le meilleur rapport signal à bruit possible. Un algorithme de détection multi-canaux a été proposé dans ce cadre [Mugnier-p-07a,Mugnier-p-07b]. Les premiers résultats extrêmement encourageant de cette approche nous ont conduit à la proposer comme base de départ pour le traitement de données de l’instrument SPHERE. Cette approche sera combinée avec les travaux menés sur le différentiel angulaire et complétée dans les mois qui viennent par l’introduction d’un modèle d’image coronographique (voir le paragraphe suivant).

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4.4.3 Imagerie coronographique En plus de l’aspect différentiel, l’autre spécificité majeure de l’imagerie à haut contraste est la présence d’un coronographe qui a pour but d’atténuer le flux de l’étoile avant la détection pour s’affranchir du bruit de photons qui lui est associé. Ce coronographe modifie de manière fondamentale le processus de formation d’image. Il faut donc le prendre en compte dans le modèle direct si l’on veut pouvoir effectuer un traitement a posteriori pertinent et appliquer l’ensemble des approches proposées dans les paragraphes précédents au cas de l’imagerie coronographique. Dans ce cadre, toujours en collaboration avec J.-F. Sauvage et L. Mugnier, nous avons développé un modèle analytique complet décrivant le processus coronographique en identifiant les termes liés aux résidus de correction par l’OA et les termes liés aux aberrations de la voie d’imagerie (qu’elles soient en amont ou en aval du coronographe). Cette description complète du modèle direct permet de définir simplement un critère pour l’inversion et d’utiliser les outils déjà disponibles dans MISTRAL pour effectuer cette dernière. Les premiers résultats d’inversion sont très encourageants et sont présentés (de même que l’expression analytique du problème direct) dans la thèse de J.-F. Sauvage (soutenue en décembre 2007). Comme dans le cas de l’imagerie différentielle spectrale, la modélisation du problème direct permet de faire apparaître fonction de structure de la phase résiduelle et aberrations non communes et donc, pour le futur, de rendre extrêmement simple la généralisation "‘myope"’ de l’approche considérée.

Chapitre 5 Perspectives

Les domaines de l’OA et du traitement d’image pour l’astronomie sont extrêmement actifs et regorgent de nouvelles problématiques à traiter (nouveaux besoins observationnels). Ainsi, les sujets abordés au cours des dix dernières années, loin d’être clos, foisonnent au contraire de pistes à explorer, de systèmes à dimensionner, de concepts à étudier, d’expérimentations à mener, d’algorithmes à proposer. Il est par nature impossible de dresser une liste exhaustive de ces futures activités mais on peut toutefois dégager quelques axes majeurs qui sont, et seront dans les prochaines années, les piliers des recherches que je souhaite mener. Le premier de ces axes est sans nul doute le développement du futur télescope géant Européen (ou E-ELT pour European Extremely Large Telescope). L’instrumentation associée à ce télescope présente nombre de défis à relever. L’OA est au cœur de cette entreprise, à la fois en ce qui concerne le télescope lui-même, qui sera adaptatif, et de l’instrumentation associée qui, pour la majeure partie des systèmes envisagés, intègre un système d’OA en complément de celui du télescope. Ces systèmes d’OA seront de plus en plus complexes et de plus en plus exigeants (en terme de qualité de correction et de champ) et demanderont des composants toujours plus performants alliés à des concepts toujours plus innovants. Dans ce cadre, des études de phase A, sont en passe d’être lancées (MASSIVE, EAGLE, EPICS) et serviront de cadre à mes recherches dans les prochaines années. A coté de l’aspect système global, les thématiques que j’aborderai au travers de ces projets sont les suivantes : – analyse de front d’onde pour l’étoile laser (gestion de l’étendue du spot, etc.) et pour la XAO (analyse et correction multi-étages, développement d’analyseurs plan focal précis et rapides) ; – la commande pour corriger de grands champs de vue (critère de performances sous contrainte d’uniformité) - commande à grand nombre de degrés de liberté, la commande de systèmes complexes avec en particulier la gestion des interactions entre l’OA du télescope et les OA intégrées dans les instruments. en mettant l’accent, comme pour ce qui a déjà été fait, sur la complémentarité des études théoriques, de la simulation et des validations expérimentales. Le deuxième grand axe est centré autour du projet SPHERE et basé sur l’intégration, la validation et la mise en œuvre sur le ciel de l’instrument. Les ajustements 37

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et les optimisations (des procédures de calibration par exemple) qui seront effectués lors de ces phases d’intégration et de tests seront les éléments essentiels à la réussite du projet et à l’obtention des performances attendues. Si cet aspect semble plus technique que le précédent, il reste le socle indispensable à la bonne compréhension de l’OA et à la maturation de nouvelles idées pour améliorer les performances et répondre aux difficultés pratiques rencontrées. Un des points essentiels que je souhaite aborder dans le cadre de SPHERE (et à plus longue échéance pour EPICS) est la mesure post-coronographique (obtenue durant l’acquisation de l’image scientifique) des aberrations statiques afin de s’approcher de la limite ultime de détection pour un système donné. Le dernier grand axe de recherche est centré autour des activités de traitement de données. Cet aspect, souvent négligé lors de la conception d’instruments, va devenir (il l’est déjà dans le cadre de SPHERE) incontournable étant donnée la complexification des systèmes d’OA et le niveau de performances requis pour répondre aux besoins observationnels. Un traitement de donnée optimisé (et intégré lors de la conception de l’instrument) sera un atout majeur pour la réussite des futurs projets tant sur le VLT que sur l’ELT. Dans ce cadre, les domaines de recherche qu’il me parait important d’approfondir sont : • la reconstruction de FEP pour les systèmes d’OA à grand champ et les méthodes de traitement a posteriori associées ; • le traitement d’images coronographiques et différentielles (dans le cadre du projet SPHERE notamment) ; • la problématique du traitement d’images 3D combinant des informations spatiales et spectrales fournis par les spectrographes à intégrale de champ. Enfin, en complément des ces activités principales, il convient de mentionner les études, présentant des liens forts avec mes grands thèmes de recherche et pouvant présenter des débouchés intéressants et permettre le developpement de nouvelles idées : • l’installation d’une optique adaptative à grand champ en Antarctique (Dôme C) ; • l’application de techniques utilisées pour l’imagerie à haute dynamique pour des missions spatiales (analyse de front d’onde et traitement de données). Enfin il est important de mentionner les thèmes de recherche non reliés à l’astronomie et notamment dans le cadre de l’OA pour l’ophtalmologie, des télécommunications optiques, de l’imagerie endo-atmosphérique et des activités laser (imagerie ou focalisation de faisceau laser). Ces activités regroupent des problématiques portant sur la mesure et la précompensation du volume de turbulence (pour corriger des effets d’anisoplanétisme mais aussi de scintillation), sur les problèmes d’analyse de front d’onde et de commande mais aussi sur le traitement de données qui sont en forte synergie avec les activités menées dans le contexte astronomique.

Chapitre 6 Bibliographie

[Ayers-a-88]

G. R. AYERS ET J. C. DAINTY, Iterative blind deconvolution and its applications, Opt. Lett., 13 :547–549, 1988.

[Babcock-53]

H. W. BABCOCK, The possibility of compensating astronomical seeing, Pub. Astron. Soc. Pacific, 65 :229, 1953.

[Beckers-p-88]

J. M. B ECKERS, Increasing the size of the isoplanatic patch with multiconjugate adaptive optics, in Very Large Telescopes and their Instrumentation, ESO Conference and Workshop Proceedings, Garching Germany, Mars 1988, ESO, pp. 693–703.

[Bello-a-03]

D. B ELLO , J.-M. C ONAN , G. ROUSSET, ET R. R AGAZZONI, Signal to noise ratio of layer oriented measurements for multiconjugate adaptive optics, Astron. Astrophys., 410 :1101–1106, Nov. 2003.

[Beuzit-p-07]

J.-L. B EUZIT, M. F ELDT , K. D OHLEN , D. M OUILLET , P. P U GET, J. A NTICI , P. BAUDOZ , A. B OCCALETTI , M. C AR BILLET, J. C HARTON , R. C LAUDI , T. F USCO , R. G RATTON , T. H ENNING , N. H UBIN , F. J OOS , M. K ASPER , M. L AN GLOIS , C. M OUTOU , J. P RAGT , P. R ABOU , M. S AISSE , H. M. S CHMID , M. T URATTO , S. U DRY , F. VAKILI , R. WATERS , ET F. W ILDI, SPHERE : A Planet Finder Instrument for the VLT, in Proceedings of the conference In the Spirit of Bernard Lyot : The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., P. Kalas, ed., Juin 2007.

[Blanc-a-03b]

A. B LANC , L. M. M UGNIER , ET J. I DIER, Marginal estimation of aberrations and image restoration by use of phase diversity, J. Opt. Soc. Am. A, 20(6) :1035–1045, 2003.

[Blanc-t-02]

A. B LANC, Identification de réponse impulsionnelle et restauration d’images : apports de la diversité de phase, Thèse de doctorat, Université Paris XI Orsay, Juil. 2002. 39

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S. B RETTE ET J. I DIER, Optimized single site update algorithms for image deblurring, in Proceedings of the International Conference on Image Processing, Lausanne, Switzerland, 1996, IEEE, pp. 65–68.

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P. RECONSTRUCTION FOR NAOS-CONICA, Clenet, y. and kasper, m. and gendron, e. and fusco, t. and rousset, g. and gratadour, d. and lidman, c. and marco, o. and ageorges, n. and egner, s., in Advances in Adaptive Optics II, L. Ellerbroek, B. et D. Bonaccini Calia, eds., vol. 6272, Soc. Photo-Opt. Instrum. Eng., 2006.

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Y. C LENET, C. L IDMAN , E. G ENDRON , T. F USCO , G. ROUS SET, M. K ASPER , N. AGEORGES , ET O. M ARCO , Psf reconstruction for naco : Current status and perspectives, in Adaptive Optics : Analysis and Methods, OSA, 2007. Date conférence : Juin 18–20, 2007, Vancouver (Canada).

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J.-M. C ONAN , L. M. M UGNIER , T. F USCO , V. M ICHAU , ET G. ROUSSET, Myopic deconvolution of adaptive optics images by use of object and point spread function power spectra, Appl. Opt., 37(21) :4614–4622, Juil. 1998.

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J.-M. C ONAN, Étude de la correction partielle en optique adaptative, Thèse de doctorat, Université Paris XI Orsay, Oct. 1994.

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G. D EMOMENT, Image reconstruction and restoration : Overview of common estimation structures and problems, IEEE Trans. Acoust. Speech Signal Process., 37(12) :2024–2036, Dec. 1989.

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F. E ISENHAUER , R. A BUTER , K. B ICKERT, F. B IANCATM ARCHET , H. B ONNET, J. B RYNNEL , R. D. C ONZELMANN , B. D ELABRE , R. D ONALDSON , J. FARINATO , E. F EDRIGO , R. G ENZEL , N. N. H UBIN , C. I SERLOHE , M. E. K AS PER , M. K ISSLER -PATIG , G. J. M ONNET, C. ROEHRLE , J. S CHREIBER , S. S TROEBELE , M. T ECZA , N. A. T HATTE , ET H. W EISZ, SINFONI - Integral field spectroscopy at 50 milli-arcsecond resolution with the ESO VLT, in Instrument Design and Performance for Optical/Infrared Ground-based Telescopes. Edited by Iye, Masanori ; Moorwood, Alan F. M. Proceedings of the SPIE, Volume 4841, pp. 1548-1561 (2003)., M. Iye et A. F. M. Moorwood, eds., vol. 4841 of Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, Mars 2003, pp. 1548–1561.

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B. L. E LLERBROEK, First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensa-

41

[Fried-a-66]

[Fried-a-82] [Fusco-a-00a]

[Fusco-a-05c]

[Fusco-a-99a]

[Fusco-a-99b]

[Fusco-p-01b]

[Fusco-p-02d]

[Fusco-p-03]

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tion in extended-field-of-view astronomical telescopes, J. Opt. Soc. Am. A, 11(2) :783–805, Fev. 1994. D. F RIED, Optical resolution through a randomly inhomogeneous medium for very long and very short exposures, J. Opt. Soc. Am., 56 :1372–1379, 1966. D. L. F RIED, Anisoplanatism in adaptive optics, J. Opt. Soc. Am., 72(1) :52–61, Jan. 1982. T. F USCO , J.-M. C ONAN , L. M UGNIER , V. M ICHAU , ET G. ROUSSET , Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution., Astron. Astrophys. Suppl. Ser., 142 :149–156, 2000. T. F USCO , M. N ICOLLE , G. ROUSSET, V. M ICHAU , A. B LANC , J.-L. B EUZIT, ET J.-M. C ONAN, Wavefront sensing issues in mcao, C. R. Physique 6, pp. 1049–1058, 2005. T. F USCO , J.-P. V ÉRAN , J.-M. C ONAN , ET L. M UGNIER, Myopic deconvolution method for adaptive optics images of stellar fields, Astron. Astrophys. Suppl. Ser., 134 :1–10, Jan. 1999. T. F USCO , J.-M. C ONAN , V. M ICHAU , L. M UGNIER , ET G. ROUSSET , Efficient phase estimation for large field of view adaptive optics, Opt. Lett., 24(21) :1472–1474, Nov. 1999. T. F USCO , J.-M. C ONAN , V. M ICHAU , ET G. ROUSSET , Noise propagation for multiconjugate adaptive optics systems, in Optics in atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, et T. J. Schmugge, eds., vol. 4538, Bellingham, Washington, 2002, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, pp. 144–155. Date conférence : Sept. 2001. T. F USCO , G. ROUSSET, A.-M. L AGRANGE , F. L A COMBE , E. G ENDRON , P. P UGET, N. H UBIN , R. A RSE NAULT, J. C HARTON , P. K ERN , P. G IGAN , P.-Y. M ADEC , D. M OUILLET, D. R ABAUD , P. R ABOU , E. S TADLER , ET G. Z INS, NAOS : the first AO system of the VLT, in Scientific Highlights 2002, F. Combes et D.Barret, eds., EDP Sciences, 2002, SF2A. T. F USCO , G. ROUSSET, ET A. B LANC, Calibration of AO system. Application to NAOS-CONICA, in Science with Adaptive Optics, W. Brandner et M. Kasper, eds., Springer-Verlag, 2004. Date conférence : Sept. 2003, Garching, Germany. T. F USCO , G. ROUSSET, R. C ONAN , N. N ICOLLE , C. P E TIT, ET J.-L. B EUZIT, End-to-end model for XAO simulations. Application to Planet-Finder, in Scientific Highlights 2004, F. Combes, D.Barret, T. Contini, F. Meynadier, et L. Pagani, eds., EDP Sciences, 2004, SF2A.

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CHAPITRE 6. BIBLIOGRAPHIE T. F USCO, Simulation and design of adaptive optics systems : Application ti sphere-saxo, in Adaptive Optics : Analysis and Methods, OSA, 2007. Date conférence : Juin 18–20, 2007, Vancouver (Canada). S. G EMAN ET D. G EMAN, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell., PAMI-6(6) :721–741, Nov. 1984. E. G ENDRON , Y. C LÉNET, T. F USCO ET G. ROUSSET, New algorithms for adaptive optics point-spread function reconstruction, Astron. Astrophys., 457, pp. 359–363 (2006).

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E. G ENDRON , F. A SSÉMAT, F. H AMMER , P. JAGOUREL , F. C HEMLA , P. L APORTE , M. P UECH , M. M ARTEAUD , F. Z AMKOTSIAN , A. L IOTARD , J.-M. C ONAN , T. F USCO , ET N. H UBIN , FALCON : Multi-object AO, C. R. Physique, 6 :1110–1117, 2005.

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D. G RATADOUR, Optique adaptative, traitement d’image et étude des noyaux actifs de galaxie, Thèse de doctorat, Ecole Doctorale d’Astronomie et d’Astrophysique d’Ile de France, 2005. T. J. H OLMES, Blind deconvolution of speckle images quantumlimited incoherent imagery : maximum-likehood approach, J. Opt. Soc. Am. A, 9(7) :1052– 1061, 1992. N. H UBIN, Ground layer adaptive optics for 8-10 m telescopes, vol. 6, 2005. A. L ABEYRIE, Detection of extra-solar planets, in Formation of Planetary Systems, A. Brahic, ed., 1982, pp. 883–+. F. L ACOMBE , G. Z INS , J. C HARTON , G. C HAUVIN , G. D U MONT, P. FAUTRIER , T. F USCO , E. G ENDRON , N. H UBIN , P. K ERN , A.-M. L AGRANGE , D. M OUILLET, P. P UGET, D. R ABAUD , P. R ABOU , G. ROUSSET, ET J.-L. B EUZIT, NAOS : from an AO system to an astronomical instrument, in Adaptive Optical System Technology II, P. L. Wizinowich et D. Bonaccini, eds., vol. 4839, Hawaii, USA, 2002, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE. A.-M. L AGRANGE , G. C HAUVIN , T. F USCO , E. G ENDRON , D. ROUAN , M. H ARTUNG , F. L ACOMBE , D. M OUILLET, G. ROUSSET, P. D ROSSART, R. L ENZEN , C. M OUTOU , W. B RANDNER , N. H UBIN , Y. C LENET, A. S TOLTE , R. S CHOEDEL , G. Z INS , ET J. S PYROMILIO, First diffraction limited images at VLT with NAOS and CONICA, in Instrumental Design and Performance for Optical/Infrared Ground-Based Telescopes, M. Iye et A. F. Moorwood, eds., vol. 4841, Bellingham, Washington, 2002, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, pp. 860–868.

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43 [Lane-a-92]

R. G. L ANE, Blind deconvolution of speckle images, J. Opt. Soc. Am. A, 9(9) :1508–1514, 1992.

[Leroux-a-04]

B. L E ROUX , J.-M. C ONAN , C. K ULCSÁR , H.-F. R AYNAUD , L. M. M UGNIER , ET T. F USCO, Optimal control law for classical and multiconjugate adaptive optics, J. Opt. Soc. Am. A, 21(7), Juil. 2004.

[Marchetti-p-03b]

E. M ARCHETTI , N. H UBIN , E. F EDRIGO , J. B RYNNEL , B. D ELABRE , R. D ONALDSON , F. F RANZA , R. C ONAN , M. L E L OUARN , C. C AVADORE , A. BALESTRA , D. BAADE , J.-L. L IZON , R. G ILMOZZI , G. J. M ONNET, R. R AGAZZONI , C. A RCIDIACONO , A. BARUFFOLO , E. D IOLAITI , J. FARI NATO , E. V ERNET-V IARD , D. J. B UTLER , S. H IPPLER , ET A. A MORIN, MAD the ESO multi-conjugate adaptive optics demonstrator, in Adaptive Optical System Technologies II., P. L. Wizinowich et D. Bonaccini, eds., vol. 4839, Proc. Soc. PhotoOpt. Instrum. Eng., SPIE, Fev. 2003, pp. 317–328.

[Marois-a-06]

C. M AROIS , D. L AFRENIÈRE , R. D OYON , B. M ACINTOSH , ET D. NADEAU , Angular Differential Imaging : A Powerful High-Contrast Imaging Technique, Astrophysical Journal, 641 :556–564, Avr. 2006.

[Marois-t-04]

C. M AROIS, La recherche de naines brunes et d’exoplanètes : développement d’une technique d’imagerie multibande., Thèse de doctorat, Université de Montréal, 2004.

[Mugnier-p-07a]

L. M. M UGNIER , J.-F. S AUVAGE , T. F USCO , ET G. ROUS SET, Multi-channel planet detection algorithm for angular differential imaging, in Adaptive Optics : Analysis and Methods, OSA, 2007. Date conférence : Juin 18–20, 2007, Vancouver (Canada).

[Mugnier-p-07b]

L. M. M UGNIER , J.-F. S AUVAGE , A. WOELFFLE , T. F USCO , ET G. ROUSSET, Algorithme multi-canaux pour la détection d’exo-planètes en imagerie différentielle angulaire, in 21ième Colloque sur le Traitement du Signal et des Images, GRETSI, 2007. Date conférence : Sept. 11-14, 2007, Troyes (France).

[Mugnier-p-99]

L. M. M UGNIER , C. ROBERT, J.-M. C ONAN , V. M ICHAU , ET S. S ALEM, Regularized multiframe myopic deconvolution from wavefront sensing, in Propagation through the Atmosphere III, M. C. Roggemann et L. R. Bissonnette, eds., vol. 3763, Bellingham, Washington, 1999, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, pp. 134–144.

[Neichel-p-05]

B. N EICHEL , T. F USCO , M. P UECH , J.-M. C ONAN , M. L E LOUARN , E. G ENDRON , F. H AMMER , G. ROUSSET, P. JA GOUREL , ET P. B OUCHET, Adaptive optics concept for multiobjects 3D spectroscopy on ELTs, in IAU Symposium 232,

44

CHAPITRE 6. BIBLIOGRAPHIE Science with ELTs, 2005. Date conférence : Nov. 2005, Capetown, South Africa.

[Neichel-p-06]

B. N EICHEL , J.-M. C ONAN , T. F USCO , E. G ENDRON , M. P UECH , G. ROUSSET, ET F. H AMMER, ELTs adaptive optics for multi-objects 3D spectroscopy : key parameters and design rules, in Advances in Adaptive Optics II, L. Ellerbroek, B. et D. Bonaccini Calia, eds., vol. 6272, Soc. Photo-Opt. Instrum. Eng., 2006.

[Nicolle-t-06]

M. N ICOLLE, Analyse de front d’onde pour les optiques adaptatives de nouvelle génération : optiques adaptatives à large champ et optique adaptative extrême, Thèse de doctorat, Université Paris XI, Dec. 2006.

[Petit-a-05]

C. P ETIT, J.-M. C ONAN , C. K ULCSAR , H.-F. R AYNAUD , T. F USCO , J. M ONTRI , ET D. R ABAUD, Optimal control for multi-conjugate adaptive optics, C. R. Physique 6, pp. 1059– 1069, 2005.

[Petit-t-06]

C. P ETIT, Etude de la commande optimale en Optique Adaptative et Optique Adaptative MultiConjuguée, validation numérique et expérimentale, Thèse de doctorat, Paris 13, 2006.

[Poyner-a-04]

L. A. P OYNEER ET B. M ACINTOSH, Spatially filtered wavefront sensor for high-order adaptive optics, J. Opt. Soc. Am. A, 21(5) :810–819, 2004.

[Press-l-88]

W. P RESS , B. F LANNERY, S. T EUKOLSKY, ET W. V ETTER LING , Numerical Recipes in C, Cambridge University press, 1988.

[Puech-a-07]

M. P UECH , H. F LORES , M. L EHNERT, B. N EICHEL , T. F USCO , P. ROSATI , J.-G. C UBY , Recovering the dynamical nature of distant galaxies with MOAO-fed Integral Field Spectroscopy : simulations and perspectives for the VLT and the EELT, Astron. Astrophys., submitted (2007).

[Ragazzoni-a-00]

R. R AGGAZZONI , E. M ARCHETTI , ET G. VALENTE, Adaptive-optics correction available for the whole sky, Nature (London), 403 :54–56, Jan. 2000.

[Ragazzoni-a-99b] R. R AGAZZONI, No laser guide stars for adaptive optics in giant telescopes, Astron. Astrophys. Suppl. Ser., 136 :205–209, Avr. 1999. [Rigaut-p-01]

F. R IGAUT, Ground-conjugate wide field adaptive optics for the ELTs, in Beyond Conventional Adaptive Optics, R. Ragazzoni et S. Esposito, eds., Padova, Italy, 2001, Astronomical Observatory of Padova.

[Roddier-81a]

F. RODDIER, The effects of atmospherical turbulence in optical astronomy, in Progress in Optics, E. Wolf, ed., vol. XIX, North Holland, Amsterdam, 1981, pp. 281–376.

45 [Root-a-87]

W. L. ROOT, Ill-posedness and precision in object-field reconstruction problems, J. Opt. Soc. Am. A, 4(1) :171–179, Jan. 1987. [Rousset-a-90] G. ROUSSET, J.-C. F ONTANELLA , P. K ERN , P. G IGAN , F. R I GAUT, P. L ÉNA , C. B OYER , P. JAGOUREL , J.-P. G AFFARD , ET F. M ERKLE, First diffraction-limited astronomical images with adaptive optics, Astron. Astrophys., 230 :29–32, 1990. [Rousset-p-02a] G. ROUSSET, F. L ACOMBE , P. P UGET, N. H UBIN , E. G EN DRON , T. F USCO , R. A RSENAULT, J. C HARTON , P. G IGAN , P. K ERN , A.-M. L AGRANGE , P.-Y. M ADEC , D. M OUILLET, D. R ABAUD , P. R ABOU , E. S TADLER , ET G. Z INS, NAOS, the first AO system of the VLT : on sky performance, in Adaptive Optical System Technology II, P. L. Wizinowich et D. Bonaccini, eds., vol. 4839, Bellingham, Washington, 2002, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, pp. 140–149. [Schultz-a-93] T. J. S CHULTZ, Multiframe blind deconvolution of astronomical images, J. Opt. Soc. Am. A, 10(5) :1064–1073, 1993. [Sheppard-a-98] D. G. S HEPPARD , B. R. H UNT, ET M. W. M ARCELLIN, Iterative multiframe superresolution algorithms for atmosphericturbulence-degraded imagery, J. Opt. Soc. Am. A, 15(4) :978– 992, 1998. [Tallon-92b] M. TALLON , R. F OY, ET J. V ERNIN, 3-d wavefront sensing for multiconjugate adaptive optics, in Progress in Telescope and instrumentation technologies, ESO Conference and Workshop Proceedings, Garching Germany, 1992, ESO, M.-H. Ulric, pp. 517–521. [Thiebaut-a-95] E. T HIÉBAUT ET J.-M. C ONAN, Strict a priori constraints for maximum-likelihood blind deconvolution, J. Opt. Soc. Am. A, 12(3) :485–492, 1995. [Tikhonov-l-77] A. T IKHONOV ET V. A RSENIN, Solutions of Ill-Posed Problems, Winston, DC, 1977. [Titterington-a-85] D. M. T ITTERINGTON, General structure of regularization procedures in image reconstruction, Astron. Astrophys., 144 :381–387, 1985. [Veran-a-97] J.-P. V ÉRAN , F. R IGAUT, H. M AÎTRE , ET D. ROUAN, Estimation of the adaptive optics long exposure point spread function using control loop data, J. Opt. Soc. Am. A, 14(11) :3057–3069, 1997. [Wizinowitch-p-00] P. W IZINOWICH , D. S. ACTON , O. L AI , J. G ATHRIGHT, W. L UPTON , ET P. S TOMSKI, Performance of the w.m. keck observatory natural guide star adaptive optic facility : the first year at the telescope, in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., vol. 4007, Bellingham, Washington, Mars 2000, Proc. Soc. Photo-Opt. Instrum. Eng., pp. 2–13.

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CHAPITRE 6. BIBLIOGRAPHIE

Deuxième partie Éléments de Curriculum Vitae

47

49

Thierry Fusco ONERA Dépt Optique Théorique et Appliquée BP 72, 92322 Châtillon cedex.

né le 23 Aout 1973 à Marseille. Nationalité française. Tél. : 01 46 73 47 37. Fax : 01 46 73 41 71. Mél : [email protected]

EXPÉRIENCE Depuis 2001 Ingénieur de recherche, ONERA, Département d’Optique Théorique et Appliquée. - Chef de projet SAXO : système d’optique adaptative (OA) extrême pour le projet SPHERE pour le VLT (ESO) - Simulation, conception et intégration de systèmes d’OA - Nouveaux concepts d’analyse de surface d’onde et de commande pour l’OA - Développement d’algorithmes de traitement d’images - Modélisation et caractérisation de la turbulence atmosphérique - Négociations, suivi d’études, revues de projets, collaborations internationales - Expert pour plusieurs projets astronomiques (ELT) ou militaires (focalisation de faisceaux lasers) 2000-01

Ingénieur de recherche, Observatoire de Paris-Meudon. - Intégration et tests du système d’OA NAOS pour le VLT (ESO) - Participation aux tests sur le ciel au Chili (observatoire de Paranal)

1997-00

Doctorant, ONERA, Département d’Optique Théorique et Appliquée : “Correction partielle et anisoplanétisme en Optique Adaptative : traitement a posteriori et Optique Adaptative Multiconjuguée” - Etude et caractérisation de systèmes d’optique adaptative multiconjuguée. Développement analytique, simulation numérique, estimation de performance - Développement de nouveaux algorithmes de traitement de d’images corrigées par OA

1996-97

Scientifique du contingent, ONERA, Département d’Optique Théorique et Appliquée : Développement d’algorithmes de traitement d’images - Campagne d’observation à l’Observatoire de Haute Provence avec le banc d’OA de l’ONERA (BOA)

FORMATION Oct 2000 1995/96 1993/95 Juin 1991

Doctorat en science de l’ingénieur de l’université de Nice - Sophia Antipolis Mention très honorable avec les félicitations du jury. DEA « Astronomie, imagerie et haute résolution angulaire » de l’université de Nice - Sophia Antipolis. Mention Bien. Licence et Maîtrise de Physique, Université de Nice. Mention AB Baccalauréat C, mention AB.

LANGUES Anglais

Courant.

50

Actions d’enseignement • cours (6h) d’optique adaptative M2 recherche OMEGA (université de Nice) depuis 2004 • TP (4h) Optique adaptative (avec E. Gendron) pour le M2 professionel (Observatoire de Paris - université paris VII) depuis 2005

Participation à des jurys de thèse • • • • •

J.-F. Sauvage, soutenue en 2007 (Encadrant) A. Kellerer, soutenue en 2007 (Examinateur) M. Nicolle, soutenue en 2006 (Encadrant) S. Thomas, soutenue en 2005 (Examinateur) S. Hamdani, soutenue 2003 (Invité)

Revues de projets J’ai participé au cours des dernières années à la revue et l’expertise de nombreux projets internationaux pour l’ESO, le TMT et GEMINI : – PFI (Planet Formation Instrument) pour le TMT. Pré-étude pour un instrument à haute dynamique sur le futur télescope de 30m américain (Thirty Meters Telescope - TMT) ; – HOT (High Order Testbench) pour l’ESO. Il s’agissait de la “Preliminary Design Review” (PDR) d’un banc de test pour l’OA extrême, développé par l’ESO – GRAAL (GRound-layer Adaptive optics Assisted by Lasers) pour l’ESO. Il s’agissait de la PDR de l’OA (grand champ) de l’instrument de deuxième génération du VLT, Hawk-I. – SPARTA (Standard Platform for Adaptive optics Real Time Applications) pour l’ESO. Il s’agissait de la PDR de la plateforme temps-réel commune développée par l’ESO et qui sera utilisée par l’ensemble des instrument de deuxième génération intégrant une optique adaptative. – GPI (Gemini Planet Imageur) pour GEMINI. Il s’agissait, là aussi, de la PDR de l’équivalent du projet SPHERE (système pour la détection directe de planètes extra-solaires) aux Etats-Unis.

Comités d’organisation, groupes de travail, expertises Je participe actuellement à deux groupes de travail de l’INSU portant sur – les ELT – l’optique adaptative

51 Je suis membre du comité scientifique pour deux conférences internationales : – OCS’08 (2nd international conference on optical complex systems) qui aura lieu à Cannes en mars 2008 – SPIE Astronomical Telescopes and Instrumentation, qui aura lieu à Marseille en juin 2008.

Manifestations scientifiques, séminaires En marge de ces activités, j’ai participé à pusieurs évènements et séminaires pour le grand public : – les journées du Ciel et de l’Espace (1999) ; – la fête de la science (2004, 2005) ; – une conférence du collège de France sur la détection directe de planètes extrasolaire (2005) ; – l’émission « C’est pas sorcier » sur le VLT - participation à la rédaction du scénario de l’émission (2004).

52

Troisième partie Activités d’encadrement

53

55

Encadrement de doctorants Je liste ci-dessous les doctorants que j’ai encadrés (en tant que responsable ou simplement par une participation plus ou moins importante) en précisant leur sujet de thèse et leur position actuelle. • Responsabilité de l’encadrement – Jean-Francois Sauvage. "Calibrations et méthodes d’inversion en imagerie haute dynamique pour la détection directe d’exoplanètes". Thèse soutenue en décembre 2007. Embauché à l’ONERA dans l’équipe HRA. – Magalie Nicolle. "Analyse de front d’onde pour les optiques adaptatives de nouvelle génération : optiques adaptatives à large champ et optique adaptative extrême". Thèse soutenue en decembre 2006. Actuellement chez SAGEM (Argenteuil). – Benoît Neichel. "Optique adaptative à grand champs pour l’observation des galaxies lointaines avec les futurs télescopes géants". En troisième année de thèse. • participation active à l’encadrement (20% ou plus) – Brice Leroux. "Commande optimale en optique adaptative classique et conjuguée". Thèse soutenue en 2003. Maitre de conférence, LAM, observatoire de Marseille. – Cyril Petit. "Etude de la commande optimale en Optique Adaptative et Optique Adaptative MultiConjuguée, validation numérique et expérimentale". Thèse soutenue en décembre 2006. Embauché à l’ONERA dans l’équipe HRA. – Sandrine Thomas. "Etude d’une optique adaptative dans le visible". Thèse soutenue en 2005. Actuellement en post-doc au LAO, université de Santa Cruz. – Anne Costille. "Analyse de front d’onde et commande en optique adaptative multiconjuguée". En deuxième année de thèse. • Participation plus “ponctuelle” ( 10/20%) – Amandine Blanc. "Identification de réponse impulsionnelle et restauration d’images : apports de la diversité de phase". Thèse soutenue en 2003. Professeur en collèges/Lycés. – Johann Kolb. "Problème de calibration en optique adaptative multiconjuguée et application au demonstrateur MAD de l’ESO". Thèse soutenue en décembre 2005. En CDD à l’ESO – Damien Gratadour. "Optique adaptative, traitement d’image et étude des noyaux actifs de galaxie". Thèse soutenue en 2005. En post-doc (GEMINI) – Fernando Quiros Pacheco. "Reconstruction and control laws for multiconjugate adaptive optics in astronomy". Thèse soutenue en 2007 (Imperial College, Londres). En post-doc (Observatoire d’Arcetri) – Guillaume Montagnier. "Etude et développement d’un système d’imagerie à haute dynamique pour la détection directe de planètes extra-solaires". Fin

56 de troisième année. – Noah Schwartz. "Pre-compensation des effets de la turbulence par optique adaptative. Application aux télécommunications lasers". Fin de première année. – Carlos Correia Da Silva. "Commande optimale pour les optiques adaptatives à grand nombre de degrés de liberté. Application aux futurs télescopes géants. ". Fin de première année.

Encadrement de post-doctorants • Rodolphe Conan [2002-2004] : Simulations pour l’optique adaptative extrème. • Amandine Blanc [2003-2005] : Dévelopement d’un algorithme de diversité de phase pour l’ESO. Etude de la couverture de ciel pour l’optique adaptative à grand champ. • Magalie Nicolle [Decembre 2006-Avril 2007] : Validation experimentale pour la XAO (analyse de front d’onde) • Francois Assemat [depuis juillet 2007] : Simulations et dimensionnement d’optique adaptative à grand champ pour l’instrument EAGLE sur l’E-ELT

Encadrement de stagiaires ingénieurs et DEA En plus des post-docs et des étudiants de thèse, j’encadré ou co-encadré chaque année des stagiaires de fin d’études (écoles d’ingénieurs et master 2ime année).

Quatrième partie Liste des publications

57

Publications (revues à comité de lecture) [1] M. Puech, H. Flores, M. Lehnert, B. Neichel, T. Fusco, P. Rosati, J.-G. Cuby, Recovering the dynamical nature of distant galaxies with MOAO-fed Integral Field Spectroscopy : simulations and perspectives for the VLT and the E-ELT, Astron. Astrophys., submitted (2007). [2] G. Montagnier, T. Fusco, J.-L. Beuzit , D. Mouillet, J. Chartonet L. Jocou, Pupil stabilization for SPHERE’s extreme AO and high performance coronagraph system, Opt. Express, To be published (2007). [3] T. Fusco, M. Nicolle, C. Petit, J. Kolb, S. Oberti, E. Marchetti, N. Hubin, E. Fedrigo et V. Michau, Experimental validation of Star Orientied optimisation for Ground Layer AO on the MAD test bench, Astron. Astrophys. (soumis). [4] B. Carry, C. Dumas, M. Fulchignoni, T. Fusco, W. Merline, J. Berthier et D. Hestroffer, Near-Infrared mapping and physical properties of asteroid 1 Ceres, Astron. Astrophys. (accepté pour publication). [5] T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper et P. Puget, High order Adaptive Optics requirements for direct detection of Extra-solar planets. Application to the SPHERE instrument., Opt. Express, (17), pp. 7515–7534 (2006). [6] M. Nicolle, T. Fusco, V. Michau, G. Rousset et J.-L. Beuzit, Optimization of star oriented and layer oriented wave-front sensing concepts for ground layer adaptive optics, J. Opt. Soc. Am. A, 23 (9), pp. 2233–2245 (2006). [7] G. Chauvin, A.-M. Lagrange, C. Dumas, B. Zuckerman, E. Gendron, I. Song, J.-L. Beuzit, P. Lowrance et T. Fusco, Probing Long-Period Companions to RV Stars with Planets. VLT and CFHT Deep Coronographic AO Imaging Survey, Astron. Astrophys., pp. 1165–1172 (2006). [8] E. Gendron, Y. Clénet, T. Fusco, et G. Rousset, New algorithms for adaptive optics point-spread function reconstruction , Astron. Astrophys., 457, pp. 359– 363 (2006). [9] S. Thomas, T. Fusco, A. Tokovini, M. Nicolle, V. Michau et Rousset, Comparison of centroid computation algorithms in a Shack-Hartmann sensor, Mon. Not. R. Astr. Soc., 371 (1), pp. 323–333 (2006). 59

60 [10] T. Fusco, A. Blanc, M. Nicolle, V. Michau, G. Rousset, J.-L. Beuzit et N. Hubin, Improvement of sky coverage estimation for MCAO systems : strategies and algorithms, Mon. Not. R. Astr. Soc., 370 (1), pp. 174–186 (2006). [11] G. Montagnier, D. Ségransan, J.-L. Beuzit, T. Forveille, P. Delorme, X. Delfosse, C. Perrier, S. Udry, M. Mayor, G. Chauvin, A.-M. Lagrange, D. Mouillet, T. Fusco, P. Gigan et E. Stadler, Five new very low mass binaries, Astron. Astrophys., 460, pp. L19–L22 (décembre 2006). [12] J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, J. Antici, A. Baruffolo, P. Baudoz, A. Berton, A. Boccaletti, M. Carbillet, J. Charton, R. Claudi, M. Downing, P. Feautrier, E. Fedrigo, T. Fusco, R. Gratton, N. Hubin, M. Kasper, M. Langlois, C. Moutou, L. Mugnier, J. Pragt, P. Rabou, M. Saisse, H. M. Schmid, E. Stadler, M. Turrato, S. Udry, R. Waters et F. Wildi, SPHERE : A ’Planet Finder’ Instrument for the VLT, The Messenger, 125, pp. 29–+ (septembre 2006). [13] M. Puech, F. Hammer, P. Jagourel, E. Gendron, F. Assémat, F. Chelma, H. Flores, P. Laporte, J.-M. Conan, T. Fusco, L. A. et F. Zamkotsian, FALCON : Extending adaptive corrections to cosmological fields, New Astronomy Reviews, 50 (4-5), pp. 382–384 (juin 2006). [14] C. Robert, J.-M. Conan, V. Michau, T. Fusco et N. Vedrenne, Scintillation and Phase Anisoplanatism in Shack-Hartmann Wavefront Sensing, J. Opt. Soc. Am. A, 23 (3), pp. 613–624 (mars 2006). [15] D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron et F. Lacombe, Near-IR AO dissection of the core of NGC 1068 with NaCo, Astron. Astrophys., 446 (3), pp. 813–825 (février 2006). [16] C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco et D. Rouan, Fundamental limitations on Earth-like planet detection with extremely large telescopes, Astron. Astrophys., 447, pp. 397–403 (février 2006). [17] T. Fusco, C. Petit, G. Rousset, J.-M. Conan et J.-L. Beuzit, Closed-loop experimental validation of the spatially filtered Shack-Hartmann concept, Opt. Lett., 30, p. 1255 (2005). [18] E. Gendron, F. Assémat, F. Hammer, P. Jagourel, F. Chemla, P. Laporte, M. Puech, M. Marteaud, F. Zamkotsian, A. Liotard, J.-M. Conan, T. Fusco et N. Hubin, FALCON : Multi-object AO, C. R. Physique, 6, pp. 1110–1117 (2005). [19] C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri et D. Rabaud, Optimal Control for Multi-conjugate Adaptive Optics, C. R. Physique 6, pp. 1059–1069 (2005). [20] T. Fusco, M. Nicolle, G. Rousset, V. Michau, A. Blanc, J.-L. Beuzit et J.-M. Conan, Wavefront sensing issues in MCAO, C. R. Physique 6, pp. 1049–1058 (2005).

61 [21] G. Chauvin, A.-M. Lagrange, C. Dumas, B. Zuckerman, E. Gendron, I. Song, J.-L. Beuzit, P. Lowrance et T. Fusco, Astrometric and spectrometric confirmation of a brown dwarf companion to GSC 08047-00232 - VLT/NACO deep imaging and spectroscopic observations, Astron. Astrophys., 430, pp. 1027– 1033 (2005). [22] F. Marchis, D. Le Mignant, D. F. Chaffee, R. Prangé, T. Fusco, S. Kwok, P. Amico, F. Campbell, A. Conrad, A. Contos, R. Goodrich et I. de Pater, Keck AO survey of Io global volcanic activity between 2 and 5 microns, Icarus, 176, pp. 96–122 (2005). [23] T. Forveille, J.-L. Beuzit, P. Delorme, D. Ségransan, X. Delfosse, G. Chauvin, T. Fusco, A.-M. Lagrange, M. Mayor, G. Montagnier, D. Mouillet, C. Perrier, S. Udry, J. Charton, P. Gigan, J.-M. Conan, P. Kern et G. Michet, LP 349-25 : A new tight M8V binary, Astron. Astrophys., 435, pp. L5–L8 (mai 2005). [24] D. Rouan, F. Lacombe, E. Gendron, D. Gratadour, Y. Clénet, A.-M. Lagrange, D. Mouillet, C. Boisson, G. Rousset, T. Fusco, L. Mugnier, N. Thatte, R. Genzel, P. Gigan, R. Arsenault et P. Kern, Hot Very Small Dust Grains in NGC 1068 seen in jet induced structures thanks to VLT/NACO adaptive optics, Astron. Astrophys., 417, pp. L1–L4 (2004). [25] A.-M. Lagrange, G. Chauvin, D. Rouan, E. Gendron, J.-L. Beuzit, F. Lacombe, G. Rousset, T. Fusco, D. Mouillet, J.-M. Conan, E. Stadler, J. Deleglise et C. Perrier, The environment of the very red object Orion n - A possible low mass companion ?, Astron. Astrophys., 417, pp. L11–L14 (2004). [26] F. Lacombe, E. Gendron, Y. Clénet, D. Field, J.-L. Lemaire, M. Gustafsson, A.-M. Lagrange, D. Mouillet, G. Rousset, T. Fusco, L. Rousset-Rouvière, B. Servan, C. Marlot et P. Fautrier, VLT/NACO infrared adaptive optics images of small scale structures in OMC1, Astron. Astrophys., 417, pp. L5–L9 (2004). [27] E. Gendron, A. Coustenis, P. Drossart, M. Combes, M. Hirtzig, F. Lacombe, D. Rouan, C. Collin, S. Pau, A.-M. Lagrange, D. Mouillet, P. Rabou, T. Fusco et G. Zins, VLT/NACO adaptive optics imaging of Titan, Astron. Astrophys., 417, pp. L21–L24 (2004). [28] Y. Clénet, D. Rouan, E. Gendron, F. Lacombe, A.-M. Lagrange, D. Mouillet, Y. Magnard, G. Rousset, T. Fusco, J. Montri, R. Genzel, R. Schodel, T. Ott, A. Eckart, O. Marco et L. Tacconi-Garman, The infrared L’-band view of the Galactic Center with NAOS-CONICA at VLT, Astron. Astrophys., 417, pp. L15–L19 (2004). [29] F. Marchis, D. Le Mignant, F. H. Chaffee, A. G. Davies, S. H. Kwok, P. Prange, D. P. I., P. Amico, R. Campbell, T. Fusco, R. W. Goodrich et A. Conrad, Keck AO survey of Io Global Volcanic Activity between 2 and 5 microns, Icarus, 176, pp. 96–122 (2004). [30] M. Nicolle, T. Fusco, G. Rousset et V. Michau, Improvement of Shack-

62 Hartmann wavefront sensor measurement for Extreme Adaptive Optics, Opt. Lett., 29 (23), pp. 2743–2745 (décembre 2004). [31] L. M. Mugnier, T. Fusco et J.-M. Conan, MISTRAL : a Myopic EdgePreserving Image Restoration Method, with Application to Astronomical Adaptive-Optics-Corrected Long-Exposure Images., J. Opt. Soc. Am. A, 21 (10), pp. 1841–1854 (octobre 2004). [32] B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier et T. Fusco, Optimal control law for classical and Multiconjugate Adaptive Optics, J. Opt. Soc. Am. A, 21 (7) (juillet 2004). [33] T. Fusco et J.-M. Conan, On- and off-axis statistical behavior of adaptive optics corrected short exposure Strehl ratio, J. Opt. Soc. Am. A, 21 (7) (juillet 2004). [34] T. Fusco, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, P.-Y. Madec, A.-M. Lagrange, J. Charton, D. Rouan, H. Hubin et N. Ageorges, NAOS on-line characterization of turbulence parameters and adaptive optics performance, J. Opt. A : Pure Appl. Opt., 6, pp. 585–596 (juin 2004). [35] M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. M. Mugnier, G. Rousset et R. Lenzen, Calibration of NAOS and CONICA static aberrations. Experimental results, Astron. Astrophys., 399, pp. 385–394 (2003). [36] A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier et G. Rousset, Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique, Astron. Astrophys., 399, pp. 373–383 (2003). [37] G. Chauvin, A.-M. Lagrange, H. Beust, T. Fusco, D. Mouillet, F. Lacombe, E. Gendron, G. Rousset, D. Rouan, W. Brandner, R. Lenzen, H. N. et H. M., VLT/NACO adaptive optics imaging of the TY CrA system. A fourth stellar component candidate detected, Astron. Astrophys., 406 (3) (August II 2003). [38] G. Chauvin, M. Thomson, C. Dumas, J.-L. Beuzit, P. Lowrance, T. Fusco, A.-M. Lagrange, B. Zuckerman et D. Mouillet, Adaptive optics imaging survey of the Tucana-Horologium association, Astron. Astrophys., 404, pp. 157–162 (juin 2003). [39] D. Hestroffer, F. Marchis, T. Fusco et J. Berthier, Adaptive Optics Observations of asteroid (216) Kleopatra, Research Note, A&A, 394, pp. 339–343 (2002). [40] P. Descamps, F. Marchis, J. Berthier, R. Prangé, T. Fusco et C. Le Guyader, First ground-based Astrometric Observations of Puck, C. R. Acad. Sci. Paris, 3, pp. 121–128 (2002). [41] R. R. Howell, J. R. Spencer, J. D. Goguen, F. Marchis, R. Prangé, T. Fusco, B. D. L., G. J. Veeder, J. A. Rathbun, G. S. Orton, A. J. Groeholski et J. A. Stansberry, Ground-based observations of volcanism on Io in 1999 and early 2000, J. Geophys. Res., 106 (E12), pp. 33–129 (2002).

63 [42] G. Chauvin, F. Ménard, T. Fusco, A.-M. Lagrange, J.-L. Beuzit, D. Mouillet et J.-C. Augereau, Adaptive optics imaging of MBM 12 association. Seven binaries and edge-on disk in a quadruple system., Astron. Astrophys., 394, pp. 949– 956 (novembre 2002). [43] F. Marchis, I. de Pater, A. G. Davies, H. G. Roe, T. Fusco, D. Le Mignant, P. Descamps, B. A. Macintosh et R. Prangé, High-resolution Keck Adaptive Optics Imaging of Violent Volcanic Activity on Io, Icarus, 160 (1), pp. 124–131 (novembre 2002). [44] G. Chauvin, T. Fusco, A.-M. Lagrange, D. Mouillet, J.-L. Beuzit, M. Thomson, J.-C. Augereau, F. Marchis, C. Dumas et P. Lowrance, No disk needed around HD 199143 B, Astron. Astrophys., 394, pp. 219–223 (octobre 2002). [45] A. Coustenis, E. Gendron, O. Lai, J.-P. Véran, J. Woillez, M. Combes, L. Vapillon, T. Fusco, L. Mugnier et P. Rannou, Images of Titan at 1.3 and 1.6 micron with adaptive optics at the CFHT, Icarus, 154, pp. 501–515 (2001). [46] F. Marchis, R. Prangé et T. Fusco, A survey of Io’s volcanism by adaptive optics observations in the 3.8 micron thermal band (1996-1999), J. Geophys. Res., 106 (E12), pp. 33141–33160 (décembre 2001). [47] T. Fusco, J.-M. Conan, G. Rousset, L. M. Mugnier et V. Michau, Optimal wavefront reconstruction strategies for Multiconjugate Adaptive Optics, J. Opt. Soc. Am. A, 18 (10), pp. 2527–2538 (octobre 2001). [48] T. Fusco, J.-M. Conan, L. Mugnier, V. Michau et G. Rousset, Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution., Astron. Astrophys. Suppl. Ser., 142, pp. 149–156 (2000). [49] T. Fusco, J.-M. Conan, V. Michau, L. Mugnier et G. Rousset, Efficient phase estimation for large field of view adaptive optics, Opt. Lett., 24 (21), pp. 1472– 1474 (novembre 1999). [50] T. Fusco, J.-P. Véran, J.-M. Conan et L. Mugnier, Myopic deconvolution method for adaptive optics images of stellar fields, Astron. Astrophys. Suppl. Ser., 134, pp. 1–10 (janvier 1999). [51] J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau et G. Rousset, Myopic Deconvolution of Adaptive Optics Images by use of Object and Point Spread Function Power Spectra, Appl. Opt., 37 (21), pp. 4614–4622 (juillet 1998).

64

Communications (actes de conférences) [52] M.-T. Velluet, V. Michau, T. Fusco, J.-M. Conan, Coherent illumination for wavefront sensing and imaging through turbulence, SPIE (2007), Date conférence : août 26–30, 2007, San Diego (USA). [53] A. Costille, J.-M. Conan, T. Fusco, C. Petit, DM dimensioning for the next generation of AO systems : strategies and rules, Dans Semaine de l’astrophysique française. SF2A, EDP Sciences (2007). Date conférence : juin Jul, 2007, Grenoble (France). [54] C. Correia, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, C. Petit, T. Fusco, Fourier-domain wave-front reconstruction for large adaptive optical systems, Dans Semaine de l’astrophysique française. SF2A, EDP Sciences (2007). Date conférence : juin Jul, 2007, Grenoble (France). [55] J.-F. Sauvage, L. Mugnier, A. Woelffle, T. Fusco, G. Rousset Multi-channel algorithm for exoplanets detection by angular differential imaging., Dans Semaine de l’astrophysique française. SF2A, EDP Sciences (2007). Date conférence : juin Jul, 2007, Grenoble (France). [56] R. Foy, M. Chatagnat, D. Dubet, P. Eric, J. Eysseric, F.-C. Foy, T. Fusco, J. Girard, A. Le Van Suu,B. Messaoudi, S. Perruchot, P. Richaud, X. Rondeau, M. Tallon, E. Thiebaut, M Boer. The polychromatic Laser Guide Star : the ELP-OA demonstrator at observatoire de Provence , Dans Semaine de l’astrophysique française. SF2A, EDP Sciences (2007). Date conférence : juin Jul, 2007, Grenoble (France). [57] S. Thomas, S. Adkins, D. Gavel, B. Ellerbreok, L. Gilles et T. Fusco, Study of the parameters for a Radial CCD unsing Laser Guide Stars, Dans Adaptive Optics : Analysis and Methods. OSA (2007), Date conférence : juin 18–20, 2007, Vancouver (Canada). [58] L. M. Mugnier, J.-F. Sauvage, A. Woelffle, T. Fusco et G. Rousset, Algorithme multi-canaux pour la détection d’exo-planètes en imagerie différentielle angulaire, Dans 21ième Colloque sur le Traitement du Signal et des Images. GRETSI (2007), Date conférence : septembre 11-14, 2007, Troyes (France). 65

66 [59] L. M. Mugnier, J.-F. Sauvage, T. Fusco et G. Rousset, Multi-Channel Planet Detection Algorithm for Angular Differential Imaging, Dans Adaptive Optics : Analysis and Methods. OSA (2007), Date conférence : juin 18–20, 2007, Vancouver (Canada). [60] B. Le Roux, M. Langlois, M. Carbillet, T. Fusco, M. Ferrari et D. Burgarella, Ground Layer Adaptive Optics for Dome C : Optimisation and Performance, Dans Adaptive Optics : Analysis and Methods. OSA (2007), Date conférence : juin 18–20, 2007, Vancouver (Canada). [61] T. Fusco, Simulation and design of Adaptive Optics Systems : Application ti SPHERE-SAXO, Dans Adaptive Optics : Analysis and Methods. OSA (2007), Date conférence : juin 18–20, 2007, Vancouver (Canada). [62] R. Foy, P. Eric, J. Eysseric, F. Foy, T. Fusco, J. Girard, A. Le Van Suu, S. Perruchot, P. Richaud, R. Y., X. Rondeau, M. Tallon et E. Thiebaut, The Polychromatic Laser Guide Star for the tilt measurement. Progress report of the demonstrator at Observatoire de Haute Provence, SPIE (2007), Date conférence : août 26–30, 2007, San Diego (USA). [63] Y. Clenet, C. Lidman, E. Gendron, T. Fusco, G. Rousset, M. Kasper, N. Ageorges et O. Marco, PSF reconstruction for NACO : Current Status and Perspectives, Dans Adaptive Optics : Analysis and Methods. OSA (2007), Date conférence : juin 18–20, 2007, Vancouver (Canada). [64] D. Burgarella, M. Ferrari, T. Fusco, M. Langlois, G. Lemaitre, B. Le Roux, G. Moretto et M. Nicole, Wide Field Astronomy at Dome C : two IR Surveys Complementary to SNAP, Dans EAS Publications Series, vol. 25 de EAS Publications Series, pp. 147–154 (2007). [65] C. Verinaud, M. Kasper, J.-L. Beuzit, N. Yaitskova, V. Korkiakoski, K. Dohlen, P. Baudoz, T. Fusco, L. Mugnier et N. Thatte, EPICS Performance Evaluation through Analytical and Numerical Modeling, Dans Proceedings of the conference In the Spirit of Bernard Lyot : The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., sous la direction de P. Kalas (juin 2007), Date conférence : juin 2007,. [66] M. Kasper, C. Verinaud, J.-L. Beuzit, N. Yaitskova, N. Hubin, A. Boccaletti, K. Dohlen, T. Fusco, A. Glindemann, R. Gratton et N. Thatte, EPICS : A Planet Hunter for the European ELT, Dans Proceedings of the conference In the Spirit of Bernard Lyot : The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., sous la direction de P. Kalas (juin 2007), Date conférence : juin 2007,. [67] J.-L. Beuzit, M. Feldt, K. Dohlen, D. Mouillet, P. Puget, J. Antici, P. Baudoz, A. Boccaletti, M. Carbillet, J. Charton, R. Claudi, T. Fusco, R. Gratton, T. Henning, N. Hubin, F. Joos, M. Kasper, M. Langlois, C. Moutou, J. Pragt,

67 P. Rabou, M. Saisse, H. M. Schmid, M. Turatto, S. Udry, F. Vakili, R. Waters et F. Wildi, SPHERE : A Planet Finder Instrument for the VLT, Dans Proceedings of the conference In the Spirit of Bernard Lyot : The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., sous la direction de P. Kalas (juin 2007). [68] C. Verinaud, N. Hubin, M. Kasper, J. Antichi, P. Baudoz, J.-L. Beuzit, A. Boccaletti, A. Chalabaev, K. Dohlen, E. Fedrigo, C. Correita Da Silva, M. Feldt, T. Fusco, A. Gandorfer, R. Gratton, H. Kuntschner, F. Kerber, R. Lenzen, P. Martinez, E. Le Coarer, A. Longmore, D. Mouillet, R. Navarro, J. Paillet, P. Rabou, F. Rahoui, F. Selsis, H.-M. Schmid, R. Soummer, D. Stam, C. Thalmann, J. Tinbergen, M. Turatto et N. Yaitskova, The EPICS project for the European Extremely Large Telescope : outcome of the Planet Finder concept study for OWL, Dans Advances in Adaptive Optics II, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [69] J.-F. Sauvage, L. Mugnier, T. Fusco et G. Rousset, Post processing of differential images for direct extrasolar planet detection from the ground, Dans Advances in Adaptive Optics II, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [70] C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri et D. Raboud, First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO, Dans Advances in Adaptive Optics II, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [71] B. Neichel, J.-M. Conan, T. Fusco, E. Gendron, M. Puech, G. Rousset et F. Hammer, ELTs adaptive optics for multi-objects 3D spectroscopy : key parameters and design rules, Dans Advances in Adaptive Optics II, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [72] G. Moretto, R. Bacon, J.-G. Cuby, F. Hammer, P. Amram, S. Blais-Ouellette, P.-E. Blanc, J. Devriendt, B. Epinat, T. Fusco, P. Jagourel, O. Hernandez, J.-P. Kneib, I. Montilla, B. Neichel, E. Pecontal, E. Prieto et M. Puech, Wide field spectrograph concepts for the European Extremely Large Telescope, Dans Ground-based and Airborne Instrumentation for Astronomy, sous la direction de I. S. McLean et M. Iye, vol. 6269. Soc. Photo-Opt. Instrum. Eng. (2006). [73] T. Fusco, C. Petit, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, J.M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi et P. Puget, Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT, Dans

68 Advances in Adaptive Optics II, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [74] T. Fusco, S. Thomas, M. Nicolle, A. Tokovinin, V. Michau et G. Rousset, Optimization of center of gravity algorithms in a Shack-Hartmann sensor, Dans Advances in Adaptive Optics, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [75] T. Fusco, G. Rouset, D. Mouillet et K. Dohlen, High order Adaptive Optics requirements and feasibility for high contrast imaging, Dans IAUC 200, Direct Imaging of Exoplanets : Science & Techniques (2006), Date conférence : octobre 2005, Nice, France. [76] M. Downing, R. Arsenault, D. Baade, P. Ballard, R. Bell, D. Burt, S. Denney, P. Feautrier, T. Fusco, J.-L. Gach, J. Garcia, C. Guillaume, N. Hubin, P. Jorden, M. Kasper, M. Meyer, P. Pool, J. Reyes, M. Skegg, E. Stadler, W. Suske et P. Wheeler, Custom CCD for adaptive optics applications, Dans High Energy, Optical, and Infrared Detectors for Astronomy II, sous la direction de D. A. Dorn et A. D. Holland, vol. 6276. Soc. Photo-Opt. Instrum. Eng. (2006). [77] K. Dohlen, J.-L. Beuzit, M. Feldt, D. Mouillet, P. Puget, J. Antichi, A. Baruffolo, P. Baudoz, A. Berton, A. Boccaletti, M. Carbillet, J. Charton, R. Claudi, M. Downing, C. Fabron, P. Feautrier, E. Fedrigo, T. Fusco, J.L. Gach, R. Gratton, N. Hubin, M. Kasper, M. Langlois, A. Longmore, C. Moutou, C. Petit, J. Pragt, P. Rabou, G. Rousset, M. Saisse, H.-M. Schmid, E. Stadler, D. Stamm, M. Turatto, R. Waters et F. Wildi, SPHERE : A planet finder instrument for the VLT, Dans Ground-based and Airborne Instrumentation for Astronomy, sous la direction de I. S. McLean et M. Iye, vol. 6269. Soc. Photo-Opt. Instrum. Eng. (2006). [78] Y. Clenet, M. Kasper, E. Gendron, T. Fusco, G. Rousset, D. Gratadour, C. Lidman, O. Marco, N. Ageorges et S. Egner, PSF reconstruction for NAOS-CONICA, Dans Advances in Adaptive Optics II, sous la direction de L. Ellerbroek B. et D. Bonaccini Calia, vol. 6272. Soc. Photo-Opt. Instrum. Eng. (2006). [79] C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, P. Martinez et D. Rouan, Limitations on Earth-like planet detection with perfect and real coronagraphs on ELTs, Dans Modeling, Systems Engineering, and Project Management for Astronomy II, sous la direction de M. J. Cullum et G. Z. Angeli, vol. 6271. Soc. Photo-Opt. Instrum. Eng. (2006). [80] B. X. CARRY, C. Dumas, M. Fulchignoni, T. Fusco et W. Merline, Nearinfrared Mapping Of Ceres Surface From Keck, Dans Bulletin of the American Astronomical Society, vol. 38 de Bulletin of the American Astronomical Society, pp. 621–+ (septembre 2006). [81] A. R. Conrad, C. Dumas, W. J. Merline, R. D. Campbell, R. W. Goodrich, D. Le Mignant, F. H. Chaffee, T. Fusco, S. Kwok et R. I. Knight, Rotation

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71 [99] A. Boccaletti, D. Mouillet, T. Fusco, P. Baudoz, C. Cavarroc, J.-L. Beuzit, C. Moutou et K. Dohlen, Analysis of ground-based differential imager performance, Dans IAUC 200, Direct Imaging of Exoplanets : Science & Techniques (2005), Date conférence : octobre 2005, Nice, France. [100] A. Boccaletti, D. Mouillet, T. Fusco, P. Baudoz, J.-L. Beuzit, C. Moutou et K. Dohlen, Analysis of ground based differential imager performance, Dans IAUC 200, Direct Imaging of Exoplanets : Science & Techniques (2005), Date conférence : octobre 2005, Nice, France. [101] J.-L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, P. Puget, T. Fusco et A. e. a. Boccaletti, A planet Finder instrument for the VLT, Dans IAUC 200, Direct Imaging of Exoplanets : Science & Techniques (2005), Date conférence : octobre 2005, Nice, France. [102] M.-T. Velluet, G. Rousset et T. Fusco, Shack-Hartmann Wavefront Sensing errors induced by laser illumination, Dans Proc. SPIE Vol. 5981, Optics in Atmospheric Propagation and Adaptive Systems VIII, vol. 5981, pp. 188–196 (juin 2005). [103] R. Conan, T. Fusco, G. Rousset, D. Mouillet, J.-L. Beuzit, M. Nicolle et C. Petit, Modeling and analysis of XAO systems. Application to VLT-Planet Finder, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. PhotoOpt. Instrum. Eng. (2004), Date conférence : juin 2004, Glasgow, UK. [104] F. Zamkotsian, K. Dohlen et T. Fusco, Optical performance and phase error budgeting of future ELTs, Dans Modeling and Systems Engineering for Astronomy, sous la direction de S. C. Craig et M. J. Cullum, vol. 5497. Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2004). [105] C. Robert, J.-M. Conan, V. Michau et T. Fusco, Anisoplanatism in ShackHartmann wavefront sensing, Dans Optics in Atmospheric Propagation and Adaptive Systems VII, sous la direction de J. D. Gonglewski et K. Stein, vol. 5572. Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2004), Date conférence : septembre 2004, Spain. [106] R. Conan, T. Fusco et G. Rousset, High order Adaptive Optics simulation tool for VLTs and ELTs, Dans Science with Adaptive Optics, sous la direction de W. Brandner et M. Kasper. Springer-Verlag (2004), Date conférence : septembre 2003, Garching, Germany. [107] F. Quiros-Pacheco, C. Petit, J.-M. Conan, T. Fusco et E. Marchetti, Control law for the multiconjugate adaptive optics demonstrator (MAD), Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. Photo-Opt. Instrum. Eng. (2004), Date conférence : juin 2004, Glasgow, UK. [108] C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud et T. Fusco, Kalman filter based control for Adaptive Optics, Dans Scientific Highlights 2004, sous la direction de F. Combes, D.Barret, T. Contini, F. Meynadier et L. Pagani, EDP Sciences, SF2A (2004).

72 [109] C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco et G. Rousset, Kalman Filter based control loop for Adaptive Optics, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2004), Date conférence : juin 2004, Glasgow, UK. [110] M. Nicolle, T. Fusco, V. Michau, G. Rousset, A. Blanc et J.-L. Beuzit, Analysis of wavefront sensing concept for ground layer adaptive optics, Dans Scientific Highlights 2004, sous la direction de F. Combes, D.Barret, T. Contini, F. Meynadier et L. Pagani, EDP Sciences, SF2A (2004). [111] M. Nicolle, T. Fusco, V. Michau, G. Rousset, A. Blanc, J.-L. Beuzit et C. Petit, Ground Layer Adaptive Optics : wavefront sensing concepts, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. Photo-Opt. Instrum. Eng. (2004), Date conférence : juin 2004, Glasgow, UK. [112] J. Kolb, E. Marchetti, G. Rousset et T. Fusco, Calibration of an MCAO system, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. PhotoOpt. Instrum. Eng., SPIE (2004), Date conférence : juin 2004, Glasgow, UK. [113] T. Fusco, G. Rousset, R. Conan, N. Nicolle, C. Petit et J.-L. Beuzit, Endto-end model for XAO simulations. Application to Planet-Finder, Dans Scientific Highlights 2004, sous la direction de F. Combes, D.Barret, T. Contini, F. Meynadier et L. Pagani, EDP Sciences, SF2A (2004). [114] T. Fusco, N. Ageorges, G. Rousset, D. Rabaud, E. Gendron, D. Mouillet, F. Lacombe, G. Zins, J. Charton, C. Lidman et N. Hubin, NAOS performance characterization and turbulence parametres estimation using closedloop data, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. Photo-Opt. Instrum. Eng. (2004), Date conférence : juin 2004, Glasgow, UK. [115] T. Fusco, M. Nicolle, G. Rousset, V. Michau, J.-L. Beuzit et D. Mouillet, Optimisation of Shack-Hartmann-based wavefront sensor for XAO system, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. Photo-Opt. Instrum. Eng. (2004), Date conférence : juin 2004, Glasgow, UK. [116] T. Fusco, G. Rousset et A. Blanc, Calibration of AO system. Application to NAOS-CONICA, Dans Science with Adaptive Optics, sous la direction de W. Brandner et M. Kasper. Springer-Verlag (2004), Date conférence : septembre 2003, Garching, Germany. [117] Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler et N. Hubin, NACO performance : Status after 2 years of operation, Dans Advancements in Adaptive Optics, vol. 5490. Proc. Soc. Photo-Opt. Instrum. Eng. (2004), Date conférence : juin 2004, Glasgow, UK. [118] A. Blanc, T. Fusco, G. Rousset, V. Michau, J.-L. Beuzit et N. Hubin, Study of sky coverage for Multi-conjugate adaptive optics, Dans Scientific Highlights 2004, sous la direction de F. Combes, D.Barret, T. Contini, F. Meynadier et L. Pagani, EDP Sciences, SF2A (2004).

73 [119] F. Assémat, F. Hammer, E. Gendron, P. Laporte, M. Marteaud, M. Pueach, J. Conan, T. Fusco, A. Liotard et F. Zamkotian, FALCON : a new-generation spectrograph with adaptive optics for the ESO VLT, Dans Optics in Atmospheric Propagation and Adaptive Systems VI, sous la direction de J. D. Gonglewski et K. Stein, vol. 5237. Proc. Soc. Photo-Opt. Instrum. Eng. (2004), Date conférence : août 2004, Denver, USA. [120] F. Assémat, F. Hammer, E. Gendron, P. Laporte, M. Marteaud, M. Pueach, J. Conan, T. Fusco, A. Liotard et F. Zamkotian, FALCON : a new-generation spectrograph with adaptive optics for the ESO VLT, Dans Optics in Atmospheric Propagation and Adaptive Systems VI, sous la direction de J. D. Gonglewski et K. Stein, vol. 5237. Proc. Soc. Photo-Opt. Instrum. Eng. (2004). [121] M. Puech, F. Sayede, F. Hammer, M. Marteaud, E. Gendron, F. Assémat, P. Laporte, J.-M. Conan, T. Fusco, F. Zamkotskian et A. Liotard, FALCON : a new concept to extend adaptive optics corrections to cosmological fields, Dans Scientific Highlights 2003, EDP Sciences, SF2A (2003). [122] D. Mouillet, T. Fusco, A.-M. Lagrange et J.-L. Beuzit, "Planet Finder" on the VLT : context, goals and critical specification for adaptive optics, Dans Astronomy with High Contrast Imaging : From Planetary Systems to Active Galactic Nuclei, sous la direction de C. Aime et R. Soummer, vol. 8 de European Astronomical Society publication series (2003), Date conférence : mai 2002. [123] T. Fusco, L. M. Mugnier, J.-M. Conan, F. Marchis, G. Chauvin, G. Rousset, A.-M. Lagrange et D. Mouillet, Deconvolution of astronomical adaptive optics images, Dans Astronomy with High Contrast Imaging : From Planetary Systems to Active Galactic Nuclei, sous la direction de C. Aime et R. Soummer, vol. 8 de European Astronomical Society publication series (2003), Date conférence : mai 2002. [124] F. Assémat, F. Hammer, E. Gendron, F. Sayede, P. Laporte, M. Marteaud, M. Puech, J.-M. Conan, T. Fusco, A. Liotard et F. Zamkotskian, FALCON : a new generation spectrograph with adaptive optics for the ESO VLT, Dans Optics in atmospheric Propagation and Adaptive Systems IV. Proc. Soc. PhotoOpt. Instrum. Eng. (septembre 2003). [125] D. Mouillet, A.-M. Lagrange, J.-L. Beuzit, F. Ménard, C. Moutou, T. Fusco, L. Abé, T. Gillot, R. Soummer et P. Riaud, VLT-"Planet Finder" : Specifications for a ground-based high contrast imager, Dans Scientific Highlights 2002, sous la direction de F. Combes et D.Barret, EDP Sciences, SF2A (2002). [126] B. Le Roux, J.-M. Conan, T. Fusco, D. Bello, V. Michau et G. Rousset, Multiconjugate Adaptive Optics, Principle, Limitations, Dans Scientific Highlights 2002, sous la direction de F. Combes et D.Barret, EDP Sciences, SF2A (2002). [127] B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier et T. Fusco, Optimal control law for multiconjugate adaptive optics, Dans Adap-

74 tive Optical System Technology II, sous la direction de P. L. Wizinowich et D. Bonaccini, vol. 4839, Hawaii, USA, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002). [128] F. Lacombe, G. Zins, J. Charton, G. Chauvin, G. Dumont, P. Fautrier, T. Fusco, E. Gendron, N. Hubin, P. Kern, A.-M. Lagrange, D. Mouillet, P. Puget, D. Rabaud, P. Rabou, G. Rousset et J.-L. Beuzit, NAOS : from an AO system to an Astronomical Instrument, Dans Adaptive Optical System Technology II, sous la direction de P. L. Wizinowich et D. Bonaccini, vol. 4839, Hawaii, USA, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002). [129] M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. M. Mugnier, G. Rousset et R. Lenzen, Calibration of CONICA static aberrations by phase diversity, Dans Instrumental Design and Performance for Optical/Infrared Ground-Based Telescopes, sous la direction de M. Iye et A. F. M. Moorwood, vol. 4841, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002), Date conférence : août 2002, Waikoloa, Hawaii. [130] M. Hartung, R. Hofmann, A. Boehm, W. Brandner, T. Fusco, P. Granier, F. Lacombe, W. Laun, C. Storz et W. Wagner, CONICA design, performance and final laboratory tests, Dans Instrumental Desigtn and Performance for Optical/Infrared Ground-Based Telescopes, sous la direction de M. Iye et A. F. M. Moorwood, vol. 4841, Bellingham, Washington, Proc. Soc. PhotoOpt. Instrum. Eng., SPIE (2002), Date conférence : août 2002, Waikoloa, Hawaii. [131] F. Hammer, E. Sayede F. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J.-M. Conan, F. Courbin, H. Flores, I. Guinouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcine, F. Rigaud, D. Rouan, G. Rousset, V. Buat et F. Zamkotsian, The FALCON Concept : Multi-Object Spectroscopy Combined with MCAO in Near-IR, Dans Scientific Drivers for the ESO Future VLT/VLTI Instrumentation. ESO (2002). [132] T. Fusco, G. Rousset, A.-M. Lagrange, F. Lacombe, E. Gendron, P. Puget, N. Hubin, R. Arsenault, J. Charton, P. Kern, P. Gigan, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler et G. Zins, NAOS : the first AO system of the VLT, Dans Scientific Highlights 2002, sous la direction de F. Combes et D.Barret, EDP Sciences, SF2A (2002). [133] T. Fusco, L. M. Mugnier et J.-M. Conan, MISTRAL, a deconvolution algorithm for astronomical adaptive optics images, Dans Scientific Highlights 2002, sous la direction de F. Combes et D.Barret, EDP Sciences, SF2A (2002). [134] T. Fusco, L. M. Mugnier, J.-M. Conan, F. Marchis, G. Chauvin, G. Rousset, A.-M. Lagrange, D. Mouillet et F. Roddier, Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics, Dans Adaptive Optical System Technology II, sous la direction de P. L. Wizinowich

75 et D. Bonaccini, vol. 4839, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002). [135] P. Fautrier, G. Rousset, R. J. Dorn, C. Cavadore, J. Charton, C. Cumani, T. Fusco, N. Hubin, P. Kern, J.-L. Lizon, Y. Magnard, P. Puget, D. Rabaud, P. Rabou et E. Stadler, Performances and results on the sky of the NAOS visible wavefront sensor, Dans Adaptive Optical System Technology II, sous la direction de P. L. Wizinowich et D. Bonaccini, vol. 4839, Hawaii, USA, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002). [136] G. Chauvin, A.-M. Lagrange, F. Ménard, D. Mouillet, J.-L. Beuzit, C. Dumas, T. Fusco, P. Lowrance, M. Thomson, B. Zuckerman, J.-C. Augereau et F. Marchis, Adaptive Optics survey of Young and Nearby Associations, Dans Scientific Highlights 2002, sous la direction de F. Combes et D.Barret, EDP Sciences, SF2A (2002). [137] D. Bello, C. Verinaud, J.-M. Conan, T. Fusco, M. Carbillet et S. Esposito, Comparison of different 3D wavefront sensing and reconstruction techniques for MCAO, Dans Adaptive Optical System Technology II, sous la direction de P. L. Wizinowich et D. Bonaccini, vol. 4839, Hawaii, USA, Proc. Soc. PhotoOpt. Instrum. Eng., SPIE (2002). [138] A.-M. Lagrange, G. Chauvin, T. Fusco, E. Gendron, D. Rouan, M. Hartung, F. Lacombe, D. Mouillet, G. Rousset, P. Drossart, R. Lenzen, C. Moutou, W. Brandner, N. Hubin, Y. Clenet, A. Stolte, R. Schoedel, G. Zins et J. Spyromilio, First diffraction limited images at VLT with NAOS and CONICA, Dans Instrumental Design and Performance for Optical/Infrared Ground-Based Telescopes, sous la direction de M. Iye et A. F. Moorwood, vol. 4841, pp. 860–868, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002). [139] D. Bello, R. Conan, B. Le Roux, E. Marchetti, J.-M. Conan, T. Fusco, A. Tokovinin, M. Le Louarn, E. Viard et N. Hubin, Performance of a NGS-based MCAO demonstrator : the NGC3366 and NGC2346 simulations, Dans Beyond Conventional Adaptive Optics, sous la direction de E. Vernet, R. Ragazzoni, S. Esposito et N. Hubin, pp. 231–237, Garching-bei-München, Germany, Astronomical Observatory of Padova, ESO (2002), Date conférence : mai 2001. [140] J.-M. Conan, B. Le Roux, D. Bello, T. Fusco et G. Rousset, Optimal Reconstruction in Multiconjugate Adaptive Optics, Dans Beyond Conventional Adaptive Optics, sous la direction de E. Vernet, R. Ragazzoni, S. Esposito et N. Hubin, pp. 209–216, Garching-bei-München, Germany, Astronomical Observatory of Padova, ESO (2002), Date conférence : mai 2001. [141] T. Fusco, J.-M. Conan, V. Michau et G. Rousset, Noise propagation for Multiconjugate adaptive Optics systems, Dans Optics in atmospheric Propagation and Adaptive Systems IV, sous la direction de A. Kohnle, J. D. Gonglewski et

76 T. J. Schmugge, vol. 4538, pp. 144–155, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002), Date conférence : septembre 2001. [142] G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler et G. Zins, NAOS, the first AO system of the VLT : on sky performance, Dans Adaptive Optical System Technology II, sous la direction de P. L. Wizinowich et D. Bonaccini, vol. 4839, pp. 140–149, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2002). [143] N. Hubin, E. Marchetti, R. Conan, R. Ragazzoni, E. Diolaiti, G. Rousset, T. Fusco, P. Madec, D. Butler et S. Esposito, The ESO MCAO demonstrator : a European collaboration, Dans Beyond Conventional Adaptive Optics, sous la direction de E. Vernet, R. Ragazzoni, S. Esposito et N. Hubin, pp. 27–35, Garching-bei-München, Germany, Astronomical Observatory of Padova, ESO (2002), Date conférence : mai 2001. [144] F. Marchis, I. de Pater, D. Le Mignant, H. Roe, T. Fusco, J. R. Graham, R. Prangé et B. Macintosh, Monitoring Io volcanic activity using the Keck AO system : 2-5microns sunlit and clipse observations, Dans AGU meeting, San-Fransisco, USA (décembre 2002), à paraître. [145] L. M. Mugnier, T. Fusco, J.-M. Conan, V. Michau et G. Rousset, Deconvolution of adaptive optics corrrected images, Dans Scientific Highlights 2001, sous la direction de F. Combes, D.Barret et F. Thévenin, pp. 593–596, EDP Sciences, SF2A (2001). [146] J.-M. Conan, B. Le Roux, D. Bello, T. Fusco, L. M. Mugnier, V. Michau et G. Rousset, MultiConjugate Adaptive Optics : performance with optimal wavefront reconstruction, Dans Scientific Highlights 2001, sous la direction de F. Combes, D.Barret et F. Thévenin, pp. 541–544, EDP Sciences, SF2A (2001). [147] T. Fusco, J.-M. Conan, V. Michau, G. Rousset et F. Assémat, MultiConjugate Adaptive Optics : Comparison of phase reconstruction approaches for large Field of View, Dans Atmospheric Propagation, Adaptive Systems, and Laser Radar Technology for Remote Sensing, sous la direction de J. D. Gonglewski, G. W. Kamerman, A. Kohnle, U. Schreiber et C. H. Werner, vol. 4167, pp. 168–179, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng. (2001). [148] J.-M. Conan, T. Fusco, L. M. Mugnier, V. Michau, G. Rousset et P.-Y. Madec, Imagerie haute résolution grand champ par optique adaptative multiconjuguée, Dans Ateliers de l’optique en astronomie, sous la direction de J.-P. Lemonnier, M. Ferrari et P. Kern, pp. 31–36, Grenoble, France, INSU/CNRS (2001).

77 [149] T. Fusco, J.-M. Conan, V. Michau, G. Rousset et L. Mugnier, Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics, Dans Adaptive Optical Systems Technology, sous la direction de P. Wizinowich, vol. 4007, pp. 1044–1055, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng. (2000). [150] J.-M. Conan, T. Fusco, L. Mugnier, F. Marchis, C. Roddier et F. Roddier, Deconvolution of adaptive optics images : from theory to practice, Dans Adaptive Optical Systems Technology, sous la direction de P. Wizinowich, vol. 4007, pp. 913–924, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (2000). [151] T. Fusco, J.-M. Conan, L. Mugnier, V. Michau et G. Rousset, Postprocessing for anisoplanatic adaptive optics corrected images, Dans Propagation through the Atmosphere IV, sous la direction de M. Roggemann, pp. 108–119, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng. (août 2000). [152] T. Fusco, J.-M. Conan, V. Michau, L. Mugnier et G. Rousset, Optimal phase reconstruction in large field of view : application to multiconjugate adaptive optics systems, Dans Propagation through the Atmosphere IV, sous la direction de M. Roggemann, vol. 4125, pp. 65–76, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng. (août 2000). [153] J.-M. Conan, T. Fusco, L. M. Mugnier et F. Marchis, MISTRAL : Myopic Deconvolution Method Applied to ADONIS and simulated VLT-NAOS Images, ESO Messenger, 99, pp. 38–45 (mars 2000). [154] T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau et J.-P. Véran, Caractérisation et traitement d’images astronomiques à réponse impulsionnelle variable dans le champ, Dans 17ième Colloque sur le Traitement du Signal et des Images. GRETSI (1999). [155] T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier et G. Rousset, Phase estimation for large field of view : application to multiconjugate adaptive optics, Dans Propagation through the Atmosphere III, sous la direction de M. C. Roggemann et L. R. Bissonnette, vol. 3763, pp. 125–133, Bellingham, Washington, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE (1999). [156] M. Séchaud, F. Mahé, T. Fusco, V. Michau et J.-M. Conan, High resolution imaging through atmospheric turbulence : link between anisoplanatism and intensity fluctuations, Dans Optics in Atmospheric Propagation and Adaptive Systems V, pp. 243–249, Florence, Italy, ESO/SPIE (septembre 1999). [157] J.-M. Conan, T. Fusco, L. M. Mugnier, E. Kersalé et V. Michau, Deconvolution of adaptive optics images with imprecise knowledge of the point spread function : results on astronomical objects., Dans Astronomy with adaptive optics : present results and future programs, sous la direction de D. Bonaccini,

78 vol. 56 de ESO Conference and Workshop Proceedings, pp. 121–132, Garching bei München, Germany, ESO/OSA (février 1999). [158] L. M. Mugnier, J.-M. Conan, T. Fusco et V. Michau, Joint Maximum a Posteriori Estimation of Object and PSF for Turbulence Degraded Images, Dans Bayesian Inference for Inverse problems, vol. 3459, pp. 50–61, San Diego, CA (USA), Proc. Soc. Photo-Opt. Instrum. Eng. (juillet 1998). [159] J.-M. Conan, L. Mugnier, T. Fusco, V. Michau et G. Rousset, Deconvolution of adaptive optics images using the object autocorrelation and positivity, Dans Optical Science, Engineering and Instrumentation, vol. 3126, pp. 56–67, San Diego, CA (USA), Proc. Soc. Photo-Opt. Instrum. Eng. (juillet 1997).

Annexe A "On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio” T. Fusco & J.-M. Conan - JOSAA - 2004

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On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio Thierry Fusco and Jean-Marc Conan Office National Etudes et de Recherches Ae´rospatiales, B.P. 72, 92322 Chaˆtillon, France Received July 12, 2003; revised manuscript received December 5, 2003; accepted February 12, 2004 Statistical behavior of the adaptive-optics- (AO-) corrected short-exposure point-spread function (PSF) is derived assuming a perfect correction of the phase’s low spatial frequencies. Analytical expressions of the Strehl ratio (SR) fluctuations of on- and off-axis short-exposure PSFs are obtained. A theoretical expression of the short SR angular correlation is proposed and used to derive a definition of an anisoplanatic angle for AOcorrected images. Several applications of the analytical expressions are proposed: AO performance characterization, postprocessing imaging, light coupling into fiber, and exoplanet detection from a ground-based telescope. © 2004 Optical Society of America OCIS codes: 010.1080, 010.1330.

1. INTRODUCTION Atmospheric turbulence severely limits the angular resolution of ground-based telescopes. Adaptive optics1–3 (AO) is a powerful technique to overcome this limitation and to reach the diffraction limit of large astronomical telescopes. The turbulent wave front is measured by a wave-front sensor (WFS); these measurements are used to control a deformable mirror that performs a real-time compensation of the phase fluctuations. With such a system, most of the turbulence effects are corrected, which leads to a highly improved image. Nevertheless, this correction is never perfect and a residual uncorrected wave front still degrades the performance of scientific instruments,4–7 such as an imaging camera, a spectrograph or coronographic device, or a single-mode fiber for interferometry.8 The limitation is essentially due to the finite number of deformable-mirror actuators (or, in other words, of WFS subapertures), the aliasing effects, the finite system bandwidth, the guide star (GS) magnitude, and, of course, the turbulence parameters such as the C n2 profile and the wind speed distribution as a function of height. In the case of bright reference stars [an essential condition to achieve a very high Strehl ratio (SR)], the main part of the residual wave front is due to both the system fitting (plus aliasing) and the anisoplanatic errors. Indeed, since astronomical objects are usually too faint to be used as references for the WFS, a nearby bright star has to be considered to measure the turbulent wave front. In that case, the AO correction is also degraded by the socalled anisoplanatic effects that are the consequence of the atmospheric turbulence distribution more than 20 km in height. Thus wave fronts coming from two separated sources (the GS and the object of interest) do not cross the same portion of the turbulent atmosphere and therefore are not identically disturbed. A correction optimized for a given direction (the GS direction in the AO case) is no longer valid in another. This leads to a degradation of 1084-7529/2004/071277-13$15.00

the AO correction as a function of angle. The so-called isoplanatic field in which the correction is efficient is only approximately a few arcseconds at visible wavelength.9 These anisoplanatic effects are the main limitation of the classical AO-system field of view (FOV) and sky coverage.10–12 System fitting and anisoplanatic errors, of course, affect the shape of the corrected long-exposure point-spread function (PSF) as well as the short-exposure PSF statistics. Long exposures have been extensively studied during the past ten years.3 Current AO applications require, however, as will be discussed below, a precise knowledge of the short-exposure PSF statistics. The objective of this paper is to study several key parameters related to short exposures. The results obtained by numerical simulation will be associated with analytical developments. The validity domain of the analytical expressions, and the underlying approximations, will be discussed. The analytical expressions both help the physical interpretation and give useful and simple expressions to evaluate the relevant parameters. We study in particular the evolution of the instantaneous SR (ISR) statistics as of function of the number of corrected modes and of the turbulence strength. The anisoplanatism effects induced by the angular distance between the source and the GS are accounted for. The present paper is complementary to recent publications on short-exposure statistics.13,14 The originality of our approach is to propose closed-form analytical expressions to account for anisoplanatism effects, all these aspects being studied in a wide range of turbulence conditions. Before going further, let us stress the importance of AO-corrected PSF statistics by briefly presenting a few potential applications. • Postprocessing of AO-corrected images. Knowledge of the PSF statistic is an important issue for deconvolution of AO-corrected images. Generally, the PSF is not known accurately, which makes deconvolution difficult. © 2004 Optical Society of America

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T. Fusco and J.-M. Conan

Myopic deconvolution15–17 is an approach that allows one to overcome imprecise knowledge of the PSF by using knowledge of the PSF statistic. • Light coupling into a fiber, hole, or slit for stellar interferometry or spectroscopy. Another application takes place in the spatial-filtering process with a single-mode fiber at a telescope focus. In a first approximation, assuming a perfect single-mode system, the estimator of coupling efficiency is close to the ISR for weak phase fluctuations.8 Therefore, for a given system, several atmospheric conditions, and several object magnitudes, a comparison between photon noise and SR fluctuations can easily be made. • Study of residual speckle evolution for a very-highorder AO system. This is directly linked to the search for extrasolar planets.18–20 One of the main limitation of the detection is the residual speckle boiling. A study of the AO long-exposure PSF statistics is essential to assess future system performances. The paper is structured as follows. AO image formation is briefly recalled in Section 1, and then definitions of the SR and coherent energy (CoEn) are recalled in Section 2 with a particular highlight on differences between these two estimators of the AO image quality. Sections 3 and 4 are dedicated to the development of an analytical expression of on-axis ISR statistics. A generalization to off-axis ISR fluctuations is proposed in Sections 5 and 6. Finally, the extension to the whole PSF statistics is proposed in Section 7.

2. ADAPTIVE OPTICS SHORT-EXPOSURE POINT-SPREAD FUNCTION In the case of a ground-based telescope with AO, the short-exposure image of an unresolved star, that is, the instantaneous PSF of the system (PSF), is given by I ␣ 共 ␳兲 ⫽ 兩 FT兵 P共 r兲 A 共 r, ␣ 兲 exp关 i⌽ res共 r, ␣ 兲兴 其 兩 2 ,

(1)

where P(r) is the telescope pupil, A(r, ␣ )exp关i⌽res(r, ␣ ) 兴 is the complex amplitude in the pupil after correction for the direction ␣ in the FOV 关 ⌽ res(r, ␣ ) is the residual phase that has not been corrected by the AO system], and FT is the Fourier transform operator. For the sake of simplicity, let us assume that atmospheric turbulence induces only phase fluctuations21 [i.e., a(r) ⫽ const.]. For infrared imaging and a typical C n2 profile, such an approximation is well verified.11,22 The center of the FOV ( ␣ ⫽ 0) is defined as the AO GS direction. If the object of interest is bright enough, it can be used as a GS, but, in most cases, one has to consider a nearby bright star to estimate and then to correct the turbulent wave fronts. Let us define ␣ as the distance between such a GS and the object of interest. On the other hand, ␳ stands for the location of a given point of the PSF shape in the detector focal plane. For a given distance ␣ between the GS and the direction of interest, we define a PSF I ␣ ( ␳) in the whole instrument focal plane. After correction by AO, the residual phase is nothing but the difference between the turbulent phase

⌽ turb(r, ␣ ) coming from the object of interest and the correction phase ⌽ corr(r, 0) (obtained by use of the measurement on the on-axis GS): ⌽ res共 r, ␣ 兲 ⫽ ⌽ turb共 r, ␣ 兲 ⫺ ⌽ corr共 r, 0兲 .

(2)

Using a Zernike modal decomposition of the turbulent phase and assuming that the AO system provides a correction of each Zernike mode up to i 0 , we obtain the following relation10: i0

⌽ res共 r, ␣ 兲 ⫽

兺 关a

i,turb共 ␣ 兲

⫺ a i,corr共 0 兲兴 Z i 共 r兲

i⫽2

⬁

⫹

兺

i⫽i 0 ⫹1

a i,turb共 ␣ 兲 Z i 共 r兲 .

(3)

Let us assume that the GS is bright enough (and the system is fast enough) to neglect both temporal and noise propagation errors in comparison with the fitting and the anisoplanatic ones. In that case, a i,corr(0) ⯝ a i,turb(0) ᭙ i ⭐ i 0 . In addition, if we consider on-axis AO correction ( ␣ ⫽ 0), Eq. (3) becomes ⬁

⌽ res共 r, 0兲 ⫽

兺

i⫽i 0 ⫹1

a i,turb共 0 兲 Z i 共 r兲 .

(4)

In that case, the residual variance obtained from Eq. (4) is nothing but the so-called fitting error. It is important to recall here that we deal with a simple but not completely realistic system, since a perfect correction of the low-order modes is assumed. In particular, this first-order simplification means that noise, temporal, and aliasing effects have been neglected in comparison with fitting and anisoplanatism errors. In other words, that means that we consider a system temporal bandwidth far larger than the turbulence temporal cutoff frequency23,24 associated with a very bright GS. This very first approximation of an AO system is required to obtain simple expressions that are essential for a good understanding of the behavior of AO system performances.

3. STREHL RATIO AND COHERENT ENERGY There are many ways to quantify the correction quality of an AO system; one of the most currently used is the SR, which can be defined as the AO-corrected PSF value at the center of the focal plane normalized by the value of the Airy pattern at the same position. If the AO system could perfectly correct all the turbulence effects, its SR would therefore be equal to 1. On the other hand, CoEn is defined5 as follows: 2 CoEn共 ␣ 兲 ⫽ exp关 ⫺␴ res 共 ␣ 兲兴 ,

(5)

2 ( ␣ ) stands for the residual phase variance in where ␴ res the direction ␣. Note that in the case of good correction, 2 ␴ res Ⰶ 1 and Eq. (5) can be approximated by CoEn( ␣ ) 2 ( ␣ ). ⯝ 1 ⫺ ␴ res The SR characterizes the energy concentration in the PSF coherent peak and thus is a relevant parameter for coupling problems behind AO (interferometry, spectroscopy). On the other hand, CoEn is an estimator of the

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Vol. 21, No. 7 / July 2004 / J. Opt. Soc. Am. A

attenuation of the optical transfer function’s high frequencies; it is therefore the relevant parameter for highresolution imaging applications. Nevertheless, the SR and CoEn are strongly linked as shown below. A. Long-Exposure Case Let us first consider long-exposure AO correction. In that case, following Goodman25 [Eq. (8.3-22)], we can decompose the long-exposure PSF into two different parts, I ␣ 共 ␳兲 ⯝ I ␣coh共 ␳兲 ⫹ I halo ␣ 共 ␳兲 ,

冉冊 ␭

D

2

I ␣共 0 兲 ⯝

冉冊 ␭

2

D

2 I ␣coh共 0 兲 ⫹ L halo I ␣halo共 0 兲 ,

SR共 ␣ 兲 ⫽ CoEn共 ␣ 兲 ⫹ 关 1 ⫺ CoEn共 ␣ 兲兴

where

Note that such a decomposition in two terms is only an approximation. It does not completely represent the PSF shape, but rather it only allows one to introduce and compare two classical performance metrics used in AO: the SR and CoEn. In particular, we want to stress the difference (often neglected) between the two quantities. We illustrate the decomposition proposed in expression (6) in Fig. 1 where a corrected PSF (36 Zernike modes fully compensated, D/r 0 ⫽ 40) is plotted. A coherent core and halo are clearly seen. In addition, the halo is well fitted by an uncorrected PSF obtained for D/r ( 0,eq) ⫽ 23, which clearly illustrates the halo-width-reduction effect brought by the correction.26 Both the coherent core and the halo can be approximated (following Roddier)21 by a gate function with, respectively, ␭/D and ␭/r 0,eq widths. Thus, from expression (6), we can write

(7)

where L halo stands for the FWHM of the halo pattern and is related to ␭/r ( 0,eq) , which depends on the AO correction degree and type (number of corrected modes, for instance).26 It is interesting to note here that, by using expressions (6) and (7), we can find an analytical expression between the CoEn and the long-exposure SR:

(6)

• I ␣coh( ␳) ⯝ CoEn( ␣ )Airy( ␳) is a coherent peak. It is nothing but the peak of the diffraction-limit pattern 2 [Airy(␳)] with an exp关⫺␴res ( ␣ ) 兴 attenuation. halo • I ␣ ( ␳) is a halo pattern that is due to the residual noncorrected turbulence. The halo width is roughly equal to ␭/r 0,eq , which is somewhat narrower than for uncorrected PSF. r 0,eq is related to the low-frequency lobe of the corrected optical transfer function.26 Because of the AO correction, r 0,eq is greater than r 0 with the gain depending on the AO correction degree.

1279

共 ␭/D 兲 2 2 L halo

.

(8)

In all cases, SR is greater than CoEn. The often used approximation SR ⯝ CoEn is valid in only two cases: • If CoEn is close to 1, that is, in the case of good correction. For a typical AO system on an 8-m class telescope and working at near-infrared wavelengths, the approximation is valid for SR ⭓ 30%. • If L halo is large enough, for instance, in the case of a good correction of a large number of modes by AO but with very strong D/r 0 (the case, for example, of an AO system, designed for the near infrared but working in the visible with a very bright GS).27 Without any AO system, the SR is roughly equal to (r 0 /D) 2 . For partial correction cases, its value depends both on D/r 0 and on the AO correction degree. B. Short-Exposure Case: The Instantaneous Strehl Ratio As presented above, the SR (or CoEn) are usually defined on long exposures, but the definition is still valid for short ones; the ISR obtained in this way can be written as ISR␣ ⫽

I ␣共 0 兲 Airy共 0 兲

.

(9)

ISR␣ can be seen as a short-exposure correction quality for a given observation angle. Let us assume that, for typical near-infrared D/r 0 (less than a few tens), for classical AO systems and for a GS not too far from the object of interest, we have ␸ (r, ␣ ) Ⰶ 1. Under this small phase perturbation assumption, a second-order expansion of the short-exposure PSF [Eq. (1)] gives

冋

冏 再

I ␣ 共 ␳ 兲 ⯝ FT ⌸ 共 r, ␣ 兲 1 ⫹ i ␸ 共 r, ␣ 兲 ⫺

␸ 共 r, ␣ 兲 2 2

册冎 冏

2

,

␳

I ␣ 共 ␳ 兲 ⯝ 兩 FT关 ⌸ 共 r兲兴 ␳ ⫹ i 兵 FT关 ⌸ ␸ 共 r, ␣ 兲兴 ␳ 其 ⫺ 兵 FT关 ⌸ ␸ 共 r, ␣ 兲 2 兴 ␳ 其 /2兩 2 ,

(10)

and, for ␳ ⫽ 0 (center of the focal plane), we obtain I ␣共 0 兲 ⯝ Fig. 1. Cross section of the long-exposure PSF: full correction of 36 Zernike modes with D/r 0 ⫽ 40 (solid curve), no correction with D/r 0 ⫽ 40 (dashed curve), and no correction with D/r 0,eq ⫽ 23 (dotted curve).

再冋 冕 冕 ⫺

册 冋冕 冕 2

⌸ 共 r兲 dr

⌸ 共 r兲 dr

⫹

⌸ 共 r兲 ␸ 共 r, ␣ 兲 dr

冎

册

2

⌸ 共 r兲 ␸ 2 共 r, ␣ 兲 dr ,

I ␣ 共 0 兲 ⯝ Airy共 0 兲 兵 1 ⫺ var关 ⌽ 共 r, ␣ 兲兴 其 ,

(11)

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where Airy is the aberration-free PSF 关 Airy(0) ⫽ S 2 兴 , var关 ⌽(r, ␣ ) 兴 ⫽ (1/S) 兰 ⌸(r)⌽ 2 (r, ␣ )dr, ⌽ is the pistonremoved phase 关 ⌽(r, ␣ ) ⫽ ␸ (r, ␣ ) ⫺ (1/S) 兰 pup ⫻ ␸ (r, ␣ )dr兴 , and S is the pupil surface 关 S ⫽ 兰 ⌸(r)dr兴 . The piston has been neglected because it does not affect image formation. var关 ⌽(r, ␣ ) 兴 is not a variance in the statistical sense of the word but rather an instantaneous spatial variance. It is a random variable giving the instantaneous spatial mean-square value of the phase in the pupil.

var共 ⌽ 兲 ⫽

• The mean value 具 ISR␣ 典 ⫽ 具 I ␣ (0) 典 /Airy(0), which is nothing but the classical SR defined on the long-exposure PSF, 2 • The variance ␴ ISR . ␣

共 second-order expansion兲 ,

具 I ␣ 共 0 兲 2 典 ⯝ Airy共 0 兲 2 兵 1 ⫺ 2 具 var关 ⌽ 共 r, ␣ 兲兴 典 ⫹ 具 var关 ⌽ 共 r, ␣ 兲兴 2 典 其 共 fourth-order expansion兲 .

(12)

Note that, for one to obtain the following expressions, a fourth-order expansion of Eq. (1) is actually necessary. For the sake of clarity, we do not detail the calculation; nevertheless, it is interesting to underline the fact that all the high-order terms cancel out, which finally leads to the simple expressions presented in expressions (13) and (14):

⫽

冕

a j Z j 共 r兲 dr

j⫽i 0 ⫹1 ⫹⬁

1

兺 兺

a ia j

i⫽i 0 ⫹1 j⫽i 0

2

which leads to the standard deviation of the ISR␣ : 共 fourth-order expansion兲 .

(14)

5. ON-AXIS FLUCTUATIONS OF THE INSTANTANEOUS STREHL RATIO A. Analytical Expressions Let us first focus on on-axis ISR fluctuation ( ␣ ⫽ 0), that is, we assume that the object of interest is also used for the WFS, or, in other words, we neglect the anisoplanatic effects. For the sake of simplicity, let us define, for this section, ␸ (r, ␣ ) ⫽ ␸ (r, 0) ⫽ ␸ (r). Assuming a pupil with no central occultation and an AO correction described in Eq. (3), the instantaneous spatial variance of the phase can be rewritten as

S

冕

Z i 共 r兲 Z j 共 r兲 dr,

since 兵 Z i 其 is an orthonormal basis: ⫹⬁

兺

var共 ⌽ 兲 ⫽

i⫽i 0 ⫹1

a i2 .

(15)

The mean and variance calculations are as follows. • Mean calculation: ⫹⬁

具var共 ⌽ 兲 典 ⫽

兺

i⫽i 0

具 a i2 典 ⫽ ␴ ⌽2 ,

(16)

2 where ␴ ⌽ stands for the residual variance of the phase. It is shown in Appendix A that, for n 0 Ⰷ 1, the classical fitting error can be expressed as 2 ␴⌽ ⫽ 0.458* 共 n 0 ⫹ 1 兲 ⫺5/3共 D/r 0 兲 5/3. (17) This expression is very close to the empirical law given by Noll.28 Note that i 0 is nothing but the last Zernike polynomial in the radial order n 0 .

• Variance calculation: We calculate 具 var(⌽) 2 典 by assuming that intercorrelations between coefficients are negligible. This assumption is valid, since the AO correction removes the lowest Zernike polynomials, which are the most intercorrelated modes:

冓兺 ⫹⬁

具var共 ⌽ 兲 典 ⫽ 2

⬁

兺

a i2 a j2

i⫽i 0 ⫹1 j⫽j 0 ⫹1

⫹⬁

⯝ 兵 1 ⫺ 2 具 var关 ⌽ 共 r, ␣ 兲兴 典 ⫹ 具 var关 ⌽ 共 r, ␣ 兲兴 典 其 , (13)

␴ ISR␣ ⯝ ␴ var关 ⌽ 共 r, ␣ 兲兴

册

册

Z i 共 r兲 Z j 共 r兲 dr ⫽ ␦ ij ,

具 ISR␣ 典 ⯝ 兵 1 ⫺ 具 var关 ⌽ 共 r, ␣ 兲兴 典 其 , 具 ISR␣2 典

冋兺

⫹⬁

These two first moments of ISR␣ can be written as

具 I ␣ 共 0 兲 典 ⯝ Airy共 0 兲 兵 1 ⫺ 具 var关 ⌽ 共 r, ␣ 兲兴 典 其

a i Z i 共 r兲

i⫽i 0 ⫹1

⫻

S

ISR␣ is a statistical process that can be completely characterized by its probability law. In this paper we focus only on the two first moments of this law:

S

⫹⬁

1

4. INSTANTANEOUS STREHL RATIO STATISTIC

冕冋 兺

⫹My

1

⯝

冔

⫹⬁

兺

i⫽i 0 ⫹1

具 a i4 典

⫹

⫹⬁

兺 兺 具 a 典具 a 典 . 2 i

j⫽i 0 ⫹1 i⫽j

2 j

(18)

Let us consider the turbulent atmosphere as a sum of thin turbulent layers.21 Each layer disturbs the incoming wave front, and then the phase on the telescope pupil can be seen as a sum of independent variables. By the central limit theorem, the probability law of the turbulent phase on the pupil is Gaussian; therefore Zernike coefficients also follow a Gaussian probability law. Using properties of the second- and fourth-order moments of such a statistic, we obtain ⫹⬁

⫹⬁

具var共 ⌽ 兲 2 典 ⯝ 3

兺

i⫽i 0 ⫹1 ⫹⬁

具var共 ⌽ 兲 典 ⫽ 2

␴ a4 i ⫹

兺 兺

i⫽i 0 ⫹1 j⫽i 0 ⫹1

兺

i⫽i 0 ⫹1

兺 兺␴

j⫽i 0 ⫹1 i⫽j

2 2 ai␴ ai,

(19)

⫹⬁

⫹⬁

⫽

⫹⬁

␴ a2 i ␴ a2 j ⫹⬁

␴ a4 i

⫹

兺兺␴ i⫽j

4 ai.

(20)

T. Fusco and J.-M. Conan

Vol. 21, No. 7 / July 2004 / J. Opt. Soc. Am. A

1281

Considering Eqs. (16) and (20), we can obtain the following expression: ⫹⬁

2 ␴var⌽

⯝2

兺

i⫽i 0 ⫹1

␴ a4 i .

(21)

Because all the Zernike coefficients in the same radial m 2 have the same variance ␴ m , this equation can be rewritten as ⫹⬁

兺

2 ␴var⌽ ⯝2

4 . 共 m ⫹ 1 兲␴m

(22)

m⫽n 0⫹1

This leads to (see Appendix A for the complete calculation) 2 ␴ var⌽ ⯝ 0.218共 n 0 ⫹ 1 兲 ⫺16/3

冉冊 D

4 ␴⌽

10/3

⯝ 1.04

r0

, 共 n0 ⫹ 1 兲2 (23)

4 4 where ␴ ⌽ is given in Appendix A: ␴ ⌽ ⯝ 0.2098(n 0 ⫺10/3 10/3 ⫹ 1) (D/r 0 ) . We have now a simple and analytical expression of 2 ␴ var⌽ , which allows us to express the ISR statistic on the optical axis (that is, for ␣ ⫽ 0). The two first moments of the random variable ISR0 can be deduced from expression (23),

冋

具 ISR0 典 ⯝ 共 1 ⫺ ␴ ⌽2 兲 ⯝ 1 ⫺ 0.458共 n 0 ⫹ 1 兲 ⫺5/3

冉冊册 D

r0

5/3

,

(24) which is the classical approximation of the SR in the case of good AO corrections, that is, for small residual phase amplitudes, and 2 2 ␴ ISR ⯝ ␴ var⌽ ⯝ 1.04 0

4 ␴⌽

共 n0 ⫹ 1 兲2

,

(25)

2 with the residual variance ␴ ⌽ given in Appendix A. By use of expressions (24) and (25) and the small phase amplitude assumption, the ISR fluctuation rate can be defined as

␴ ISR0

具 ISR0 典

⯝ 1.02

2 ␴⌽

共n0 ⫹ 1兲

,

(26)

2 where ␴ ⌽ is defined in Eq. (17). It is interesting to highlight here that the ISR0 fluctuation rate depends on (D/r 0 ) 5/3 and (n 0 ⫹ 1) ⫺8/3:

␴ ISR0

具 ISR0 典

⯝ 0.467共 n 0 ⫹ 1 兲 ⫺8/3

冉冊 D

r0

5/3

.

(27)

B. Comparison between Theory and Simulation In this subsection, numerical simulations are presented to validate and illustrate the theoretical expressions derived in the previous sections. Wave fronts are simulated by use of the Roddier approach.29 It is based on a ran-

Fig. 2. Intensity fluctuation rate ( ␴ ISR0 / 具 ISR0 典 ) as a function of D/r 0 for different kinds of correction (one to seven corrected radial orders). Symbols, computed ISR0 rates from simulation; solid lines, predicted the ISR0 fluctuation rate from the analytical expression. One can note that all the simulated curves saturate to 1 for large D/r 0 values that are large residual phase fluctuations.

dom generation of Zernike polynomials following a Kolmogorov statistic. Only the first 861 coefficients are considered, but the error due to this limited number of polynomials is shown to be negligible for the D/r 0 considered here.26 Turbulent phase screens of 128 ⫻ 128 pixels and their associated Shannon-sampled PSFs (256 ⫻ 256 pixels) are created by use of these 861 Zernike polynomials (i.e., the first 40 radial orders). It corresponds to a pupil diameter of 8m. It has been checked that the turbulent phases follow the well-known statistic in terms of a spatial power spectrum. It has also been checked that the number of pixels in the pupil (128 ⫻ 128) is large enough to appropriately create high Zernike polynomial orders (up to 861) and then high spatial frequencies. In the following simulations, different correction levels are considered, from one to six corrected radial orders. For each correction degree, a large scale of D/r 0 values is considered (from 0.1 to 100). To obtain an accurate statistic and to reduce the convergence noise, we simulate 1000 decorrelated phase screens and then short-exposure AO-corrected PSFs for each correction case. From these 1000 corrected short exposures, the fluctuation rate of ISR0 ( ␴ ISR0 / 具 ISR0 典 ) is computed. Its evolution as a function of n 0 as well as D/r 0 is plotted in Fig. 2 and compared with the analytical expression derived in Section 3. The curves presented in Fig. 2 present two distinct behaviors. 1. For small residual phase fluctuation 关 ␸ (r) Ⰶ 1 兴 , that is, for small D/r 0 or large correction degrees or both, the behavior predicted by the analytical expressions is well observed with the expected (D/r 0 ) ( 5/3) and (n 0 ⫹ 1) ( ⫺8/3) behaviors. Even for n ⫽ 1, the agreement between theory and simulation is quite good. Nevertheless, some differences can be seen for D/r 0 values around 1. In that case, the Zernike coefficient correlations are significant [the approximation made in expression (18) is no longer valid] and lead to a transition range. Considering the two asymptotic laws, one can note

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• An overestimation by the theory for a small value of D/r 0 (the case of a small perturbation approximation), • An underestimation for strong D/r 0 . These can be seen as a competition between the residual statistic of the angle of arrival and the blurring.21 As soon as n 0 is greater than 1 and for D/r 0 values compatible with the small perturbation phase approximation (the larger n 0 , the larger the range of D/r 0 values), the analytical expression perfectly fits the simulated values. 2. For large residual phase fluctuations, the ISR0 fluctuation rate saturates to 1. In this particular case, there is no coherent peak in the long-exposure PSF any more, even if the first turbulent modes are corrected. The only effect of AO is to reduce the number of speckles. The PSF statistic is therefore completely dominated by the well-known speckle boiling phenomena.21 Such a boiling statistic has been already studied in the case of laser30 and stellar Speckle.31 The complex amplitude in a point M of the focal plane can be considered a sum of random phasors.25 The focal-plane-intensity probability law is therefore given by (28) PB共I兲 ⫽ 1/具 I 典 exp共⫺I/具I典兲, with a fluctuation rate of the focal-plane intensities equal to ␴I /具I典 ⫽ 1. (29) This behavior is similar to the well-known result in laser speckle: 具 I 典 ⫽ ␴ I . Note that the main result of this subsection, ␴ I / 具 I 典 ⫽ 1, is valid for each point of the focal plane and in particular for the intensity at the center of the focal plane 关 I(0) 兴 . Then for the normalized intensity I 0 we also obtain the relation

␴I0 /具I0典 ⫽ 1.

(30)

In conclusion, the theoretical expressions derived in Subsection 5.A have been validated by simulation. It is shown that the slight hypothesis made to obtain these simple and easily usable expressions does not introduce any quantifiable errors as soon as the correction degree is greater than a simple tip–tilt correction. All the analytical developments presented here have been made in the case of a full correction of low-order modes. This could be extended to the more general case of a partial correction of low-order modes. Of course, such an extended approach needs to account for a larger number of parameters (GS magnitude, detector noise, sampling frequency, system bandwidth, etc.), which leads to more complex expressions. However, because of the residual low-order modes, one would have to check if the approximation made in expression (18) (intercorrelations neglected) is still valid.

fluence of the angular separation between the object of interest and the GS used by the WFS for the computation of the AO compensation. A. Theoretical Expression As explained in Section 4, we still consider here a AO system providing a perfect correction of the i 0 first Zernike polynomials. The main difference is that this perfect correction is obtained in the GS direction (0) and not for the object-of-interest position (␣). The residual phase in this direction is therefore given by the following relation:

(31) where a i,0 stands for the ith Zernike coefficient for the onaxis GS ( ␣ ⫽ 0) and a i, ␣ stands for the coefficient in the observing direction (␣). As already proposed in Subsection 5.A, one can therefore compute both the statistical mean and the variance of var⌽ 关 (r, ␣ ) 兴 . • Mean calculation We recall here the definition of var⌽ 关 (r, ␣ ) 兴 : ⬁

i0

var关 ⌽ 共 r, ␣ 兲兴 ⫽

兺

i⫽2

共 a i, ␣ ⫺ a i,0兲 2 ⫹

In the previous sections, we have focused on the effect on the ISR statistic of the partial AO correction due to the limited number of corrected modes. Let us now introduce the influence of anisoplanatic effects, that is, the in-

a i,2 ␣ . (32)

Again, a computation of the statistical mean and variance of the instantaneous spatial variance [see Eq. (B4)] in the telescope pupil can be obtained: ⬁

i0

具var关 ⌽ 共 r, ␣ 兲兴 典 ⫽ 2

兺共 i⫽2

␴ a2 i

⫺ 具 a i, ␣ a i,0典 兲 ⫹

兺

i⫽i 0 ⫹1

␴ a2 i .

(33) If we denote 具 a i, ␣ a i,0典 as C i ( ␣ ), we obtain (see Appendix B) ⬁

i0

具var关 ⌽ 共 r, ␣ 兲兴 典 ⫽ 2

兺关 i⫽2

⫺ C i 共 ␣ 兲兴 ⫹

␴ a2 i

兺

i⫽i 0 ⫹1

␴ a2 i .

(34) Note that we have here the analytical expression of the angular evolution of the residual phase variance.10 • Variance calculation Let us first develop the second-order moment as follows:

具 兵 var关 ⌽ 共 r, ␣ 兲兴 其 2 典

冓兺 ⬁

⫽

⬁

兺

冔 冓兺 兺 i0

i⫽i 0 ⫹1 j⫽i 0 ⫹1

a i,2 ␣ a j,2 ␣

i0

⫹

i⫽2 j⫽2

2 ⫺ 2a i, ␣ a i,0兲共 a j,2 ␣ ⫹ a j,0 ⫺ 2a j, ␣ a j,0兲

6. OFF-AXIS FLUCTUATIONS OF THE INSTANTANEOUS STREHL RATIO

兺

i⫽i 0 ⫹1

冓兺 冓兺 ⬁

⫹2

⫹2

冔

i0

兺a

i⫽i 0 ⫹1 j⫽2 ⬁

2 共 a i,2 ␣ ⫹ a i,0

2 ⫹ a j,0 ⫺ 2a j,0a j, ␣ 兲

2 2 i, ␣ 共 a j, ␣

2 ⫹ a j,0 ⫺ 2a j,0a j, ␣ 兲 .

i0

兺a

j⫽i 0 ⫹1 i⫽2

冔 冔

2 2 i, ␣ 共 a j, ␣

(35)

T. Fusco and J.-M. Conan

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1283

By use of Eqs. (35) and (34) the variance expression reads

再兺 ⬁

2 ␴var 关 ⌽ 共 r, ␣ 兲兴 ⫽ 2

i⫽i 0 ⫹1

冎

i0

␴ a4 i ⫹ 4

兺 关␴ i⫽2

2 ai

⫺ C i 共 ␣ 兲兴 2 . (36)

Assuming that we are still in the small perturbation hypothesis and using relations (13) and (14), we can express the two first moments of the ISR␣ as

再兺 ⬁

具 ISR␣ 典 ⯝ 1 ⫺

i⫽i 0 ⫹1

再兺

⫹2

⬁

2 ␴ ISR ⯝2 ␣

i⫽i 0 ⫹1

冎

i0

␴ a2 i

兺 关␴ i⫽2

i0

␴ a4 i ⫹ 4

兺 关␴ i⫽2

2 ai

2 ai

⫺ C i 共 ␣ 兲兴 , (37)

冎

⫺ C i 共 ␣ 兲兴 2 . (38)

These two last expressions simply mean that the greater ␣, the smaller the average SR and the greater the variance around this average value. This is the expected behavior, since it is well known that the correction quality decreases when the distance between the GS and the object of interest increases. But the main interest here is to have quantitative results in addition to qualitative ones. B. Comparison between Theory and Simulation Again let us compare the analytical expression found in Subsection 6.A with numerical simulations. We use the same simulation procedure presented in Subsection 5.B. In addition, we consider here a turbulence profile composed of two turbulent layers located at 0 and 10 km and with the same strength (C n2 * ⌬h) to simulate the anisoplanatic effects. For a given angle ␣, the corresponding part of the upper layer is added to the groundlayer contribution and gives the resulting phase in the telescope pupil for the ␣ direction. We plot in Fig. 3 the ISR␣ evolution for D/r 0 ⫽ 14 and three kinds of AO correction (35, 66, and 120 corrected Zernike polynomials) and in Fig. 4 the ISR␣ evolution for

Fig. 4. SR Strehl ratio fluctuation rate as a function of angle. Three D/r 0 values are considered: 14 (asterisks), 10 (pluses), and 5 (diamonds). For each case, the correction degree is equal to 66 corrected Zernike polynomials. The theoretical curves are plotted (solid curves) for comparison.

various D/r 0 (5, 10, and 14) and a given AO correction degree (66 corrected Zernike polynomials) These two figures show the good agreement between the simulation and the analytical results for small angular separations (and therefore weak phase fluctuations). As soon as the angular separations increase, the ISR␣ tends (and saturates) to 1 when the small phase perturbation hypothesis is no longer valid. In that case, the PSF statistic is dominated by the speckle boiling phenomena as already shown in Subsection 5.B.

7. ANGULAR CORRELATION OF THE INSTANTANEOUS STREHL RATIO In Section 6 we have studied the influence of an angular separation between the WFS GS direction and the objectof-interest direction. But we have considered only onaxis effects (in term of ISR fluctuations). Let us now study the angular decorrelation of the ISR as a function of the FOV position. This study can be seen as a generalization, in the case of AO correction, of the Roddier et al. study of the uncorrected short-exposure angular decorrelation.32

A. Theoretical Expression Let us consider the correlation function of the ISR as

C共 ␣ 兲 ⫽

具 I 0 共 0 兲 I 0 共 ␣ 兲 典 ⫺ 具 I 0 共 0 兲 典具 I 0 共 ␣ 兲 典 关 ␴ I2

0共 0 兲

Fig. 3. SR fluctuation rate as a function of angle. Three correction degrees are considered: 35 (asterisks) 66 (pluses) and 120 (diamonds) Zernike polynomials. In all the cases D/r 0 is equal to 14. The theoretical curves are plotted (solid curves) for comparison.

␴ I2

0共 ␣ 兲

兴 1/2

,

(39)

where I 0 ( ␣ ) stands for the intensity at the center of the focal plane for a given angular direction (on the sky) ␣. It is interesting to note that such a definition is similar to the one used by Roddier et al. in Ref. 32 for the definition of uncorrected short-exposure isoplanatic angles. We still consider here an on-axis perfect correction of the first i 0 Zernike coefficients. In that case, assuming a

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T. Fusco and J.-M. Conan

weak phase fluctuation case, the ISR angular correlation [defined in Eq. (39)] is equal to

C共 ␣ 兲 ⫽

具 var关 ⌽ 共 r, 0兲兴 var关 ⌽ 共 r, ␣ 兲兴 典 ⫺ 具 var关 ⌽ 共 r, 0兲兴 典具 var关 ⌽ 共 r, ␣ 兲兴 典 2 2 1/2 兵 ␴ var 关 ⌽ 共 r,0兲兴 ␴ var关 ⌽ 共 r, ␣ 兲兴 其

Let us first consider 具 var关 ⌽(r) 兴 var关 ⌽(r, ␣ ) 兴 典 :

具 var关 ⌽ 共 r, 0兲兴 var关 ⌽ 共 r, ␣ 兲兴 典

册冔 冓 冉 兺 冊冋 兺 兺 冓兺 兺 冔 冓兺 兺 冔 冓兺 兺 冔 冓 兺 兺 冔 ⬁

⫽

⬁

i⫽i 0 ⫹1 ⬁

⫽

2 a i,0

i0

j⫽i 0 ⫹1

a j,2 ␣ ⫹

j⫽2

⬁

⬁

共 a j, ␣ ⫺ a j,0兲 2

⬁

i⫽i 0 ⫹1 j⫽i 0 ⫹1

⫹

the decorrelation of the SR fluctuations is very fast it is only in regard to very small fluctuations around high

2 a i,0 a j,2 ␣

⫹

i⫽i 0 ⫹1 j⫽2 ⬁

i0

i⫽i 0 ⫹1 j⫽2

i0

2 2 a i,0 a j,0

⫺

2 a i,0 a j,2 ␣

.

(40)

mean SR values. An interesting limit case appears when a perfect on-axis correction is considered. In that case, on-axis ISR is equal to 1 (without any fluctuation), but, as soon as an off-axis direction is considered, the ISR becomes less than 1 with some small fluctuations. Even if the off-axis ISR is close to 1 and the corresponding fluctuation is close to zero, the normalized ISR angular correlation is equal to zero, since it is nothing but the correlation of a constant with a fluctuating signal.

i0

2

i⫽i 0 ⫹1 j⫽2

2 a i,0 a j,0a j, ␣ .

(41) Using this expression we obtain for C( ␣ ) (see Appendix C for more details) ⬁

2 C共 ␣ 兲 ⫽

兺

i⫽i 0 ⫹1

关 C i 共 ␣ 兲兴 2

2 2 1/2 兵 ␴ var 关 ⌽ 共 r,0兲兴 ␴ var关 ⌽ 共 r, ␣ 兲兴 其

.

(42)

Because C i ( ␣ ) is proportional to (D/r 0 ) 5/3 as well as 2 ␴ var 关 ⌽ ( r, ␣ ) 兴 , the numerator and denominator have the same (D/r 0 ) 10/3 factor. Thus, in the weak fluctuation hypothesis, the ISR angular correlations do not depend on D/r 0 (owing to the definition of the normalized correlation function) but depend only on • The number of corrected modes that is the system correction degree, • The telescope diameter and the turbulence repartition as a function of height.10 B. Comparison between Theory and Simulation We consider in this section the same simulation parameters as presented in Sections 5 and 6. We plot in Fig. 5 the angular correlation of the ISR for various D/r 0 and the same number of corrected modes. As predicted by the analytical expression, the normalized correlation does not depend on D/r 0 . On the other hand, we plot in Fig. 6 the normalized angular correlation as a function of the number of corrected modes for a given D/r 0 . It is interesting to note here that the ISR decorrelation becomes more and more important as a function of the correction degree. The correlation tends to a Dirac function for a perfect on-axis correction, which means that, if we can perfectly correct the phase on axis, the ISR fluctuations at various positions in the field are completely uncorrelated. Nevertheless, we have to be very careful in the interpretation of such a behavior. For the correction of a large number of radial order, that is, very good AO correction, the ISR tends to 1, the instantaneous fluctuations are quite small, and even if

Fig. 5. Angular correlations: comparison between simulations and analytical results. C( ␣ ) is plotted in the case of three D/r 0 —14 (asterisks), 10 (pluses), and 5 (diamonds)—and one correction degree (66 corrected Zernike polynomials). The analytical results are plotted for each case (solid curve) (the same turbulence screens are used in each correction case).

Fig. 6. ISR angular correlations: comparison between simulations and analytical results. C( ␣ ) is plotted in the case of three correction degrees: 35 (asterisks), 66 (pluses), and 120 (diamonds) corrected Zernike polynomials. For each case, D/r 0 ⫽ 14. The analytical results are plotted for each case (solid curve) (the same turbulence screens are used in each correction case).

T. Fusco and J.-M. Conan

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1285

average of the short exposures, in particular for groundbased exoplanet detections.19,20 D. Long-Exposure Adaptive Optics Statistics Ve´ran et al. have shown that it is possible to obtain a infinite exposure PSF from the WFS data.34 But, for various applications (spectroscopy, for example), it is very important to estimate the error bars on the SR estimation for finite exposure time. In that case, the long-exposure variance is directly linked to short-exposure variance by the following relation26:

␴ I2 共 T 兲 ⫽ Fig. 7. ␪ s (FWHM of the ISR correlation) as a function of the correction degree. The fit by a n ⫺6/5 law is plotted (dashed line) for comparison.

C. Definition of an Isoplanatic Angle Using correlation FWHM, one can derive an ISR isoplanatic angle ( ␪ s ), which characterizes the angular decorrelation of the ISR fluctuations. As already explained in the previous subsection, ␪ s does not depend on r 0 as long as the correction is good enough to be in the small phase approximation. As shown by Chassat10 or Sasiela,11 the Zernike angular correlation C i ( ␣ ) depends on the C n2 profile and the telescope diameter. Therefore, following Eq. (42), ␪ s depends only on the normalized distribution of the turbulence, the telescope diameter, and, of course, the number of corrected modes. The evolution of ␪ s as a function of the last corrected radial order n is plotted in Fig. 7. This evolution is perfectly fitted by a n ⫺6/5 law with the coefficient a depending on the C n2 profile and the telescope diameter. In our case, a ⯝ 80 (estimated by a fit on a curve in Fig. 7). It is interesting to note that ␪ s tends to zero when n tends to infinity. This behavior has to be commented on, since it is quite surprising. As already explained in the previous section, when the correction degree increases, the ISR onand off-axis increases but becomes more and more decorrelated from a short exposure to another. The limit is when a perfect correction is considered, and an on-axis ISR is equal to 1 (without any fluctuation). Using the Taylor hypothesis, it is interesting to make some links between angular and temporal decorrelations. In a first-order approximation,32,33 one can simply derive the temporal behavior from the angular one. In that case, one can define the AO short-exposure correlation time ␶ ISR following the same approach used for the isoplanatic angle. In that case, ␶ ISR is none other than the FWHM of the ISR temporal correlation, which can be defined by analogy with Eq. (40) if we replace the Zernike angular correlations 关 C i ( ␣ ) 兴 by their temporal correlations C it (t), which depend only on the Zernike number i, the wind speed per turbulent layer, and the C n2 profile. In that case, it is interesting to highlight the increase of temporal decorrelation as a function of correction degree. In particular, as already shown for the isoplanatic angle, the correlation time tends to zero when the AO correction tends to be perfect (i.e., the SR tends to 1). This is an important issue for all the applications that need a fast

␶ T

␴ I2 ,

(43)

where T is the integration time, ␶ is the correlation time of the residual phase (typically less than 10 ms), and ␴ I2 is the variance of the short-exposure PSF. The ratio ␶ /T represents the number of decorrelated events. To estimate the error on the SR due to finite exposure time, we find that relations (43) and (25) give

冑

␴ I 共 T 兲 ⫽ 1.02

␶

2 ␴⌽

T 共n ⫹ 1兲

.

(44)

Note that, in a first approximation, ␶ can be taken as the speckle lifetime defined by Roddier et al.32 But it seems more realistic to compute the temporal correlation of AOcorrected short exposure by use of the same scheme proposed in Subsection 7.C. In that case, we will obtain a ␶ value depending only on the wind speed and the AO correction degree, and, because of the n ⫺6/5 dependency (n is the last corrected radial order), ␶ tends to zero with the increase of the correction. That means the better the correction, the more efficient the ISR average and the less important the SR fluctuation for a given exposure time.

8. EXTENSION OF THE INSTANTANEOUS STREHL RATIO STUDY TO THE WHOLE POINT-SPREAD-FUNCTION SHAPE Since the beginning of this paper we have focused on the ISR statistic that characterizes only one particular point [that is, the center of the field value ( ␳ ⫽ 0)] of the PSF. Let us now extend it to the whole PSF shape.

Fig. 8. X-axis cut of a PSF mean and a PSF root-mean-square for a correction of four radial orders. D/r 0 ⫽ 10, SR ⫽ 30%.

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T. Fusco and J.-M. Conan

plane. The statistic found for the center of the plane is still valid in the whole coherent peak. In the halo the statistic is the same as in the case of a perfect scatterer ␴ I ⫽ 具 i 典 . These two behaviors came from the different natures of the light in these two cases: a coherent light for the coherent peak and an incoherent light for the halo. This statistical characterization of the AO-corrected short exposure is important for various applications such as AO performance characterization, PSF a priori for postprocessing, studies of light coupling into single-mode fibers, and, most important, for the detection of exoplanets from the ground.

Fig. 9. X-axis cut of ␴ I / 具 I 典 for a correction of four radial orders and for different D/r 0 ⫽ 关 5, 10, 15, 20兴 , i.e., SR ⫽ 关 68%, 30%, 11%, 3% 兴 .

To study the evolution of the two first moments of the short-exposure PSF as a function of the position in the PSF, we consider a perfect correction of first four radial orders (i.e., the first 16 Zernike polynomials) of the turbulent phase. Nevertheless, the following results can be extended whatever the number of corrected radial orders. One thousand decorrelated short exposures (I) are simulated. A x-axis cut of 具I典 and ␴ I is plotted for D/r 0 ⫽ 10 in Fig. 8. For such a D/r 0 , the long-exposure SR is equal to 30%. This figure shows the different statistical behaviors as a function of position in the PSF, i.e., the nonstationarity of the two first moments of I( ␳ ). Figure 9 shows, for different D/r 0 , the evolution of the ISR fluctuation rate as a function of the focal-plane position. Figure 9 clearly shows that the statistical behavior of the short-exposure PSFs is directly linked to two specific regions described above (coherent peak and halo). On one hand, the coherent peak statistic has the same evolution that the central pixel has [see expression (26)] and, on the other hand, the halo statistic follows a ␴ I / 具 I 典 ⫽ 1 law, which is the typical behavior of noncorrected shortexposure PSFs.

APPENDIX A: CALCULATION OF ␴ 2⌽ AND 2 AS A FUNCTION OF n ␴ var⌽ Noll28 gives a theoretical expression of the variance for each radial order:

␴ n2

⫽ 2.2698共 n ⫹ 1 兲

Adaptive optics systems are now currently used to correct atmospheric turbulence effects. Knowledge of the statistical behavior of corrected short exposure and, more particularly, of the instantaneous Strehl ratio is important for many applications. In this paper, we have derived simple expressions of this ISR statistic, that is, the center of the focal-plane intensity, for the two first moments of their probability law. These analytical expressions depend only on the correction degrees of the AO systems (residual variance and last corrected radial order) and on the turbulence strength (D/r 0 ). Extension of the analytical expression for offaxis correction is derived. Moreover, the ISR angular correlations are proposed, which lead to a definition of a corrected short-exposure isoplanatic angle. In addition, numerical simulations allow us to validate these expressions. With these simulations, we have extended the study of the ISR statistic to the whole focal

D

17

23 6

关 ⌫ 共 6 兲兴 2 ⌫ 共 n ⫹

兲 r0

5/3

. (A1)

Using an asymptotic development of the gamma function (Stirling formula), we obtain an approximation of ␴ n2 :

冉冊 冉冊 D

␴ n2 ⯝ 0.7632共 n ⫹ 1 兲 ⫺11/3

␴ n4 ⯝ 0.5824共 n ⫹ 1 兲 ⫺22/3

5/3

,

r0

(A2)

10/3

D

.

r0

(A3)

The phase residual variance after a correction of n radial orders is given by ⫹⬁

2 ␴⌽ ⫽

兺

n ⬘ ⫽n⫹1

2

共 n ⬘ ⫹ 1 兲 ␴ n⬘,

(A4)

4

(A5)

⫹⬁

2 ␴ var⌽ ⫽2

9. CONCLUSION

冉冊

⌫n ⫺ 共 65 兲

兺

n ⬘ ⫽n⫹1

共 n ⬘ ⫹ 1 兲 ␴ n⬘.

2

Replacing ␴ n ⬘ with an asymptotic expression and the discrete sum by a integral, we obtain 2 ␴⌽ ⯝ 0.458共 n ⫹ 1 兲 ⫺5/3

冉冊 冉冊

5/3

D

,

r0

2 ␴ var⌽ ⫽ 0.218共 n ⫹ 1 兲 ⫺16/3

D

(A6)

10/3

.

r0

(A7)

Note that the number of the last corrected polynomial j max is given by j max ⫽ 关(n ⫹ 1)(n ⫹ 2)兴/2; expression (A6) can therefore be rewritten, for large values of n, as 2 ␴⌽

⯝ 0.257共 j max兲

⫺5/6

冉冊 D

r0

5/3

.

(A8)

This expression is very close to the empirical law given by Noll28:

T. Fusco and J.-M. Conan

Vol. 21, No. 7 / July 2004 / J. Opt. Soc. Am. A i0

兺

共2兲 ⫽ 6

␴ a4 i

i⫽2

⫹4

i0

兺兺

␴ a2 i ␴ a2 j

i⫽j

i0

⫹4

兺

⫺ 24

兺␴ i⫽2

兺兺 i⫽j

⫹4

⫹2

兺␴ i⫽2

2 ai

i0

C i2 共 ␣ 兲

i⫽2

⫺8

1287

2 ajC i共 ␣ 兲 i0

␴ a2 i C j 共 ␣ 兲

⫹4

兺

i⫽2

i0

␴ a4 i

⫹8

兺 C 共␣兲 2 i

i⫽2

兺 兺 C 共 ␣ 兲C 共 ␣ 兲, i

j

i⫽j

再兺 i0

Fig. 10. Residual variance after a perfect correction of the n first radial orders. Noll expression [expression (A9)] (asterisks) and asymptotic expression [expression (A6)] (solid line) D/r 0 ⫽ 1.

2 ␴ ⌽,Noll ⯝ 0.2944共 j max兲

⫺冑3/2

冉冊 D

i⫽2

⫹

兺 兺 关␴ i⫽j

5/3

.

r0

关 ␴ a2 i ⫺ C i 共 ␣ 兲兴 2

共2兲 ⫽ 4 3

2 ai

冎

⫺ C i 共 ␣ 兲兴关 ␴ a2 j ⫺ C j 共 ␣ 兲兴 .

(A9) At least, the third and fourth terms are equal to

The two formulas are plotted in Fig. 10 for comparison.

再兺 ⬁

共 3, 4兲 ⫽ 4

APPENDIX B: OFF-AXIS Š(var[⌽(r, ␣ )])2‹ CALCULATION

Summing

具 兵 var关 ⌽ 共 r, ␣ 兲兴 其 2 典

冓兺 ⬁

⫽

⬁

冔 冓兺 兺 i0

兺

a i,2 ␣ a j,2 ␣

i⫽i 0 ⫹1 j⫽i 0 ⫹1

i⫽2 j⫽2

2 ⫺ 2a i, ␣ a i,0兲共 a j,2 ␣ ⫹ a j,0 ⫺ 2a j, ␣ a j,0兲

冓兺 冓兺 ⬁

⫹2

兺a

⬁

⫹2

共 a i,2 ␣

2 a i,0

⫹

2 2 i, ␣ 共 a j, ␣

兺a

j⫽i 0 ⫹1 i⫽2

2 2 i, ␣ 共 a j, ␣

冔

冔 冔

2 ⫹ a j,0 ⫺ 2a j,0a j, ␣ 兲 .

(B1)

共1兲 ⫽ 3

兺

i⫽i 0 ⫹1

⫹

兺兺 i⫽j

(B2)

冉兺 兺 i0

⫽

i0

i⫽2 j⫽2

冊

再兺

i⫽i 0 ⫹1

⫹4

兺 关␴ i⫽2

2 ai

冎

⫺ C i 共 ␣ 兲兴 2 . (B4)

册冔 冓 冉 兺 冊冋 兺 兺 冓兺 兺 冔 冓兺 兺 冔 冓兺 兺 冔 冓 兺 兺 冔 i⫽i 0 ⫹1

2 a i,0

⫹

j⫽2

共 a j, ␣ ⫺ a j,0兲 2

⬁

i⫽i 0 ⫹1 j⫽i 0 ⫹1

⫹

j⫽i 0 ⫹1

i0

a j,2 ␣

⬁

⬁

2 a j,2 ␣ 典 ⫺ 4 具 a j,2 ␣ a i,0a i, ␣ 典 具 a i,2 ␣ a j,2 ␣ 典 ⫹ 具 a i,0

⫹ 2 具 a j,0a j, ␣ a i,0a i, ␣ 典 .

subtracting

i0

␴ a4 i

⬁

⬁

As for the second term, it can be rewritten as 共2兲 ⫽ 2

then

具 var关 ⌽ 共 r, 0兲兴 var关 ⌽ 共 r, ␣ 兲兴 典

⫽

␴ a2 i ␴ a2 j .

⫽2

⬁

⬁

␴ a4 i

and

APPENDIX C: CALCULATION OF Švar[⌽(r, 0)]var[⌽(r, ␣)]‹

The first term of this expression is equal to ⬁

terms

⬁

2 ⫹ a j,0 ⫺ 2a j,0a j, ␣ 兲

i0

these

␴ a2 j ⫺ C j 共 ␣ 兲兴 .

2 a i关

2 neous spatial variance ␴ var 关 ⌽ ( r, ␣ ) 兴 :

2 ␴ var 关 ⌽ 共 r, ␣ 兲兴

i0

i⫽i 0 ⫹1 j⫽2

兺␴

i⫽i 0 ⫹1 j⫽2

具 var关 ⌽(r, ␣ ) 兴 典 2 , we obtain the expression of the instanta-

i0

⫹

all

冎

i0

2 a i,0 a j,2 ␣

⫹

i⫽i 0 ⫹1 j⫽2 ⬁

i0

i⫽i 0 ⫹1 j⫽2

i0

2 2 a i,0 a j,0

⫺

2 a i,0 a j,2 ␣

i0

2

i⫽i 0 ⫹1 j⫽2

2 a i,0 a j,0a j, ␣ .

(C1) (B3)

Assuming that the turbulent Zernike coefficient statistic is Gaussian and that two distinct coefficients a i and a j are independent if i ⫽ j, we obtain25

The sum of four terms has to be detailed. Assuming that the turbulent Zernike coefficients follow the Gaussian statistic and that two coefficients a i and a j are independent if i ⫽ j, we obtain

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T. Fusco and J.-M. Conan

First term: ⬁

REFERENCES 1.

⬁

兺 兺

i⫽i 0 ⫹1 j⫽i 0 ⫹1

2 a j,2 ␣ 典 具 a i,0

2.

⬁

⫽

⬁

兺

2 具 a i,0 典具 a i,2 ␣ 典 ⫹ 2

i⫽i 0 ⫹1

⬁

i⫽i 0 ⫹1 i⫽j

4.

2 2 i,0典具 a j, ␣ 典

5.

⬁

⫽

⬁

兺

i⫽i 0 ⫹1

3.

C i2 共 ␣ 兲

⬁

兺 兺 具a

⫹

兺

i⫽i 0 ⫹1

␴ a4 i ⫹ 2

⬁

兺

C i2 共 ␣ 兲 ⫹

i⫽i 0 ⫹1

⬁

兺 兺␴

i⫽i 0 ⫹1 i⫽j

2 2 ai␴ aj.

(C2) Second and third terms: ⬁

7. ⬁

i0

兺 兺 具a

2

i⫽i 0 ⫹1 j⫽2

6.

2 2 i,0a j, ␣ 典

⫽2

i0

兺 兺␴

i⫽i 0 ⫹1 j⫽2

2 2 ai␴ aj.

(C3) 8.

Fourth term: ⬁

⫺2

⬁

i0

兺 兺

i⫽i 0 ⫹1 j⫽2

2 a j,0a j, ␣ 典 具 a i,0

⫽ ⫺2

9.

i0

兺 兺

i⫽i 0 ⫹1 j⫽2

␴ a2 i C j 共 ␣ 兲 .

10.

(C4) 11.

And then, 12.

具 var关 ⌽ 共 r, 0兲兴 var关 ⌽ 共 r, ␣ 兲兴 典 ⬁

⫽

⬁

兺

i⫽i 0 ⫹1

关

␴ a4 i

⫹

兺 兺␴

i⫽i 0 ⫹1 i⫽j

13.

2 2 ai␴ aj

14.

i 0 ⫹1

⬁

⫹2

⫹

2C i2 共 ␣ 兲兴

兺 兺

i⫽i 0 ⫹1 j⫽2

␴ a2 i 关 ␴ a2 j ⫺ C j 共 ␣ 兲兴 .

(C5) 15.

One can also compute 具 var关 ⌽(r, 0) 兴 典具 var关 ⌽(r, ␣ ) 兴 典 , which is equal to

16.

具 var关 ⌽ 共 r, 0兲兴 典具 var关 ⌽ 共 r, ␣ 兲兴 典 17. ⬁

⫽

⬁

兺 兺

i⫽i 0 ⫹1 j⫽i 0 ⫹1 ⬁

⫹2

␴ a2 i ␴ a2 j i0

兺 兺␴

i⫽i 0 ⫹1 j⫽2

2 a i关

␴ a2 j ⫺ C j 共 ␣ 兲兴 .

(C6)

18. 19.

And we obtain the final expression of C( ␣ ): ⬁

2 C共 ␣ 兲 ⫽

兺

i⫽i 0 ⫹1

关 C i 共 ␣ 兲兴 2

2 2 1/2 兵 ␴ var 关 ⌽ 共 r,0兲兴 ␴ var关 ⌽ 共 r, ␣ 兲兴 其

.

(C7)

T. Fusco, the corresponding author, may be reached by e-mail at [email protected].

20.

H. W. Babcock, ‘‘The possibility of compensating astronomical seeing,’’ Publ. Astron. Soc. Pac. 65, 229 (1953). G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Le´na, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, ‘‘First diffraction-limited astronomical images with adaptive optics,’’ Astron. Astrophys. 230, 29–32 (1990). F. Roddier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, Cambridge, UK, 1999). F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, F. Merkle, and P. Le´na, ‘‘Adaptive optics on a 3.6-m telescope: results and performance,’’ Astron. Astrophys. 250, 280–290 (1991). G. Rousset, P. Y. Madec, and D. Rabaud, ‘‘Adaptive optics partial correction simulation for two telescopes,’’ in High Resolution Imaging by Interferometry II, J. M. Beckers and F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104. M. C. Roggemann, ‘‘Limited degree-of-freedom adaptive optics and image reconstruction,’’ Appl. Opt. 30, 4227–4233 (1991). J. M. Conan, P. Y. Madec, and G. Rousset, ‘‘Evaluation of image quality obtained with adaptive optics partial correction,’’ in Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474. C. Ruilier and F. Cassaing, ‘‘Coupling of large telescopes and single-mode waveguides: application to stellar interferometry,’’ J. Opt. Soc. Am. A 18, 143–149 (2001). D. L. Fried, ‘‘Anisoplanatism in adaptive optics,’’ J. Opt. Soc. Am. 72, 52–61 (1982). F. Chassat, ‘‘Calcul du domaine d’isoplane´tisme d’un syste`me d’optique adaptative fonctionnant a` travers la turbulence atmosphe´rique,’’ J. Opt. (Paris) 20, 13–23 (1989). R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence Evaluation and Application of Mellin Transforms (Springer-Verlag, Berlin, 1995). G. Molodij and G. Rousset, ‘‘Angular correlation of Zernike polynomials for a laser guide star in adaptive optics,’’ J. Opt. Soc. Am. A 14, 1949–1966 (1997). M. P. Cagigal and V. F. Canales, ‘‘Speckle statistics in partially corrected wave fronts,’’ Opt. Lett. 23, 1072–1074 (1998). V. F. Canales and M. P. Cagigal, ‘‘Rician distribution to describe speckle statistics in adaptive optics,’’ Appl. Opt. 38, 766–771 (1999). J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, ‘‘Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra,’’ Appl. Opt. 37, 4614–4622 (1998). T. Fusco, J.-P. Ve´ran, J.-M. Conan, and L. Mugnier, ‘‘Myopic deconvolution method for adaptive optics images of stellar fields,’’ Astron. Astrophys. Suppl. Ser. 134, 1–10 (1999). T. Fusco, L. M. Mugnier, J.-M. Conan, F. Marchis, G. Chauvin, G. Rousset, A.-M. Lagrange, D. Mouillet, and F. Roddier, ‘‘Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,’’ in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 1065–1075 (2002). S. Stahl and D. Sandler, ‘‘Optimization and performance of adaptive optics for imaging extrasolar planets,’’ Astrophys. J. Lett. 454, L153–L156 (1995). D. Mouillet, A.-M. Lagrange, J.-L. Beuzit, F. Me´nard, C. Moutou, T. Fusco, L. Abe´, T. Gillot, R. Soummer, and P. Riaud, ‘‘VLT-‘Planet Finder’: specifications for a groundbased high contrast imager,’’ in Scientific Highlights 2002, F. Combes and D. Barret, eds. (EDP Sciences, Les Ulis, France, 2002). D. Mouillet, T. Fusco, A.-M. Lagrange, and J.-L. Beuzit, ‘‘ ‘Planet Finder’ on the VLT: context, goals and critical specification for adaptive optics,’’ in Astronomy with High Contrast Imaging: From Planetary Systems to Active Galactic Nuclei, C. Aime and R. Soummer, eds., EAS Publication Series (EDP Sciences, Les Ulis, France, 2002).

T. Fusco and J.-M. Conan 21.

22.

23. 24.

25. 26.

27.

F. Roddier, ‘‘The effects of atmospherical turbulence in optical astronomy,’’ in Progress in Optics, E. Wolf, ed. (NorthHolland, Amsterdam, 1981), Vol. 19, pp. 281–376. F. Roddier and C. Roddier, ‘‘NOAO infrared adaptive optics program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,’’ in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE 628, 298–304 (1986). D. P. Greenwood, ‘‘Bandwidth specification for adaptive optics systems,’’ J. Opt. Soc. Am. A 67, 390–393 (1977). J.-M. Conan, G. Rousset, and P.-Y. Madec, ‘‘Wave-front temporal spectra in high-resolution imaging through turbulence,’’ J. Opt. Soc. Am. A 12, 1559–1570 (1995). J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985). ´ tude de la correction partielle en optique J.-M. Conan, ‘‘E adaptative,’’ Ph.D. thesis (Universite´ Paris XI, Orsay, France, 1994). J.-M. Conan, T. Fusco, L. Mugnier, F. Marchis, C. Roddier, and F. Roddier, ‘‘Deconvolution of adaptive optics images: from theory to practice,’’ in Adaptive Optical Systems Tech-

Vol. 21, No. 7 / July 2004 / J. Opt. Soc. Am. A

28. 29. 30. 31. 32. 33. 34.

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nology, P. Wizinowich, ed., Proc. SPIE 4007, 913–924 (2000). R. J. Noll, ‘‘Zernike polynomials and atmospheric turbulence,’’ J. Opt. Soc. Am. 66, 207–211 (1976). N. Roddier, ‘‘Atmospheric wavefront simulation using Zernike polynomials,’’ Opt. Eng. 29, 1174–1180 (1990). J. Goodman, ‘‘Statistical properties of laser speckle patterns,’’ in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 255–280. J. C. Dainty, ‘‘Stellar speckle interferometry,’’ in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 255–280. F. Roddier, J. M. Gilli, and J. Vernin, ‘‘On the isoplanatic patch size in stellar speckle interferometry,’’ J. Opt. (Paris) 13, 63–70 (1982). F. Roddier, J. M. Gilli, and G. Lund, ‘‘On the origin of speckle boilding and its effects in stellar speckle interferometry,’’ J. Opt. (Paris) 13, 263–271 (1982). J.-P. Ve´ran, F. Rigaut, H. Maıˆtre, and D. Rouan, ‘‘Estimation of the adaptive optics long-exposure point-spread function using control loop data,’’ J. Opt. Soc. Am. A 14, 3057– 3069 (1997).

Annexe B "NAOS on-line characterization of turbulence parameters and adaptive optics performance" T. Fusco et al. - Pure and Applied Optics - 2004

93

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS

J. Opt. A: Pure Appl. Opt. 6 (2004) 585–596

PII: S1464-4258(04)74681-4

NAOS on-line characterization of turbulence parameters and adaptive optics performance T Fusco1,8 , G Rousset1 , D Rabaud1,2 , E Gendron3 , D Mouillet4,5 , F Lacombe3 , G Zins2,5,6 , P-Y Madec1,7 , A-M Lagrange5 , J Charton5 , D Rouan3 , N Hubin6 and N Ageorges6 1

Office National d’Etudes et de Recherches A´erospatiales (ONERA), BP 72, 92322 Chatillon Cedex, France 2 Shaktiware, 27 Boulevard Charles Moretti, 13014 Marseille, France 3 Observatoire de Meudon, Place J Jansen, 92195 Meudon, France 4 Observatoire Midi-Pyrenees, 57, Avenue d’Azereix BP 826, 65008 Tarbes, France 5 Observatoire de Grenoble, BP 53, 38041 Grenoble, France 6 ESO, Karl-Schwarzschild-Straße 2, D-85748, Garching, Germany 7 CILAS, 8 avenue Buffon, 45100 Orleans, France E-mail: [email protected]

Received 9 January 2004, accepted for publication 30 March 2004 Published 23 April 2004 Online at stacks.iop.org/JOptA/6/585 DOI: 10.1088/1464-4258/6/6/014

Abstract An on-line estimation of turbulence parameters (r0 , L 0 and wind speed) and adaptive optics (AO) performance is presented. The method is based on the reconstruction of open-loop data from deformable mirror voltages and residual wavefront sensor slopes obtained in closed loop. This dedicated tool implemented in the real time computer of the NAOS (Nasmyth adaptive optics system) system (first AO of the very large telescope) allows us without any loop opening to automatically monitor and display (every 15 s) both the atmospheric conditions and the system performance. We have validated the algorithm and tested its robustness on simulated and experimental data (both in the laboratory and on sky). Using data obtained over more than one year (from January 2002 to June 2003), a statistical study on NAOS performance and turbulence characteristics is proposed. Keywords: turbulence, adaptive optics, Shack–Hartmann, Strehl ratio

1. Introduction Adaptive optics (AO) (see Roddier (1999) for a review) is now a technique widely implemented in astronomy. It allows the large ground-based telescopes to produce diffractionlimited images by real time compensation of the atmospheric turbulence effects. On-site turbulence characterization is of great importance to correctly specify AO systems and predict their performance. A number of dedicated instruments, including the Shack–Hartmann wavefront sensor (WFS), have 8 Author to whom any correspondence should be addressed.

been developed for such a purpose (Sarazin and Roddier 1990, Wilson 2002, Avila et al 1997) and they provide information on atmospheric parameters for most of the astronomical observatories (Racine 1992, Wilson et al 1999, Martin et al 2000, Conan 2002, Tokovini et al 2003). During the operation of an AO system, it is crucial to continuously monitor the atmospheric turbulence parameters and estimate the AO performance with respect to the observing conditions. For this purpose, an AO system must be capable of recording temporal sequences of raw wavefront (WF) data to be reduced afterwards. This feature was implemented very early in the development of the Come-On AO prototype

1464-4258/04/060585+12$30.00 © 2004 IOP Publishing Ltd Printed in the UK

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T Fusco et al

system (Rousset et al 1990, Rigaut et al 1991) and then in the Come-On-Plus (Rousset et al 1993) and Adonis (Beuzit et al 1994) upgrades, based on the recording of open-loop or closed-loop WF data. More recently, (Sch¨ock et al 2002) have developed the same kind of tools in quasi open loop. Such tools avoid the drawback of on-site dedicated instruments (the differential image motion monitor (DIMM) (Sarazin and Roddier 1990) for instance), which do not see exactly the same turbulence than an AO system. A new step has been achieved in NAOS (Nasmyth adaptive optics system), the first AO system of the very large telescope (VLT) (Rousset et al 2002), with the on-line characterization of turbulence parameters and system performance based on the recording of closed-loop WF measurements and deformable mirror (DM) voltages during observation (i.e. without any loop opening). In this paper, we describe our approach to estimate the turbulence parameters (i.e. Fried diameter r0 , outer scale L 0 , wind speed v and coherence time τ0 ) and the system performance (Strehl ratio) as well as the results obtained. After a brief overview of the NAOS system in section 2, a detailed description of the algorithms used to estimate both atmospheric parameters and system performance is presented in section 3. Then, laboratory measurements performed during the assembling integration and testing (AIT) of the whole NAOS system in France are presented. These tests, summarized in section 4, have validated the algorithms and have given a first estimation of their accuracy. Moreover, the possible error sources have been identified. Finally, in section 5 we present on-sky results and a statistical study on NAOS performance and turbulence characteristics using data obtained during the five commissioning periods and two guarantee time periods (from January 2002 to June 2003).

2. An overview of the NAOS system NAOS (Rousset et al 2002) compensates for turbulence in order to provide diffraction limited images at 2.2 µm on the VLT (8 m class telescope) to the high angular resolution infrared (IR) spectro-imaging instrument (CONICA) that covers the J to M band range (Lenzen et al 2002). NAOS is designed to work with natural guide stars (NGS) that can be either bright or faint stars or even extended objects. It is equipped with two Shack–Hartmann wavefront sensors (WFS), one operating in the visible (V) (Feautrier et al 2002) and the other in the infrared (IR). The IR-WFS (Gendron et al 2002) is so far a unique and useful feature that allows us to close the loop on very red sources. The two WFS are equipped with 14 × 14 and 7 × 7 square lenslet arrays. The sub-pupil FoV ranges between 2.4 and 4.8 arcsec. The detectors, an EEV CCD for the V-WFS and a Rockwell Hawaii for the IR-WFS, can be read at different frame rates (from 15 to 480 Hz for the V-WFS and up to 180 Hz for the IR-WFS) and allow various windowing and binning parameters. This leads to a large number of configurations for the AO system that are meant to match a variety of astrophysical and atmospherical conditions. The correction is provided by a two-axis tip tilt mirror (TTM) and a 185 piezo stack actuator DM. Optimized modal control (Gendron and L´ena 1994) is continuously updated and used in the real time computer (RTC), allowing the best correction in terms of turbulence strength and noise level. A dedicated 586

software tool is in charge of the on-line characterization of turbulence and system performance. The main parameters related to turbulence and performance are displayed at a 15 s refreshing rate. NAOS and CONICA were successfully installed on the fourth 8 m telescope of the VLT in December 2001 (Rousset et al 2002). All the system functionalities have been checked on sky during five commissioning periods from December 2001 to September 2002. The system became available to the astronomical community in October 2002. The first astrophysical results can be found in Lagrange et al (2002).

3. Algorithm presentation We present in this section the turbulence characterization and performance estimation tools as well as their numerical validations. These tools only use closed-loop data. They are mainly based on the approach developed by ONERA (Office National d’Etudes et de Recherches A´erospatiales) for the Come-On and Come-On-Plus projects, and subsequently for other applications (Rousset et al 1991a, Rigaut et al 1991, Madec et al 1992, N¨oel 1997, Dessenne 1998). They were substantially updated and their robustness was improved for their implementation in NAOS. In order to validate the algorithms, several sets of data have been simulated with various turbulence characteristics and WFS noise levels. Algorithm accuracy and validity ranges are given. 3.1. Open-loop data reconstruction from AO loop data AO loop data are synchronous sets of measured residual WFS slopes and computed DM and TTM voltages on 2048 time steps at frame rate up to 480 Hz, captured in closed-loop mode. The turbulence characterization tool uses these AO loop data in order to reconstruct the incoming turbulent phases for all the loop steps. First, the AO loop data are expanded in the Zernike polynomial basis (Noll 1976). The Zernike coefficients are computed for the wavelength λ = 0.5 µm. The maximum number of expansion coefficients is directly linked to the WFS geometry. In our case, we have chosen to expand the WF on 66 Zernike polynomials for the 14 × 14 lenslet array (and 36 for the 7 × 7). The Zernike coefficients are given by



ai,volt



ai,slope

i=2,66


 = V Z v j j =1,187 ,

(1)


 = S Z s j j =1,288 ,

(2)

 i=2,66

where V Z is the projection matrix of voltages {v j } onto the Zernike basis. It is computed using the measured DM and TTM influence functions. S Z is the projection matrix of slopes {s j } (in x and y measured by the WFS) onto the Zernike basis. It is computed theoretically using the WFS parameters. In order to minimize the aliasing effects (of high-order modes on computed ones) only the first 36 Zernike polynomials are effectively used for the turbulence parameter estimation in the 14 × 14 lenslet array configuration (21 for the 7 × 7).

NAOS on-line characterization of turbulence parameters and AO performance

noise variance estimation

2 Figure 2. A comparison of the simulation of theoretical σtot,noise (x 2 axis) versus estimated σtot,noise (y-axis). An x = y law is plotted for comparison.

Figure 1. An example of σa2i ,noise estimation using temporal autocorrelation data. (Dot–crosses) = computed autocorrelation, (solid curve) polynomial fit.

Secondly, at each time step the two vectors of Zernike coefficients {ai,volt } and {ai,slope } must be added in order to reconstruct the corresponding open-loop vector as proposed by Dessenne (1998). We have to account for two-frame delay in the loop due to the exposure and read-out times on the WFS detectors. This leads to the reconstructed vector at time t: ai,rec (t) = ai,volt (t) + ai,slope (t + 2).

(3)

The vector {ai,rec } is therefore the set of Zernike coefficients corresponding to the incoming turbulent WF for all t which would have been measured by the WFS in open-loop mode. From the reconstructed open-loop data {ai,rec }, the atmospheric parameters—noise on the WFS, r0 (Fried parameter), L 0 (outer scale), v (wind speed) and τ0 (coherence time)—are estimated. 3.2. Noise and turbulence variance estimation

2 σtot,noise =

Using the open-loop reconstructed data, we can compute a mode by mode variance σa2i ,rec . This variance is the sum of the turbulent variance σa2i ,turb and the measurement noise variance σa2i ,noise : σa2i ,rec = σa2i ,turb + σa2i ,noise . (4) σa2i ,noise is the propagation of the slope measurements noise (assumed to be a white noise) on each reconstructed Zernike polynomial. The estimation of σa2i ,noise is derived from the temporal autocorrelation Ci,rec (τ ) of the reconstructed Zernike coefficients following a method proposed by Rousset (1992). Ci,rec (τ ) is defined as Ci,rec (τ ) =
ai,rec (t)ai,rec (t + τ )t .

36 

σa2i ,noise .

(7)

i=2

The estimation gives very accurate results for medium to high noise variance. A saturation appears for very low noise level. The saturation threshold corresponds to the WFS signal to noise ratio (SNR) (defined as the ratio of angle of arrival turbulence variance over photon and detector noise variance) equal to 800. This roughly leads, for NAOS VWFS and typical observation conditions, to visible guide star magnitudes higher than 9. Nevertheless, the saturation level remains very small (2 × 10−2 rad2 to be compared to around 103 rad2 for turbulence) and will not induce any significant error on atmospheric parameter estimation for bright stars.

(5)

Indeed, because σa2i ,noise is temporally decorrelated, Ci,rec (τ ) can be written as Ci,rec (τ ) = Ci,turb (τ ) + σa2i ,noise δ(τ ).

five first points (with the very first point excluded) allows us to estimate Ci,turb (0) by extrapolation. It is nothing but the turbulent variance associated to the i th Zernike polynomial. The difference between Ci,rec (0) and Ci,turb (0) gives the noise variance σa2i ,noise . The main interest of such a computation is that all the noise effects are located on one single point, which leads to a robust noise variance estimation. In practice, the WFS temporal sampling time must be small enough to ensure that the Zernike turbulence autocorrelation is smooth enough. It must be much smaller than the smallest Zernike coefficient turbulence correlation time which depends on the wind speed. Typical values for the sampling frequency are larger than 60 Hz. 2 In figure 2 we plot the simulated noise variance σtot,noise as a function of the theoretical one, with

(6)

An example of autocorrelation on simulated data is shown in figure 1. A polynomial fit of the autocorrelation on the

3.3. r0 and L 0 estimation Knowing that Ci,turb (0) only depends on the radial order (Noll 1976), we compute a radial order by radial order turbulence 2 , which is the average of all the Ci,turb (0) variance σturb,n over a radial order. Considering the analytical expression of 587

T Fusco et al

Figure 3. The estimated average turbulence variance per Zernike radial order (circles) and fit (on r0 and L 0 ) using equation (8) (solid 2 line). The corresponding estimated noise variance σtot,noise is 2 3.3 rad in this case. 2 σturb,n (Winker 1991, Chassat 1992),

 D 5/3 = 2.34 × 10 r0        2π D 2 2π D 7/3 D × 1 − 0.39 + 0.27 +o L0 L0 L0 (8)  5/3  [n − 5/6] D 2 σturb,n
3 = 0.756(n + 1)  [n + 23/6] r0      0.38 D 2π D 2 × 1− +o L0 (n − 11/6) (n + 23/6) L 0

2 σturb,n=2

−2



a least-squares fit is performed both on the outer-scale(L 0 ) and the Fried parameter (r0 computed at λ = 0.5 µm). o LD0 represents the high-order terms of the development which are neglected. Only radial orders 2–7 (5 for the 7 × 7 lenslet array) are used. Tip-tilt coefficients are excluded because of the possible additional errors caused by telescope vibrations (Rousset et al 2002) and tracking errors. An example of the results obtained by simulation is plotted in figure 3. Here, the fit performance is limited only by the noise estimation accuracy (see section 3.2). Figure 4 presents the influence of the WFS noise on r0 and L 0 estimations. The accuracy is very good for σtot,noise
30 rad2 . A 100 rad2 noise variance corresponds to an SNR on the WFS equal to 0.2 (considering r0 = 20 cm), which roughly corresponds to a 14.5 magnitude on a 14 × 14 configuration and a 16 magnitude on a 7 × 7 configuration (both with a 100 Hz sampling frequency and considering NAOS V-WFS characteristics (Feautrier et al 2002)). Table 1 presents the estimation error for both r0 and L 0 in different SNR conditions. For r0 , the error remains lower than 1 cm, and for L 0 lower than a few metres. The same order of magnitudes have been obtained for r0 between 5 and 30 cm. Note that the accuracy on the L 0 measurements decreases with the increase of L 0 (the least-squares fit becomes less sensitive to this parameter as the ratio D/L 0 gets smaller). 588

Figure 4. The influence of noise variance (as defined in equation (7)) on r0 (top) and L 0 (bottom.) estimations. The true values are 20 cm and 25 m for r0 and L 0 , respectively. Table 1. The rms error on r0 and L 0 estimation computed on simulation (Monte Carlo). The true r0 and L 0 are respectively 20 cm and 25 m. WFS SNR 2

Noise variance (rad ) r0 estimation error (cm) L 0 estimation error (m)

100

10

1

0.1

0.17 ±0.02 ±0.1

1.7 ±0.04 ±0.3

17 ±0.13 ±1.0

170 ±0.42 ±3.2

3.4. Wind speed and τ0 estimation The wind speed estimation is derived from the Zernike coefficient temporal autocorrelation. 1/e width (τi ) is computed for each Zernike autocorrelation function. For each radial order (n) an average of τi is obtained (τn ). From τn we estimate a temporal cut-off frequency f n : f n = 1.15πτn−1 .

(9)

The 1.15 coefficient has been determined by the choice of the 1/e criterion on simulated autocorrelation functions using a fit of the Zernike temporal power spectral densities (Conan et al 1995). Under the Taylor hypothesis, a simple relation exists between f n and the wind speed v for n  2 (Conan et al 1995): v f n = 0.3(n + 1) . (10) D

NAOS on-line characterization of turbulence parameters and AO performance

From equation (10), a least-squares estimator of v is performed using all the f n values (for n = 2 to n max = 7 with a 14 × 14 array). It becomes   n max

v 2 ∂  f mes (n) − 0.3(n + 1) = 0, (11) ∂v n=2 D which leads to v=

1.15π D (n + 1)τn−1 . 0.3 (n + 1)2

(12)

Following Roddier et al (1982), one can compute the coherence time τ0 from v: r0 τ0 = 0.31 . (13) v Note that the derivation described above is based on the Taylor hypothesis. In the case of a multi-layer profile the estimated v is nothing but an average wind speed value (v) ¯ characterizing the temporal evolution of the phase in the telescope pupil.

the telescope pupil, i.e. we neglected the effect of the small central obscuration of the VLT. Hence, 2 σres =

∞ 


ai2 ,

(16)

i=2

where the
ai2  are given by either the turbulence statistics (Noll 1976) or the AO loop error statistics. We considered that the system is able to compensate for the first N Zernike polynomials, that is, all the polynomials whose radial degree n ranges from 1 to n N . For n sufficiently large, typically n > 3, the turbulence variance
ai2 turb is given by Conan (1994)
ai2 turb = 0.7632 (n + 1)−11/3



D r0

5/3 ,

(17)

which does not depend on the azimuthal degree m (Noll 1976). The fitting error is simply derived from 2 = σfitting

∞ 


ai2 turb .

(18)

i=N +1

3.5. Performance estimation The goal of the performance estimation tool is to give an order of magnitude of NAOS efficiency. The AO efficiency is usually determined by the Strehl ratio (SR). The SR will be approximated here by the coherent energy (CoEn), which represents the percentage of the energy (photons collected by the telescope) gathered in the diffraction-limited central core of the point spread function (PSF) (Rousset et al 1991b):     0.5 2 2 , CoEn = exp −σres λim

(14)

2 is the statistical mean of the residual phase spatial where σres variance at 0.5 µm and λim is the imaging wavelength on CONICA. It is known that such a quantity is a good approximation of the classical SR. In all cases, the SR is greater than the CoEn (Conan 1994). The approximation SR  CoEn is valid if the CoEn is not too small, that is, in the case of good correction. For a typical AO system on a 8 m class telescope working at near infrared wavelengths, the approximation is valid for an SR typically greater than 10%. For lower SR values, the CoEn gives a pessimistic estimation of the SR. 2 . It can be divided in The goal is therefore to estimate σres two parts: 2 2 2 = σAO (15) σres loop + σfitting , 2 where σAO loop stands for the residual wavefront due to AO loop errors (measurement noise, temporal error, aliasing effect) and 2 σfitting stands for the fitting error due to the finite number of actuators on the DM (Roddier 1999). 2 2 Both σAO loop and σfitting can be computed using parameters obtained in the previous sections.

3.5.1. The estimation of σ 2f itting . As is well known (Born and Wolf 1980), the phase spatial variance can be computed by the sum of the squares of the Zernike coefficients. In our case, we assumed that the polynomials were orthonormal on

Knowing there are (n + 1) polynomials per radial degree n, the fitting error can be expressed by Noll (1976), Conan (1994)  2 σfitting = 0.458 (n N + 1)5/3

D r0

5/3 .

(19)

Let us note that this first term is directly determined by the r0 value derived in section 3.3 and n N (see section 3.5.3). 2 The estimation of the 3.5.2. The estimation of σ AO loop . 2 second term σAO is more complex. This term is written loop as N  2 σAO = σa2i,res , (20) loop i=2

σa2i,res

must be extracted from the closed-loop WFS where measurements. Assuming that the noise in the measurements is independent of the measured residual phase, it becomes σa2i,res = σa2i,slopes − σa2i,noise ,

(21)

where σa2i,noise is estimated by the method presented in section 3.2. But only the first 36 polynomials for the 14 × 14 array (21 for the 7×7) are reconstructed. We have first to fit the σa2i,res behaviour on these polynomials and then to extrapolate this behaviour to the whole set of compensated polynomials. σa2i,res has three main contributors: • the noise filtered by the noise temporal transfer function of the AO loop, which has a (n + 1)−2 dependency (Rigaut and Gendron 1992), • the temporal error corresponding to the filtering of the turbulence by the AO loop error transfer function, which has a (n + 1)−5/3 dependency (Conan 1994), and • the aliasing error, which is significant only on the compensated polynomials of highest radial degrees. 589

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Figure 5. An example of residual phase on experimental data for the V-WFS and theoretical behaviours versus Zernike polynomial number (top) and radial order n (bottom). Seeing = 0.6 arcsec, wind speed = 10 m s−1 , bright star.

This last contributor is neglected in the fit. The first and second contributions have very similar dependences on (n + 1), so we did not try to distinguish between the two power laws in the experimental data. Therefore, only a (n + 1)−2 dependency was fitted. This is illustrated in figure 5 where a Zernike decomposition of NAOS residual phase is plotted. These data were obtained during laboratory tests (see section 4) using the V-WFS with a 209 Hz sampling frequency. The (n + 1)−2 behaviour is well fitted. One can also see the high level of the two last Zernike coefficients of each radial order as predicted by Rigaut and Gendron (1992). Also note the perturbed behaviour of highest orders due to aliasing effects. Thus, as a first approximation, we perform an (n + 1)−2 fit of σa2i ,res :  1 σ 2  a (n + 1)−2 , n + 1 on radial degree n ai,res

(22)

where a is a constant to be fitted in the data. a fully defines 2 σAO loop , knowing n N . 2 3.5.3. The estimation of σres . On the one hand, modal optimization ensures that for each corrected mode σa2i ,res will always be smaller than σa2i ,turb . On the other hand, because of the finite number of sub-apertures, one can define a maximum

590

Figure 6. An example of residual variance Zernike (dots) versus Zernike radial order for two correction cases. Dashed line: (top) fit made on radial order
14 (good SNR), (bottom) fit made on radial order
5 (low SNR). Continuous line:
ai2 turb for n > n N .

radial order for the correction: n max = 14 for the 14 × 14 lenslet array and n max = 7 for the 7 × 7 one. These two points lead to the definition of a cut-off radial order n N in the AO loop error depending on the observing conditions:      0.763 3/5 D − 1 , n max , (23) n N = min E a r0 where E [x] stands for the entire part of x. Using these n N 2 values, the residual variance σres is given by  5/3 n N  D 2 σres = a (n + 1)−1 + 0.458 (n N + 1)−5/3 , (24) r0 n=1 to be used in equation (14) for the computation of CoEn. Examples of the residual variance behaviour are plotted in figure 6 for both high (up) and low (down) WFS SNR. In the good SNR case, the fit of σa2i ,res is performed on the 36 first Zernike coefficients and σa2i ,res is extrapolated up to n N = n max . We see that σa2i ,res are well below
ai2 turb (only plotted for n > n N ). In the bad SNR case, a new cut-off radial order is determined (n N = 5) where σa2i ,res 
ai2 turb .

4. Laboratory tests Before its installation at Paranal in November 2001, the whole NAOS + CONICA system was integrated in the laboratory

NAOS on-line characterization of turbulence parameters and AO performance

in France, for one year. All the system functionalities were tested and validated, with special attention paid to the system performance and turbulence parameter estimators.

(a)

4.1. Turbulence generator tool description A simulator unit (SU) was used to feed NAOS with an F/15 turbulent beam (Arsenault et al 1999). The SU is composed of two phase screens which can emulate various kinds of atmospheric conditions (seeing and wind speed). The incoming beam intercepts a small part of the phase screens (the phase screen diameter is roughly equal to ten times the beam diameter). The wind speed is emulated by the rotation of the phase screens. In the following, we will consider two typical seeing conditions obtained with two different phase screens. Both of them have been precisely spatially characterized (using a high precision Shack–Hartmann WFS with 64 × 64 sub-apertures):

(b)

• a 0.6 arcsec seeing for the medium turbulence phase screen, and • a 0.93 arcsec seeing for the strong turbulence phase screen. For both phase screens, L 0 is equivalent to 25 m at the entrance pupil of the telescope. 4.2. Experimental validation of the atmospheric parameters estimator 4.2.1. The validation of r0 and L 0 estimation. r0 and L 0 were estimated using the procedure described in section 3 and implemented in the RTC. We consider a constant acquisition time on the V-WFS, that is, we use the same sampling frequency (209 Hz) and the same number of frames (2048). With such parameters we obtain a complete phase screen rotation for an equivalent 10 m s−1 wind speed on one set of WF data. Modifying the wind speed allows us to obtain different realizations of phase. The limited accuracy of r0 and L 0 is shown in figure 7. We find an average r0 equal to 18 ± 2 cm rms (that is a seeing equal to 0.57 ± 0.08 arcsec) and an average L 0 equal to 23 ± 2 m rms. The estimation is in good agreement with the phase screen calibration. Note that the same calibration was made for the 0.93 arcsec seeing phase screen, and a value of 11.5 ± 2 cm was found (that is a seeing of 0.9 ± 0.2 arcsec), also in agreement with the calibration. The ±2 cm accuracy on r0 is due to (i) the fitting precision of Ci,turb (0), because of noise on the measurements and the finite number of samples, which leads to a statistical error on both r0 and L 0 estimations, (ii) phase screen deviation from purely Von-Karman statistics, (iii) system mis-calibration (WFS pixel scale misknowledge, precision on DM influence function measurements). Given the system’s complexity and all the required calibrations, it seems that this ±2 cm accuracy on r0 is more than acceptable for the turbulence parameter estimator.

Figure 7. The estimation of (a) r0 (in cm) at 0.5 µm and (b) L 0 (in m) from RTC data as a function of wind speed (in m s−1 ).

Figure 8. Wind speed evolution as a function of SU phase screen rotation speed. The loop is closed with a 14 × 14 V-WFS, a 209 Hz frame frequency.

4.2.2. Wind speed calibration. The SU phase screen rotation speed was not calibrated in terms of absolute wind speed. We present in figure 8 the estimated wind speed using NAOS data as a function of the screen rotation speeds (in arbitrary units). Only one phase screen is used, therefore the Taylor hypothesis is perfectly verified and a linear correspondence 591

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Figure 9. The difference between SRraw , SRloop and CoEn for various AO loop gain (from 0.001 to 0.5). Data were obtained with the 0.6 arcsec seeing phase screen, a wind speed of 10 m s−1 , the 14 × 14 V-WFS (444 Hz) and a 2.166 µm imaging wavelength.

between rotation speed and estimated wind speed is found in a reasonable range of speeds. Note that, as soon as the phase screen motor is switched on, a slight rotation appears (even for a corresponding 0 on the tabulation). 4.3. The validation of the AO loop performance estimator Now, let us validate the AO loop performance estimator implemented in the RTC. In the following, we will compare the RTC estimated CoEn with the CONICA derived SRloop . This SRloop is the SR computed on CONICA images (SRraw ) and corrected from all the unseen aberrations located in the NAOS-CONICA non-common path (that is NAOS dichroic + CONICA optics aberrations). The WFS path aberrations are calibrated by the reference artificial source at the entrance of the WFS channel and are used as reference slopes in the loop. For the CONICA set-up considered (13S camera and Brγ filter), the contribution of the unseen aberrations is calibrated on an artificial point source placed at the entrance of NAOS, and corresponds to an SR  92% on CONICA images (Blanc et al 2003, Hartung et al 2003). In figure 9, we compare SRloop and CoEn for a 0.6 arcsec seeing and a 10 m s−1 wind speed. The 14 × 14 V-WFS is considered with a 444 Hz sampling frequency. Several AO loop gains are considered (from 0.001 to 0.5), in order to simulate various correction levels. The behaviour of both CoEn and SRloop are very close. Nevertheless, as predicted by the theory (Conan 1994), CoEn underestimates SRloop for very bad correction. On the other hand, one can note that for the very good correction CoEn overestimates SRloop (around 7% at maximum). The overestimation error is caused by: the following. • The limited accuracy of the algorithm used to estimate the SR from CONICA images (SR estimation error  4% (Hartung et al 2003)). • Possible saturation effects on the CONICA detector as pointed out later after our tests were performed. • The limited accuracy of the turbulence parameter estimations, in particular r0 , which determines directly 592

Figure 10. A comparison between estimated r0 from RTC data (x axis) and estimated r0 from the ASM (y axis). Data have been obtained from January 2002 to June 2003. 2 the value of the term σfitting . Indeed, for the larger gains shown in figure 9, the WF error is dominated by the fitting error.

5. On-sky results During the instrument commissioning period and the first year of routine use, a wide range of data has been obtained. All these data—saved in the header of the CONICA image file— provide RTC computed turbulence parameters (r0 , L 0 , τ0 ) and performance (CoEn). 5.1. Data selection: the criteria From 2984 available sets of data obtained from both the commissioning runs (from January 2001 to September 2002) and the guarantee time period (March 2003 and June 2003), only 1075 sets have been selected using the following criteria: • data obtained in a NAOS WFS mode with a sampling frequency 100 Hz (187 data rejected), in order to obtain a good fit of the temporal autocorrelation of the WF data, • data obtained for a CONICA exposure time 15 s (1722 data rejected), to ensure the validity of the WF data set captured by the RTC. The rejected data correspond mainly to short exposure CONICA data (in chopping mode, for instance). 5.2. r0 estimation: comparison with the ASM measurements During NAOS-CONICA observations, the Paranal Atmospheric Seeing Monitor (ASM) has regularly recorded seeing data. ASM is a differential image motion monitor (Sarazin and Roddier 1990) which provides an estimation of the seeing (λ/r0 ) at zenith obtained from the measurements of differential image motion given by two sub-pupils of a telescope on a bright star. It is important to note that this bright star is NOT the same one used by the NAOS WFS. Thus, different parts of the atmosphere are sensed by NAOS and ASM. After correction of the zenithal angle for the NAOS guide star, a comparison between the two r0 estimations can be made. In figures 10 and 11, we plot r0 estimated both with

NAOS on-line characterization of turbulence parameters and AO performance

Figure 11. A comparison between estimated r0 from RTC data (solid curve) and estimated r0 from the ASM (dashed curve). Data have been obtained from January 2002 to June 2003.

ASM and NAOS. A good correlation is clearly seen in both figures. A bias of 1.7 cm is found between NAOS and ASM measurements. ASM underestimates r0 in comparison to NAOS. This behaviour can be explained by considering the position in height of both the UT4 primary mirror (12 m) and the ASM (6 m). Thus ASM ‘sees’ a more important part of the ground layer turbulence, which can explain the systematic r0 underestimation in comparison to NAOS. In figure 10 the standard deviation is 3.8 cm (which is two times bigger than laboratory results). This error can be due to the following. • Orography and instrument location: depending on the wind speed direction, the local seeing around ASM and telescope can be different. • Turbulence statistic isotropy: even if computed r0 are corrected for zenith angle effects, two different columns of turbulent volume are sensed by NAOS and ASM. An isotropic hypothesis on r0 is therefore assumed. • Turbulence temporal stationarity: ASM measurements are usually obtained at the beginning of the AO loop exposure while NAOS results correspond to average values computed during the whole exposure. A temporal stationarity of the turbulence statistics during the two data acquisitions is assumed. Considering the results previously obtained in the laboratory (a standard deviation on r0 estimation of ±2 cm as presented in section 4.2.1) and all the possible error sources, the results obtained on sky and the comparison with ASM values are very encouraging. 5.3. Strehl ratio estimation: comparison with the SR measured on CONICA images We compare in this section the estimated performance of NAOS using closed-loop data and the Strehl ratio directly estimated on a CONICA image of an unresolved star. The data used for this study were specifically acquired during the fifth commissioning run (September 2002) by F Lacombe after a full calibration of the instrument. CONICA images are pre-compensated for the noncommon path aberrations (Blanc et al 2003, Hartung et al 2003). Nevertheless, a residual mis-calibration error still degrades the CONICA internal SR. After CONICA aberration pre-compensation, this internal SR is estimated (on average, since it depends on the chosen camera configuration) to be 92% (for K narrow-band filters).

Figure 12. A comparison of CoEn estimated by the NAOS RTC and the SR measured on CONICA images. CoEn is compensated for non-common path residual aberrations (92% SR).

The NAOS performance estimation does not include these non-common path residual aberrations, thus we have to correct the estimated CoEn from this value. Figure 12 shows the good agreement between the measured SR on CONICA images and the estimated CoEn using the NAOS data (after correction of non-common path residual aberrations). A 7.3% standard deviation is found between the NAOS estimator and CONICA images. This last value can be seen as the sum of the errors of the CONICA SR measurement and the NAOS CoEn estimation. In addition a slight bias of 2.6% is visible (an overestimation of the SR using NAOS measurements). This bias can be due to a slight defocalization of CONICA (the precision on CONICA focalization is estimated to ±2.5% (in SR) as presented in Hartung et al (2003)). Note that this bias is much smaller than the one found in laboratory (see section 4.3). 5.4. NAOS performances versus atmospheric parameters The first application of turbulence parameter and system performance estimators is to study the influence of seeing and coherence time on the AO system performance in order to obtain a better global understanding of the system behaviour. Figure 13 shows the system performance evolution (CoEn) versus seeing. The dispersion of data is due to both wind speed and GS magnitude, but the global behaviour clearly shows the performance evolution (degradation) with the increase of the seeing. For comparison only the theoretical SR behaviour due 593

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Figure 13. CoEn evolution as a function of the seeing. The data were collected from January 2002 to June 2003. (Dots) NAOS performance estimation data, (solid curve) theoretical value (only fitting and aliasing effects considered).

Figure 15. Seeing statistics on 1075 data. These data were obtained from January 2002 to June 2003. A comparison between the NAOS and the ASM estimated seeing histograms is proposed.

Figure 14. CoEn evolution as a function of the coherence time. The data were collected from January 2002 to June 2003.

Figure 16. L 0 statistics on 1075 data obtained from January 2002 to June 2003.

to fitting and aliasing error is plotted. This curve only accounts for one item of the global NAOS error budget and thus must be seen as an upper bound. Nevertheless, it shows that NAOS performance is coherent with the theory. Figure 14 shows the system performance (CoEn) versus coherence time. Once again the dispersion of the data is due to both r0 and GS magnitudes. This plot highlights the major influence of τ0 on system performance for fast turbulence (typically τ0 < 3 ms). For τ0
1 ms, the decrease of CoEn is very sharp due to the limited servo-loop bandwidth of the system. Let us recall that the specifications of NAOS were elaborated for τ0  3 ms.

We present in figure 15 a comparison of the seeing histogram from ASM and NAOS data (the same as figure 10). The histograms are very close (with a slight bias already explained in section 5.2) as well as the median (1.05 and 0.97 for ASM and NAOS, respectively) and the mean (1.12 and 1.03 for ASM and NAOS, respectively). NAOS also gives information on outer-scale values. The histogram of L 0 is plotted in figure 16. The results are fully coherent with measurements made at Paranal (Martin et al 2000, Conan 2000) using the generalized seeing monitor (Ziad et al 2000) with a log-normal distribution law and a median value of 19.3 m (22.0 m measured by the GSM). These results show the robustness of the outer scale determination with a WFS on a 8 m class telescope. Finally, figure 17 shows the coherence time τ0 histogram. From this histogram, it can be shown that the mean is equal to 6.1 ms and the median is equal to 4.4 ms. Let us underline that τ0 is lower than 3 ms for 40% of the time, which is higher than usually admitted values. This figure proves the importance of this third key parameter, which is rarely measured on astronomical sites. The results produced by the NAOS tool will be crucial for the specification of the new generation AO for the VLT.

5.5. Statistical study of atmospheric parameters at Paranal using NAOS This last section is dedicated to a first statistical study of atmospheric parameters based on NAOS data. One has to keep in mind that the main purpose here is to show the potential of the NAOS atmospheric parameter estimation tool. It should not be seen as a comprehensive description of Paranal atmospheric conditions. 594

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and S Meimon for helpful discussions. We also thank all our colleagues of the NAOS and CONICA teams for making the NAOS-CONICA system happen and work. We also thank ESO staff who participated in the development of the instrument and helped us during the commissioning periods. This work was performed through ESO contract (4932/INS/95/7454/GWI) and ONERA’s own funds.

References

Figure 17. τ0 statistics on 1075 data obtained from January 2002 to June 2003.

6. Conclusion We have presented a dedicated tool, implemented in the real time computer of the NAOS system, allowing us without any loop opening to automatically monitor and display (every 15 s) both the atmospheric conditions (r0 , L 0 and τ0 ) and the system performance (CoEn). Our method is based on the reconstruction of open-loop data from DM and TTM voltages and residual WF slopes closed-loop data. We are able to extract the noise and then to fit the turbulence data by its relevant parameters. The system performance estimation is based on an extrapolation of the residual phase variance deduced from the residual WF slopes and the estimated noise. We have validated the algorithms and tested their robustness using both simulated and experimental data obtained from laboratory and on-sky sessions. This unique tool is essential for closed-loop diagnostics and for the validation of the observations. In particular, all the computed parameters which represent the observation conditions are saved in the CONICA header files and may be used afterwards during data reduction. The turbulence parameter estimator could also be used to perform statistical studies on r0 , L 0 and τ0 at Paranal as a complement to ASM data. In addition, the combined estimation of turbulence conditions, WFS noise and AO performance allows a better understanding of the behaviour of NAOS under different observing conditions. The first key parameter is r0 (or the seeing), as expected. The second one is τ0 . Very small values of τ0 lead to a dramatic fall of system performance. This highlights the need for precisely taking into account turbulence parameter statistics (also including L 0 ) during the specification of future AO systems for VLT second generation instrumentation and future extremely large telescopes (ELT).

Acknowledgments We are grateful to G Dumont who coded the performance and turbulence estimator in NAOS RTC. We want to warmly thank M Sarazin, R Conan, J-M Conan, F Quiros-Pachequo

Arsenault R et al 1999 NAOS Simulator Unit Design Report, Technical Report VLT-TRE-NAO-11650-1-400000-0001 Avila R, Ziad A, Borgnino J, Martin F and Agabi A 1997 J. Opt. Soc. Am. 14 3070 Beuzit J-L, Hubin N, Chazallet F, Demailly L, Gendron E, Gigan P, Lacombe F and Rabaud D 1994 Adaptive Optics in Astronomy ed M A Ealey and F Merkle; Proc. SPIE 2201 Blanc A, Fusco T, Hartung M, Mugnier L M and Rousset G 2003 Astron. Astrophys. 399 373 Born M and Wolf E 1980 Principles of Optics 6th edn (Oxford: Pergamon) Chassat F 1992 PhD Thesis Universit´e Paris XI, Orsay Conan J-M 1994 PhD Thesis Universit´e Paris XI, Orsay Conan J-M, Rousset G and Madec P-Y 1995 J. Opt. Soc. Am. A 12 1559 Conan R 2000 PhD Thesis Universit´e de Nice-Sophia Antipolis, Nice Conan R et al 2002 Astron. Astrophys. Suppl. Ser. 396 723 Dessenne C 1998 PhD Thesis Universit´e de Paris VII, Paris Feautrier P et al 2002 Adaptive Optical System Technology II; Proc. SPIE 4839 250–8 Gendron E and L´ena P 1994 Astron. Astrophys. 291 337 Gendron E et al 2002 Adaptive Optical System Technology II; Proc. SPIE 4839 195–205 Hartung M, Blanc A, Fusco T, Lacombe F, Mugnier L M, Rousset G and Lenzen R 2003 Astron. Astrophys. 399 385 Lagrange A-M et al 2002 Instrumental Design and Performance for Optical/Infrared Ground-based Telescopes; Proc. SPIE 4841 Lenzen R, Hartung M, Brandner W, Finger G, Lacombe F, Lagrange A-M, Lehnert M D, Moorwood A F and Mouillet D 2002 Instrumental Design and Performance for Optical/Infrared Ground-based Telescopes; Proc. SPIE 4841 Madec P-Y, Conan J-M and Rousset G 1992 Progress in Telescope and Instrumentation Technologies (Garching, Germany: ESO) Martin F, Conan R, Tokovinin A, Ziad A, Trinquet H, Borgnino R, Agabi A and Sarazin M 2000 Astron. Astrophys. Suppl. Ser. 144 39 No¨el T 1997 PhD Thesis Universit´e Paris VI, Paris Noll R J 1976 J. Opt. Soc. Am. 250 280 Racine R and Ellerbroek B L 1995 Adaptive Optical Systems and Applications; Proc. SPIE 2534 248–57 Rigaut F and Gendron E 1992 Astron. Astrophys. 261 677 Rigaut F, Rousset G, Kern P, Fontanella J-C, Gaffard J-P, Merkle F and L´ena P 1991 Astron. Astrophys. 250 280 Roddier F (ed) 1999 Adaptive Optics in Astronomy (Cambridge: Cambridge University Press) Roddier F, Gilli J M and Lund G 1982 J. Opt. (Paris) 13 263 Rousset G, 1992 private communication Rousset G, Fontanella J-C, Kern P, Gigan P, Rigaut F, L´ena P, Boyer C, Jagourel P, Gaffard J-P and Merkle F 1990 Astron. Astrophys. 230 29 Rousset G, Madec P-Y and Rabaud D 1991b High Resolution Imaging by Interferometry II (ESO Proc. vol 39) (Garching, Germany: ESO) Rousset G, Madec P-Y and Rigaut F 1991a ICO Toptial Mtg on Atmospheric, Volume and Surface Scattering and Propagation ed A Consortini pp 77–80 Rousset G et al 1993 ESO Proc. vol 48 (Garching, Germany: ESO) Rousset G et al 2002 Adaptive Optical System Technology II; Proc. SPIE 4839 140–9 and references herein

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Sarazin M and Roddier F 1990 Astron. Astrophys. 227 294 Sch¨ock M, Le Mignant D, Chanan G and Wizinowich P 2002 Adaptive Optical System Technology II; Proc. SPIE 4839 813–24 Tokovinin A, Baumont S and Vasquez J 2003 Mon. Not. R. Astron. Soc. 340 52

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Wilson R W 2002 Mon. Not. R. Astron. Soc. 337 103 Wilson R W et al 1999 Mon. Not. R. Astron. Soc. 309 379 Winker D M 1991 J. Opt. Soc. Am. A 8 1568 Ziad A, Conan R, Tokovinin A, Martin F and Borgnino J 2000 Appl. Opt. 39 5415

106ANNEXE B. NAOS ON-LINE CHARACTERIZATION OF TURBULENCE PARAMETERS AN

Annexe C "High order adaptive optics feasibility and requirements for direct detection of extra-solar planets" T. Fusco et al. - Opt. Express - 2006

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High-order adaptive optics requirements for direct detection of extrasolar planets: Application to the SPHERE instrument T. Fusco1 , G. Rousset2,1 , J.-F Sauvage1 , C. Petit1 , J.-L Beuzit3 , K. Dohlen4 , D. Mouillet5 , J. Charton3 , M. Nicolle1 , M. Kasper6 , P. Baudoz2 , and P. Puget2 1

ONERA, Optics Department, BP 72, F-92322 Chatillon, France LESIA, Observatoire de Paris, 5 Place Jules Janssen 92195 Meudon, France 3 LAOG, Observatoire de Grenoble, BP 53, F-38041 Grenoble, France 4 LAM, Observatoire de Marseille, BP 8, F-13376 Marseille, France Observatoire Midi-Pyr´en´ees, 57 Avenue d’Azereix, BP 826, F-65008 Tarbes, France 6 ESO, Karl-Schwarzschild-Straße 2, Garching, D-85748 Germany [email protected] 2

5

Abstract: The detection of extrasolar planets implies an extremely high-contrast, long-exposure imaging capability at near infrared and probably visible wavelengths. We present here the core of any Planet Finder instrument, that is, the extreme adaptive optics (XAO) subsystem. The level of AO correction directly impacts the exposure time required for planet detection. In addition, the capacity of the AO system to calibrate all the instrument static defects ultimately limits detectivity. Hence, the extreme AO system has to adjust for the perturbations induced by the atmospheric turbulence, as well as for the internal aberrations of the instrument itself. We propose a feasibility study for an extreme AO system in the frame of the SPHERE (Spectro-Polarimetry High-contrast Exoplanet Research) instrument, which is currently under design and should equip one of the four VLT 8-m telescopes in 2010. © 2006 Optical Society of America OCIS codes: (010.1080) Adaptive optics; (120.1880) Detection

References and links 1. J.-L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, P. Puget, T. Fusco, and A. Boccaletti, “A planet finder instrument for the VLT,” in Proceedings of IAU Colloquium 200, Direct Imaging of Exoplanets: Science & Techniques (Cambridge University Press, 2005), pp. 317–323. 2. R. Racine, G. A. Walker, D. Nadeau, and C. Marois, “Speckle noise and the detection of faint companions,” Pub. Astron. Soc. Pacific 112, 587–594 (1999). 3. R. Lenzen, L. Close, W. Brandner, M. Hartung, and B. Biller, “NACO-SDI: A novel simultaneous differential imager for the direct imaging of giant extra-solar planets,” in Proceedings of Science with Adaptive Optics, W. Brandner and M. Kasper, eds. (Springer-Verlag, 2004), pp. 47–50. 4. J.-F. Sauvage, L. Mugnier, T. Fusco, and G. Rousset “Post processing of differential images for direct extrasolar planet detection from the ground,” in Advances in Adaptive Optics II, Proc. SPIE 6272, pp. 753–763. 5. H. M. Schmid, J.-L. Beuzit, M. Feldt, D. Gisler, R. Gratton, T. Henning, F. Joos, M. Kasper, R. Lenzen, D. Mouillet, C. Moutou, A. Quirrenbach, D. M. Stam, C. Thalmann, J. Tinbergen, C. Verinaud, R. Waters, and R. Wolstencroft, “Search and investigation of extra-solar planets with polarimetry,” in Proceedings of IAU Colloquium 200, Direct Imaging of Exoplanets: Science & Techniques (Cambridge University Press, 2005), pp. 165–170. 6. R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J. 618, 161-164 (2005).

#69303 - $15.00 USD (C) 2006 OSA

Received 23 March 2006; revised 20 June 2006; accepted 20 June 2006 21 August 2006 / Vol. 14, No. 17 / OPTICS EXPRESS 7515

7. A. Boccaletti, P. Riaud, P. Baudoz, J. Baudrand, D. Rouan, D. Gratadour, F. Lacombe, and A.-M. Lagrange, “The four-quadrant phase-mask coronagraph IV. First light at the very large telescope,” Pub. Astron. Soc. Pacific 116, 1061–1071 (2004). 8. J.-F. Sauvage, T. Fusco, G. Rousset, C. Petit, A. Blanc, B. Neichel, and J.-L. Beuzit, “Fine calibration and precompensation of non-common path aberrations for high performance AO system,” in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 88–95 (2005). 9. F. Rigaut, J.-P. V´eran, and O. Lai, “Analytical model for Shack-Hartmann-based adaptive optics system,” Proc. SPIE 3353, 1038–1048 (1998). 10. L. Jolissaint and J.-P. V´eran, “Fast computation and morphologic interpretation of the Adaptive Optics Point Spread Function,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., Vol. 58 of European Southern Observatory Conference and Workshop Proceedings (ESO, 2002), pp. 201– 205. 11. R. Conan, T. Fusco, G. Rousset, D. Mouillet, J.-L. Beuzit, M. Nicolle, and C. Petit, “Modeling and analysis of XAO systems. Application to VLT-Planet Finder,” in Advancements in Adaptive Optics, Proc. SPIE 5490, 602–608 (2004). 12. C. Robert, J.-M. Conan, V. Michau, T. Fusco, and N. Vedrenne, “Scintillation and phase anisoplanatism in Shack– Hartmann Wavefront Sensing,” J. Opt. Soc. Am. A 23, 613–624 (2006). 13. O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005). 14. F. Roddier and C. Roddier, “NOAO Infrared Adaptive Optics Program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE 628, 298–304 (1986). 15. E. Masciadri, M. Feldt, and S. Hippler, “Scintillation effects on a high-contrast imaging instrument for direct extrasolar planets’ detection,” Astrophys. J. 613, 572–579 (2004). 16. F. Mah´e, V. Michau, G. Rousset, and J.-M. Conan, “Scintillation effects on wavefront sensing in the Rytov regime,” in Propagation through the Atmosphere IV, M. Roggemann, ed., Proc. SPIE 4125, 77–86 (2000). 17. C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, and D. Rouan, “Fundamental limitations on Earth-like planet detection with extremely large telescopes,” Astron. Astrophys. 447, 397–403 (2006). 18. P.-Y. Madec, “Control techniques,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University Press, 1999), pp. 131–154. 19. C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” in Advancements in Adaptive Optics, Proc. SPIE 5490, 1414–1425 (2004). 20. E. Gendron and P. L´ena, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337–347 (1994). 21. L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004). 22. T. Fusco, M. Nicolle, G. Rousset, V. Michau, J.-L. Beuzit, and D. Mouillet, “Optimisation of Shack-Hartmannbased wavefront sensor for XAO system,” in Advancements in Adaptive Optics, Proc. SPIE 5490, 1155–1166 (2004). 23. T. Fusco, C. Petit, G. Rousset, J.-M. Conan, and J.-L. Beuzit, “Closed-loop experimental validation of the spatially filtered Shack-Hartmann concept,” Opt. Lett. 30, 1255–1257 (2005). 24. M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack-Hartmann wavefront sensor measurement for extreme adaptive optics,” Opt. Lett. 29, 2743–2745 (2004). 25. T. Fusco, G. Rousset, and A. Blanc, “Calibration of AO system. Application to NAOS-CONICA,” in Science with Adaptive Optics, W. Brandner and M. Kasper, eds. (Springer-Verlag, 2004), pp. 103–107. 26. M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti, “Fast calibration of high-order adaptive optics systems,” J. Opt. Soc. Am. A 21, 1004–1008 (2004). 27. A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003), URL mailto:[email protected]. 28. M. S. Robbins and B. J. Hadwen, “The noise performance of electron multiplying charge coupled device,” IEEE Trans. Electron. Devices 50, 1227–1232 (2003).

1.

Introduction

Direct detection and spectral characterization of exoplanets is one of the most exciting but also one of the most challenging areas in the current astronomy. In that framework, the SPHERE (Spectro-Polarimetry High-contrast Exoplanet Research) instrument is currently under design and should equip one of the four 8-m telescopes of the European Southern Observatory Very Large Telescope (ESO VLT) at Paranal (Chile). The main scientific objective of SPHERE [1] is #69303 - $15.00 USD (C) 2006 OSA

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the direct detection of photons coming from giant extrasolar planets (between 1 and 20 Jupiter masses). Any detection will then be followed by a first characterization of the planet atmosphere (clouds, dust content, methane, water absorption...). In addition, the survey of an extended number of stars (typically a few hundreds) is mandatory for performing meaningful statistical studies. Such extremely challenging scientific objectives directly translate into a relatively complex high-contrast instrument. Coronagraphic and smart imaging capabilities are essential for reaching the high contrast (close to the optical axis) required for direct extrasolar planet detection. From the ground, the core of any high-contrast instrument is an extreme adaptive optics (XAO) system. Such a system must be capable of making corrections for the perturbations induced by the atmospheric turbulence as well as for the internal aberrations of the instrument itself. In the following sections, we mainly focus on the main AO loop design. After a brief description of the SPHERE instrument in Section 2, a presentation of the coronagraphic image profile is posed, and its impacts on system performance are highlighted. The global error budget of the AO system is presented in Section 3, and details of each term are given in Sections 4 to 9. After the required trade-offs in terms of system design, a global presentation of the SPHERE extreme AO system (SAXO) is proposed in Section 10. 2.

SPHERE and SAXO

The SPHERE system aims at detecting extremely faint sources (giant extrasolar planets) in the vicinity of bright stars. Such a challenging goal requires the use of a very-high-order performance AO system, a coronagraphic device to cancel out the flux coming from the star itself, and smart focal plane techniques to calibrate any coronagraph imperfections and residual uncorrected turbulent or static wavefronts. The detection limit for the SPHERE instrument is 10−6 (i.e., 15 magnitudes between star and the planet) with a goal around 10−8 . There is no direct link between the AO system performance and the final detectivity of the instrument; nevertheless, the impact of AO on the final performance is related to the performance of the coronagraph. A better AO correction leads to a better coronagraph extinction and therefore leads to the following improvements in system performance: • a reduction of the photon and flat-field noises (i.e., a gain in Signal-to-Noise Ratio for a given integration time), • a reduction of the pinned speckle (through the reduction of airy pattern intensity due to the coronagraph optimization). These reductions are important from the global system performance point of view, and the optimization of the coronagraph rejection is a main goal of the SPHERE system. It of course requires the use and the optimization of an XAO system, as presented in the following. Nevertheless, the ultimate detection limit will be achieved through an extreme control of system internal defects (noncommon path aberrations (NCPAs), optical axis decentering, vibrations, coronagraph and imaging system imperfections, and so on). This ultimate control will also be partially ensured by the AO system through the use of additional devices in the AO concept (see Sections 8 and 9). To meet the requirements (and hopefully the goal) in terms of detection, the proposed design of SPHERE is divided into four subsystems, namely, the common path optics and three science channels. The common path includes pupil-stabilizing foreoptics (tip-tilt and derotator) where insertable polarimetric half-wave plates are also provided, the SAXO XAO system with a visible wavefront sensor, and near infrared (NIR) coronagraphic devices in order to feed the infrared dual-imaging spectrograph (IRDIS) and the integral field spectrograph (IFS) with a #69303 - $15.00 USD (C) 2006 OSA

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highly stable coronagraphic image in the NIR. The three scientific channels gather complementary instrumentation to maximize the probability of exoplanet detection and to give us access to a large range of wavelengths and information (e.g., imaging, spectra, polarization). The first instrument is an imaging camera (IRDIS). It is based on the principle of differential imaging initially proposed by R. Racine [2] and recently demonstrated on the VLT NACO instrument with the SDI device [3]. The idea is to record simultaneously two images at two close wavelengths. Assuming that there is a spectral feature in the object (absorption in one of the wavelengths), it is therefore possible to distinguish the speckle pattern (which has the same contribution at the two wavelengths) and the faint object. This distinction can be done by using a simple subtraction of the two images or considering more clever signal processing approaches [4]. The second focal plane instrument will be an IFS working from 0.95 µ m to 1.7 µ m and providing low spectral resolution (R ∼ 30) over a limited, 3” × 3”, field of view. The last scientific channel contains a visible dual-imaging polarimeter (ZIMPOL), working between 0.65 µ m and 0.95 µ m. Due to its innovative lock-in technique [5], it can achieve polarimetric precisions better than 10−5 on a localized signal measured differentially against a smooth background. ZIMPOL shares the visible channel with the wavefront sensor and includes its own coronagraphic system. The concept behind this very challenging instrument is illustrated in Fig. 1, where the common NIR-Vis beam is indicated in orange, the exclusively NIR beam is indicated in red, and the exclusively Vis beam is indicated in blue.

Fig. 1. Global concept of the SPHERE instrument, indicating the four subsystems and the main functionalities within the common path subsystem. Optical beams are indicated in red for NIR, blue for Vis, and orange for common path.

The foreoptics system, originally dedicated to pupil stabilization (lateral and rotational), also accommodates two insertable half-wave plates (required during ZIMPOL observations) and a polarizer for its calibration. A photon-sharing scheme has been agreed on between IRDIS and IFS, thus allowing IFS to exploit the NIR range up to the J-band. The H-band, optimal for the dual-band imaging (DBI) mode, is required for IRDIS during the main observation program. This multiplexing optimizes observational efficiency. However, the additional requirements in #69303 - $15.00 USD (C) 2006 OSA

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terms of broadband Atmospheric dispersion (ADC) and coronagraph efficiencies still need to be fully investigated. The instrument will be mounted on the Nasmyth platform rather than directly attached to the telescope Nasmyth rotator. Indeed, this one is not adapted for carrying the full charge of the instrument bench. All the subsystems will be mounted onto the bench, which is likely to be actively damped by a pneumatic servo-controlled system and equipped with a dust cover. The extreme AO system (SAXO) is the core of the SPHERE instrument, and is essential for reaching the extremely high contrast requirements. In this framework, the SAXO must fulfill the following three high-level requirements: • Ensure the measurement and correction of the turbulent phase perturbations of the telescope and system common optics aberrations and of the NCPAs (main AO loop); • Ensure an extremely high stability (at low temporal frequency) of the optical axis at the level of the coronagraphic mask [auxiliary sensor (AS)]; • Ensure the measurement and the correction of any pupil motion [pupil motion sensor (PMS)]. In keeping with the three main high-level requirements and in close collaboration with astronomers, we have performed a detailed optimization of the SAXO system, which is summarized hereafter. 3.

Coronagraphic profile and detection signal-to-noise ratio

The first and critical point to be addressed for any AO system optimization is the performance estimation parameter. Unlike classical AO systems, residual variance and Strehl ratio are not sufficient anymore for optimizing the system and deriving the pertinent trade-offs. They have to be replaced by a more accurate parameter that can provide information on the coronagraphic image shape in the focal plane. During the past few years, a large number of coronagraphic devices have been proposed, ranging from modified Lyot concepts (with apodization for instance [6]) to interferometric devices such as the four quadrants coronagraph [7]. Each approach has its own advantages and drawbacks, and it is likely that more than one device will be implemented in the SPHERE instrument. In any case, the purpose of the coronagraph is to remove the coherent light coming from the on-axis guide star (GS). Therefore one can analytically define a ”perfect coronagraph” using the following equations: h i 2 √ (1) Cres (ρ ) = h FT P(r)A(r)eiϕres (r) − Ec P(r) i.

Cres (ρ ) corresponds to the image intensity in the focal plane after the coronagraphic process. ρ stands for the focal plane position, r for the pupil plane coordinates, and h.i for a statistical average; and, with A(r) the wavefront amplitude, ϕres (r) the residual phase after AO correction, P(r) the pupil function and Ec the short exposure coherent energy defined as follows: h i 2 Ec = exp −σϕ2 − σlog(A) , (2) with

σϕ2

1 = S

2 σlog(A)

1 = S

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(Z 

1 ϕ (r ) − S

S

  log A(r′ ) −

S

(Z 

′



Z

S



ϕ (r)dr

1 S

Z

S

2

dr

′

)

log [A(r)] dr

(3) 2

dr

′

)

.

(4)

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It is interesting to note that when Ec = 1, i.e., when all the phase and amplitude effects have been corrected, all the light coming from the star has been canceled out. When only a partial correction is performed, the coherent peak is removed and only the incoherent light (the residual uncorrected speckles) remains. In that case, it is easy to show that, as a first approximation (first order expansion) the coronagraphic image intensity is proportional to the residual phase power spectral density: Cres (ρ ) ∝ h|FT [ϕres (r)]|2 i. (5) This is illustrated in Fig. 2 where both classical and coronagraphic images (cut off along the x-axis) are plotted in the case of a 41 x 41 actuator system with a 1.5 kHz sampling frequency. A sufficiently bright GS is considered, so the noise measurement effects can be disregarded. The average wind speed value is 12.5 m/s, and the seeing is 0.85 arcsec.

Fig. 2. Comparison of focal plane intensity repartition (expressed in terms of contrast versus the center of the FoV) for AO corrected PSF (without coronagraph) and for a perfect coronagraph, as defined in Eq. (1). The x-axis is in milli-arcsec in the focal plane.

The final performance of the instrument depends on the accuracy of system internal calibrations; that is: • the calibration of NCPAs [8], the differential aberrations between the two (or more) channels in differential imaging. • the imperfection of the coronagraphic device itself • the science detector calibration (flat field), the level of sky background, etc. Nevertheless, the ultimate limit is given by the photon noise level. In that case, one can show that the total integration time required for achieving a given signal-to-noise ratio (between the planet signal and star residual light) is directly proportional to the shape of Cres , as shown in Eq. (6): p √ Tint ∗Cres (ρ ) ∗ Ns SNR ∝ 2 ∗ , (6) D2 ∗ S ∗ Np #69303 - $15.00 USD (C) 2006 OSA

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where Np and Ns are the numbers of photons per m2 and per s on the telescope pupil for the planet and the star. S stands for the Strehl ratio and D the telescope pupil diameter. It is interesting to note that the integration time decreases as 1/D4 for a given contrast. 4.

Balancing an error budget

The whole AO study is performed with the aim of achieving a balanced error budget.A first approximation (first-order expansion) of this coronagraphic shape is given by the residual phase power spectral density (PSD) as shown in Section 3. Nevertheless, in the coronagraphic shape of the final image, second-order terms can play a non-negligible role depending on the system characteristics (see Subsection 7.1 for instance). Therefore, the global error budget for the AO system has to account for these second error terms, and the global error budget can be summarized as follows: Cres (ρ ) =Cscint +Cdi f f +Cchrom +Cre f rac +Caniso {z } | atmospheric limitation

+C f it +Ctemp +Calias +Cnoise +Ccalib +Caberr . {z } {z } | | calibration errors

low order residual error

|

{z

AO loop residual error

(7)

}

Cres is expressed in terms of residual focal position in order to highlight the domains that are affected by each error item. Each error item is described in the following. More generally, the AO error budget can be divided into three main items: atmospheric limitations, AO loop residual errors, and calibration errors. The optimization of this error budget will be performed to meet three main criteria: • corrected area, i.e., the focal plane area where the image contrast is significantly improved by the AO system. It mainly drives the choice of the number of actuators (the correction area is equal to λim /d in diameter, where d is the actuator spacing). Considering the typical targets that will be observed by SPHERE and the imaging wavelengths (J-, H-, and K-bands), this area has to be larger than 0.8 arcsec in diameter. If a perfect coronagraph is considered, the corrected area can go as close as possible to the optical axis. In practice, a limit will be set by the characteristics and defects of the coronagraph device. For an efficient coronagraph such as the four quadrants phase mask, a reasonable limit should be set around a few (two or three) diffraction angles (λ /D), which is typically 100 mas for an 8 m telescope in the H-band. • detectivity level, i.e., the capability of the whole system to detect the planet signal. This level is affected by the AO loop errors (temporal, noise, aliasing, etc.), which evolve rapidly with time and can be calibrated using differential imaging and a reference PSF. It can also be degraded by the telescope and the system’s high spatial frequencies and NCPAs, which slowly evolve with time and represent the ultimate limitation for the differential imaging and reference PSF subtraction techniques. The minimization of the slowly varying defects implies the measurement and the correction of NCPAs (see Section 8), as well as the stabilization of the optical beam during a whole observation sequence (see Subsection 9.2). • system sensitivity, i.e., the limiting magnitude of the natural GS used to close the AO loop. This criterion is driven by the number of stars to be observed, but it depends highly on the detectivity level and the corrected area size. Indeed, the larger the corrected area, the smaller the available flux per individual measurement zones (subaperture in the case #69303 - $15.00 USD (C) 2006 OSA

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of an SH device, for instance). In addition, increasing the detectivity level implies a reduction in terms of temporal and noise errors, which leads to a faster system working on brighter GSs for wavefront sensing. One can also note that, considering the required level of performance in terms of AO correction, the laser GS solution has been excluded. A first trade-off between scientific goals and system requirements has led to a limiting magnitude of 8 in the H-band (corresponding to magnitudes 10–11 for the visible band, depending on the GS types) for the system, implying that the detectivity capabilities have to remain optimum up to this magnitude. In addition to these three scientific criteria, other constraints have to be taken into account during the instrument design: • The use of well-proved technologies, if possible. • New developments for critical issues only, with associated experimental validations. • A tight schedule (typically 5 years) with finite manpower and budget for building the system. These last three points are essential for minimizing the risk factor during the instrument realization. 5.

Simulation tools

Two main classes of simulation tools have been used for the analysis and design of SAXO: • a PSD-based simulation tool, based on the generalization of an approach first introduced by F. Rigaut and J.-P. V´eran [9, 10]. Analytical expressions of spatial PSD are obtained for various errors affecting the AO system (fitting, aliasing, temporal, noise, anisoplanatism, differential refraction, and so on) and used to directly compute AO residual phase screens, • an end-to-end simulation tool based on a complete and exhaustive (as far as possible) simulation of the AO loop [11]. It includes the Fresnel propagation through the atmosphere (including spherical and laser propagation [12]); an accurate model of the correcting devices (with various influence functions, nonlinear and hysteresis effects, and so on); the Wave Front Sensor (WFS) devices (including diffraction effects, focal plane filtering, chromatism effects, various signal-processing algorithms for both SH and Pyramid); and the control laws (optimal modal gain integrator, Kalman filtering, ... ). The calibration processes are also simulated with their possible error sources (noise, misalignments, NCPAs, and so on). The PSD-based tool mainly enables rapid reduction of the parameter space in order to make the first system trade-offs [13]. It is also used to feed focal plane instruments (coronagraphs, differential imaging, IFU, etc.) with corrected wavefronts. On the other hand, only an end-to-end model allows for in-depth study of each subsystem behavior and, therefore, the optimization of all the AO parameters (fine design of the WFS, choice of DM characteristics, control law optimization, and so on.). Using the simulation tools described above, we have studied the effects of all error sources that may degrade the final performance of the AO system. In addition, critical points and news ideas have been validated experimentally by the use of the ONERA AO bench. A short summary of these studies is presented below.

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6.

Atmospheric limitations

The first kind of error is a gathering of all the propagation effects: scintillation, diffraction, and differential refraction effects. These atmospheric limitations [14] represent the ultimate limitation of a classical AO system, since they cannot be corrected using a single-pupil conjugated DM. The only way to mitigate them is to change the wavefront-sensing wavelength. A morecomplete correction of these effects would require a multi-DM system in order to compensate for both phase and amplitude effects. In the following subsection, the different atmospheric effects are studied, and the choice of a classical AO system versus a more-complex multistage AO system is justified. 6.1.

Scintillation and diffraction effects

Let us first consider the scintillation and diffraction effects, i.e., the modification of the phase and amplitude of the turbulent wavefront after its propagation through the atmosphere. These effects depend on the turbulence profile (Cn2 profile) and on the wavelength. They affect both the focal plane images [15] (as illustrated in Fig. 3) and the WFS accuracy [16]. Figure 3

Fig. 3. Comparison of coronagraphic images in the case of a fitting error only (without scintillation error) and in the case of a perfect phase correction but with a scintillation error. [Left] two Cn2 profiles are considered: a typical Paranal case (θ0 = 2.5 [email protected]µ m) [solid curve] and a pessimistic profile (θ0 = 1.2 [email protected]µ m). The fitting case [dotted curve] is plotted for comparison. [Right] Effect of the imaging wavelength (from 2.2 to 0.7 µ m). The typical Cn2 profile is considered.

highlights the fact that scintillation effects are negligible in comparison with the residual phase effects, even in the ideal case where only a fitting error is considered (no temporal, noise, or aliasing effects). It also demonstrates that, for a given system (i.e., a given number of actuators) the scintillation effects become more and more negligible (in comparison with phase effects) when the imaging wavelength decreases. The global effects in variance increase in λ 7/6 for scintillation and in λ 2 for phase effects. Moreover, it is shown that the scintillation effects on the coronagraphic images are barely chromatic in the ”corrected area” (wavelength impacts both on the image shape and on the focal plane position in arcsec). In addition, using a complete Shack–Hartmann (SH) and Pyramid WFS model, it has been shown that the scintillation effects on the WFS accuracy are lower than typically 20 nm rms for a

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pessimistic Cn2 profile and monochromatic wavelength at 0.7 µ m, which is certainly pessimistic since broadband effects will reduce this error. 6.2.

Chromatism effects on refraction index

The optical path of the incoming beam through the atmosphere depends on imaging and WFS wavelengths. Indeed, ignoring the effects of humidity and pressure fluctuations, one can express the index fluctuation by the following relation:    45.47 P 6839.4 −6 ∆n(λ ) = 10 + ∆T, (8) 23.7 + −2 −2 130 − (λ ) 38.9 − λ T2 with λ in microns, P the mean atmospheric pressure, T the mean atmospheric temperature and ∆T the temperature fluctuations. Note that it is assumed that the index n is equal to n(λ ) = naverage + ∆n(λ ).

(9)

Hence, the difference between the wavefront sensor effective wavelength and the imaging wavelength induces two effects: differential refraction and correction chromatism. 6.2.1.

Differential refraction effects

The first effect is due to the wavelength dependency of the refraction index of the atmosphere. Such an effect induces a differential refraction, that is a beam shift for two different wavelengths (each beam propagates in a slightly different part of the turbulence, as shown in Fig. 4). This can be seen as an anisoplanatic degradation: the beam for a given wavelength comes from an equivalent GS position angle of deviation θ :  
θ = ∆n λw f s − ∆n [λim ] tan (zenith) , (10)

with ∆n [λ ] the refraction index fluctuations at the wavelength λ , and zenith the zenithal angle. It is therefore clear that the degradation depends on the zenithal angle, the Cn2 profile, the wavefront sensor effective wavelength, and the imaging wavelength. The effects onto the coronagraphic image are illustrated in Fig. 4 for various zenithal angles (right plot). A typical Cn2 profile is considered with a 1.65 µ m imaging wavelength and a 0.65 µ m wavefront sensing wavelength. It is interesting to note that the modification of the zenith angle (Fig. 4 [left]) impacts both on the corrected area (due to the differential refraction effects) and on the uncorrected region (due to the increase of the seeing values and thus the fitting errors). Because differential refraction effects become more and more important when the wavelength difference between imaging and wavefront sensing increases, they will be included in the choice of the WFS wavelength. This trade-off will be made considering that, from a science point of view (number of available targets), SPHERE has to be able to observe up to 40 (with a goal at 50) degrees from zenith. 6.2.2.

Chromatism effects on the wavefront correction

The second effect of the difference between imaging and WFS wavelengths is due to the small but yet-existing dependency of the refraction index fluctuations with respect to the wavelength. Hence, the wavefront at a given wavelength λ1 is modified with respect to the wavefront at another wavelength λ0 by the relation:

φturb (λ1 ) =

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∆n(λ1 ) φturb (λ0 ). ∆n(λ0 )

(11)

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λ2

λ 1 : WFS wavelength γ

λ1 d

λ 2 : Imaging wavelength γ : Zenith angle

d : maximum displacement

Fig. 4. [Left] Schematic representation of the differential refraction effect. [Right] Influence of the zenith angle for a 1.65µ m imaging wavelength and a 0.65µ m WFS wavelength (a perfect correction at WFS wavelength is assumed in each case). A typical Cn2 profile has been considered. The fitting error of a 40x40 subapertures system has been plotted for comparison).

Because the correction provided by the DM is proportional to n(λ ) and not to ∆n(λ ), the correction phase obtained from λ0 measurements and expressed at the λ1 wavelength is equal to n(λ1 ) φcorr (λ1 ) = φcorr (λ0 ). (12) n(λ0 ) Hence, even if the turbulence is perfectly measured and corrected at one given wavelength, the residual wavefront for another wavelength is not null and is directly proportional to the input signal itself, with an attenuation coefficient depending on the index values for the two wavelengths: (α − 1) (13) φturb (λ1 ), φres (λ1 ) = φturb (λ1 ) − φcorr (λ0 ) ≃ α where α = ∆n(λ1 )/∆n(λ0 ). Therefore, the coronagraphic error term induced by the chromatism error on wavefront correction is directly proportional to the turbulent power spectral density, that is, with a ρ −11/3 dependency at least for ρ larger than a few tens of arcseconds (not affected by outer scale effects). The proportionality coefficient is equal to (α − 1)2 /α 2 . It is interesting to mention here that this effect is fully predictable (if average atmospheric parameters T and P are known). It can be included in a control law scheme in order to dramatically reduce its effects. Nevertheless, as shown in Fig. 5, error remains negligible in comparison with differential refraction effects at least in the inner working area (between 100 and 3000 mas). The conclusion is surely different if a larger telescope is considered. In that case the diffraction pattern becomes smallerm and one would want to work closer to the optical axis where the chromatism effects cannot be neglected anymore. In that case a modification of the control law should be required to compensate for the chromatism effects.

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Fig. 5. Impact of the WFS wavelength on chromatic errors (differential refraction and wavefront correction). The imaging wavelength is 1.6µ m. The Zenith angle is equal to 40o . A typical Cn2 profile has been considered (a perfect correction at WFS wavelength is assumed in each case). The fitting error of a 40x40 subapertures system has been plotted for comparison).

6.3.

Conclusion in terms of system design

Full correction of scintillation effects would only lead to an equivalent reduction of the phase variance smaller than 20 nm rms. In comparison with the small gain in performance, the system complexity is highly increased (two DMs, measurement devices, reconstruction process, calibration issues, etc.). In consideration of the small expected gain, even if the scintillation were fully corrected (which is far from being manifest), it has been decided not to consider a scintillation corrector for the SAXO system. It has been shown that the main limitation in terms of atmospheric errors would be the differential refraction rather than scintillation. In that case, the ways to mitigate these effects would be a modification of the WFS wavelength in order to be as close as possible to the imaging wavelength, or to limit the portion of accessible sky (observe only close to zenith). 7.

AO loop residual errors

The AO loop residual errors gather all the errors related to the AO system itself. These errors can be divided in two main types: high-order errors, which affect the high spatial frequencies only, i.e., mainly the focal plane area located far from the optical axis (> λ /2d, with d the interactuator distance); and low-order errors, which affect the low spatial frequencies–mainly the focal plane area located close to the optical axis. 7.1.

Fitting error

Concerning the top level specifications for SPHERE, the minimization of the global error budget is not the only pertinent criterion; the spatial repartition of the errors also has to be taken into account. In particular, the detectivity performance is also linked to the capability of the AO #69303 - $15.00 USD (C) 2006 OSA

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system to “clean up the PSF” (i.e., to be as close as possible to the diffraction pattern) in an area wider that 150 arcsec around the optical axis. This corrected area (corr) in the focal plane (see Fig. 6) is directly linked to the interactuator distance (d = D/(nact − 1), where nact is the linear number of actuators in the telescope diameter D) with the following relation: corr = λim /2d. In that case, the residual phase power spectral density (PSD) is equal to 0 in the 0 − λ /2d frequency domain. Nevertheless, when a coronagraphic image is consider, the approximation coronagraphic image equal the phase PSD is only valid in a first order approximation (the classical approximation eiϕ ≃ 1 + iϕ ). When a general coronagraphic image formation is considered, the second-order terms of this approximation can no longer be neglected. Their effect is nothing but a spread of the residual uncorrected phase in the whole focal plane (see eq. (7) and Ref. [17]). It induces a plateau in the focal plane domain for small separation (between 0 and λ /2d, that is, in the corrected area). Therefore, decreasing d, i.e., increasing the number of actuators, reduces the low-frequency plateau on the coronagraphic image. Nevertheless, decreasing d has some consequences in terms of the system limiting magnitude. Indeed, the larger the number of DM actuators is, the larger the number of WFS subapertures and, hence, the smaller the number of available photons per subaperture and per frame becomes.

Fitting error

Temporal error

Fig. 6. [Left] Effect of the number of actuators on the coronagraphic image. Four actuator grids (20x20, 40x40, 60x60 and 80x80) have been considered. The imaging wavelength is set to 1.65 µ m. Only the fitting error is considered. [Right] Effects of temporal errors on the coronagraphic images. Only atmospheric perturbations are introduced (no vibrations). The average wind speed is equal to 12.5 m/s.

Concerning the science goals and all the error sources, a 40 x 40 actuator DM is a good compromise in terms of corrected area size, detectivity issues, and system limiting magnitude. In addition, such a number of actuators can be achieved by using well-known piezo-stack technologies with the required performance in terms of actuator and interactuator strokes (to correct for turbulence and system phase defects), bandwidth (higher than a few kHz), hysteresis (a few percent), etc. A larger number of actuators would require the use of microdeformable mirrors. Even if these promising technologies are in widespread development, they represent an important risk for a fast track system (less than 5 years). The use of existing (or soon-to-be) technologies would imply a woofer–tweeter configuration (one DM for low spatial frequency #69303 - $15.00 USD (C) 2006 OSA

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correction, another for high frequency correction), which represents a non-negligible risk from a system point of view that includes calibration issues, ultimate performance of the device, side effects, and so on. 7.2.

Temporal error

Another important item is the temporal error. Its behavior is dependent on turbulence characteristics and on each AO component: detector integration time, readout noise, real-time computations of commands from WFS data, numerical corrector, digital–analog converter, high voltage amplifier, and DM actuator temporal response. Choosing the temporal sampling frequency results in trade-offs among the following: • the atmospheric turbulence to correct for (seeing and wind speed). The system bandwidth, which can be defined as Fsamp /12.5 (for maximum gain under stability and robustness constraints [18] and for a two-frame-delay system), has to be larger than the turbulence temporal evolution of the faster mode to be corrected. • the expected limiting magnitude of the whole system (detectivity issues and trade-offs between measurement noise and temporal effects); the higher the sampling frequency, the lower the limiting magnitude. • the system parameters (detector technologies, correcting devices, real time computer performance). • the telescope and system vibration issues. With a classical integrator law, the correction of vibrations requires a system bandwidth much larger than the vibration frequency. This may be an extremely tight specification and may be incompatible with the limiting magnitude requirements. Another (and optimal way) to deal with all these parameters (for a given sampling frequency) is to design a Kalman-filter–based control algorithm [19]. Nevertheless, practical implementation of such a control law is complex and requires more computing power, especially for high-order systems. This leads to a significant increase of the RTC complexity. A hybrid solution has been considered for dealing with this problem. An optimal modal gain integrator [20] has been chosen to control high-order modes while a Kalman filter will be considered for tip-tilt modes to optimally correct turbulence and vibration effects. Figure 6 (right) shows the evolution of the coronagraphic image as a function of the temporal frequency of the WFS device. A two-frame delay is considered (one for the detector integration and one for the detector read-out and voltage computation) along with a classical integrator law. 7.3.

Aliasing error

These effects are the result of high spatial frequencies that are seen as low ones by the WFS device. The uncorrected high-frequency signal is translated in low-frequency measurements by the WFS device itself. These aliased high-frequency measurements are added to the real lowfrequency signal and thus induce a measurement error. Therefore this aliasing error is directly linked to the fitting error (the greater the residual uncorrected signal, the larger the aliasing effects). It dramatically increases the PSF residuals in its corrected area (because, as explained before, aliasing effects translate uncorrected high-frequency signals in low-frequency errors). Of course the total amount of error depends on the WFS concept. As an example in the specific case of classical SH WFS, it corresponds to roughly 40% of the total fitting error variance. Some aliasing-free focal plane sensors have been proposed to significantly reduce these effects (see Fig. 7). One of them is the spatially filtered SH, proposed by Poyneer and Macintosh #69303 - $15.00 USD (C) 2006 OSA

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[21, 22]. It is based on a focal plane filtering before the wavefront sensing device. The filtering process is performed using a pinhole device located in a focal plane before the SH lenslet array. This device has been studied in depth and optimized with respect to the system and turbulence characteristics (spectral bandwidth, WFS sampling, turbulence, etc.). Its optimal size ranges between 0.85 and 1.6 arcsec, typically. In addition, an experimental validation of the concept has been conducted using the ONERA AO bench (see Ref. [23]) in closed loop and with turbulence. The gain brought by a filtering device has been clearly demonstrated, and the experimental results have been found in good agreement with the simulations, which validates the potentiality of the concept and its use in the SAXO design.

Fig. 7. Impact of aliasing effects on coronagraphic images. The case of a classical SH, a spatially filtered SH, and an aliasing-free WFS have been considered.

7.4.

WFS measurement error

7.4.1.

WFS wavelength

The choice of the analysis wavelength is based on the available number of photons in a given spectral range (it is therefore linked to the GS type, as shown in Fig. 8 [left]) and on the detector characteristics. These two points allow us to compare WFS noises. Without any other consideration, it is interesting to note that the number of available photons becomes more important in IR bands only for very red stars (typically later than M5). A global comparison has been made between IR and VIS WFS. The main advantages and drawbacks of each type of WFS are summarized below. • IR-WFS – Advantages: (1) no differential refraction between WFS and imaging path and smaller scintillation effects on the WFS, inducing an increase of the overall system performance at very high flux and leading to a simpler system (only one ADC at the entrance for the system, which can be seen and corrected by the AO loop); (2) accessibility to faint red targets (M5 and redder). #69303 - $15.00 USD (C) 2006 OSA

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Fig. 8. [Left] Comparison of system zero-points for various GS types and assuming an IR or a VIS-WFS. [Right] Comparison of coronagraphic images in the case of a Pyramid (perfect and with diffraction effects) and a SH WFS.

– Drawbacks: (1) lower flux for GS up to M5 (decrease of the number of accessible targets); (2) flux separation between WFS and scientific paths (need for several beam splitters optimized for each scientific instrument purpose); (3) increase of system complexity; (4) need for a complex cryostat; (5) high RON values for high frame rates; (6) high background noise; (7) complex detector calibration (cosmetics, etc.). • VIS-WFS
2 – Advantages: (1) gain in sensitivity (in λim /λw f s , i.e., a factor 8 between 2 and 0.7 µ m); (2) no flux separation (all the VIS photons are available for WFS); (3) low RON level (even close to 0 for new EMCCD technologies); (4) very low background noise (> 22 mag/arcsec2 ). – Drawbacks: (1) differential refraction effects (i.e., limitation to zenith angles smaller than 50o ); (2) scintillation effects on WFS; (3) limited access to faint red targets; (4) two ADC in the two separated paths. With consideration given to the system requirements and the VLT-PF targets, the choice of a VIS-WFS has been made, with a spectral bandwidth going from 0.45 to 0.95 µ m. This choice assumes that an EMCCD with read-out noise lower than 1 e- at 1.5 kHz will be available in time. The development of such a device has been funded by the European Community (FP6Opticon Program), and a first detector should be available within the next two years. 7.4.2.

Pyramid versus SH

A comparison of the SH and Pyramid WFS in the frame of the VLT-PF AO system has been conducted. In both cases, performance, required calibrations and optimizations, as well as fundamental limitations have been identified and quantified. Theoretically, i.e., without modulation and with its expected noise propagation terms (as shown in Fig. 7 [left]), the Pyramid WFS provides a better shape of the coronagraphic image. Nevertheless, more accurate end-to-end simulations have identified some yet unsolved problems concerning the Pyramid WFS. In particular, #69303 - $15.00 USD (C) 2006 OSA

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in a partial correction domain (case of a visible WFS), the pattern structure introduces a “natural” modulation that evolves with time, introducing a variability of the linearity coefficient of the WFS sensor and then a corresponding gain fluctuation of the global AO loop. To overcome this effect, an instrumental modulation must be introduced, which will affect the propagation terms and thus make the Pyramid results comparable (and even poorer for medium-to-bad atmospheric conditions) to those of the SH. Hence considering the WFS performance, stability, complexity, and risk evaluation, a spatially filtered SH WFS, combined with a new optimized slope estimation algorithm [24], has been chosen as a baseline for the SAXO system. It allows us to reach a limiting magnitude of approximately 10-11, depending on GS type and CCD performance [22]. 8.

Calibration errors

The calibration errors [25] gather the AO loop (interaction matrix [IM] and reference slopes) and NCPA miscalibrations. 8.1.

AO calibration

The effects of misalignment on system performance have been studied, leading to tight specifications on system stability (pupil conjugation between DM and WFS, pupil motion, noise of IM, and reference slopes measurements, among other things). To illustrate, Table 1 shows the impact of a misregistration between DM and WFS, i.e., a differential pupil shift between these two components. Table 1. Impact of pupil shift between DM and WFS.

Pupil Shift between DM and WFS (in sub-aperture %) error in nm rms

5 4

10 9

15 15

20 23

25 32

This effect is critical in terms of system performance. To avoid it, in the SAXO optical design both DM and TTM are located in a pupil plane, and there are no moving optics located outside the pupil between the DM and the WFS. To reduce noise effects on IMs, we will consider Hadamar approaches [26] for IM measurements. 8.2.

Calibration of NCPAs

Each NCPA will be measured using a phase diversity approach and corrected in a closed-loop scheme through a modification of the WFS references [27, 8]. A detailed global analysis has shown that correcting an NCPA is mandatory for achieving the contrast goals. Of course, the low-order modes are critical, and the error budget balance leads us to consider only the precompensation of, typically, the first 100 modes. These modes have to be corrected with an accuracy of less than 10 nm rms (that means less than 1 nm per mode). An example of the optimized procedure developed at ONERA and applied on our AO bench (BOA) is proposed in Fig. 9. Twenty-five Zernike coefficients have been measured by using phase diversity on the imaging camera, and precompensated through the modification of the WFS reference slopes by using an iterative process to account for uncertainties on the WFS model. The residual wavefront error on the corrected static aberration is smaller than 2 nm rms with such a procedure. It can be noticed that next 50 uncorrected Zernike coefficients have a

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SR 72.3%

SR 93.4%

Experimental validation of the NCPA measurement and precompensation process Fig. 9. [left] Wavefront error of static aberration on the ONERA AO bench before and after measurements and precompensation. Twenty-five Zernike polynomials have been measured and precompensated. [Right] PSF before and after precompensation. The SR goes from 72 ± 2% up to 93 ± 2% at 633 nm.

total rms error of 25 nm, which is fully coherent with the SR estimation made directly on the image (93% at 633 nm, see Fig. 9). 9. 9.1.

Auxiliary devices IR tip-tilt sensor

The average image position (in other words, the optical axis position) on the coronagraphic mask is a main specification for the VLT-PF system. The required accuracy for the mean image position is 0.5 mas or better. The global error for the image position depends on the following: • the residual uncorrected tip-tilt fluctuations at very high frequency (considered here as a noise). From the AO residual errors, these fluctuations have to be lower than 2 mas (at 1 kHz frame rate). Considering a 100 ms integration time, this leads to a residual error of 0.2 mas; • the differential refraction effect (between VIS and IR wavelengths). Such an effect has been estimated to 0.16 mas per second in the more pessimistic case of a 60 degrees zenith angle; • the differential thermal or mechanical effects (between WFS and imaging paths). Such effects have been estimated to 0.031 mas per second. Considering the requirement, we believe that an open-loop model of each differential evolution will not be accurate enough (considering all the possible parameters involved) to reach the absolute position performance. As an example, a 10% error on the model will lead to a residual shift of the mean image position of 0.02 mas per second; and therefore, the specification can be kept only for 25 s of observation time. #69303 - $15.00 USD (C) 2006 OSA

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To ensure that the specification will be fulfilled, an auxiliary IR tip-tilt sensor (AS) at the level of the coronagraphic mask has been proposed. This sensor will be coupled with a differential tip-tilt mirror (DTTM) located in a pupil plane in the WFS arms. The auxiliary loop can be summarized as follows: (1) image acquisition with AS (typically 10 Hz); (2) computation of residual image motion from AS data; (3) application of DTTM voltages from AS data; and (4) the differential motion induced by the DTTM is seen by the VIS-WFS and thus corrected by the main tip-tilt mirror, which leads to an image recentering on the coronagraphic mask. 9.2.

Pupil motion sensor

Pupil stability is a major issue in ensuring the VLT-PF performance. The pupil must remain motionless during the entire observation process. When the pupil is located behind the Nasmyth focus of the telescope, this stability requirement implies a pupil derotator and a pupil recentering device. It has been shown on simulations that a pupil shift of 1% of the pupil diameter or a pupil rotation of 1 degree will reduce by a factor of 1.5 to 2 (for typical conditions at the VLT) the detection capability of a coronagraphic + differential imaging system. It has led to impose

differential image I(λ1 ) − I(λ2 )

motionless pupil pupil motion = 0.6% pupil motion = 1.2% Differential image + calibration on reference star [I(λ1 ) − I(λ2 )] − [Re f (λ1 ) − Re f (λ2 )]

Fig. 10. [Left] Differential coronagraphic (4-quadrant) image (λ1 = 1.56µ m, λ2 = 1.59µ m). [Right] differential coronagraphic image + reference subtraction: pupil shift between object and reference star = 0, 0.6, and 1.2 % of the full pupil. The companion (∆m = 15, separation = 0.6 arcsec) is clearly distinguishable from residual fixed speckles for a fixed pupil.

a pupil stability in translation better than 0.2% (goal 0.1%) of the full VLT pupil. This performance is achieved using a pupil tip-tilt mirror located close to the entrance focal plane of the VLT-PF. This mirror is controlled by a pupil motion sensor (PMS). The PMS directly uses the SH-WFS data to measure pupil motion. Because pupil motion is rather slow, a measurementcorrection process has to be performed typically every minute, which ensures a good SNR on the PMS data. 10.

Global system design

A global trade-off from all the points mentioned above (combined with optical design, technological aspects, cost, and risk issues) leads to the following main characteristics of the AO system: • 41x41 actuator DM (roughly 1300 useful actuators) of 180 mm diameter, located in a pupil plane with an interactuator stroke > ±1µ m (mechanical), a maximum stroke > ±3.5µ m (mechanical), and a temporal transfer function phase shift lower than 5o at 80 Hz.

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• 2-axis TTM (70 mm diameter) located in a pupil plane, with a resolution of ±0.5 mas and a transfer function phase shift lower than 8o at 80 Hz. • 40x40 SH WFS with a spectral range covering 0.45 to 0.95 µ m; 6x6 pixels per subaperture (Shannon sampling @ 0.65 µ m); a focal plane filtering device with variable size (from λ /d to 3λ /d at 0.7µ m); and a temporal sampling frequency of 1kHz (goal 1.5 kHz). The foreseen detector is a 256x256 pixels Electron Multiplication CCD detector with a read-out-noise < 1e− and a 1.4 excess photon noise [28, 22]. • Mixed numerical control laws with a Kalman-filter law for tip-tilt control and an optimal modal gain integrator law for DM control. The global AO loop delay has to be lower than 1 ms (goal 666 µ s). • NCPAs with off-line measurements and on-line compensation using a phase diversity algorithm and an on-line modification of the WFS reference slopes. • Pupil motion sensor that uses SH data directly (no additional sensor) and controls a slow (typically lower than 1 Hz) tip-tilt mirror located near the system entrance focal plane. • Auxiliary IR tip-tilt sensor and associated differential tip-tilt mirror located in a pupil plane on the WFS arm. This auxiliary sensor corrects for optical axis position and should allow an average position of the optical axis with respect to the coronagraphic device better than 0.5 mas (for optical axis fine centering). 11.

Conclusion

The SAXO system, but more generally any AO system for a planet finder instrument, represents a large step forward with regard to system components and calibration procedures. Nevertheless, a complete analysis with a detailed error budget has shown that an AO system fulfilling all the requirements that are mandatory for the direct detection of giant extrasolar planets is feasible on a reasonable time scale (4 years) and with existing and proven technologies. The realization of the SPHERE instrument is a critical step toward a future 30- to 60-m Extremely Large Telescope (ELT), both from the conceptual and a technological points of view. The next steps will be the realization of the first AO systems for ELT (foreseen to be within 10–15 years), and on a larger time scale (probably the second generation of ELT instrumentation), the realization of a planet finder instrument for these telescopes with the ultimate goal of the direct imaging of an extrasolar terrestrial planet. Acknowledgments The authors warmly thank all the members of the SPHERE consortium for their contributions to this study. ONERA and LESIA will be involved in the development of the SAXO system in the frame of the newly established “Groupement d’Intˆeret Scientifique” PHASE.

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128ANNEXE C. HIGH ORDER ADAPTIVE OPTICS FEASIBILITY AND REQUIREMENTS FOR

Annexe D "Closed loop experimental validation of the spatially filtered Shack-Hartmann concept” T. Fusco et al. - Opt. Letters - 2005

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Closed-loop experimental validation of the spatially filtered Shack–Hartmann concept T. Fusco, C. Petit, G. Rousset, and J.-M. Conan Office National d’Etudes et de Recherche Aerospatiales, Département d’Optique Théorique et Appliquée, 29 Avenue de la Division Leclerc, 92322 Chatillon Cedex, France

J.-L. Beuzit Laboratoire d’Astrophysique de Grenoble, BP 53, F-38041 Grenoble, Cedex 9, France Received December 3, 2004 An experimental validation of the spatially filtered Shack–Hartmann concept (F-SH) is presented that allows aliasing effects to be decreased. This effect is studied by use of an adaptive optical testbed with a focal plane pinhole in front of the wave-front sensor. First, the ability to close the loop on turbulence is demonstrated. Then the performance gain conferred by the device is quantified and compared with end-to-end simulation results. Another property of the F-SH concept, i.e., the reduction of the waffle modes, is highlighted. © 2005 Optical Society of America OCIS codes: 010.1080, 010.7350.

Direct detection and spectral characterization of exoplanets is one of the most exciting but also one of the most challenging areas of today’s astronomy. Using ground-based telescopes for such a purpose automatically implies the use of an extremely high performance adaptive optics (XAO) system.1 A key component of such a XAO system is the wave-front sensor (WFS). One of the major limitations of classic WFSs is due to the aliasing effect linked to the spatial sampling of the wave front: high spatial frequencies are measured by the WFS device as low spatial frequencies. This error (in the specific case of a Shack– Hartmann WFS) corresponds roughly to one third of the total variance caused by fitting error.2 A method for canceling the largest part of the aliasing error was recently proposed.3 It is based on locating a filtering pinhole in a focal plane at the entrance of the WFS. The goal of such a device is to remove all wavefront components that are larger than the pinhole. Indeed, considering a weakly perturbed wave front, a pinhole of diameter ␪ 共arc sec兲 put in the focal plane permits the filtering of wave-front spatial frequencies greater than 2 / ␪ 共arc sec−1兲. Here we present an experimental validation of the concept, using a real adaptive-optics (AO) system. It allows us to highlight the key points for a real application of this theoretical concept. In particular, the ability to close the loop on turbulence is demonstrated. The gain in terms of aliasing reduction was computed and showed good agreement between experimental results and simulations. We also underline an additional interesting feature of the filtered Shack–Hartmann concept (F-SH), which consists of a reduction of the waffle modes. This experimental validation is a first step toward implementation of future Shack–Hartmann-based XAO systems. The laboratory bench operates with a fiber laser diode source working at 630 nm located in the entrance focal plane of the turbulence generator. The turbulence is generated by a rotating mirror reproducing a von Kármán spectrum and inserted in a collimated 0146-9592/05/111255-3/$15.00

beam. It reproduces seeing conditions that correspond to D / r0 ⯝ 6 at 630 nm, where D is the telescope diameter and r0 is the Fried parameter. The V / D ratio is set to 0.8. The wave-front corrector includes a tip-tilt mirror and a 9 ⫻ 9 actuator deformable mirror. The Shack–Hartmann (SH) WFS is composed of a 128 ⫻ 128 pixel Dalsa camera [readout noise, 43 共e − / pixel兲 / frame] and a square 8 ⫻ 8 lenslet array. These conditions allow us to mimic, for each subaperture, the behavior of an XAO system (d / r0 ⬍ 1, where d is the subaperture size). The WFS temporal sampling frequency is set to 270 Hz. The imaging camera is a 512 ⫻ 512 pixel Princeton camera with a 4 共e − / pixel兲 / frame readout noise. Two circular pinholes are used as filtering devices: a 150 ␮mdiameter pinhole, corresponding to a size of 1.5␭ / d, and a 200 ␮m-diameter pinhole, corresponding to a size of 2.0␭ / d. Particular attention was paid to ensure good positioning of the pinhole at the entrance focal plane of the WFS. After the pinhole was centered, the AO loop was successfully closed on the turbulence. It allowed us to obtain our first critical validation of the F-SH: the possibility of closing the loop with the device by using small pinhole sizes. No problems were identified in the numerical simulations; nevertheless, it was important to confirm our conclusion by testing it on a real AO system. We present in

Fig. 1. Closed-loop images of a SH subaperture (left) without filtering (before thresholding) and (right) with filtering. © 2005 Optical Society of America

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Table 1. Strehl Ratios Obtained with and without a Filtering Device for Two Runsa Date of Acquisition Type of Device Ratio for classic SH Sensor Ratio for F-SH 共2␭ / d兲 Ratio for F-SH 共1.5␭ / d兲

08/09/04

21/09/04

60.6 ± 1.6% 64.0 ± 1.5% 67.0 ± 1.5%

61.2 ± 1.7% 64.3 ± 1.5% 67.1 ± 1.0%

a

SRs were computed directly for the five PSFs recorded by an imaging camera under the same conditions.

and the PSF integration. We repeated this operation five times to compute Strehl ratio (SR) error bars. The results clearly show the gain conferred by the filtering device, as presented in Table 1. The SR (computed on the image) goes from 61% for a classic SH effect to 67% for the smallest pinhole 共1.5␭ / d兲. Two runs, separated in time by two weeks, led to similar results. The PSF profiles are plotted in Fig. 2. One can see that the two PSFs have the same level of uncorrected turbulence (with the expected −11/ 3 decrease), which means that the same amount of turbulence was considered in each case. In the AO loop correction domain, the attenuation of the PSF shape owing to the reduction of aliasing by the F-SH is clearly visible. This attenuation is more important for the 1.5 ␭ / d pinhole than for the 2 ␭ / d, as predicted. After accurate calibration of all the AO bench elements (deformable mirror influence functions, WFSs, and imaging camera pixel scales …), we used an endto-end model4 to simulate the whole experiment. The low-order aberrations of the imaging path were measured in a phase diversity approach.5 Then they were applied during image formation in the numerical simulation such that the SR obtained without turbulence with the model (=81% at 633 nm) is coherent with the SR measured directly on the imaging camera (79%). The slight difference is due to high-order aberrations that have not been measured by the phase diversity tool (and thus not included in the end-to-end model) and to uncertainties in SR measurements made with experimental data. When the loop is closed on turbulence, simulation and experimental results remain comparable, in terms both of SR (64.4% for the classic SH sensor, 66.3% for the 2␭ / d F-SH, and 69.3% for the 1.5␭ / d F-SH to be compared to the results listed in Table 1) and of PSF profiles (as shown in Fig, 3 which can be compared with Fig. 2). The SRs obtained in simulations are always slightly larger than the measured SRs. This result is essentially due to imaging-path aberrations (espe-

Fig. 2. Top, log–log representation of the average PSF profiles for classic and 1.5 ␭ / d filtered SH WFSs. Bottom, zoom of the top figure when 2 ␭ / d filtering is added.

Fig. 1 a closed-loop WFS subaperture image for a classic SH sensor and a 1.5␭ / d F-SH. The subaperture point-spread function (PSF) is far more symmetrical for a F-SH than for the classic SH sensor, proving that the main part of the wave front’s high spatial frequencies is filtered. Before closing the loop, we recorded an interaction matrix for each system configuration. After the command matrix computation, the loop was closed on the turbulent phase screen in the three SH configurations (classic, 1.5 ␭ / d, and 2 ␭ / d pinholes). To have the same turbulence statistics we started the closed loop for the three configurations at roughly the same position of the turbulent phase screen such that we could see the same turbulence during the phase-screen rotation

Fig. 3. Comparison of PSF profiles with classic SH and 2 ␭ / d and 1.5 ␭ / d filtered SH sensors obtained by end-toend simulations with all parameters of the AO bench (in particular, D / r0 = 6 and imaging-path low-order aberrations) taken into account.

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cially to high-order modes, above Zernike mode 21), which contribute to SR degradation but are not taken into account in the numerical simulations. To prevent these effects, we consider only SR gain [as defined in Eq. (1)] rather than the absolute SR: SRgain = SRfilterd

SH/SRclassic SH −

1,

共1兲

which is the gain from the F-SH relative to the classic SH sensor. SRgain obtained by experiment and by numerical simulation is 5.3% and 2.9% (filtered 2␭ / d) and 10.1% and 7.6% (filtered 1.5␭ / d), respectively. In this case it is interesting to note that the gains that result from F-SH are underestimated by the numerical simulation (for instance, 7.6% versus 10.1% for a 1.5␭ / d filtering pinhole). At first glance such results are surprising because the reduction in aliasing seems to be more efficient for real data than for simulations. In fact, this result can be explained by focusing on an additional advantage of the filtered SH concept, which is the reduction of the waffle mode effects. Indeed, this high-frequency mode is due to the deformable mirror’s high-frequency pattern, which is theoretically not seen by the SH WFS. But in the experiment the pure waffle mode is barely measured (owing to misalignment effects), resulting in a contamination of the interaction matrix eigenmodes, which makes it impossible to filter the waffle mode fully in the command matrix and then results in a continued noise propagation into this mode. With a filtered SH sensor, two combined effects reduce the influence of the waffle mode. First, the interaction matrix is less contaminated by the waffle mode because of the optical filtering, leading to a command matrix of better quality. Second, the aliasing error is attenuated, along with its propagation, in a closed loop and thus has a smaller influence on the final image. This effect is illustrated in Fig. 4, where a closed-loop PSF without turbulence is shown for a classic SH and for a 1.5␭ / d filtered SH sensor. If the two PSF shapes are extremely similar, a waffle pattern is clearly visible (four secondary peaks located at the corners of a 2␭ / d size square) for the classic SH WFS and is significantly attenuated for the 1.5␭ / d filtered SH sensor (the peak values are reduced by a factor of 3). Thus, even without turbulence, the SR goes from 76.5% for the classic SH sensor to 79% for the filtered SH sensor. Because in simulations the system is perfect, waffle modes do not occur. This explains that the gain between classic and filtered SH sensors is reduced compared with that in experimental data. Adding

Fig. 4. Closed-loop PSFs obtained without turbulence with (left) a classic and (right) a filtered 共1.5 ␭ / d兲 SH sensor. PSF peaks have been saturated to focus on waffle-mode features (four secondary peaks are located at the corners of a 2 ␭ / d size square). The SRs are, respectively, 76.5% and 79%.

waffle mode effects (i.e., the SR gain obtained for experimental data without turbulence) to the simulation results leads to the following modified values for the F-SH gain: 4.9% for 2␭ / d pinhole size and 11.1% for a 1.5␭ / d pinhole size, which have to be compared with the experimental results of 5.3% and 10.1%, respectively. We have presented an experimental validation of the filtered Shack–Hartmann concept for adaptive optics with the loop closed on simulated turbulence. We have demonstrated an ability actually to close the loop with a filtered SH device and the gain in terms of performance that results. An additional interesting feature of the filtered SH sensor, i.e., the reduction of the waffle modes, has been highlighted. Good agreement between experimental and simulated results has been demonstrated. Our results give confidence about the relevance of a filtered SH sensor for extremely high-performance adaptive-optics systems. T. Fusco’s e-mail address is [email protected]. References 1. D. Mouillet, T. Fusco, A.-M. Lagrange, and J.-L. Beuzit, in Astronomy with High Contrast Imaging: From Planetary Systems to Active Galactic Nuclei, C. Aime and R. Soummer, eds., Vol. 8 of EAS Publications Series (European Astronomical Society, Geneva, 2002), pp. 193–200. 2. F. Rigaut, J.-P. Véran, and O. Lai, Proc. SPIE 3353, 38 (1998). 3. L. A. Poyneer and B. Macintosh, J. Opt. Soc. Am. A 21, 810 (2004). 4. R. Conan, T. Fusco, G. Rousset, D. Mouillet, J.-L. Beuzit, M. Nicolle, and C. Petit, Proc. SPIE 5490, 602 (2004). 5. A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, Astron. Astrophys. 399, 373 (2003).

Annexe E "Improvement of Shack-Hartmann wave-front sensor measurement for extreme adaptive optics” M. Nicolle et al. - Opt. Letters - 2004

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Improvement of Shack–Hartmann wave-front sensor measurement for extreme adaptive optics M. Nicolle, T. Fusco, G. Rousset, and V. Michau Département d’Optique Théorique et Appliquée – Office National d’Etudes de Recherche Aérospatiale, 29 Avenue de la Division Leclerc, 92322 Châtillon Cedex, France Received April 19, 2004 The development of high-performance adaptive optics systems requires the optimization of wave-front sensors (WFSs) working in the high-order correction regime. We propose a new method to improve the wave-front slope estimation of a Shack – Hartmann WFS in such a regime. Based on a detailed analysis of the different errors in the slope estimation with a classical centroid and with the new method, the gain in terms of wave-front-sensing accuracy in both the detector and the photon noise regimes is stressed. This improvement is proposed without major system disruption. © 2004 Optical Society of America OCIS codes: 010.1080, 010.7350, 110.2990, 220.2740.

Adaptive optics (AO; see Roddier1 for a review) is now widely used in astronomical ground-based telescopes. One of the most challenging next-generation AO systems is extreme AO, the goal of which is to achieve an overall Strehl ratio of better than 90%. Development of such a system requires performance improvement of each of its components and particularly of its wave-front sensor (WFS). Different WFSs are candidates for use in extreme AO systems. Among them is the Shack –Hartmann (SH) WFS.2 In such a sensor the local wave-front (WF) slope directly drives the image spot position in each subaperture. Position estimation is generally done by computing the spot’s center of gravity (COG). This measurement is corrupted by both photon and detector noise. Windowing or thresholding before the COG calculation may reduce the noise effects, but these are not the optimal processes for improving the WF estimation.3 However, they are the most widely used, since they are easy to implement on a real-time computer.2,4,5 The behavior of the different position estimators is driven by the WFS [detector sampling and field of view (FOV)] and by the guide star magnitude. Assuming a Gaussian spot in a square subaperture to approximate windowing or thresholding effects on the spot, Rousset et al.6 showed analytically by deriving the COG that the noise variance in the local WF estimate, given in terms of phase difference at the borders of a subaperture (in rad2 ), is expressed by 2 COG 兲ph, th 苷 共sDf

µ ∂ NT 2 p2 2 ln共2兲Nph ND

(1)

µ ∂ µ ∂ p 2 sdet 2 NS 2 2 3 Nph ND

(2)

for photon noise and 2 COG 兲det, th 苷 共sDf

for detector noise, where Nph is the number of detected photons contributing to the COG computation after thresholding and windowing, NT is the full 0146-9592/04/232743-03$15.00/0

width at half-maximum (FWHM) of the subaperture point-spread function (PSF) in pixels, and ND is the FWHM of the diffraction-limited PSF in pixels (ND corresponds to l兾d, where d is the subaperture size; for Nyquist sampling ND 苷 2), NS 2 is the total number of pixels used in the COG computation (after thresholding and windowing), and sdet is the standard deviation of the detector noise, expressed in electrons per pixel. In Eq. (1) the true Gaussian FWHM is considered, which leads to a slightly different numerical coefficient from the one given in Refs. 2 and 6. COG According to Eq. (1), 共sDf 2 兲ph, th is independent of the size of the COG computation window (assuming thresholding process). On the contrary, Eq. (2) shows COG that 共sDf 2 兲det, th increases dramatically with NS . These behaviors have been numerically simulated. The simulation computes images in the focal plane of a square lenslet by use of incoming turbulent wave fronts. Shannon-sampled PSFs are first computed in a large f ield 共64 3 64 pixels兲. Integration of intensity on the pixel size is not taken into account. Then the PSFs are windowed and binned to obtain the effective FOV and sampling. The reported results correspond to d兾r0 苷 1 (where r0 is the Fried parameter), a subaperture size d of 20 cm, a WFS wavelength of 0.7 mm, and a detector noise of 3e2 per pixel. In this case we have NT 苷 ND . To take into account the closed-loop AO correction, we subtract from the incident WF its angle of arrival and add a random residual angle of arrival f luctuation corresponding to a phase difference variance on one subaperture sDFinc 2 艐 1022 rad2 . This variance mimics the WF residuals for very good AO correction. The measurement error is computed between the estimated spot position and the true position. For each simulation case the number of photons reported in the f igures is Nph , the averaged total number of detected photons within the lenslet aperture. Figure 1 shows the simulated measurement error versus the total number of incident photons in the subaperture. Figure 1 shows two centroid position estimators commonly used for SH WFS in AO7,8: (1) the COG estimator, for two pixel sizes l兾2d and l兾d and a FOV of 4l兾d, without any additional windowing or © 2004 Optical Society of America

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Fig. 1. WF variance of the COG 共2 2 2, pixel size l兾2d; 2 ? 2, pixel size l兾d) and four-quadrant estimators (2, pixel size 2l兾d); 2 · · · 2, four-quadrant theoretical detector noise (low f lux) and photon noise (intermediate f lux) asymptotes.

thresholding, and (2) the four-quadrant estimator, for a quad-cell detector with 2l兾d-wide pixels and a FOV of 4l兾d. Theoretical errors given by the adaptation of Eqs. (1) and (2) to the four-quadrant are plotted. Figure 1 reveals three domains of the error variance curves. At low photon f lux the error is dominated by detector noise, at moderate f lux photon noise prevails, and at high f lux a saturation of accuracy is observed. At high f lux the error saturates whatever the photon f lux level is. There is no gain in accuracy by increasing the number of detected photons. This saturation of the accuracy has already been reported.4 It is linked to the fact that the instantaneous image is not perfectly symmetrical with respect to the center of the window because of its residual movement and because the turbulence distorts its shape. Any static aberration will also introduce such an effect. Therefore the window induces an error in the computation of the centroid by image truncation. We have verified that this effect diminishes when the size of the f ield corresponding to the computation window increases: the saturation level decreases, providing better accuracy. For FOVs of 4l兾d and 8l兾d, the error is close to 0.015 and 0.003 rad2 , respectively. In the case of four-quadrant detection undersampling of the image is an additional contribution to the saturation of the accuracy, which leads to a higher error at high f lux than in the two COG cases for the same FOV. Hence increasing the FOV of the quad cell does not lead to a substantial reduction of this error. For the quad cell and the COG this error contributor at high f lux limits the accuracy. It is in fact correlated to the turbulence and the residual angle of arrival. When the turbulence is stronger, the error increases. For pure turbulent incoming WFs, we have found an error proportional to 共d兾r0 兲5兾3 . At moderate f lux the error dependence is close to the Nph 21 theoretical behavior [Eq. (1)], where Nph is the number of photons effectively detected in the computation window. Note that, for a FOV of 4l兾d, Nph is of the order of 90% of Nph . This behavior is observed

near 102 photons for the quad cell but is not clearly seen for the classical COG. This is due to the relatively low photon noise level given by Eq. (1). Indeed, the photon noise domain is extremely limited because of the relatively small size of the computation window (inducing the saturation of the accuracy) and the detector noise level (3e2 per pixel). At low f lux the error dependence is close to the theoretical behavior of Nph 22 Ns 4 [Eq. (2)]. Reducing the number of pixels to 2 pixels in the quad-cell case significantly decreases the error variance. In Fig. 1, Nph 苷 600 typically corresponds to a visible magnitude equal to 11 for a WFS exposure time of 2 ms (500-Hz frame rate, SH WFS similar to the one in Ref. 8). For stars with a magnitude fainter than 11, the four-quadrant estimator is much better than the COG because it uses fewer pixels, leading to a lower detector noise contribution. For brighter stars, improving image sampling (and using the COG) allows us to achieve better accuracy for a given FOV. We have already underlined that increasing the FOV also reduces the error in this case. Hence the number of pixels in the subaperture image appears to be one of the key parameters determining the noise performance of the SH WFS with respect to the number of incident photons. To cover a large range of photon numbers as required in astronomy, one image sampling and one FOV are not sufficient. For such a WFS a CCD is a useful device, allowing efficient binning and windowing for adaptation of the sensor to the observing conditions, as implemented in NAOS.8 Thresholding, windowing, and binning induce a loss of information in the image and are not optimal.3 To account for all the illuminated pixels without amplifying noise, we propose to consider a new centroid estimator [weighted center of gravity (WCOG)], using the whole subaperture image and a penalizing function weighting (but not canceling) low signal-to-noise ratio pixels: P i, j xij Wij Iij , (3) 共CWCOG 兲x 苷 P i, j Wij Iij where xij is the abscissa of the 共i, j 兲 pixel, Wij is its weighting coeffQ icient, and Iij is its intensity. If Wij 苷 Q ij , where NS ij is the centered window function NS of width NS and is equal to 1, we achieve a windowing process and Eq. (3) reduces to the COG estimate. Using the same analytical approach as reported in Ref. 6, we can derive the new photon and detector noise errors on the local WF estimation: 2 WCOG 兲ph, th 共sDf

µ ∂ µ ∂ NT 2 NT 2 1 NW 2 2 , p2 苷 2 ln共2兲Nph ND 2NT 2 1 NW 2

(4)

2 WCOG 兲det, th 共sDf

µ ∂ µ ∂ sdet 2 NT 2 1 NW 2 2 . p3 苷 32关ln共2兲兴2 Nph ND

(5)

Equations (4) and (5) are found by use of Gaussian prof iles for both the spot and the weighting functions. The weighting function is fully characterized by its FWHM, NW . Equations (4) and (5) stress that a

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Fig. 2. WF variance for the four-quadrant 共2兲 and the WCOG (NW 苷 NT 关? ? 1 ? ?兴, NW 苷 3NT 关? ? 䊐 ? ?兴 and NW 苷 8NT 关? ? D ? ?兴, pixel size l兾2d for the three curves). 2 · · · 2, Eqs. (4) (intermediate f lux) and (5) (low f lux); 2 2 2, COG saturation level.

suitable choice of the width of the weighting function has to be made. It can be shown9 that, in the case of Gaussian noise statistics, the WCOG is the maximum likelihood estimate if the weighting function is the mean spot intensity. In this case, NW 苷 NT . We compare in Fig. 2 the WCOG performance with that of the four-quadrant COG, using the same numerical simulation as presented in Fig. 1 and Eqs. (4) and (5). The WCOG curves are plotted for the FWHM of three weighting functions (NW 苷 NT , NW 苷 3 NT , and NW 苷 8 NT , with ND 苷 2). One can clearly observe the gain achieved with our approach at a low light level (weighting the spot with the mean spot, i.e., NW 苷 NT ) even compared with the four-quadrant case. The error variance is reduced by a factor of 2 at 100 incident photons. As for the classical COG, the photon noise domain is limited. Equation (5) is valid only for NT 2 1 NW 2 # NS 2 . Indeed, Eq. (5) has been derived assuming an infinite subaperture size, the weighting function acting as a smooth windowing. When the weighting function FWHM becomes comparable with the FOV, the WCOG, error is close to the classical COG error. When the number of photons is increased (i.e., when truncation effects dominate), Fig. 2 shows that one must increase the weighting function FWHM to preserve the efficiency of the WCOG computation from NW 苷 NT to NW 苷 8NT . Enlarging the weighting function of the WCOG improves the WF estimation at high f lux for the same reasons as the ones given above to explain the improvement of COG performance when the computation window is increased. Therefore we find the same error behavior (proportional to 共d兾r0 兲5兾3 for pure turbulence) for the WCOG estimator as for the classical COG. Note that at high photon f lux the only way to improve the performance further will be to increase the FOV per subaperture.

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The weighting function is introduced into SH WFS image real-time processing by use of weighting coefficients [see Eq. (3)]. The weighting function allows easy adaptation of the values of the coefficients with regard to the guide star magnitude and drifts in atmospheric conditions. There is no need here to have different readout modes of the CCD to cover the large photon f lux range, as there is, for instance, in Ref. 8. Only one readout mode is required, simplifying the AO system, and adaptation to the observing conditions is accomplished by the choice of the weighting function FWHM, i.e., the weighting coeff icients. In conclusion, estimating the spot position in the focal plane of a SH WFS from the PSF intensity weighted by the PSF mean value brings a significant improvement in wave-front measurement accuracy for low and moderate photon f lux. At high f lux the weighting function FWHM must be adapted to obtain the best performance. It has been shown that for close-to-zero WF f luctuations (closed-loop mode) the phase slope error budget can be reduced. The Strehl ratio of large ground-based telescopes using AO systems can therefore be improved without major disruption of implementation or real-time computing. The weighted center of gravity estimate should be seriously considered for next-generation AO systems, especially in the case of extreme AO, where the considered photon f lux is relatively high. Ms. Nicolle’s e-mail address is magali.nicolle@ onera.fr. References 1. F. Roddier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, Cambridge, England, 1999). 2. G. Rousset, in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, England, 1999), Chap. 5, pp. 91 – 130. 3. M. Van Dam and R. Lane, J. Opt. Soc. Am. A 17, 1319 (2000). 4. T. Noël, “Caractérisation spatiale et temporelle de la turbulence atmosphérique par analyse de surface d’onde,” Ph.D. dissertation (Université Paris VI, Paris, 1997). 5. J. J. Arines, Opt. Lett. 27, 497 (2002). 6. G. Rousset, J. Primot, and J.-C. Fontanella, in Workshop on “Adaptive Optics in Solar Observations,” Lest Foundation Tech. Rep. 28 (University of Oslo, Oslo, Norway, 1987), pp. 17 – 33. 7. P. Wizinowich, D. S. Acton, O. Lai, J. Gathright, W. Lupton, and P. Stomski, Proc. SPIE 4007, 2 (2000). 8. G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, Proc. SPIE 4839, 140 (2002). 9. T. Fusco, M. Nicolle, G. Rousset, V. Michau, J.-L. Beuzit, and D. Mouillet, “Optimisation of Shack – Hartmann wave front for extreme adaptive optics systems,” in Advancement of Adaptive Optics, Proc. SPIE 5490 (SPIE, Bellingham, Wash., to be puslished).

Annexe F "Comparison of centoid computation algorithms in a Shack-Hartmann Sensor” S. Thomas et al. - MNRAS - 2006

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Comparison of centroid computation algorithms in a Shack-Hartmann sensor

Journal: Manuscript ID: Manuscript Type: Date Submitted by the Author: Complete List of Authors:

Keywords:

Monthly Notices of the Royal Astronomical Society MN-06-0231-MJ.R1 Main Journal n/a Thomas, Sandrine; AURA/NOAO Fusco, Thierry; ONERA, DOTA Tokovinin, Andrei; AURA/CTIO Nicolle, Magalie; ONERA, DOTA Michau, vincent; ONERA, DOTA Rousset, Gerard; Observatoire de Paris, LESIA instrumentation: adaptive optics < Astronomical instrumentation, methods, and techniques, instrumentation: high angular resolution < Astronomical instrumentation, methods, and techniques

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Comparison of centroid computation algorithms in a Shack-Hartmann sensor S. Thomas1? , T. Fusco2, A. Tokovinin1, M. Nicolle2 , V. Michau2 and G. Rousset2 1 Cerro

Tololo Inter-American Observatories, Casilla 603, La Serena, Chile

2 ONERA,

BP 72, 92322 Chatillon France

Accepted . Received ; in original form

ABSTRACT Analytical theory is combined with extensive numerical simulations to compare different flavors of centroiding algorithms: thresholding, weighted centroid, correlation, quad cell. For each method, optimal parameters are defined in function of photon flux, readout noise, and turbulence level. We find that at very low flux the noise of quad cell and weighted centroid leads the best result, but the latter method can provide linear and optimal response if the weight follows spot displacements. Both methods can work with average flux as low as 10 photons per sub-aperture under a readout noise of 3 electrons. At high flux levels, the dominant errors come from non-linearity of response, from spot truncations and distortions and from detector pixel sampling. It is shown that at high flux, CoG and correlation methods are equivalent (and provide better results than QC) as soon as their parameters are optimized. Finally, examples of applications are given to illustrate the results obtained in the paper. Key words: Atmospheric turbulence, Adaptive Optics, Wavefront sensing, ShackHartmann

1 INTRODUCTION Adaptive Optics (AO) is nowadays a mature astronomical technique. New varieties of AO, such as multi-object AO (MOAO) (Gendron et al. 2005) or extreme AO (ExAO) (Fusco et al. 2005) are being studied. These developments, in turn, put new requirements on wave-front sensing de?

E-mail: [email protected]

c 0000 RAS

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vices (WFS) in terms of their sensitivity, precision, and linearity. New ideas like pyramid WFS (Ragazzoni & Farinato 1999; Esposito & Riccardi 2001) are being proposed. Yet, the classical Shack-Hartmann WFS (SHWFS) remains a workhorse of astronomical AO systems now and in the near future. A SHWFS samples the incident wavefront by means of a lenslet array which produces an array of spots on a detector. The wavefront is then analyzed by measuring, in real time, the displacements of the centroids of those spots which are directly proportional to the local wavefront slopes averaged over sub-apertures. A good estimation of the wavefront distortion is therefore obtained from a good measurement of the spots positions. The accuracy of such measurements depends on the strength of the different noise source as well as on a non-negligible number of WFS parameters such as the detector size, the sampling factor, Field of View (FoV) size, etc. The goal of this study is to compare quantitatively different estimators of spot positions and suggest best suitable methods in cases of low and high photon fluxes. In this paper, three main classes of algorithms are considered: quad-cell estimator (QC), center of gravity approaches (CoG) and correlation methods (Corr). Centroid measurements are usually corrupted by the coarse sampling of the CCD, photon noise from the guide star, readout noise (RON) of the CCD, and speckle noise introduced by the atmosphere. In case of strong RON and weak signal, the spot can be completely lost in the detector noise, at least occasionally. This presents a problem to common centroid algorithms like thresholding that rely on the brightest pixel(s) to determine approximate center of the spot. When the spot is not detected, the centroid calculation with such methods is completely wrong. Thus, the lowest useful signal is determined by spot detection rather than by centroiding noise. Concerning the CoG approaches, different types of algorithms have been developed to improve the basic centroid calculation in a SHWFS: Mean-Square-Error estimator (van Dam & Lane 2000), Maximum A Posteriori estimator (Sallberg et al. 1997) or Gram-Charlier matched filter (Ruggiu et al. 1998). Arines & Ares (2002) analyzed the thresholding method. On the other hand, many parameters are involved in the estimation of the centroid calculation error as explained below, and therefore there are many ways to approach this problem. For example, Irwan & Lane (1999) just considered the size of the CCD and the related truncation problem. They concentrated on photon noise only using different models of spot shape (Gaussian or diffraction-limited spot), neglecting readout noise or other parameters. In this paper, we choose to focus on various causes of errors such as detector, photon noises or turbulence as well as WFS parameters. Most previous studies remained theoretical or compared performance to simple centroiding. Despite a large number of c 0000 RAS, MNRAS 000, 000–000

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papers on this subject, the ultimate performance of SHWFS and a recipe of the best slope estimation method (at least in the framework of an AO loop) are still debated. Moreover, there are no clear results concerning detection limits and linearity.

As for the linearity, it is often assumed that an AO system working in a closed loop scheme keeps the spots well centered, hence linearity is not essential. However, in any real AO instrument the spots can be offset intentionally to compensate for the non-common-path aberrations (Blanc et al. 2003; Hartung et al. 2003; Sauvage et al. 2005). In critical applications such as planet detection by means of ExAO with coronagraph, a linearity of the response becomes important. It is interesting to add thatit will also drive the selection of the sensing technique towards a SHWFS, as opposed to curvature sensing or pyramid (Fusco et al. 2004). The same is true for MOAO, where turbulence will be compensated in open loop relying on a SHWFS with a perfectly linear response. Our study is thus of relevance to these new developments.

Finally, an AO system working with faint natural guide stars (for low or medium correction level) requires a WFS with highest possible sensitivity, whereas linearity becomes less critical. Here, simple quad-cell centroiding is often used (e.g. Herriot et al. (2000)), despite its nonlinearity. Are there any better options? As shown below, the weighted centroiding method (Nicolle et al. 2004) offers comparable noise performance while being linear.

Correlation approaches are another way to measure a spot position. Such approaches are particularly well adapted when complex and extended objects are considered (c.f. Michau et al. (1992); Rimmele & Radick (1998); Poyneer et al. (2003)). They have been widely used in solar AO for more than 10 years. Here we apply correlation centroiding to the case of a point source, select best variants of calculating the position of the correlation peak, and compare it to also other algorithms.

We begin by introducing relevant parameters and relations and by describing our technique of numerical simulations in Sect. 2 . Different techniques of centroid measurements are discussed in detail in Sect. 3 (for the simple CoG), Sect. 4 (for the improved CoG algorithms) and Sect. 5 (for the correlation method). Finally, a comparison of the different methods for an ideal case and for more realistic systems is given in Sect. 6. c 0000 RAS, MNRAS 000, 000–000

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P(x,y)

(x0 ,y ) 0

I(x , y ) i

N

y

j

T

W x

Figure 1. Some notations used in the paper. Intensity distribution in the spot P (x, y) is transformed by the detector into a discrete array of pixel values Ii,j .

2 THE METHOD OF STUDY 2.1 Definitions Throughout the paper, we consider only one spot in one sub-aperture of a SHWFS (Fig. 1). The spot is sampled by the detector within a Field-of-View (FoV) of the width W p pixels. A spot intensity distribution P (x, y) is first transformed into an array of pixel intensity values I i,j of the size Wp × Wp , and then corrupted by photon and detector noise. These data are used to compute

the spot centroid (ˆ x, yˆ), whereas the true centroid of P (x, y) is located at (x 0 , y0 ). The Full Width at Half Maximum (FWHM) of the spot is NT pixels. Let d be the size of a square sub-aperture and λ the sensing wavelength. The parameter Nsamp = (λ/d)/p conveniently relates the angular pixel size p to the half-width of the diffractionlimited spot, λ/d. The condition Nsamp = 2 corresponds to the Nyquist sampling of the spots and is used throughout this paper, unless stated otherwise. It means that for diffraction-limited spots NT = Nsamp = 2. Such sampling is close to optimum at medium or high flux (Winick 1986). When spot images are very noisy, the optimum sampling corresponds to a spot FWHM of 1 to 2 pixels. Selecting an even coarser sampling, Nsamp < 1, only increases the error. Over-sampling (Nsamp > 2) does not bring any additional information but increases the effect of detector noise and thus the final centroid error. In the following, Wp is used for the FoV of the subaperture expressed in pixels and W for the same FoV expressed in λ/d. c 0000 RAS, MNRAS 000, 000–000

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2.2 Spot profile Two spot models were used. The first is a 2-dimensional Gaussian function:   Nph (x − x0 )2 + (y − y0 )2 P (x, y) = exp − , (1) 2 2 2πσspot 2σspot where (x0 , y0 ) is the true centroid position. We introduce a random jitter of the spot center with rms amplitude σt = 0.1 pixels, with the average spot position centered in the field. The FWHM of such √ spot is NT = 2 2 ln 2σspot ≈ 2.3548σspot . Gaussian spots are convenient for analytical derivations

and as a benchmark for comparison with our simulations. In the following, we assimilate N T with λ/d for convenience, using the fact that the first rough approximation of a diffraction spot is a Gaussian. Diffraction spots formed by a square d × d sub-aperture and distorted by atmospheric turbu-

lence represent a second, more realistic model. In this case, P (x, y) becomes a random function

and its parameters, like NT , can be defined only in statistical sense. Atmospheric phase distortion was generated for each realization from a Kolmogorov power spectrum with a Fried parameter r 0 (at the sensing wavelength λ). The overall tilt was subtracted, and a monochromatic spot image was calculated. The true centroids (x0 , y0 ) were computed for each distorted spot. We call this realistic spot model throughout the paper. The strength of spot distortion depends on the ratio d/r 0 : for d/r0 < 1 the spots are practically diffraction-limited (hence NT = Nsamp ),     x y 2 2 P (x, y) = Nph sinc sinc , (2) Nsamp Nsamp where sinc(y) = sin(πy)/(πy). On the other hand, for d/r0 > 3 the central maximum begins to split randomly into multiple speckles.

2.3 Measurement error As mentionned in the introduction, light intensity in each detector pixel is first corrupted by the photon noise (following a Poisson statistics) and by the additive Gaussian readout noise (RON) with variance Nr2 . Moreover, centroiding errors arise from coarse sampling of the spot by CCD pixels, from truncation of the spot wings by finite size of the sub-aperture field, from the spot distortions produced by atmospheric turbulence, etc. The error variance of the estimated centroid position (ˆ x) can be expressed as

 σx2 = (ˆ x − x 0 )2 ,

(3)

where h.i represents a statistical (ensemble) average, x0 is the real centroid position in pixels and

xˆ is the centroid position estimated by a given algorithm. We write the estimate in a general form c 0000 RAS, MNRAS 000, 000–000

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as (4)

xˆ = αr x0 + fnl (x0 ) +  + noise,

where αr is a response coefficient (it remains the same whatever the spot motion), f nl describes the non-linearity of the centroid algorithm, and  gathers all errors due to spot shape, including truncations by the finite FoV (we often refer to these effects as “turbulence”). Finally, noise stands for errors caused by the readout and photon noise. Including Eq. 4 in Eq. 3 leads to

 σx2 = [(αr − 1)x0 + fnl (x0 ) +  + noise]2 .

(5)

The function fnl (x) can be represented as Taylor expansion around x = x0 . The constant and

linear terms are zero by definition. Assuming that fnl (x) is symmetric around x = x0 , the quadratic term vanishes as well, and we can model the non-linearity by a cubic term with a coefficient β, (6)

fnl (x0 ) ≈ βx30 .

We can safely assume that photon and detector noises are not correlated with the shape and position of the spot. We make further, less certain assumptions that  and x 0 are uncorrelated (not strictly true for truncation error) and that the response coefficient α r is constant (true for undistorted spots, d/r0 < 1, and for Gaussian spots). To simplify further, we will assume here that the centroid calculation algorithm is always adjusted to unit response, α r = 1. Then Eq. 5 becomes

  2 σx2 = β 2 x60 + 2 + σnoise . (7)

If the residual spot motion in a SHWFS is Gaussian with zero average and variance σ t2 = hx20 i

2 contains two independent terms related to detector pixels2 , then hx60 i = 15σt6 . The variance σnoise

(σN2 r ) and photon (σN2 ph ) noises. Hence

(8)

σx2 = 15β 2 σt6 + σ2 + σN2 r + σN2 ph .

2 It is important to remind that σnoise is defined here for unit response, αr = 1. The relative

importance of each noise source changes depending on the conditions of use of the SHWFS. The strategy was to identify major noise contributors in each case and to select most appropriate centroiding algorithms. Whenever possible, the modelization of different terms in Eq.8 is provided. For example, a well-known theoretical result is that the minimum possible centroid noise for an un-biased estimator, considering a Gaussian spot with rms size σ spot pixels and pure photon p noise, is equal to σNph = σspot / Nph (Irwan & Lane 1999; Winick 1986), where Nph is the average number of photons per spot and per frame. Moreover, this boundary is reached by a simple centroid (Rousset 1999), which is thus the maximum likelihood estimator in this case. c 0000 RAS, MNRAS 000, 000–000

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Here we also adopt the common practice of expressing centroid errors in units of phase difference across sub-aperture (in radians). Thus, the relation between the rms phase error σ φ and the centroid error σx is σφ =

2πdp 2π σx = σx , λ Nsamp

(9)

where λ is the sensing wavelength, d is the sub-aperture size and p is the angular size of the CCD pixel. We caution the reader that σφ is computed in radians for one sub-aperture at the WFS wavelength. To be used in calculations of AO performance, it has to be re-scaled to the imaging wavelength and to the full telescope aperture, and filtered by the AO loop rejection transfer function (Madec 1999). Thus, errors exceeding 1 rad are perfectly acceptable for IR imaging when a visible-band WFS is considered. However, when, for a Gaussian spot, we reach a condition σx > σspot , centroid measurements become meaningless because they fail to localize spots better than the spot size. Given that Nsamp = NT = 2.355σspot , this condition corresponds to σφ = 2π/2.355 = 2.67 rad, or a variance of 7.1 rad2 .

2.4 Simulations Our main technique to study various centroid algorithms consists in numerical simulation. We generate series of 1000 independent spot realizations. The intensity distribution P (x, y) (either Gaussian or realistic) is computed without noise first. The pixel signals are then replaced by the Poisson random variable, with the average of Nph photons per spot on a FoV Wp pixels. A zeromean normal noise with variance Nr2 is added to simulate the RON. At very low light level, each simulated spot is tested for detectability. The first possible check is to have the maximum well above the RON, Imax > 2Nr . A second one is to reject the centroids with measured |ˆ x − x0 | > σspot or |ˆ y − y0 | > σspot as spurious (outside the spot). If one of those checks is not passed, the measure is assumed to have failed and therefore rejected. The number of

rejected cases gives us information on the detectability limit: when more than a certain fraction of images are rejected, we consider that the centroid measurements have failed and are not reliable for those light conditions. Otherwise, the rms centroid error σx is computed on the retained images. A certain fraction of frames with undetected spots is acceptable because an AO system will then simply use centroid measurements from previous frames (this only leads to an additional delay in the closed loop scheme). We set this fraction to 50% and determine for each method the minimum number of photons Nph,min when this limit is reached. Adopting a somewhat more strict c 0000 RAS, MNRAS 000, 000–000

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criterion (say, 10% invalid frames) would increase the detection threshold. The robustness of each centroiding method is characterized here by Nph,min . In the following we will describe the centroid algorithms considered in this study. For each method, we give a short explanation and then we focus on their advantages and drawbacks.

3 SIMPLE CENTROID (COG) The Center of Gravity (CoG) is the simplest and most direct way to calculate the position of a symmetric spot: P xIx,y xˆCoG = P . (10) Ix,y This formula is widely used in AO. However, it has some limitations when using a real spot (diffraction or seeing limited) and in presence of readout noise. In the following, those limitations are described. 3.1 Centroid noise Let us first recall well-known results concerning the noise of the CoG estimator. Rousset (1999) shows that for a Gaussian spot, the photon-noise and RON contributions to phase variance are, respectively, 2 σφ,N ph 2 σφ,N r

 2 NT π2 1 , = 2 ln 2 Nph Nsamp π 2 Nr2 Ns4 , = 2 2 3 Nph Nsamp

(11) (12)

where NT is the FWHM of the spot in pixels, Nr is the readout noise, and Ns2 the total number of pixels used in the CoG calculation. It is interesting to note here that for N T = Nsamp , the photon2 reaches its Cram´er-Rao bound (Sect. 6.1). At low light levels, the RON noise contribution σφ,N ph

contribution is dominant. It can be decreased by using the smallest number of pixels N s2 possible in the CoG calculation. This leads to the quad-cell method and to other modifications of the CoG considered below. Two main hypothesis have been used to derive Eq. 11. Firstly, a Gaussian spot shape has been considered whereas diffraction spots are described by Eq. 2. Compared to the Gaussian function, the sinc2 function decreases slower in the field. In the case of a Poisson statistics (photon noise case), it can be shown that (c.f. appendix): 2 σφ,N ≈2 ph

W , Nph

(13) c 0000 RAS, MNRAS 000, 000–000

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Figure 2. Error variance as a function of the FoV in presence of photon noise for a spot with a sinc 2 distribution.

Figure 3. Validity of the analytical expression (11) for the noise influence on CoG measurements in case of Gaussian spots. Error variance as a function of the spot sampling (i.e the number of pixels per FWHM). 2 pixels corresponds to a Nyquist sampling. Photon-noise case (100 photons per sub-aperture and per frame).

where W is the sub-aperture FoV expressed in λ/d units (W = Wp .dp/λ, with p the pixel size in radians). In this context and in presence of photon noise only, this means that the size of the window will become important for a realistic spot, while the error variance does not depend on the size of the window for a Gaussian spot. When taking into account the diffraction, the error variance increases linearly with the FoV of one lenslet. Thus, for an infinite window size we get an infinite error variance. This result can be explained by the non-integrability of the function x 2 sinc2 (x). It is therefore obvious that the noise depends on the structure of the spot. This structure changes for different configurations – Gaussian spot, diffraction spot, turbulent spot, – which adds some difficulties in the determination of a general theoretical expression. On the other hand, in presence of RON, the noise does not depend on the structure of the spot, since when the FoV is increased, the error due to the RON dominates the photon error. Secondly, a Nyquist sampling criterion (Nsamp > 2) is implicitly assumed. As shown in Fig. 3, increasing Nsamp (typically Nsamp > 1.25) does not modify the noise variance. On the other c 0000 RAS, MNRAS 000, 000–000

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Figure 4. Linearity of the simple CoG as a function of the spot motion for different FoV sizes. The unit used here is N T , FWHM of the spot.

hand, taking Nsamp smaller than 1.25 induces an additional error related to the under-sampling effect. This error can be explained by non-linearity for under-sampled images, as shown in the next section and in Fig. 3. 3.2 Response coefficient and non-linearity Non-linear CoG response appears when the FoV is too small in comparison to the spot size. Indeed, it is straightforward to show that the CoG is perfectly linear for infinite FoV and adequate sampling. The smaller the FoV, the larger the deviation from linearity (Fig. 4). We can easily quantify this effect for a Gaussian spot of rms size σspot centered on (x0 , 0). The CoG estimate xˆ calculated on the finite window of size Wp pixels is r 2 eζ − e η xˆ = x0 − σspot , (14) π Φ(ζ) + Φ(η) √ √ √ Rt where ζ = (Wp /2−x0 )/( 2σspot ), η = (Wp /2−x0 )/( 2σspot ), and Φ(t) = 2/ π 0 exp(−u2 ) du.

It is interesting to note that if Wp → ∞ then xˆ = x0 , in other words the CoG algorithm is √ asymptotically unbiased. For moderately large FoV size, Wp /2 > 2 2σspot , we can develop (14) in series around x0 = 0, up to the 3rd order:

xˆ ' x0 − σspot

r

2 − e π

(Wp /2)2 2 2σspot

W x 1 √ p 0 + 2σspot 6



W x √ p 0 2σspot

3 !

(15)

We recognize here a small deviation of the response coefficient from 1 (2nd term) and a nonlinearity proportional to x30 (3rd term). The difference of αr from 1 is less than 2.5% for a W = 2 (or Wp = 4 for a Nyquist sampling) and < 0.04% for a W = 3 (or Wp = 6 for a Nyquist sampling) . Apart from the FoV size, the linearity of response is affected by sampling (Hardy 1998). This c 0000 RAS, MNRAS 000, 000–000

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Figure 5. Illustration of the dependence of the CoG error variance on the turbulence and the sub-aperture FoV. The different curves represent different turbulence strengths (i.e various d/r0 ). A Nyquist-sampled spot is considered.

error is periodic, with the period of 1 pixel. We verified that CoG non-linearity is extremely small for reasonably well sampled spots, Nsamp > 1, and becomes obvious only for very coarse sampling, Nsamp = 0.5

3.3 Atmospheric effects The behavior of all centroiding methods changes when we go from the Gaussian spot to a realistic diffraction spot distorted by atmospheric turbulence. In order to isolate the contribution of the atmosphere itself, we simulated recentered and noiseless spots. All methods effectively truncate outer parts of the spots and thus lead to a difference between calculated position and real CoG. This difference mainly depends on the Shack-Hartmann FoV size W relative to λ/d and on the turbulence strength. It can be modeled as: 2 σφ,Atm = K W −2 (d/r0 )5/3 ,

(16)

Fits to our simulation results (Fig. 5) show that K ' 0.5 for well-sampled (N samp > 1.5) spots

(the error increases for coarser samplings).

The origin of atmospheric centroid error can be understood as follows. The maximum intensity in the spot corresponds to a position in the FoV where the waves from sub-aperture interfere constructively or, in other words, to a minimum rms residual phase perturbation. The best-fitting plane approximating a given wave-front corresponds to the Zernike tilt, while the true spot centroid corresponds to the average phase gradient over the sub-aperture, called G-tilt (Tyler 1994). Formulae for both tilts and their difference are well known in case of circular apertures and lead 2 to the expression σφ,Atm = 0.241(d/r0 )5/3 (see Eq. 4.25, Sasiela 1994). Most of this difference is

related to the coma aberration in the wavefront over each sub-aperture, never corrected by the AO. c 0000 RAS, MNRAS 000, 000–000

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Thus, even in the worst case where only the spot maximum is measured (e.g. high threshold), the atmospheric error should not exceed 0.25(d/r0 )5/3 . The residual fitting error in an AO system with sub-aperture size d is of the order 0.3(d/r 0 )5/3 rad2 (Noll 1976). It might seem that reducing the WFS measuring error much below this quantity is useless as it will not improve the Strehl ratio. However, fitting errors contain mostly high spatial frequencies and hence scatter light into the distant wings of the compensated PSF. The residual PSF intensity at distances from λi /D to λi /d (where D is the telescope diameter and λi is the 2 imaging wavelength) is directly related to the WFS errors, dominated by σ φ,Atm in case of bright

stars and unfortunate choice of centroiding algorithm. 3.4 CoG: the necessary trade-offs The trade-offs needed to optimize a simple CDG concern the FoV and the sampling factor. In presence of photon noise only, Nyquist-sampled images are required. Then a trade-off in terms of FoV W is needed. This trade-off depends on the photon noise error, which increases with W (cf. Sect. 3.1), and the atmospheric error, which decreases as W −2 . In this case, one can define the optimal FoV size Wopt by minimizing the sum of Eqs. 13 and 16. W 2 σnoise+atm =2 + 0.5W −2 (d/r0 )5/3 . Nph This leads to an analytical expression for Wopt :

(17)

1/2

(18)

Wopt = 1.26 Nph (d/r0 )5/9 . Taking, for example, d/r0 = 2 and Nph = 50, we obtain the best FoV of 6.8 λ/d.

In presence of detector noise, we want to decrease the number of pixels to minimize the noise at low flux. However, this configuration is not optimal at high flux, as explained earlier. It is therefore interesting to improve the simple CoG algorithm, adapting it to a large range of flux and readout noise. We present some optimization methods in the following.

4 IMPROVED CENTER-OF-GRAVITY ALGORITHMS Centroid errors due to detector noise can be reduced if we take into account only pixels with signal above certain threshold. The thresholding approach is detailed in Sect. 4.1. Recently, it has been proposed to weight pixels depending on their flux and readout noise. This method is called the weighted CoG (WCOG) (Nicolle et al. 2004) and is detailed in Sect. 4.2. c 0000 RAS, MNRAS 000, 000–000

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4.1 Thresholding (TCoG) Compared to the simple CoG, the thresholding method is used to follow the spot and therefore avoids the non-linearity problems (if the threshold value is not too high). However, the truncation effects are still present since we take into account only a fraction of all pixels. There are many ways to select pixels with high flux. We could consider a fixed number of brightest pixels or choose pixels with values above some threshold. We use the following method. The pixel with the maximum value Imax is first determined, then the threshold IT is set to IT = T Imax , where T is a parameter to optimized. IT is then subtracted from the spot image and the centroid is computed using only pixels with non-negative values, such as

xˆT CoG

P I>I x(I − IT ) = P T . I>IT (I − IT )

(19)

It is important to subtract the threshold before the CoG calculation. Indeed, it can be shown that otherwise, the response coefficient αr will be less than 1, i.e. the estimate xˆT CoG will be intrinsically biased. In the low-flux regime, it may be difficult to detect spot maximum against the RON. Therefore, we add the following condition: the threshold is set to IT = max(T Imax , mNr ). The choice of m depends on a trade-off between robustness (m > 3 required) and sensitivity (m ∼ 1 is better). We choose m = 3

The noise of thresholded CoG can still be computed with Eqs. 11, 12, where N s2 represents the 2 but increase the average number of pixels above threshold. By reducing N s2 , we diminish σφ,N r

error due to the atmospheric turbulence. Hence, there is a compromise to find for T to optimize the threshold IT as a function of Nph . In conclusion, thresholding resolves only part of our problems, such as the noise at medium flux. It is also very simple to implement. However, it is not optimal as it is difficult for example to choose pixels to be considered and their number (Arines & Ares 2002). In the next section we present a more efficient method proposed recently. 4.2 The Weighted Center of Gravity (WCoG) The idea of the Weighted Center of Gravity (WCoG) is to give weight to the different pixels depending on their flux – a kind of “soft” thresholding. The contribution of the noisy pixels with very little signal – outside the core of the spot – is attenuated but not eliminated. Let’s define c 0000 RAS, MNRAS 000, 000–000

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(Fw )x,y the weighting function of FWHM Nw . Then the WCoG centroid is computed as P xIx,y (Fw )x,y xˆW CoG = γ P . (20) Ix,y (Fw )x,y The coefficient γ is needed to ensure unit response, αr = 1. We can simply specify a circular p window of radius r, (Fw )x,y = 1 for x2 + y 2 < r, or a square window. However, a better

choice of the weighting function (Fw )x,y ( i.e different from a constant) is needed to optimize the performance of the centroid algorithm, as shown by Nicolle et al. (2004).

The WCoG method exists in two versions. The weight (Fw )x,y can either be fixed, or recentered on the spot (“following weight”), in a manner similar to thresholding. The properties of these two algorithms are different. Here, we consider only a fixed weight, centered on the most likely spot position, which can be seen as an a priori information for centroid measurement. This WCoG flavor is well suited for closed-loop AO systems where the spots are always centered. 4.2.1 Response coefficient and non-linearity The result of the estimation of the spot position using the WCoG with fixed window would be biased if we set γ = 1. For a Gaussian spot and Gaussian weight, the WCoG response can be calculated analytically, similarly to the CoG: xˆW COG

2 σeq = γ 2 x0 − γσeq σspot

r

2 eζ − e η , π Φ(ζ) + Φ(η)

(21)

2 2 2 where if σw2 is the rms size of the weighting function, σeq is defined as σeq = σspot σw2 /(σspot + σw2 ),

and the variables ζ and η are the same as in Eq. 14 with σspot replaced by σeq . In order to obtain unit response, we have to set 2 σspot N2 + N2 γ = 2 = T 2 w. σeq Nw

(22)

4.2.2 Noise of WCoG The noise of WcoG with unit response is obtained from the study of Nicolle et al., corrected by the factor γ 2 : 2 4 NT (NT2 + Nw2 ) (23) 2 Nsamp (2NT2 + Nw2 ) Nw4  2 4 Nr (NT2 + Nw2 ) π3 2 (24) σNr ,W CoG = 2 Nsamp Nw4 32 (ln2)2 Nph Eqs. 23 and 24 were derived by assuming a Gaussian spot, a Gaussian weight, and a good sam2 σN ph ,W CoG

π2 = 2ln2Nph



pling. We see from those formulae that for photon noise only and Nw = NT , the error variance is c 0000 RAS, MNRAS 000, 000–000

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15

Figure 6. Error variance of WCoG as a function of the sampling, for photon noise (dashed line) and RON with N r = 3 (full line). Nph = 20 and NT = Nw . The analytical expressions are plotted in dotted lines.

Figure 7. Influence of sub-aperture FoV W and of Nw on the photon noise variance with a WCoG, in presence of a diffraction spot. The number of iterations is 10000 and we took the same set of noise for each value of N w , explaining the absence of fluctuations in the curves. The analytical curve corresponds to the Eq. 23 derived for a Gaussian spot.

1.78 times larger than for simple CoG (11). This factor tends to 1 when N w increases. Therefore there is no real interest to use this method when the spot is Gaussian and in presence of photon noise only. This ideal case is only useful as an illustration. In the following we will present the advantages of WCoG. 4.2.2.1 Sampling. The analytical formulae have been obtained for a well-sampled spot. A comparison between theory and simulation is given in Fig. 6. A good match is obtained for N samp > 2, but for coarser samplings the errors are larger than given by Eqs. 23 and 24. The error due to RON, 2 σN , reaches a minimum for Nsamp = 1.5. Our simulations show that by reducing sampling r ,W CoG 2 from Nsamp = 2 to 1.5, the σN is reduced by 1.35 times, not 1.77 as predicted by Eq. 23. r ,W CoG

4.2.2.2 Spot shape. The contribution of RON noise to WCoG (Eq. 24) does not depend on the spot shape, as for the simple CoG. However, the photon noise (Eq. 23) does. We observe c 0000 RAS, MNRAS 000, 000–000

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Figure 8. The variance of atmospheric error of WCoG centroids depends on the width of the weighting function N w and on the FoV size W .

differences between analytical formulae and simulations, both for a Gaussian and for diffraction spots, and the noise depends on the FoV size (Fig. 7). The behaviors of WCoG and simple CoG with respect to photon noise are radically different. By weighting pixels with a Gaussian function Fw , we reduce the dependence of photon noise on the FoV size (as long as it is large enough, W > 2). There is an optimal width of the weighting function, Nw,opt ' 4.5 pixel (for Nyquist sampling Nsamp = 2), cf. Fig. 7, that ensures the minimum photon-noise centroid variance of 2 σN ph ,lin,opt

7.1 ≈ = Nph



2π Nsamp

2

2 σspot . Nph

(25)

Therefore, by using the WCOG algorithm with diffraction spot, we can reach the same level of photon noise as for a Gaussian spot of the same FWHM. We suspect that this is close to Cram´erRao bound, although we did not optimize the shape of weighting function explicitly for the case of diffraction spot. The gain of WCoG over simple CoG depends on the FoV size and can be dramatic.

4.2.3 Atmospheric effects As for the CoG, the error variance due to the atmosphere depends on the subaperture FoV, W . It also depends on the weighting function FWHM Nw . A first approximation of this error variance is given by: 2 −2 −2 σatmo,W )(d/r0 )5/3 CoG ≈ K(4Nw + W

(26)

where Nw is in pixels and W is the FoV in λ/d units. The coefficient K is equal to 0.5 for Nyquistsampled spot. c 0000 RAS, MNRAS 000, 000–000

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17

Figure 9. Trade-off for the FHWM of the weighting function. Photon-noise (dashed line) and atmospheric (dotted line) variances are plotted as a function of Nw (in pixels) for the following conditions: d/r0 = 1, Nph = 100 and W = 8λ/d. The variance sum (solid line) shows a minimum around Nw = 7. Nyquist-sampled spot is considered.

4.2.4 Optimization As for the CoG, there is an optimum width of the weight, Nw , depending on the noise parameters, d/r0 , and the number of photons. For example, in the case of a detector without RON, the optimum 2 2 Nw is found by minimizing the sum σN + σatmo,W CoG . Figure 9 gives the optimum for ph ,W CoG

d/r0 = 1 and Nph = 100.

4.3 Quad cell (QC) A quad-cell (QC) is the specific case of the CoG for a 2x2 pixels array (N s2 = 4). In this case, the FoV is given by the pixel size and by definition, the spot is under-sampled. QC is widely used in astronomical AO systems, e.g. in Altair (Herriot et al. 2000), because the weak signal from guide stars is better detected against RON and because, with a small number of pixels, the RON can be further reduced by a slow CCD readout. A quad-cell algorithm calculates the centroid xˆ QC in each direction from the differential intensity between one half of the detector and the other, xˆQC = πγ

Il − I r , Il + I r

(27)

where Il and Ir are the intensities, respectively, on the left and the right halves of the detector and γ is the coefficient, given in pixels, translating intensity ratio into displacement, depending on the √ spot shape and size. For a Gaussian spot we found that γ = σ/ 2π pixels c 0000 RAS, MNRAS 000, 000–000

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4.3.1 Noise of QC Expression (27) leads to the following variance for photon and detector noises Rousset (1999) (28)

σN2 ph ,QC = π 2 κ2 /Nph

(29)

σN2 r ,QC = 4π 2 κ2 (Nr /Nph )2 .

For a diffraction-limited spot Nyquist-sampled, κ = 1. For a Gaussian spot, κ depends on the rms size of the spot. Following our definition for a Gaussian spot, the rms size is defined by σ and is given in radians in this case (and in this paper for this case only). Therefore κ = γ 2π/(λ/d) = √ 2π σ/(λ/d). In real AO systems γ is usually variable, and considerable effort has been invested in developing methods of real-time QC calibration (Veran & Herriot 2000). In the following, we will assume that we are able to correct for γ fluctuations. In our simulation, the estimation of γ has been made by first calculating the long-exposure PSF, considering the turbulence and the spot parameters. From this point several methods have been investigated. The best method is linear fitting by Fourier method. Firstly, the Fourier Transform allows to oversample the PSF and then, we use a linear extrapolation. It is also interesting to note that nowadays, detectors in SHWFS can be photon-noise limited. In that case, the error variance ratio of the QC and the simple CoG is: √ σN2 ph ,QC 2π = 2ln2 √ . 2 σNph ,CoG 2 2ln2 The QC’s error variance is 1.48 times greater than the simple CoG’s one.

(30)

4.3.2 Non-linearity of QC The response of the QC algorithm is non-linear. Elementary calculation for a Gaussian spot leads √ 2 to β = −1/σspot . Hence, the non-linearity centroid error (in pixels) is σNL = 15 σspot (σt /σspot )3

(cf. eq. 8) It may quickly dominate other error sources even at moderate N ph . On top of that, if the FoV size becomes smaller than the spot, additional non-linearity appears, similar to simple CoG. 4.3.3 Atmospheric noise As for the previous algorithms, we study the atmospheric component of QC noise by ignoring both RON and photon-noise contributions, as well as any residual spot motion. Figure 10 shows the error variance as a function of d/r0 for different FoV. A fit of the data leads to the following expression: 2 σφ,Atm ≈ KW (d/r0 )5/3 ,

(31) c 0000 RAS, MNRAS 000, 000–000

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19

Figure 10. Atmospheric error variance of QC centroids: simulations (symbols) and models (lines).

with KW depending on the FoV. When the FoV is very small, the error is higher (K W = 0.2) because the spot is truncated. At larger FoV, the error variance saturates at K W = 0.07. This model, however, works only for low turbulence, d/r0 < 1. As we see in Fig. 10, the dependence on d/r0 becomes steeper than 5/3 power under strong turbulence. One explanation comes from the noise of γ’s fluctuations when d/r0 > 1. The atmospheric error barely depends on the FoV size (as soon as it is larger than a few λ/d). Even in the low-turbulence case (KW = 0.07), it is 1.4W 2 times larger than for simple CoG (cf. Eq. 16). In conclusion, using QC centroiding is only efficient for a noisy detector under low-flux conditions and considering small d/r0 values. For accurate wavefront measurement and photonnoise-limited detectors other CoG methods are better.

5 CORRELATION ALGORITHM The use of the correlation in imagery is not new and has been already proposed for AO systems that use extended reference objects, e.g. in solar observations (Rimmele & Radick 1998; Michau et al. 1992; Keller et al. 2003; Poyneer et al. 2003). In this study, we apply correlation algorithm (COR) to a point source. First, we compute the cross-correlation function (CCF) C between the spot image I and some template Fw . C(x, y) = I ⊗ Fw =

X

Ii,j Fw (xi + x, yj + y)

(32)

i,j

and then determine the spot center from the maximum of C(x, y). The methods of finding this position are discussed below. Since the COR method is not based on the centroid calculation, it appears to be very good at suppressing the noise from pixels outside the spot. Moreover, correlation is known to be the best method of signal detection (“matched filtering”). We note that the c 0000 RAS, MNRAS 000, 000–000

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Figure 11. Error variance of x ˆcorr determined by estimation of correlation maximum using a thresholded center of gravity with Gaussian shape spots (with FWHM = 2 pixels). The spot motion is 0.1 pixel rms. Both photon and readout (N r = 3) noises are considered. Various values of threshold are used. The template image is a noise-free Gaussian (P(x)).

coordinates x, y are continuous, unlike discrete image pixels, hence C(x, y) can be computed with arbitrarily high resolution. A correlation template Fw (x, y) can be either a mean spot image, some analytical function or the image in one subaperture, like in solar AO systems (Keller et al. 2003). In practice, the CCF has been calculated using Fourier Transform (FT). In that case, the image has to be put in a support at least twice as big as the image size to avoid aliasing effects. The sampling of the computed CCF can be made arbitrarily fine. One way to do this is to plunge the FT product into a grid Ke times larger, where Ke is the over-sampling factor. The behavior of correlation centroiding with respect to the image sampling is not different from other centroid methods. At low flux – for example Nph = 30 for Nr = 3e− – it is slightly better to use Nsamp = 1.5. However, for higher flux, under-sampling (Nsamp < 2) leads to a worse error variance, while for over-sampling (Nsamp > 2) the error variance stays the same. We remind the reader that NT is the FWHM of the spot image, Nw the FWHM of the template Fw . Furthermore, we will call δ the FWHM of CCF. 5.1 Determination of the CCF peak Once the CCF is computed, it is not trivial to determine the precise position of its maximum xˆ corr . To do this, we studied four methods: simple CoG with threshold (No¨el 1997), parabola fitting and Gaussian fitting. For the thresholding method, xˆcorr is computed by Eq. 19 where I is replaced by C and IT by CT = Tcorr max(C). The value of Tcorr has been optimized in parallel with Ke . Figure 11 shows the behavior of the error variance for different thresholds (from T = 0 or T = 0.9). For the parabola fitting, the determination of xˆcorr was done separately in x and y. Three points c 0000 RAS, MNRAS 000, 000–000

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Figure 12. Comparison of three different cases of peak determination for a Gaussian spot: the thresholding, the parabola fitting and the Gaussian fitting. All methods use an oversampling factor Ke = 8. There is photon noise and readout noise (Nr = 3). The theory corresponds to the sum of Eq.B7 and Eq.35.

around the maximum x∗ , y∗ of C along either x or y define a parabola, and its maximum leads to the xˆcorr estimate (Poyneer et al. 2003) as xˆcorr = x∗ −

0.5[C(x∗ + 1, y∗ ) − C(x∗ − 1, y∗ )] . C(x∗ + 1, y∗ ) + C(x∗ − 1, y∗ ) − 2C(x∗ , y∗ )

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In this case, a resampling of Ke = 4 is necessary and sufficient. For the Gaussian fitting, the one-dimensional cut through the maximum C(x∗ , y∗ ) is fitted by a Gaussian curve to find xˆcorr . We found that all methods of peak determination are almost identical and linear (Fig. 12) when using a Gaussian spot. However, while the determination of the correlation peak using a few pixels around the maximum position is the best method to use in presence of readout noise, it is not the case in presence of atmospherical turbulence. In this latest case, information contained in the wings of the CCF is important as well. For this reason, the methods using a given function fit, such as the parabola fitting, will not be optimum. The thresholding method on the other hand can be optimized by adapting the threshold value in function of the dominant noise. If the readout noise is dominant, we will use a high value of threshold to be sensitive only to the information contained in the peak. If the atmospherical noise is dominant, we will use a very low threshold to also be sensitive to the information contained in the wings of the CCF and therefore the image itself. Thus, for comparison with other centroid algorithms, we will use the thresholding method with a adaptable threshold. c 0000 RAS, MNRAS 000, 000–000

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Figure 13. Error variance as a function of the number of photons for the correlation method, in presence of photon noise only, for different threshold values. The spot and template are Gaussian.

5.2 Noise of correlation centroiding It is possible to derive a theoretical expression of the error variance for the correlation method (Michau et al. 1992). The derivation is given in Appendix. A simplified expression of Eq. B7 is: 2 σx,COR,N = r

4δ 2 Nr2 , 2 Nph

(34)

2 where δ is the FWHM of ChFw i , the autocorrelation function of Fw . σx,COR,N is given in pixels2 . r

The photon noise derivation is more complex, we studied is by simulation only. Fig. 13 shows the behavior of the correlation error variance in presence of photon noise only. A fit of the curve corresponding to the best thresholding gives:  2 π2 NT ≈ (35) 2 ln(2)Nph Nsamp This expression is equivalent to the error variance found in presence of photon noise only for the 2 σx,COR,N Ph

simple CoG. It can be shown (Cabanillas 2000) that the correlation is similar to the CoG, the optimal centroid estimator (Maximum likelihood) for a Gaussian spot in presence of photon noise only, if the template is the logarithm of the same Gaussian Fw . We found however that using the Gaussian distribution or the logarithm of this distribution gives very similar results. 5.3 Response coefficient and non-linearity The response coefficient of COR used with any of the optimized peak determination methods is equal to 1. Moreover, the linearity is very good, even when using the thresholding. Indeed, C(x, y) is a function which can be resampled, as explained before. This allows to increase the value of the threshold to select only the region very close to the maximum without considering only one pixel. c 0000 RAS, MNRAS 000, 000–000

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Figure 14. Error variance due to the atmospheric noise as a function of the correlation window size in λ/d for different value of d/r 0 . The model (Eq. 36) is overplotted as full lines. In this case T=0.01

5.4 Atmospheric noise In this section, we study the behavior of the correlation method in function of the size of the window used in the CCF peak determination Wcor and the FWHM of the template, in presence of atmospheric turbulence only. The CCF peak is determined by thresholding with T = 0.01, which is the best method in this particular case; it gives lower errors at high flux. It is important to also notice that using the thresholding method to determine the CCF peak allows the adaptation of the threshold value depending on the flux and readout noise. Hence, the results are more accurate than for other CCF peak determination method for any flux. We find the same dependence of the error variance on the window size W cor and the strength of the turbulence as for the simple CoG, which is: 2 σφ,cor,Atm = KWcor −2 (d/r0 )5/3 .

(36)

A fit to the data of Fig. 14 gives K ' 0.5 for well-sampled spots (Nsamp = 2). These results are valid as long as the correlation function is not truncated (due to a high threshold for example). This K value depends obviously on the method used to determine the peak position of the correlation function. For example, the parabolic fitting is worse than the thresholding with a low threshold by a factor 2. Indeed, the parabolic fitting take into account only the few pixels around the maximum. We saw in section 5.1 that this was not the optimum method. The main conclusion here is that in presence of only atmospheric turbulence, the correlation method is identical to the CoG. c 0000 RAS, MNRAS 000, 000–000

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Figure 15. Comparaison of different methods. Variance of the error in rad 2 in presence of photon noise and readout noise (Nr = 3). The Gaussian spot moves randomly of 0.1 pixels rms in each direction. The plain line is the Cram´er-Rao bound. The values of N w and the FoV are given in λ/d for reference. If Tcorr = 0.6 for the correlation, the error variance decreases at low flux but the noise at high flux is higher. The best way is therefore to adapt the threshold as a function of the flux and noise.

6 COMPARISON BETWEEN CENTROID METHODS 6.1 The Cram´er-Rao Bound The Cram´er-Rao bound (CRB) is a lower boundary for the error variance of any unbiased statistical estimator. Winick (1986) applied this powerful tool to the case of SHWFS. He assumes the CCD noise to be Poisson-distributed shot noise generated by both the image spot and the detector dark current. In theory, no un-biaised method can give better results than this limit. Therefore, we computed and used it in our study as a lower boundary, to compare with our simulations. Moreover, it also indicates whether the best method has been found or not. However, it is really important to be careful with the validity of assumptions inherent to the CR method. The estimator has to be unbiased, which is the case of all the estimators considered in this study except the QC. Indeed, QC is non-linear and therefore the response coefficient α r becomes rapidly different from 1. This explains why the QC curve goes slightly below the CR bound (e.g. Fig. 15). This is also the case for WCoG method with γ = 1.

6.2 Robustness at low flux In Fig. 15 we compare centroid methods for a Gaussian spot, at different levels of flux. The upper limit of reliable centroiding corresponds to the centroid variance equal to the rms spot size, or phase variance of 7.1 rad2 . It turns out that the most robust method is the Quad Cell, which can give a good estimation of the spot position at 10 photons when N r =3. We recall that detection test has been implemented at low flux, explaining the behavior (saturation) of the curves. c 0000 RAS, MNRAS 000, 000–000

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Table 1. Parameters of the study. Note that the minimum number of photons is lower for PF because of the lower detector RON value for this system. d/r0

Nr

Nph

Nsamp

1 2

0.5 5

[2-104 ] [10-104 ]

2 1

PF SAM

6.3 Comparison for a Gaussian spot After studying in detail centroiding methods independently, the methods are inter-compared. The conclusions are: • In presence of photon noise only, all methods after optimization are equivalent, with the

exception of the QC where the error variance saturates at high flux due to the non-linearity. The

motion of the spot is 0.1 pixel rms, leading to a non-linearity component equal to about 5 10 −4 rad2 . • In presence of both readout and photon noise however, the best method depends on the flux,

as shown in Fig. 15. At low flux, the QC is the optimum method. Then, its error variance saturates at around 5 10−4 rad2 for higher flux. At higher flux, the photon noise dominates and all methods except the QC are identical. Fig. 15 shows a comparison with the CRB as well. The sampling is equal to Nsamp = 2, which is the optimum for all methods at high flux. At low flux, N samp = 1.5 gives only marginally better results, which explains why we did not consider this value. It is possible to adapt the values of the different method’s parameter as a function of flux and readout noise to get lower error variances. In the following, we used this adaptation in the comparison of the different methods in presence of atmospherical turbulence.

6.4 Example of results for real AO systems Considering the high number of parameters of this study, using one method with one set of parameters only is an utopia. The solution is to adapt the parameters of each method depending on the turbulence strength, the readout noise and the flux. We will give a comparison of performance for two real systems, Planet Finder (PF) (Fusco et al. 2005) and the SOAR Adaptive Module (SAM) (Tokovinin et al. 2004), working in two different configurations (Table 1). PF is a second generation extreme AO for the VLT and SAM is an AO being built for the SOAR telescope using the Ground Layer AO concept. The first example (PF) almost can be assimilated to the case of photon noise only, which can be reached by using a new type of CCD detectors with internal multiplication, L3CCD (Basden c 0000 RAS, MNRAS 000, 000–000

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Figure 16. PF: Variance of the error in rad2 in presence of the atmospheric turbulence (d/r0 = 1) for different centroid methods. Only photon noise is considered. The vertical dotted line (Nph = 8) represents the limit for which the signal in the brightest pixel is always greater than 2N r . The correlation method gives the same results as the WCoG when the same paramters are used.

et al. 2003). The second example (SAM) shows the case of common detectors where both photon and readout noise are present. The last difference is the sampling: Nyquist for PF and half-Nyquist for SAM. For each case, we compared the different methods described before, adapting their parameters in order to reach the best performance. We will first comment on the detectability limit. Just as a reminder, the detectability limit is calculated from the occurence of the maximum signal being higher than 2N r over 1000 iterations 2.4. Here we will show the limit for which the maximum signal is lower than 2N r at least once. To give an idea for SAM, the maximum intensity of an average image is equal to about 8% of the total flux when Nyquist sampled and 14% when half Nyquist sampled (assuming that the center of the spot is in between 4 pixels). We then conclude that for Nph,min = 70, the maximum intensity of the spot is too low compared to the readout noise, and all methods relying on this maximum (like the thresholding) are useless. This limit is equal to Nph,min = 8 for the case of PF. The values of Nph,min decrease to a few photons for FP and to about 20 photons when we set the purcentage of occurence to 50%. In the following we will concentrate on the WCoG, the correlation and the QC. On the figures, we disgarded thresholding and simple CoG to avoid confusion. Those two methods are not optimal at low flux and similar to correlation and WCoG at high flux. Case of PF. Without RON (Fig. 16), all method are equivalent except the QC, as expected. Considering the computation time down-side of the correlation, it is better to use the simple or weighted CoG. The vertical dotted line (Nph = 8) represents the limit for which the signal in the brightest pixel is always greater than 2Nr . If the signal happens to be lower for one iteration, we remind that the measure is not taking into account (see section 2.4). c 0000 RAS, MNRAS 000, 000–000

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Figure 17. SAM: Variance of the error in rad2 in presence of atmospheric turbulence (d/r0 = 2) for SAM with a sampling of Nsamp = 1. Both photon and readout noise are present. The vertical dotted line (Nph = 70) represents the limit for which the signal in the brightest pixel is always greater than 2Nr .

Case of SAM. If the RON increases (Fig. 17), the results do not change when the number of photons is high enough – NP h > 300 for Nr = 5 – since the error variance is limited by the atmospheric turbulence. To reduce the impact of the noise, we used a positivity constraint (use a theshold such as T = 0) on the images before applying any method. For low flux, the QC is 1.5 times better, assuming the optimistic case where the FWHM and hence the response coefficient γ are known. This is linked to the high readout noise and the undersampling. The two other methods however either do not rely on the knowledge of the FWHM or allow to measure the FWHM with accuracy since we have a direct access to the shape of spot, which is not the case for QC. Moreover, the results are better at high flux. Therefore, if the error budget allows it, the simplest and more reliable method, which is the WCoG, must be chosen, even if it is not the best at low flux.

6.5 Implementation issues From those results, it is essential to consider the implementation issues of the WFS before making a final choice. In the following we will give some comments for the different methods. • The QC. The advantage of this method is the low number of pixels (only 2 × 2), even if in

practice we often use more pixels. However, there is an unknown which is the exact FWHM of the

spot in presence of turbulence. This FWHM must be known to get a response coefficient γ = 1. Considering the poor sampling of the image, it is difficult to calculate the spot size from the data. Moreover, this method is non-linear. The QC method can be interesting though in case of a very high readout noise and small values of d/r0 . c 0000 RAS, MNRAS 000, 000–000

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S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau and G. Rousset • The CoG and its improvements. The subaperture field is typically of 6 × 6 or 8 × 8 pixels.

The pure CoG is very sensitive to noisy pixels when the signal is low compared to the readout

noise or when the turbulence is high. Improvement can be achieved by optimizing the threshold or by using a WCoG. The last one gives the best results when adapting the rms size of the F w as a function of flux and readout noise. Moreover, the response coefficient γ depends on the FWHM of the spot like for the QC. However, in that case it will be easier to determine γ since we have a direct measurement of the spot. The procedure will be to recenter and average off-line individual frames in order to obtain a recentered long exposure image and then derive the FWHM value from this direct data. Then, this value of FWHM will be used in the next measurement. • The correlation. The computation for this method is more complex, especially when over

sampling is needed to estimate the peak position of the correlation function. The thresholded CoG

gives the best estimation of the peak position of the correlation function, since the adaptation of the threshold allows to deal with either high readout noise or high atmospheric turbulence. Other methods are not as robust and give higher error in presence of atmospheric turbulence. The advantage of COR is that the response coefficient does not depend on the spot size and shape.

7 CONCLUSIONS AND FUTURE WORK In this paper, we gave a practical comparison of different methods of centroid calculation in a Shack-Hartmann wavefront sensor. We studied some variations of the center of gravity such as WCoG, thresholding and quad cell. We also considered the use of the correlation to determine the spot positions. The first part of the paper was focused on the simplified case of Gaussian spot or diffraction-limited spot without atmospheric turbulence, while the second part considered all sources of error. We are not presenting here a real recipe but a methodology to calculate the error of a SH wavefront sensor. This study can be applied to different domains by changing the parameters and the shape of the spot. We first show a good understanding of the theory in the case of a Gaussian spot. For this particular case, the formulae can be used directly to estimate the noise in the WFS. For diffraction-limited spot and spots distorted by atmospheric turbulence, the derivation of such formulae is more challenging. Therefore, we have studied the methods mainly by simulation. The comparison is given in two different cases: with and without readout noise for a strength of c 0000 RAS, MNRAS 000, 000–000

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turbulence equal to d/r0 = 2. The best method would be the WCoG, with adapting the size of the weighting function, which does not require a complicated implementation. The correlation gives also good results and good detectability by adapting the threshold value to the flux and the RON considered. One can notice as well that at high flux, the correlation and the simple CoG give smallest errors and are similar. It is safe to say that for point sources the correlation method is not worth using, considering the complexity of its implementation. However, in presence of elongated spots it is the best method (Poyneer et al. 2003). We are planning to continue this study. For lower turbulence, the QC is very efficient for high readout noise and leads to simpler detectors. The conclusion though is that we do not have a magic method. However, the WCoG gives the optimum results independently of the signal to noise ratio when adjusting the FWHM of the weighting function. The study was made in the context of detailed designs and trade-offs where simplified analytical formula do not apply in the prediction of the WFS behavior. We also show the complexity of the problem and the importance of the contribution of each error in the budget for the comparison of different methods (or more generally different type of WFS).

ACKNOWLEDGEMENTS This study has been financed by NOAO. We thank Lisa Poyneer and Rolando Cantarutti for useful discussions. Jean-Marc Conan and Laurent Mugnier as well were very helpful in this study.

APPENDIX A: NOISE EXPRESSION IN THE CASE OF A DIFFRACTION LIMITED SPOT In the case of a diffraction limited spot and a Poisson statistics (photon noise case), the eq. 11 is no longer valid. Indeed, there is some signal contained in the wings of the PSF leading to photon noise. The size of the window has to be optimized. The photon noise variance (for one direction) can be expressed for this case as 2  Z W/2 Z W/2 2πd 1 2 x2 hP 2 (x, y)idxdy σφ,Nph = 2 λ Nph −W/2 −W/2 c 0000 RAS, MNRAS 000, 000–000

(A1)

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with P (x, y) the sinc2 shape of the spot (cf Eq. 2). Because we are in presence of photon noise only in this case, hP 2 (x, y)i = P (x, y) and therefore we can write:  2   Z W/2 2πd 1 x 2 2 2 σφ,Nph = x sinc dx λ Nph −W/2 λ/d   Z W/2 y 2 dy. sinc λ/d −W/2

(A2)

Thus, 2 σφ,N ph

4 W = (1 − sinc (πW )) π Nph

Z

πW

sinc2 (y)dy,

(A3)

0

where W is the sub-aperture FoV expressed in λ/d units (W = Wp .dp/λ, with p the pixel size in R πW radians). The function 1 − sinc (πW ) tends to 1 (with some oscillation) and 0 sinc2 (y)dy tends

to π/2 when W tends to infinity. In that case, Eq. A3 can be approximated by 2 σφ,N ≈2 ph

W . Nph

(A4)

APPENDIX B: NOISE CALCULATION FOR THE CORRELATION METHOD Here, we consider the case where the reference Fw is a known, deterministic function. Let’s s be the threshold and D the domain where C(x, y) > s, and Dc the domain where the functions are defined. Then R x [C(x, y) − s] dxdy Ng xcorr = RD = , Dg [C(x, y) − s] dxdy D

(B1)

where Ng is the numerator and Dg the denominator. If we assume that the fluctuations of Dg are negligible compared to those of Ng , we find that: 2 σx,COR

where hNg2 i

2

− hNg i =

Z Z D

hNg2 i − hNg i2 = , hDg i2

D

(B2)

xx0 σC2 (x, y, x0 , y 0 )dxdydx0 dy 0 ,

(B3)

where σC2 (x, y, x0 , y 0 ) is the variance of the correlation function. We can show that σC2 (x, y, x0 , y 0 ) = Z Z Fw (u, v)Fw (u0 , v 0 ) [hI(u + x, v + y)I(u0 + x0 , v 0 + y 0 )i Dc

Dc

− hI(u + x, v + y)i hI(u0 + x0 , v 0 + y 0 )i] dudvdu0 dv 0 .

(B4) c 0000 RAS, MNRAS 000, 000–000

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Since [hI(x, y)I(x0 , y 0 )i − hI(x, y)i hI(x0 , y 0 )i] =   σ 2 (x, y), if (x, y) = (x0 , y 0 ) b

(B5)

0, if (x, y) 6= (x0 , y 0 )



where σb (x, y) is the noise density. In presence of readout noise, which is a white noise, σ b is a constant equal to Nr . σC2 (x, y, x0 , y 0 )

=

Nr2

Z

Dc

Fw (u, v)Fw (u + x − x0 , v 0 + y − y 0 )dudv

Therefore, a development of Eq. B2 gives:   R R 0 0 0 0 0 2 2 D D xx ChFw i (x − x , y − y ) dxdydx dy σx,COR,Nr = Nr , hDg i2

(B6)

(B7)

2 where ChFw i is the autocorrelation function of Fw . σx,COR,N is given in pixels2 . r

A simplified expression of Eq. B7 can be derived using the following approximation: ChFw i (x, y) ≈ CI (x, y)(Dc ⊗ Dc )(x, y) close to the maximum. Then, we can fit the function near the maximum by a parabola as:   x2 + y 2 2 CI (x, y) = Nph 1 − 2δ 2 where δ is the FWHM of ChFw i .

(B8)

(B9)

We finally find 2 σx,COR,N = r

4δ 2 Nr2 . 2 Nph

(B10)

REFERENCES Arines J., Ares J., 2002, Opt. Let., 27, 497 Basden A. G., Haniff C. A., Mackay C. D., 2003, MNRAS, 345, 985 Blanc A., Fusco T., Hartung M., Mugnier L. M., Rousset G., 2003, A&A, 399, 373 Cabanillas S., 2000, Thesis from Aix-Marseille University Esposito S. , Riccardi A., 2001, A&A, 369, L9 Fusco T., Nicolle M., Rousset G., Michau V., Beuzit J.-L., Mouillet D., 2004, Proc. SPIE, 5490, p. 1155 Fusco T., Rousset G., Beuzit J.-L., Mouillet D., Dohlen K., Conan R., Petit C., Montagnier G., 2005, Proc. SPIE, 5903, p.148 Gendron E. et al., 2005, C. R. Physique 6, in press Guillaume M., Melon P., Refregier P., Llebaria A., 1998, J.Opt.Soc.Amer(A), 15, p.2841 c 0000 RAS, MNRAS 000, 000–000

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S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau and G. Rousset

Hardy J.W., Adaptive Optics for Astronomical Telescope, 1998, Oxford University Press, p.147 Hartung M. , Blanc A., Fusco T., Lacombe F., Mugnier L. M., Rousset G., Lenzen R., 2003, A&A, 399, 385 Herriot G., Morris S., Anthon A., 2000, Proc. SPIE, 4007, p. 115 Irwan R., Lane R. G., 1999, Appl. Opt., 38, 6737 Keller Ch. U., Plymate C., Ammons S.M., 2003, Proc. SPIE, 4853, p. 351 Madec P.Y., 1999, in F. Roddier, ed., Adaptive optics in Astronomy. Cambridge University press, p. 131 Michau V., Rousset G., Fontanella J.C., 1992, in Real time and post facto solar image correction, Sunspot, New Mexico, p. 124 Nicolle M., Fusco T., Rousset G., Michau V., 2004, Opt. Let., 29, 2743 No¨el T., 1997, Thesis ONERA, Paris Noll R.J., 1976, J.Opt.Soc.Amer(A), 66, p. 207 Poyneer L. A., LaFortune K., Awwal A.A.S., 2003, Lawrence Livermore National Lab Report (Livermore, CA 94551) Ragazzoni, R., Farinato, J., 1999, A&A, 350, L23 Rimmele T. R., Radick R. R., 1998, Proc. SPIE, 3353, p.72 Rousset G., 1999, in F. Roddier, ed., Adaptive optics in Astronomy, Cambridge University press, p.115 Ruggiu J.-M., Solomon C. J.,Loos G., 1998, Opt. Let., 23, 235 Sallberg S.A., Welsh B.M., Roggemann M.C., 1997, J.Opt.Soc.Amer(A) , 14, 1347 Sandler D., 1999, in F. Roddier, ed., Adaptive optics in Astronomy. Cambridge University press, p.294 Sasiela R.J., Electromagnetic wavepropagation in turbulence, 1994, Springer-Verlag, Berlin Sauvage J.-F., Fusco T., Rousset G., Petit C., Neichel B., Blanc A., Beuzit J.-L., 2005, Proc. SPIE, 5903, 100 Tokovinin A., Thomas S., Gregory B., van der Bliek N., Schurter P., Cantarutti R., Mondaca E., 2004, Proc. SPIE, 5490, 870 Tyler G.A., 1994, J.Opt.Soc.Amer(A), 11, 358 van Dam M. A. & Lane R. G., 2000, J.Opt.Soc.Amer(A), 17, 1319 V´eran J.-P., Herriot G., 2000, J.Opt.Soc.Amer(A), 17, 1430 Winick K.A., 1996, J.Opt.Soc.Amer(A), 3, 1809

c 0000 RAS, MNRAS 000, 000–000

Annexe G "First laboratory validation of vibration filtering with LQG control law for Adaptive Optics” C. Petit et al. - Opt. Express - 2007

171

First laboratory validation of vibration filtering with LQG optimal control law for Adaptive Optics Cyril Petit, Jean-Marc Conan, Thierry Fusco Office National d’Etudes et de Recherches A´erospatiales, BP 72, 92322 Chˆatillon, France

Caroline Kulcs´ar, Henri-Francois Raynaud Universit´e Paris 13, Institut Galil´ee, L2TI, 93430 Villetaneuse, France [email protected]

Abstract: We present a first experimental validation of vibration filtering with a Linear Quadratic Gaussian (LQG) optimal control law in Adaptive Optics (AO). A quasi-pure mechanical vibration is generated on a classic AO bench and filtered by the control law, leading to an improvement of the Strehl Ratio and image stability. These results are of particular interest for the SPHERE AO design. They also comfort our optimal control approach, already shown to be very attractive for MultiConjugate Adaptive Optics. © 2007 Optical Society of America OCIS codes: (010.1080) Adaptive optics; (100.3190) Inverse problems

References and links 1. G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou,E. Stadler, G. Zins, “ NAOS, the first AO system of the VLT: on sky performance”. Adaptive Optical System Technology II, vol. 4839, pp 140-149, Bellingham, Washington, 2002. Proc. SPIE. 2. C. Dessenne, P. Y. Madec, G. Rousset, “ Optimization of a predictive controller for closed-loop adaptive optics”, Appl. Opt., 37,21, pp 4623–4633, 1998. 3. C. Petit, J.-M. Conan, et al., “ Optimal Control for Multi-conjugate Adaptive Optics” , Comptes Rendus de l’Acad´emie des Science, Physique 6, pp 1059–1069, 2005 (http://www.sciencedirect.com/science/journal/16310705). 4. B. Le Roux, J.-M. Conan, C. Kulcs´ar, H.-F. Raynaud, L. Mugnier, T. Fusco,“ Optimal control law for classical an multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004). 5. C. Petit, F. Quiros-Pacheco, et al., “ Kalman Filter based control loop for Adaptive Optics” , Advancements in Adaptive Optics, D. Bonaccini Calia, B.L. Ellerbroek, R. Ragazzoni , Proc. SPIE 5490, (2004). 6. C. Kulcs´ar,H.-F. Raynaud, C. Petit,J.-M. Conan, and P. Viaris de Lesegno,”Optimal control, observers and integrators in adaptive optics”, Opt. Express, vol.14, number 17, pp. 7464–7476,2006 7. C. Mohtadi. “Bode’s integral theorem for discrete-time systems.” Proceedings of the IEE, vol 137, pp 57–66, 1990. 8. T. Fusco, C. Petit, et al., “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” vol. AWB 1, Optical Society of America, OSA, 2005. 9. C. Dessenne, “ Commande modale et pr´edictive en optique adaptative”, Ph. D. thesis, Univ. Paris VII, 1998 10. C. Petit, “Etude de la commande optimale en Optique Adaptative et Optique Adaptative MultiConjugue, validation numrique et exprimentale”, Ph. D. thesis, Univ. Paris XIII, 2006, 1059–1069, 2005 (http://www.sciencedirect.com/science/journal/16310705).

1.

Introduction

Adaptive Optics (AO) is now mature and allows to perform a real-time correction of the atmospheric turbulence effects on image formation. Nevertheless, AO performance is limited, due to

different error terms. One of them is mechanical vibrations which can be of particular concern [1], all the more as very high performance is expected, such as in eXtreme AO (XAO). These systems require indeed a particular attention on all error terms, especially on tip tilt mode stability. In the context of the XAO project SPHERE (Spectro-Polarimetric High-contrast Exoplanet REsearch), vibration filtering for tip and tilt modes stabilisation is thus considered. Passive filtering, performed through smart experimental equipment can reduce vibrations, but fails to filter them entirely. Then, active filtering should be considered, particularly through the control loop. Integrators are not able to filter vibrations specifically, and they may even amplify them depending on the vibrations frequencies. Predictors, such as proposed by Dessenne [2], can filter specific vibrations, nevertheless optimality and stability are difficult to ensure. A Linear Quadratic Gaussian (LQG) control law (see [3, 4] and references therein) can provide an optimal (in the sense of residual phase minimal variance) correction of the vibrations [5], and is thus considered for the SPHERE AO called SAXO. We propose here an experimental validation of this optimal control law. We briefly recall our general LQG control solution and describe the priors used to add vibration filtering. We propose a numerical simulation of the expectable performance on the AO bench with a multiple vibrations pattern. We then focus on experimental validation of vibration filtering in AO. We present the AO bench used for the tests. Then, a single vibration is generated on the bench and filtered with the control law. Results are compared to numerical simulation. We finally discuss the application to the SPHERE project. 2.

LQG control and vibration filtering

2.1. LQG control brief overview The implemented optimal control law is fully described in [5]. The basic assumptions are the following. The DM provides a linear and instantaneous response (no dynamics), constant over a frame period T (zero-order hold) so that the correction phase φ cor n−1 during time period [(n − 2)T, (n − 1)T ] is: φ cor (1) n−1 = Nun−2 , where N is the influence matrix. The WFS is linear and integrates the residual phase signal over one frame period. It provides a measurement yn during the time interval [(n − 1)T, nT ] defined by :
(2) yn = D φ n−1 − φ cor n−1 + wn ,

where D is a matrix characterizing the WFS, φ n−1 is the average turbulent signal over time period [(n − 2)T, (n − 1)T ] and wn is the measurement noise. We assume that CCD read-out and slope computation use one frame period, computation and DM control representing a negligible amount of time. It is thus a two frame period delay system. In this context, and defining an optimality criterion in the sense of residual phase minimal variance, we have proposed an optimal control law that we briefly recall hereafter. We first exhibit a linear time-invariant state-space model of the system in the form : xn+1 = A xn + Bun + ν n , yn = C xn + wn ,

(3) (4)

where xn represents the state vector of the system at instant n and gathers all the knowledge needed at instant n to compute next state xn+1 and output (WFS measurement) yn . ν n and wn are assumed to be decorrelated zero-mean white gaussian noises with covariance matrix Σν and Σw . A convenient choice of state vector is :
(5) xtn = φ tn , φ tn−1 , utn−1 , utn−2 ,

where t stands for transposition. We assume that the dynamics of the turbulent phase (which usually follow a Taylor’s hypothesis) can be approximated by a one-order auto-regressive model : φ n+1 = Atur φ n + νn , (6) where νn is zero-mean white gaussian noise with covariance matrix Σν and Atur is the matrix defining the dynamic characteristics of the turbulence. Details on computation of Atur can be found in [3, 4]. Thus taking into account this model and the equation 2, equation 3 and 4 can be written:       Id 0 Atur 0 0 0      Id 0 0 0   xn +  0  un +  0  ν n , xn+1 =  (7)      0 0  Id 0 0 0 0 0 0 0 Id 0
yn = D 0 Id 0 −N xn + wn , (8)

Based on the separation principle, the control issue is then split into first an optimal estimation of the incoming phase and then phase correction [6]. Estimation is based on a Kalman filter and provides a reconstruction of the phase on a truncated Zernike basis of N modes. It realizes a temporal prediction accounting for time delay. More precisely, optimal estimation and prediction of the turbulence is provided by: xˆ n+1/n = A xˆ n/n−1 + Bun + L n (yn − C xˆ n/n−1 ),

(9)

The Kalman optimal observer corresponds to a particular value of the gain L n given by:

L n = A Σn/n−1 C t (C Σn/n−1 C t + Σw )−1

(10)

where Σn/n−1 is the covariance matrix of the state vector and is obtained by solving the following Riccati matrix equation: Σn+1/n = A Σn/n−1 A t + Σv − A Σn/n−1 C t (C Σn/n−1 C t + Σw )−1 C Σn/n−1 A t .

(11)

This equation does not depend on measurement and can be therefore computed off-line, and even replaced by its constant asymptotic solution L ∞ (by letting Σn+1/n in (11) converge to its asymptotical value) with non-significant loss of optimality, as in [4]. Correction is then deduced thanks to a classic least-square projection of the estimated turbulent phase onto the DM modes. The state feedback form is then: un = Kˆxn+1/n ,

(12)

where K = (P, 0, 0, 0) and P = (Nt N) Nt . This control law, called standard LQG control henceforth, has been already thoroughly simulated and experimentally validated in classic AO, off-axis AO, and MCAO [3]. Note that the turbulent phase being estimated on N modes, 2 × N modes are estimated in the state vector (voltages are known and not estimated, and thus could be removed from the state vector for computing efficiency). −1

2.2. Vibration filtering Now dealing with additive vibrations, it is rather straightforward to include these perturbations in the state vector and estimate them just as the turbulence. We only need to modify our models

to describe explicitly the impact of vibrations on the phase. Estimation then takes into account additive priors on the vibrations, but the overall structure of the control law is identical. Assuming for instance the existence of one vibration on a particular mode of the phase, a discrete-time domain second order auto-regressive model can be used to describe the vibration: vk = a1 vk−1 + a2vk−2 + ξk , where ξk is a forcing function, and the coefficients a1 , a2 are defined by: p a1 = 2e−K ω0 Te cos(ω0 Te 1 − K 2), a2 = −e−2K ω0 Te ,

(13)

(14)

where K is the damping coefficient (related to the vibration bandwidth), ω0 = 2π fvib is the natural pulsation of the vibration (of frequency fvib ). The forcing function ξk stands for the excitation source. It is modeled as a white noise of variance σξ2 . For a given (ω0 , K), the power of the vibration is proportional to σξ2 . This model defines our new priors on the vibration from which a vibration filtering LQG control is derived. The state vector can then take the following structure:
(15) xtn = vtk , vtk−1 , φ tn , φ tn−1 , utn−1 , utn−2 ,

The various matrices of the state space model in (3) and (4) are easily modified accordingly and the estimation of the vibration and turbulence is still provided by an equation of the form of Equation (9). Correction is also performed similarly, by projecting onto the DM modes both the turbulence and the vibration. Note that another application of this approach could be both vibration and turbulence estimation but only turbulence correction. This approach would be usefull when vibrations affect the WFS arm and not the imaging arm. We have considered here the estimation and correction of one vibration on one mode, but we could account similarly for p vibrations on M different modes of the phase, dealing with each mode separately and assuming vibrations are not coupled. This leads to the estimation of 2 × p × M components in addtion to the 2 × N turbulent components but with 2p × M 2500, where ∆ f is the full width at half max of the vibration. The peak-valley amplitude of the vibration is A = 0.24 λ /D. This vibration mainly affects the tip and tilt modes, and so the x and y average slope measurements. Closing the AO loop with standard control laws (either integrator or LQG), leads to a drop of the long exposure SR down to 86% (compared to 91% without vibrations), due to image jitter. Meanwhile, the x and y average slopes measured on the WFS show an increased variance and a resonance peak at frequency fvib . On Fig. 2, the Cumulated Temporal PSD (CTPSD) of the x average slope, in closed-loop, shows a steep step due to the vibration. Indeed with the selected frequency, fvib = 15.2 Hz, the vibration is not damped, and even amplified by standard control laws. The loss of SR and the average slope variance increase are consistent, proving that the vibration is common to both imaging and WFS arms and should be corrected. Optimization of the integrator gain is useless as the power of the vibration is too weak compared to the turbulence power. Such a scenario is rather realistic compared to the vibration effects measured on systems like NAOS [1]. We now consider implementing our vibration filtering LQG control. We have stressed that the tip and tilt modes mainly suffer from the vibration. Vibration filtering is thus applied only on these 2 modes, leading to a negligible computational cost increase (N = 120, p × M = 2). The parameters are estimated thanks to open-loop measurements and CTPSD, leading to fvib = 15.2 Hz, K = 10−4 and σξ2 = 5, 4 10−6 rad2 . 4.3. Vibration filtering results When closing the AO loop with vibration filtering LQG control, the vibration is strongly attenuated. The SR increases up to 90%, close to the vibration free performance of the standard control laws (91%). Comparison of the CTPSD of the x average slope with or without vibration for standard control and for vibration filtering LQG control is given in Fig. 3. It shows that the steep step due to the vibration is almost completely damped. In fact, a slight residual step is still visible since a compromise is made between the filtering of the vibration and the correction of the turbulence. Small differences can be seen on this figure between the different CTPSD in particular at low frequencies. They are mainly due to experimental variability, all the more as the AO bench is not stabilized. Experimental error bound is lower than ±2.5 10−4(λ /D)2 . Bode’s integral theorem also participates to these differences. We provide on Fig. 4 the x average slope transfer function as defined in Eq. (16) for either the integrator (compared to theory), or the LQG control (compared to simulation). As for the integrator, a good agreement is found between experimental transfer and theory. A strong damping is observed on the LQG control transfer at fvib , proving the vibration filtering provided by LQG control. The peculiar low frequency behaviour on the experimental PSD is consistent with simulation results. It is characteristic of the LQG control, and so is the high frequency (close to fe /2) loss of gain of the simulated PSD. This loss of gain is not observed on experimental data probably due to high frequency defects of the experimental set-up. Finally, experimental curves of Fig. 4 show that some small peaks appear between 8 Hz and 12 Hz, due to non stationary vibrations present on the bench. 5.

Conclusion

This is the first demonstration in AO of optimal control with vibration filtering. It demonstrates the significant improvement brought by correction of such components. Multiple vibrations can also be filtered out as shown numerically. We have not addressed here the problem of robustness. Irrespective of the particular choice of controller structure and/or design method, vibration filtering ultimately boils down to lowering the controller gain for the frequencies that need to be filtered out. Obviously, lowering this

Fig. 3. Experimental CTPSD of the x average slope for a LQG control law, without vibration (dotted line), with vibration and no specific filtering (dashed), and with vibration filtering (solid). Experimental error bound is lower than ±2.5 10−4 (λ /D)2 .

gain over too large a frequency range is bound to degrade overall performance. Thus, a delicate balance needs to be achieved between robustness (measured by the width of the frequency band where filtering is effective) and performance. In the LQG approach presented in this paper, this engineering trade-off is essentially embodied in the choice of the dampening coefficient K, a low value of which results in a controller which effectively filters out vibrations only in a narrow frequency band around the vibration frequency. Nevertheless the impact of parameters modification and the resulting engineering trade-off have not been evaluated so far. This issue is currently under investigation. Indeed, these promising results have led us to consider a Kalman based control for the tip and tilt modes correction for the SPHERE project, the higher order modes being corrected thanks to a classic optimized modal gain integrator. It would allow correcting both the turbulence and vibrations on the tip and tilt modes, which is of particular concern for coronographic applications. Preliminary results are encouraging. So as to deal with robustness issues, the control procedure should be based on a periodic identification of the vibration parameters and a regular update of the LQG control. The effort should be then made on real-time identification procedures. These results extend our previous work on optimal control. Recent studies based on experimental validations [3, 10] have also proved its interest for MultiConjugate AO and Laser Tomographic AO (LTAO).

Fig. 4. Comparison of the experimental x average slope transfer. Top is obtained with integrator (dotted) and compared to theory (solid). Bottom is obtained with vibration filtering LQG control (dotted) and compared to numerical simulation (solid).

Annexe H "Calibration of NAOS and CONICA static aberrations Application of the phase diversity technique” A. Blanc et al. - A&A - 2003

& “Calibration of NAOS and CONICA static aberrations Experimental results” M. Hartung et al. - A&A - 2003

181

ASTRONOMY AND ASTROPHYSICS Calibration of NAOS and CONICA static aberrations Application of the phase diversity technique A. Blanc1,2 , T. Fusco1 , M. Hartung3 , L. M. Mugnier1 , and G. Rousset1 1

2 3

Office National d’Etudes et de Recherches A´erospatiales, D´epartement d’Optique Th´eorique et Appliqu´ee, BP 72, F-92322 Chˆ atillon cedex, France E-mail: [email protected] Laboratoire des Signaux et Syst`emes, Ecole Sup´erieure d’Electricit´e, Plateau de Moulon, F-91192 Gif-sur-Yvette, France Max-Planck-Institut f¨ ur Astronomie, K¨ onigstuhl 17, D-69117 Heidelberg, Germany

Abstract. We describe and evaluate the performance of a phase diversity wavefront sensor used to measure the static aberrations of the VLT instrument NAOSCONICA. The main limitations of this phase diversity technique are compiled. We investigate the systematic errors due to the experimental implementation and the design restrictions. Further error sources stem from the imperfect knowledge of the system, and from limitations of the algorithm. The influence of these errors on the wavefront estimation is evaluated on numerical and experimental data. This study highlights the essential verifications and calibrations needed to obtain accurate results and gives a practical guideline for the application of a phase diversity wavefront sensor. The comprehensive calibration results and the final gain in optical performance are presented and discussed in a complementary paper (Hartung et al., this issue). Key words: Wavefront sensing, phase diversity, active optics, adaptive optics 1. Introduction The VLT instrument NAOS-CONICA comprised of the adaptive optics (AO) system NAOS1 (Rousset et al., 1998; Rousset et al., 2000) and the high resolution camera CONICA2 (Lenzen et al., 1998; Hartung et al., 2000), aims at providing very high quality images on one (UT4-Yepun) of the 8-m telescopes of the Cerro Paranal observatory. To achieve its specification, NAOS has been designed to provide high Strehl ratios (≥ 70%) under good observing conditions. But even if the AO system ideally corrects for atmospheric turbulence, the optical train of CONICA as well as a part of the optical train of NAOS which is not sensed by the wavefront sensor also contribute to the over1 2

NAOS stands for Nasmyth Adaptive Optics System CONICA stands for COud´e Near Infrared CAmera

all degradation of the instrument performance. Indeed, the optical train is not free of aberrations and consequently reduces the Strehl Ratio (SR) on the images. In order to reach the ultimate performance with the instrument, it is necessary to pre-compensate these down-stream static aberrations by the AO system. This is a feature provided by NAOS which is able to introduce known static aberrations on the deformable mirror (DM) in closed loop. In the NAOS-CONICA instrument it is not possible to introduce an additional wavefront sensor in the optical train of the imaging device CONICA to estimate the wavefront. Hence, the CONICA images themselves must be used to derive the unknown aberrations. In this paper, we present a phase diversity (PD) (Gonsalves, 1982) approach to calibrate these unseen aberrations using a couple of focused and defocused images obtained on the CONICA detector. Because of the huge number of observation modes of NAOS-CONICA, the calibration of the unseen aberrations must be split in several parts: NAOS dichroics, i.e. the beam splitters between the wavefront sensor and imaging paths, and CONICA aberrations, i.e. the different filters and objectives. Detailed explanations of the global calibration procedures as well as a comprehensive presentation of the results are presented in (Hartung et al., this issue) (hereafter called Paper II). In the present paper we focus on the validation of the phase diversity approach used to estimate these aberrations. In Section 2, a description of the phase diversity concept is given along with the specific algorithm used. In Section 3, a brief presentation of the experimental setup is proposed. A more complete description can be found in Paper II. In Section 4, a validation of our PD algorithm is proposed on simulated data. It allows us to assess qualitatively and quantitatively the accuracy of the aberration estimation. Finally, we exploit the simulations and the experimental results to present the limitations and the overall performance of the aberration calibration procedure in Sec-

2

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

tion 5. We list, analyze and quantify all the possible error sources which may induce a loss of accuracy in the PD results. This Section allows us to define the optimal parameters to calibrate the whole system. In Section 6, we present a practical example and give a detailed description of the global procedure to estimate NAOS-CONICA static aberrations. This procedure is quite general but the particularities of CONICA and NAOS-dichroic aberration measurements are underlined.

Kendrick et al., 1994; Lee et al., 1997; Thelen et al., 1999; L¨ofdahl et al., 2000) and also to restore images, as in solar imaging through turbulence (Restaino, 1992; L¨ofdahl and Scharmer, 1994; Seldin and Paxman, 1994). It uses a low-cost, optically simple wavefront sensor which consists in the imaging camera itself, but it requires a complex numerical and iterative processing to restore the unknowns from the images. 2.2. Imaging model

2. Estimation of static aberrations by Phase Diversity 2.1. Phase diversity principle The estimation of the aberrations from the sole focused image does not ensure the uniqueness of the solution. This indetermination is due to the relationship between the point spread function (PSF) h and the aberrated phase: a couple (φ, φ0 ) exists such that h(φ) = h(φ0 ). Phase diversity (Gonsalves, 1982; Paxman et al., 1992) was proposed to add information and removes this indetermination. The idea is to collect at least one additional image, which differs from the focused one by a known phase variation. Figure 1 illustrates the phase diversity principle. One can note that there are several ways to introduce the known aberration. A possibly non comprehensive list of the possibilities is the following : – simultaneously using a beam splitter and two detectors as presented in Figure 1, – introducing a beam-splitter and a prism and recording the focused and defocused images on the same detector (Gates et al., 1994), – sequentially using a translation of the detector or introducing the known aberration in the optical path (see Subsection 2.3.2). In this case, it is assumed that the aberrations and the object do not evolve between the two acquisitions (which is the case of the NAOSCONICA static aberrations for instance). telescope

beam splitter focused image

φ

defocused length defocused image

Fig. 1. Phase diversity principle

This technique has been successfully used by some authors to determine aberrations (Carreras et al., 1994;

In the isoplanatic patch of the imaging system, the image is the noisy convolution of the PSF h in the observation plane and the object o: i(r) = (h ∗ o)(r) + n(r)

(1)

where ∗ denotes the convolution product, r is a twodimensional vector in the image plane and n is an additive noise. The PSF associated with the focused image is given by: h1 (r) = |F T −1 {P (u). exp[iφ(u)]} |2

(2)

where u is a two-dimensional vector in the pupil plane, φ is the unknown aberrated phase function, P is the aperture function and F T −1 denotes the inverse Fourier transform. The phase function is expanded on a set of Zernike polynomials. Indeed, aberrations in an optical system can be mathematically represented by Zernike polynomials (Noll, 1976). φ(u) =

k X

ai Zi (u)

(3)

i=2

Theoretically, k should tend to infinity to describe any wave form, but in the particular case of static aberration estimation, the first polynomials (typically the first twenty) are enough to describe the aberrations arising from misalignment of systems made mostly of spherical surfaces or of surfaces with a modest degree of asphericity. In the following, we will note a = (a1 , ..., ak )T the unknown aberration coefficients, where T denotes transposition. In the defocused plane, the PSF is given by : h2 (r) = |F T −1 (P (u). exp {i[φ(u) + φd (u)]} )|2

(4)

where φd is the known diversity phase function. In our case, φd (u) = ad4 Z4 (u) where Z4 is the defocus Zernike polynomial. In practice, data are discrete arrays because of the spatial sampling of the images. Equation 1 takes the form: i = Ho + n

(5)

where H is the Toeplitz matrix corresponding to the convolution by h (Ekstrom and Rhoads, 1974), and where i, o and n are the discrete forms of the previous variables.

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

The problem is to estimate the aberration parameters a (the set of ai ) from the data (focused i1 and defocused i2 images) and the defocused distance, without knowing the object o. 2.3. Aberration estimation principle Using a Joint Maximum A Posteriori (JMAP) approach (Little and Rubin, 1983), an estimator can be defined as ˆ )JMAP = arg max p(i1 , i2 , o, a) (ˆ o, a o,a

= arg max p(i1 |o, a)p(i2 |o, a)p(o)p(a)

(6)

3

where ˜. stands for a Discrete Fourier Transform (DFT), f the spatial frequency and So the power spectral density (PSD) of the object. ˆ (a) can be Using the same periodicity approximation, o written as : ˜ ∗ (f ) ˜ı2 (f ) ˜ ∗ (f ) ˜ı1 (f ) + h h 2 1 ˜oˆ(a) = . (11) ˜ 1 (f )|2 + |h ˜ 2 (f )|2 + σ 2 /So (f ) |h

˜oˆ(a) can be introduced in Equation (10) so that the new 0 (a) only depends on the unknown phase criterion JGML aberrations (a) and can be minimized using an iterative algorithm (for instance a conjugate gradient).

o,a

where p(i1 , i2 , o, a) is the joint probability density function of the data (i1 ,i2 ), of the object o and of the aberrations a. p(i1 |o, a) and p(i2 |o, a) denote the likelihood of the data i1 and i2 , p(o) and p(a) are the a priori probability density function of o and a. The noise is modeled as a stationary white Gaussian noise with a variance σ 2 (the same for the two images). In our case, we do not use an explicit probability density function for a (p(a) = 1) since the regularization is brought by the small number of estimated Zernike coefficients. Consequently, this joint estimation will be called a Generalized Maximum Likelihood (GML) approach. We choose a Gaussian prior probability distribution for the object with a covariance matrix Ro . The maximization of the a posteriori probability law defined in Equation 6 is equivalent to the minimization of its neg-logarithm (JGML ) defined as 1 1 JGML (o, a) = 2 ki1 − H1 ok2 + 2 ki2 − H2 ok2 2σ 2σ 1 t −1 (7) + o Ro o 2 It is important to note that the derivative of JGML (o, a) with respect to the object gives a closed-form expression ˆ (a) that minimizes the criterion for a given for the object o a (Paxman et al., 1992). This expression is that of a biframe Wiener filter:
−1 ˆ (a) = Ht1 H1 + Ht2 H2 + σ 2 R−1 (Ht1 i1 + Ht2 i2 ) (8) o o ˆ (a) into the criterion yields a new criterion Substituting o that does not depend explicitly on the object: 0 (a) = JGML (ˆ o(a), a) JGML

(9)

2.3.1. Object regularization The a priori information required on the object consists in the choice of the object power spectral density model. In our case a focused and a defocused image are acquired with high Signal to Noise Ratio (SNR) and the object is close to be a Dirac function. Consequently the phase estimation can be obtained without any knowledge or estimation of the prior object spectral density. The dimensionless ratio σ 2 /So (f ) is simply set to a small arbitrary constant (10−6 in our case) in order to avoid numerical problems due to computer precision. Notice that, in the case of lower SNR, the use of a marginal estimator which estimates the sole aberration parameters, gives better results than this ad-hoc regularization (Blanc et al., 2000). 2.3.2. Choice of the defocus distance The choice of the known defocus distance is essential to obtain accurate results. The RMS defocus coefficient ad4 depends on the defocus distance d of the second image, the telescope diameter D and the focal length F through: πd (in radian) ad4 = √ 8 3λ(F/D)2

(12)

The corresponding peak-to-valley optical path ∆ is equal to √ d 3λad4 ∆= = (13) π 8(F/D)2

In the following sections, ad4 will be given in nanometers. The interest of this new criterion is its drastic reduction For NAOS+CONICA, F/D = 15. It has been shown (Lee et al., 1999; Meynadier et al., 1999) that a defocus ∆ equal of the solution space. Furthermore, using a periodic approximation (which to λ provides accurate results. That is a defocus distance corresponds to approximate a Toeplitz by a circulant ma- equal to 4 mm for λ = 2.2 µm. In fact the “optimal” detrix) (Demoment, 1989), the criterion can be expressed in focus distance depends on the object structure, the phase amplitude φ and the SNR on the images. In practice a the discrete Fourier domain: " large domain around this value (typically λ ± λ/2) still X |h˜1 (f ) o˜(f ) − ˜ı1 (f )|2 JGML (o, a) ∝ provides accurate results (Meynadier et al., 1999). σ2 f In the case of NAOS-CONICA static aberration esti# mation, two different procedures are applied to introduce |˜ o(f )|2 |h˜2 (f ) o˜(f ) − ˜ı2 (f )|2 + + (10) the defocus. They are explained briefly in Section 3 and σ2 So (f ) in detail in Paper II.

4

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

3. Practical implementation 3.1. CONICA calibration The estimation of CONICA stand-alone aberrations (objectives and filters) is obtained through the use of pinholes (diameter 10µm) located at different defocused positions in the camera entrance focal plane (0, 1, 2 and 4 mm). The telescope pupil is simulated by a cold pupil placed inside CONICA. A detailed explanation of the experimental setup is given in Paper II. The defocus distances induced through the use of different pinhole pairs are summarized in Table 1. pinhole pairs Defocus distance (mm)

0-1 1.0

0-2 2.0

0-4 4.0

1-4 3.0

2-4 2.0

Table 1. Defocus distances induced by the use of various pinhole pairs. 0-2 and 0-4 pairs are recommended for J-H and K filters respectively.

The necessity of introducing enough diversity between the two images and the higher SNR of images obtained with the pinhole 0 (which is in the focal plan) lead us to choose the pair 0-2 for J and H filters and the pair 0-4 for K filters. Note that the use of pinholes in the entrance focal plane is not optimal for the phase diversity algorithm since – the known aberration is not a pure defocus (a longitudinal translation in the entrance focal plane of the camera is not completely equivalent to a detector translation in the imaging focal plane) – see Subsection 5.1.5, – the use of different pinholes may induce errors in the aberration estimation (the PD algorithm assumes that the same object is used to obtain focused and defocused images, see Subsection 5.1.4). Shape differences between two pinholes can induce phase estimation errors, – the focused and defocused images are not at the same position on the detector and need to be re-centered – see Subsection 6.2 – since PD can not estimate relative tip-tilt greater than λ between the two images. Indeed, the phase is estimated modulo 2π (see Equations 2 and 4), – lastly, there may be field aberrations due to the different pinhole position in the beam – see Subsection 5.1.6. The procedure described in Subsection 2.3 allows us to estimate a set of aberrations for each CONICA configuration, that is a filter plus a camera objective. 3.2. Dichroic calibration The estimation of the NAOS dichroic aberrations is obtained through the use of the AO system. A focused image of a fiber source, located in the entrance focal plane

of NAOS, is recorded in closed-loop in order to avoid the common-path aberrations from the optical train between the source and the dichroic. Then a given defocus is introduced on the DM with the AO loop still closed to record the defocused image (See Paper II for a complete description of this procedure.). This approach gives the input data for the estimation of the NAOS dichroic aberrations together with the CONICA aberrations. The value of the separated dichroic aberrations is obtained by subtracting the value of the previously estimated CONICA aberrations. The introduction of a defocus by the DM avoids the difficulties of object defocussing highlighted above for the CONICA calibration. Now, the same object is considered and thus the two images are located at the same position on the detector. 4. Simulation results In order to validate the algorithm and to quantify its precision, we first consider simulated images. The conditions for this simulation are given by a point-like object, an imaging wavelength of 2.166 µm and a pure defocus equal to λ (peak to valley) between the two images, corresponding to a defocus coefficient ad4 of 641.5 nm RMS. We degrade the PSF by a wavefront deformation described by its Zernike coefficients. The coefficients are arbitrary but chosen to have comparable values as observed in the calibration procedure (see Paper II). The phase is generated with the first 15 Zernike polynomials, note that the estimated phase will be expanded on the same polynomials. We add white noise to each image in order to obtain a SNR of 200 which corresponds roughly to the SNR of the CONICA data (this is the typical average SNR corresponding to the time exposure used, which has been estimated on several measurements). The SNR is defined as the ratio of the maximum flux in the focus image over the RMS noise. The same noise statistics is applied to the defocused image. That results in a lower SNR on this image since the defocus spreads the PSF and reduces its maximum value. The focused and defocused images are presented in Figure 2. In this example only two possible limiting parameters (see Section 5) have been taken into account: noise and image re-centering. The system is assumed to be perfectly adjusted and the images to be perfectly pre-processed without having any residual background features. Figure 3 shows a comparison between the input images of the simulation and the PSFs being reconstructed by the estimated aberrations and visualizes the quality of the aberration estimation. This visualization is helpful to judge real calibration data when only in-focus and outof-focus images are available. Table 2 quantifies the performance of wavefront estimation by comparing the true and the estimated Zernike coefficients. On each coefficient a good accuracy is obtained: the errors are less than a

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

Fig. 2. Simulated focused and defocused images (logarithmic scale). Noise is added to obtain a SNR of 200 in the focused image.

few nanometers. The maximum error is for the two astigmatisms. The influence of SNR on estimation results is analyzed in Section 5. The slight tip-tilt introduced between the two input images (a2 =70 nm RMS and a3 =-103 nm RMS) are estimated with a high precision, too. The error amounts to less than 1 % (the tip-tilt values are not shown in Table 2.). In the next section, we focus on the possible sources for losses of estimation accuracy.

5

Zernike number

true (nm)

estimated (nm)

4 5 6 7 8 9 10 11 12 13 14 15

60.5 -39.3 58.1 -16.2 -14.1 -2.5 13.7 -24.3 0.5 -3.2 2.8 -2.4

61.5 -46.7 61.7 -15.9 -10.9 -3.6 12.9 -26. 1.2 -4.9 3.3 -2.2

error (nm) 1 7.4 3.6 0.3 3.1 1.1 0.7 1.7 0.7 1.7 0.4 0.2

Table 2. Comparison between true and estimated Zernike coefficients applying phase diversity to simulated data with an image SNR equal to 200. The absolute value of the error is given for each coefficient. The total error is equal to 9.3 nm RMS.

5. Limitations The PD algorithm is based on several assumptions which must be well verified to obtain a good accuracy on the results. A list of possible error sources is given below. Quantitative results are essentially given on experimental data and with additional simulation data when necessary. The global procedure of data reduction can be found in Section 6. The error sources can be decomposed in three parts: the errors due to a non-perfect knowledge of the system (calibration errors or uncertainties), the errors due to the image acquisition and pre-processing (noise, residual background, etc.) and the errors due to limitation of the algorithm (spectral bandwidth, amplitude of the estimated aberrations, etc.). 5.1. System limitations Let us first focus on the errors due to the imperfect system knowledge. 5.1.1. Defocus distance

Fig. 3. Comparison between images (left) and reconstructed PSFs (right) from estimated aberrations. On top the focused images, at the bottom the defocused images (logarithmic scale and zoom×2 are considered for each image).

The major assumption of the PD principle is the addition of a known distortion (defocus in our case) between two images. An error on the defocus induces an error on the coefficients of radially symmetric aberrations, with a main part on the estimated defocus itself (see Figure 4). The difference between the maximal and the minimal estimated value of the defocus coefficient is 55 nm. For the spherical aberration, it is 10 nm and for the other ones, it is less than 5 nm. In addition one can show in Figure 5 that for reasonable errors on the known defocus, the propagation error coefficient is equal to one. Hence, the uncertainty on the known defocus distance yields directly the uncertainty on the estimated defocus aberration.

6

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

– the precision of a focus adjustment with the NAOS deformable mirror for the dichroic aberration estimation. This error is estimated to less than a few percent. All these items lead to a precision of the estimated defocus roughly equal to a ±30 nm. It will be shown in the following that the error on the defocus distance is by far the dominant error for our application. 5.1.2. Camera pixel scale

Fig. 4. Influence of the error on defocus distance on estimated aberration by phase diversity. The CONICA camera C50S and the narrow band filter Brγ are used. The focused and defocused images are obtained using the 0-4 pinhole pair (that is a theoretical defocus of 4 mm between the two images). For the same couple of images several defocus distances (from 3.8 to 4.2 mm) serve as input parameters.

Fig. 5. Influence of the error on defocus distance on the estimation of the defocus. Experimental data have been used. The results are given for the CONICA camera C50S and two narrow band filters: FeII (1.527 µm) and Brγ (2.166 µm). The pinhole pair 0-2 (that is a theoretically defocus of 2 mm) and 0-4 (that is a theoretically defocus of 4 mm) are respectively used for FeII and Brγ filters. For each couple of images several defocus distances (respectively from 1.5 to 2.5 mm and from 3.5 to 4.5 mm for FeII and Brγ filters) serve as input parameters.

This uncertainty on the defocus distance can be due to: – an uncertainty on the physical position of the CONICA pinholes in the entrance focal plane. This uncertainty is estimated to ±0.15 mm corresponding to ∆a4 = ±24 nm RMS, – a systematic uncertainty on the F/D ratio (estimated to be less than two percent),

The camera pixel scale is needed to calculate the oversampling factor. An error on this factor induces an error on the coefficients of all radially symmetric aberrations (defocus, spherical aberration ...) as shown in Figure 6.

Fig. 6. Influence of the pixel scale error on estimated aberration by phase diversity. Experimental data have been used. The CONICA camera C50S and the narrow band filter Brγ are used. The focused and defocused images are obtained using the 0-4 pinhole pair. For the same couple of images several pixel scales (from 13.05 to 13.45 mas) serve as input parameter for the PD algorithm.

An error of the pixel scale is essentially propagated to the defocus aberration estimation. The difference between the maximal and the minimal estimated value of the defocus coefficient is 17 nm. A slight error of 6 nm can be seen on spherical aberration but remains negligible in comparison to the one on the defocus. In Figure 7 the evolution of the estimation error of the defocus coefficient is plotted as a function of a pixel scale measurement error. It is assumed that the true value is 13.25 mas measured during the firs on-sky tests of the AO system. Since the accuracy on the pixel scale measurement is better than 0.2 mas, one can estimate the wavefront error (WFE) due to this uncertainty to be less than a few nanometers and therefore to remain negligible.

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

7

tion for different pinhole pairs. It shows a good agreement between all the pinholes, except for the pinhole pair 0-1. The discrepancy in the estimated aberrations can probably be attributed to the small defocus distance. Disregarding the pair 0-1, the good agreement of all the ohter pairs leads us to assume the pinholes close to be identical. Indeed, the main part of the WFE is due to the defocus coefficient and highlights the uncertainties on pinhole positions as already mentioned in Subsection 5.1.1. 5.1.5. Translation in the entrance focal plane

Fig. 7. Influence of the pixel scale error. The reference value is set to 13.25 mas. For this value it is assumed that the error on the defocus coefficient estimation is zero (the experimental conditions are the same as in Figure 6).

5.1.3. Pupil shape An exact knowledge of the pupil shape (diameter, central obstruction, global shape) is required. In particular, few percent of mis-alignment of the pupil leads to an error of a few tens nanometers on the phase estimation. 5.1.4. Differential object structures The algorithm is based on the assumption that the same object is used to obtain the focused and the defocused image. If two different objects are used (case of CONICA pinholes), errors could be induced if they are not completely identical. Figure 8 shows for the narrow band filter FeII

The maximal detector translation along the optical axis is not enough to introduce significant diversity between focused and defocused images. For the calibration of CONICA aberrations, we introduce defocus by translating “the object” in the entrance focal plan. As we said before (Subsection 3.1), this implementation does not create exactly a pure defocus but also in first order, some spherical aberration. We have quantified the deviation from a pure defocus by using the optical design software ZEMAX and shown that it can be neglected (a translation of 4 mm in the entrance focal plan induces defocus and a negligible spherical aberration a11 = 0.14 nm). 5.1.6. Field aberrations The images obtained by the pinhole 1 and 2 are not located at the same position on the detector than those obtained by the pinhole 0 and 4 (separation of 100 pixels in y). It induces that some focused and defocused images (for example pair 0-1 and 0-2) do not see exactly the same aberrations. The evolution of the field aberrations has been evaluated by optical calculations using ZEMAX. This study has shown that the main influence concerns the astigmatism a5 but in a negligible way (its variation is less than 5 nm for a separation of 100 pixels in x and 100 pixels in y). 5.2. Image limitations 5.2.1. Signal to Noise ratio

Fig. 8. Comparison of the estimated aberrations of various pinhole pairs. Camera C50S with FeII narrow band filter (1.644 µm).

(1.64 µm) and the objective C50S the estimated aberra-

The accuracy of the PD algorithm is directly linked to the signal to noise ratio in the images (the definition of SNR is given in Section 4). We present in Figure 9 the WFE error evolution as a function of image focal plane SNR. This figure, obtained on simulated data, shows the perfect agreement between simulation and expected (theoritical) 1/SNR behavior (Meynadier et al., 1999). For the NAOSCONICA aberration estimation it has been checked that the SNR is high enough to make this error source negligible in comparison to the others since the typical values of SNR are greater than a few hundred. That leads to a WFE of a few nanometers due to signal to noise ratio. This has been experimentally checked on one set of data.

8

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

a pre-processing algorithm which allows to remove these residual background features.

Fig. 9. Evolution of the wavefront error as function of SNR on focal plane image. Simulation presented in Section 4 have been used. A 1/SNR theoretical behavior is plotted for comparison. Fig. 10. Evolution of the wavefront error as a function of the image size. Experimental data have been used.

5.2.2. Residual background features The principle of the PD method is the minimization of a criterion (Equation 10) which is based on a convolution image model (Equation 1). Thus the image should be perfectly corrected for all instrumental features (background, dead pixels, etc.) in order to match the model. In practice, residual features are still present. In particular, in the case of CONICA images, a background fluctuation due to pick-up noise can induce residual features on the images even after a proper background calibration (see Subsection 6.2). These features are interpreted as signal by the phase-diversity algorithm. Therefore they induce bias on the aberration estimation. The effect of such fluctuations is highlighted in Figure 10 on experimental data. The difference of the PD results obtained with and without residual background features yield the WFE which is plotted as a function of the image size. This can be understood as a function of the residual background influence, too, because it obviously depends on the image size: the smaller the images, the less important the residual background in comparison to the signal. Nevertheless, the image size should be large enough to contain the whole signal. Furthermore, the modelisation of the pupil shape (see Subsection 5.3.3) must also be taken into account to choose the right image size. In Subsection 6.2 we describe

5.2.3. Number of estimated Zernike polynomials As presented in Subsection 2.3, the phase regularization in our algorithm is provided by a truncation of the solution space through the use of a finite (and small) number of unknowns (typically the first twenty Zernike coefficients). Figure 11 shows, on experimental data, the influence of the number of estimated Zernike on the reconstruction quality. Note that, in the case of measurements of CONICA standalone aberrations, the pupil is unobscured and thus the Zernike polynomials are strictly orthogonal.

Zernike value (in nm rms)

Four pairs of focused-defocused images have been recorded sequentially in the same CONICA configuration (FeII filter and camera C50S). The SNR on these images is 400. Aberrations are then estimated for each couple and the WFE fluctuation on this four set of coefficients is equal to 2.2 nm RMS, which is in perfect agreement with Figure 9 (obtained on simulation).

number of estimated Zernike

Fig. 11. Evolution of the aberration estimation as a function of the number of Zernike in PD algorithm. Experimental data have been used.

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

There exists a limit to the number of Zernike polynomials that can be estimated with a reasonable accuracy. Of course this limit depends on the signal to noise ratio on the images. In the present case, this number is equal to 36. Note that if a more sophisticated regularization term is introduced in the PD algorithm both on the object and the aberrations this limitation should be overcome. Nevertheless such a regularization is not needed here since the aberration amplitudes are negligible (less than a few nanometers) for Zi above i = 11 (i.e the spherical aberration). The WFE between estimated Zernike coefficients 4-15 and estimated Zernike coefficients 4-36 is about 1.3 nm RMS. This WFE is very small and thus shows that the aliasing of the Zernike polynomials above 15 on the estimated coefficients 4-15 is negligible. 5.3. Algorithm limitations 5.3.1. Spectral bandwidth The phase diversity concept proposed here is a monochromatic wave-front sensor (theoretically the concept can be applied on polychromatic images but it induces an important modification of the algorithm to model the data (Seldin and Paxman, 2000)). Nevertheless it has been shown (Meynadier et al., 1999) that the use of broadband filters does not significantly degrade the accuracy as long as ∆λ λ is lower than a few tens of percents (typically ∆λ ≤ 0.15). λ

9

Fig. 12. Pupil shape in the PD algorithm for three numbers of pixel used in the pupil sampling [8, 32, 64 and 256]. The oversampling factor of 2 in K band leads to corresponding image sizes equal to [32, 128, 256 and 1024].

of residual background unless the images are tapered outside their central region. – a computation time problem. Because of the iterative resolution of criterion defined in Equation 7, the increase of images size will lead to a increase of computation time. This highlights the choice of a good compromise between pupil model in the PD algorithm to avoid phase reconstruction error and a reasonable image size. The evolution of the reconstruction error as a function of pupil sampling is presented in Figure 13. To minimize the residual background effects, all the background pixels (that is pixels with no PSF signal) has been put to zero in the images.

5.3.2. Image centering As mentioned above, the PD algorithm can not estimate a tip-tilt between the two images larger than 2π. Therefore, a fine centering between focused and defocused images must be done before the aberration estimation (see Subsection 6.2). 5.3.3. Pupil model Since we consider here experimental data (see Section 6), the modelisation of the pupil shape in the algorithm is critical, in particular the pixelisation effects. Indeed, in PD algorithm the pupil definition depends on the image size and on the oversampling factor. For example, images of camera C50S in K band oversample with a factor of 2 and a 32×32 image will lead to a pupil diameter of 8 pixels (see Figure 12). In this case, the pixelisation effects on the shape of the pupil will induce aberration estimation error. These effects are illustrated in Figure 13. Therefore, large images are recommended to well model the pupil and to obtain accurate results. Nevertheless, two problems may occur with the processing of large images: – a residual background problem (see Subsection 5.2.2). The larger the image, the more important the effects

Fig. 13. Evolution of the wavefront reconstruction error as function of pupil sampling in PD algorithm. The x axis gives the pupil diameter in pixel in the PD algorithm.

Considering the results shown in Figures 13 and 10 along with the computation load lead us to choose an image size equal to 128×128 pixels for the K band and 64×64 pixel for the J band. 5.4. Conclusion In this part, we have analyzed and quantified, on experimental and simulated data, the possible sources of errors in the static aberration estimation for NAOS-CONICA. It is shown that the main source of errors is due to an imperfect knowledge of the system (that is calibration errors).

10

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

In particular a precise knowledge of the defocus distance between focused and defocused planes is essential. If very high precisions are required on the estimation and on the correction of static aberrations (for instance in the case of a future very high SR AO for exo-planet detection), the PD must be taken into account in the early design of the system in order to optimize with respect to the constraints and error sources listed above. 6. Practical example In this section we give a detailed description on a practical example of the global procedure used to estimate NAOSCONICA static aberrations. This procedure is quite general but the particularities of the CONICA and NAOS dichroic aberration measurements are underlined. All the illustrations are obtained for the following configuration of CONICA: objective C50S (pixel size equal to 13.25 mas) and Brγ filter. 6.1. Input data The input data are a focused and a defocused image (i1,2 ) with their associated backgrounds (b1,2 ) and a known defocus distance expressed in mm in the entrance focal plane of CONICA. This distance is given by the pinhole choice in the case of CONICA measurements (see Table 1) or by the defocus introduced by the DM in the case of the NAOS dichroic aberration measurements. In the example we consider the first approach and introduce the defocus by the pinhole choice. 6.2. Pre-processing The pre-processing of the images is required before the wavefront can be estimated. We split the pre-processing in several steps: – Conventional background subtraction in order to remove the main part of detector defects (bad pixels, background level, possible background features, etc.): icorr 1,2 = i1,2 − b1,2

(14)

The division by a flat-field pattern is recommended to increase the accuracy of the results even if it is not done in this example. – The focused and defocused images are re-centered by a correlation procedure. For each filter we computed the relative shifts of the PSFs against each other. The median of these shifts was determined and serves to recenter the defocused images. This procedure ensures that re-centering is accurate enough to obtain a relative tip-tilt between the two images lower than 2π. – Removing of the residual background feature: the most important feature is a residual sine function in vertical direction due to pick-up noise. Its amplitude is

Fig. 14. Comparison of focused and defocused images before (on top) and after (at the bottom) the application of an algorithm which removes residual background features (log scale). The estimated SNR is equal to 400.

greater than the noise level. An estimation of the residual background features is necessary and performed directly on the images using a median filter in horizontal direction applied to each image column. A comparison of the images before and after this residual feature removal is presented in Figure 14. – Image windowing: the image size is a trade-off between a good numerical pupil modeling and a reasonable computation time. We achieve reliable results by using 128×128 frames. Note that we have assumed that all the bad pixels have been removed by the background subtraction. If some of them are still present in the pre-processed images, they must be removed by hand to ensure that they do not induce reconstruction errors in the PD algorithm. 6.3. CONICA aberration estimation When both focused and defocused images have been preprocessed as described above, the PD algorithm can be applied. The inputs of the PD algorithm are: – pixel scale: 13.25 mas (camera C50S), – central obstruction given by the fraction of the pupil diameter: 0 (full pupil), – wavelength (in µm) : 2.166 (Brγ filter), – highest estimated Zernike number: 15 (estimation of Zernike polynomials from 4 up to 15),

A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

11

– defocus coefficient ad4 (in nm RMS): in the present example, d = 4 mm which leads to ad4 = −641.5 nm, – the focused and defocused images obtained after preprocessing. Zernike

4

5

6

7

8

9

aberration (nm) raw aberration (nm) corrected

91

-27

48

3

-9

-4

112

-24

47

1

-5

-1

Zernike

10

11

12

13

14

15

19

-20

-5

-1

1

-2

17

-19

-3

-2

-3

-3

aberration (nm) raw aberration (nm) corrected

Table 3. Measured aberrations (in nm RMS) for the CONICA camera C50S and the Brγ filter. The defocus distance between the two images is 4 mm and the estimated SNR is 400. Only the 12 first Zernike 4-15 are given. The raw values are obtained without residual background subtraction. The corrected ones are obtained after subtraction of the residual background features.

The results obtained are summarized in Table 3. A bad background correction leads to an important error on the defocus (w 20 nm). A comparison between focused and defocused images and reconstructed PSFs from the estimated Zernike is proposed in Figure 15. The estimated SR on the 12 estimated Zernike is equal to 87 %. It compares nicely to the SR directly computed on the focal plane image, which is equal to 85 %. 7. Conclusion We have given a precise description of the phase diversity algorithm developed at ONERA and of its use for the calibration of NAOS-CONICA static aberrations. The concept of phase diversity has first been recalled, then the expression of the criterion to be minimized has been given. Guidelines for a practical implementation have been proposed. The essential parameters of the algorithm have been studied. This has allowed us to highlight the essential verifications and calibrations needed to obtain accurate results with PD. First, a simulated example of utilization has been proposed in order to demonstrate the precision of the estimation under typical SNR conditions of acquisition of NAOSCONICA images. It shows that for a SNR of 200, the error on the aberration estimation is less than few nanometers per polynomial. This simulation gives the ultimate performance for a given SNR since it does not account for experimental uncertainties and bias. The limitations of the experimental approach have been listed and studied. It has been shown that the main

Fig. 15. Comparison between images (left) and reconstructed PSF from estimated aberrations (right). [up] focused image, [down] defocused image (log scale are considered for each image).

source of error is induced by the uncertainty on the pinhole distances which leads to an error on the defocus estimated to ±24 nm. An other source of degradation is due to the bad knowledge of the pixel scale which contributes for a few nanometers on the defocus. Furthermore, an exact knowledge of the pupil alignment is required to ensure a good estimation of the aberrations. The residual background after compensation in the images is another important source of degradation, estimated to a few tens nanometers. In the case of the images of NAOS-CONICA, the SNR is high enough to make this error source negligible in comparison to the others. All these error sources lead to an uncertainty on the aberration estimation: ± 35 nanometers on the defocus and around few nanometers for the high order Zernike polynomials. Finally, an example of processing of experimental data obtained on CONICA during commissioning at the VLT has been given. Pre-processing required to minimized the Camera defects has been described and a comparison of focal plane PSFs with reconstructed PSFs using estimated aberrations has been made. It shows good agreements both in term of PSF shape and SR measurements. It has been shown that Phase Diversity is a simple and powerful approach to compute unseen aberration of an AO system. This is of a great interest for future very high SR system in which an accurate estimation and correction of such aberrations is essential to achieve the ultimate

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A. Blanc et al.: Calibration of NAOS and CONICA static aberrations

performance and to reach the scientific goals (exo-planet detection for instance) Acknowledgements. We wish to thank V. Michau and M.-T. Velluet for fruitful discussions and helpful comments on this work. And we are thankful to F. Lacombe for his help during the acquisition and the interpretation of NAOS-CONICA data. This research was partially supported through the Marie Curie Fellowship Association of the European Community and by European Southern Observatory contract.

References Blanc, A., Idier, J., and Mugnier, L. M., 2000, in J. B. Breckinridge and P. Jakobsen (eds.), UV, Optical, and IR Space Telescopes and Instruments, Vol. 4013, pp 728–736, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, Bellingham, Washington Carreras, R. A., Restaino, S., and Duneman, D., 1994, in Image Reconstruction and restoration, Vol. 2302, pp 323–328, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE Demoment, G., 1989, IEEE Trans. Acoust. Speech Signal Process. 37(12), 2024 Ekstrom, M. P. and Rhoads, R. L., 1974, J. Comput. Phys. 14, 319 Gates, E. L., Restaino, S. R., Carreras, R. A., Dymale, R. C., and Loos, G. C., 1994, in Image Reconstruction and Restoration, Vol. 2302, pp 330–339, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE Gonsalves, R. A., 1982, Optical Engineering 21(5), 829 Hartung, M., Bizenberger, P., Boehm, A., Laun, W., Lenzen, R., and Wagner, K., 2000, in M. Iye and A. F. Moorwood (eds.), Optical and IR Telescope Instrumentation and Detectors, Vol. 4008, pp 830–841, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE Hartung, M., Blanc, A., Fusco, T., Lacombe, F., Mugnier, L. M., Rousset, G., and Lenzen, R., In this issue, Astron. Astrophys. Kendrick, R. L., Acton, D. S., and Duncan, A. L., 1994, Appl. Opt. 33(27), 6533 Lee, D., Welsh, B., and Roggemann, M., 1997, Opt. Lett. 22(13), 952 Lee, D. J., Roggemann, M. C., and Welsh, B. M., 1999, J. Opt. Soc. Am. A 16(5), 1005 Lenzen, R., Hofmann, R., Bizenberger, P., and Tusche, A., 1998, in Infrared Astronomical Instrumentation, Vol. 3354, pp 606–614, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE L¨ ofdahl, M. G. and Scharmer, G. B., 1994, Astron. Astrophys. 107, 243 L¨ ofdahl, M. G., Scharmer, G. B., and Wei, W., 2000, Appl. Opt. 39(1), 94 Little, R. J. A. and Rubin, D. B., 1983, The American Statistician 37(3), 218 Meynadier, L., Michau, V., Velluet, M.-T., Conan, J.-M., Mugnier, L. M., and Rousset, G., 1999, Appl. Opt. 38(23), 4967 Noll, R. J., 1976, J. Opt. Soc. Am. 66(3), 207 Paxman, R. G., Schulz, T. J., and Fienup, J. R., 1992, Journal of the Optical Society of America A 9(7), 1072 Restaino, S., 1992, Appl. Opt. 31(35), 7442 Rousset, G., Lacombe, F., Puget, P., Gendron, E., Arsenault, R. Kern, P., Rabaud, D., Madec, P.-Y., Hubin, N., Zins,

G., Stadler, E., Charton, J., Gigan, P., and Feautrier, P., 2000, in P. L. Wizinowich (ed.), Adaptive Optical Systems Technology, Vol. 4007, pp 72–81, Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, Bellingham, Washington Rousset, G., Lacombe, F., Puget, P., Hubin, N., Gendron, E., Conan, J.-M., Kern, P., Madec, P.-Y., Rabaud, D., Mouillet, D., Lagrange, A.-M., and Rigaut, F., 1998, in D. Bonaccini and R. K. Tyson (eds.), Astronomical Telescopes & Instrumentation, Vol. 3353, Proc. Soc. Photo-Opt. Instrum. Eng., Kona, Hawaii Seldin, J. and Paxman, R., 2000, in Imaging Technology and Telescopes, Vol. 4091-07, SPIE Seldin, J. H. and Paxman, R. G., 1994, in Image Reconstruction and Restoration, Vol. 2302, SPIE Thelen, B. J., Paxman, R. G., Carrara, D. A., and Seldin, J. H., 1999, J. Opt. Soc. Am. A 16(5), 1016

Astronomy & Astrophysics manuscript no. aa2914 (DOI: will be inserted by hand later)

Calibration of NAOS and CONICA static aberrations Experimental results M. Hartung1 , A. Blanc2,3 , T. Fusco2 , F. Lacombe4 , L. M. Mugnier2 , G. Rousset2 , and R. Lenzen1 1

2

3 4

Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, 69117 Heidelberg, Germany e-mail: [email protected] Office National d’Etudes et de Recherches A´erospatiales, D´epartement d’Optique Th´eorique et Appliqu´ee, BP 72, 92322 Chˆatillon Cedex, France ´ ´ Laboratoire des Signaux et Syst`emes, Ecole Sup´erieure d’Electricit´ e, Plateau de Moulon, 91192 Gif-sur-Yvette, France Observatoire Paris-Meudon, LESIA, Place Jules Janssen, 92195 Meudon Cedex, France

Received 16 July 2002 / Accepted 25 October 2002 Abstract. NAOS is the first adaptive optics (AO) system of the Very Large Telescope and will provide CONICA with

diffraction-limited images. CONICA is a near infrared camera that offers a variety of imaging and spectroscopic observing modes. A technique will be described to benefit of the AO system to correct not only for atmospheric turbulence but also for the internal optical aberrations of the high-resolution camera and the beam splitters of NAOS. The aberrant optical components in the light path of CONICA as well as the beam splitters are outside of the AO loop and therefore no self-acting correction is possible. Independently of the AO wavefront sensor, a separate measurement of these aberrations using a method called phase diversity allows one to predict for a certain instrument configuration the corresponding aberrations. They are quantified by sets of Zernike coefficients that are rendered to the adaptive optics. This technique turns out to be very flexible and results in a further improvement of the optical overall performance. The application of phase diversity to the instrument is investigated in a preceding paper (Blanc et al. 2002). In the present paper we present in detail the instrumental implementation of phase diversity, the obtained calibration results, and the achieved gain in optical performance. Key words. instrumentation: adaptive optics

1. Introduction NAOS-CONICA is the first adaptive optics (AO) system of the Very Large Telescope (VLT) and saw its first light in November 2001 (Brandner et al. 2002). The Nasmyth Adaptive Optics System NAOS (Rousset et al. 1998; Rousset et al. 2000) delivers diffraction-limited images to the Coud´e1 Near Infrared CAmera CONICA (Lenzen et al. 1998; Hartung et al. 2000). To retrieve the maximum possible performance of the system in terms of Strehl ratio2 (SR) a method has been developed to calibrate the remaining degradation of the image quality induced by its optical components. Defaults of the wavefront attributed to any degradation within the AO loop (common path) are seen directly by the AO wavefront sensor (WFS) and thus the AO system can correct for these aberrations automatically. Send offprint requests to: T. Fusco, e-mail: [email protected] 1 The notation Coud´e has historical origins as CONICA originally was planned for the Coud´e focus. 2 The SR ratio is a common way to describe the quality of the point spread function. It is given by the ratio of the measured and the theoretical diffraction-limited peak intensity.

This is not the case for a degradation of image quality induced by components outside the AO loop. An experimental setup has been applied which allows one to sense the wavefront of the light which has passed the whole system without making use of the AO wavefront sensor. Therefore we draw on a well-known method called phase diversity (Gonsalves 1982; Paxman et al. 1992). It turns out that a number of theoretical and experimental constraints have to be examined before reliable results can be obtained in sensing the wavefront via phase diversity (PD). We focused on this in a precedent paper (Blanc et al. 2002), hereafter Paper I. In this second paper we first give a brief description of the instrument (Sect. 2). Then we focus on the experimental setup which enables us to calibrate the variety of beam splitters, filters and camera objectives. The design constraints for the implementation of PD are illustrated, and the resulting setup as well as the procedure to obtain the appropriate input data for PD are described (Sect. 3). Because of the huge number of instrument modes it is not feasible to perform the PD calibration for each possible configuration. We explain how the wavefront degradations of the different optical components are disentangled. Then, the individual parts of the optical train can be calibrated separately and it

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M. Hartung et al.: Calibration of NAOS and CONICA static aberrations

is no longer required to do this for every possible combination. In detail, we will allocate the wavefront error to the dichroic mirrors of NAOS (beam splitter between wavefront sensor and imaging path), to the CONICA filters and camera objectives (Sect. 4). Thereafter, the sensed wavefront errors are used to calculate the corresponding SRs. These are compared to the SRs directly determined from the images and the consistency is verified (Sect. 5). Finally, after presentation of the complete calibration procedure and its results, the measured wavefront errors are rendered in terms of Zernike coefficients to the AO system to demonstrate the gain in overall performance after closed loop correction (Sect. 6).

Detector plane

CONICA

Analysing Optics Entrance focal plane ADC Dichroic

DM TTM

2. The instrument This section is dedicated to a brief description of the instrument NAOS-CONICA. In particular, we emphasize the subdevices as much as the aspects of design which are important with respect to the static wavefront error estimation by phase diversity. Figure 1 gives an overview of the VLT instrument. NAOS is installed at one of the Nasmyth foci. It picks up an f/15 beam, corrects for atmospheric turbulence and hands on again an f/15 beam providing CONICA with diffraction-limited images. Having passed the Nasmyth focal plane, the beam is led to a first collimating parabola. Then it is reflected successively onto the tip-tilt (TTM) and the deformable mirror (DM). The following dichroic mirror separates the optical train into the imaging path and the wavefront sensing path. NAOS offers five different dichroic beam splitters to adapt for the flux and the spectral characteristics of the guide star. In the imaging path the light is refocused onto the entrance focal plane of CONICA, which is located behind the entrance window in the cold cryostat. Between NAOS and CONICA an atmospheric dispersion compensator (ADC) can be slid in in case of a high zenith angle. The wavefront sensing path consists of a field selector (Spanoudakis et al. 2000) and two wavefront sensors. They are located between the dichroic mirror and the WFS input focus. For the sake of clarity these components are not shown in Fig. 1. The two wavefront sensors, one in the visible and one in the near infrared spectral range, enhance the sky coverage of possible guide stars. The field selector chooses the guide star in a 2 arcmin field of view and allows differential object tracking, pre-calibrated flexure compensation and counter-chopping. In combination with the deformable mirror it is also able to correct for a certain amount of defocus, as needed when the prisms of the atmospheric dispersion compensator are shifted into the beam. Note that this ability of focus correction offers a possibility to perform PD measurements that we will refer to later on (Sect. 3.2). The high angular resolution camera CONICA is equipped with an Aladdin array (1K × 1K) covering the 1–5 µm spectral region. Splitting the wavelength region into two parts (1 to 2.5 µm and 2.0 to 5 µm) allows us to keep the light path achromatic. Therefore the four different pixel scales are realized by seven cameras (Table 1). To each pixel scale a camera is associated with the short wavelengths region (S-camera) and another

WFS Input Focus

NAOS

VLT Nasmyth Focus

Fig. 1. Outline of the VLT instrument NAOS-CONICA. Table 1. The required defocus distances for a phase diversity of 2π rad (peak to valley) (λ = 1 µm) are listed corresponding to the f-ratios (pixel scales). Camera C50S C25S/C25L C12S/C12L C06S/C06L Entrance focal plane

f /D 51 25.5 12.8 6.38 15

Pixel scale 13.3 mas/pixel 27.1 mas/pixel 54.6 mas/pixel 109 mas/pixel 1.72 mas/µm

d (mm) 20.8 5.2 1.3 0.33 1.8

one with the long wavelengths region (L-camera). The only exception is the camera with the highest magnification (C50S). There is no long wavelength counterpart needed3. A variety of different observing modes is provided by the analyzing optics: chronography, low resolution long slit spectroscopy, imaging spectroscopy by a tunable cold Fabry-Perot, polarimetry by wire-grids or Wollaston prisms, and about 40 broad- and narrow-band filters can be chosen.

3. Phase diversity setup Our input for phase diversity wavefront estimation are two images: one of them in focus and the other one out of focus. In this manner we introduce the well-known phase diversity which is an obligatory input parameter for PD. One should recall that best phase diversity estimates are to be expected applying a peak to valley phase diversity ∆φ between 1π and 3π and the input images must be at least Nyquist sampled4 (Paper I). The 3

The Nyquist criterion for L and M-band is already fulfilled for a camera with lower magnification (C25L). 4 Indeed, it is possible to model the wavefront error even at undersampled images (Jefferies et al. 2002). In our case there is no cost to be constrained by the Nyquist criterion because we can rather select

M. Hartung et al.: Calibration of NAOS and CONICA static aberrations Collimator Pupil stop Filter

Camera objectiv

Zernike tool

Detector

Fig. 2. The Zernike tool with its pinholes in the light path of CONICA.

corresponding defocus distance d depending on the applied wavelength is obtained by d=

4λ ( f /D)2 ∆φ. π

(1)

In the next two subsections we describe in detail two ways of introducing this phase diversity. Both ways are essential to enable us to separate the wavefront error and to assign it to different contributors. This disentanglement is described in Sect. 4.2.

3.1. CONICA stand-alone: focus shift by object First, we regard the possibilities to obtain the necessary input images with CONICA stand-alone. The CONICA detector is mounted on a tunable stage which is software controlled and can be driven in the cold environment. This allows us in principal to obtain a defocused image but the focus drive spans only a region of 2 mm. Using Eq. (1) we compile for all available camera objectives the necessary defocus distances in the detector plane corresponding to a diversity of 2π at a wavelength of 1 µm in Table 1. Only for the low magnification cameras (C06, C12) is the defocus distance sufficient. But these very cameras undersample in K and at shorter wavelengths so that the focus stage mechanism finally fails in every case. For that reason we swerve to the entrance focal plane. Here a phase diversity of 2 π corresponds to 1.8 mm at a wavelength of 1 µm or 3.6 mm at 2 µm, which is small enough to be implemented in the entrance focal plane. In this plane a wheel is located carrying different field limiting masks, coronographic masks and the slits for spectroscopy. On the wheel we implement four different pinholes at different axial positions. The pinhole diameter is 10 µm. One pinhole is placed exactly in the entrance focal plane and yields a focused image on the detector (0 mm), and three other pinholes are located 1 mm, 2 mm and 4 mm out of the entrance focal plane. The four pinholes are mounted onto a plate fitting in a socket of the mask wheel. This device will be referred to as a Zernike tool later on. Impacts on the PD estimation due to the mechanical precision of the pinhole positions and possible deviations of their shape are investigated in Paper I. Figure 2 depicts the setup for the CONICA internal phase diversity measurements. The leftmost component carries the four pinholes, which are shifted against each other, with the values given above. In rotating the wheel holding the Zernike tool we are able to select a pinhole in the field of view. After a collimating lens and a pupil stop, a filter selects the wavelength range and finally the camera objective forms the object image the appropriate filter or else we deal with a camera used in the L- and M-band where static wavefront aberrations are negligible.

3

on the detector. The chosen camera objective determines the f-ratio and the pixel scale. To center the image of the pinholes on the detector, the whole pinhole mount is shifted by turning the mask wheel. In principal PD needs the input images to be on the same spot to ensure that the same aberrations are sensed. The horizontal position of the pinholes can be controlled by adjusting the rotation angle of the wheel. In vertical direction there is no degree of freedom, but the four pinholes are mounted circularly to compensate for the circular movement. By this means a vertical precision of 50 mas (C50S) can be reached5. This is easily sufficient not to see any influence due to field aberration effects. PD measurements taken at different detector positions and calculations performed with an optical design software showed that even at the corner of the field of view (13 arcsec) the field aberration is negligible (Paper I). Note that for some measurements in Paper I an earlier version of the Zernike tool was used with a design not optimized for the circular movement of the pinholes. The worst separation that could occur with the former Zernike tool was about 1.3 arcsec. But even with this tool no relevant impact on the precision of wavefront sensing was detected. Apart from the fact that the Zernike tool with its pinholes at the entrance focal plane provides the required focus shifts, it is convenient that the required focus shifts do not depend on the camera objective (pixel scale) anymore. But note: defocusing by moving an object in the entrance focal plane does not correspond exactly to a defocus due to a shifted detector plane. An investigation of this effect is done in Paper I and turns out to be negligible. To summarise this section: the PD input data to derive the total CONICA internal aberrations are obtained by object defocusing in the CONICA entrance focal plane. The object defocusing is realized by four 10 µm pinholes at different axial positions. Note that since the entrance focal plane of CONICA is located inside the cold cryostat, aberrations accrued from the CONICA entrance window are not included in this wavefront estimation.

3.2. NAOS-CONICA: Focus shift by the deformable mirror Now, we describe how the PD input images are obtained which are used to sense the wavefront aberrations of the whole instrument, i.e., the adaptive optics NAOS together with its infrared camera CONICA. In this case we can take advantage of the AO system’s capabilities to itself introduce an adequate focus shift and thus there is no need for the implementation of a special tool or a modifaction of the design. In the entrance focal plane of NAOS, which coincides with the VLT Nasmyth focal plane, a calibration point source can be slid in and imaged by CONICA. This point source is realized by the output of a fiber with a diameter of 10 µm fixed on a movable stage. On the same stage a second source much larger in diameter (400 µm) is mounted. It is only seen by the WFS and serves as a reference source to close the loop. This 5

1 pixel corresponds to 13.3 mas (C50S).

4

M. Hartung et al.: Calibration of NAOS and CONICA static aberrations

Light coming from NAOS entrance focal plane

1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 DM 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111

Dichroic

CONICA

Field Selector

11111 00000 00000 11111 00000 11111 00000 11111

defocus

Real Time Computer

11111 00000 00000 11111 00000 11111 00000 11111

0110 1010 10

Focused and defocused images

WFS

Fig. 3. Defocusing in closed loop using the NAOS Field Selector

extended source is needed for technical reasons. In the case of no atmospheric turbulence the more extended source provides a much better feed-back signal to the WFS than the small one. By this means the AO control loop is adjusted for any aberrations emerging in the common path. To obtain the focus shift affecting the entire instrument, we introduce the desired amount of defocusing in the WFS path by moving the mirrors of the field selector. During this process the loop is kept closed. Instantly, the arising focus shift is detected by the WFS. Correspondingly, the real time computer commands the DM to compensate for the detected defocus. Finally, the spots on the Shack-Hartmann WFS are centered again, but the defocus of the DM takes effect in the imaging path. For a pure defocus the DM will take a parabolic shape. The maximum achievable defocus by this method is limited by the DM’s stroke and turns out to be about 20 mm. Refering to Table 1, this is enough to introduce the needed diversity for an f/15 beam. The procedure is shown in Fig. 3 and provides us with the PD input data to estimate the NAOS-CONICA overall wavefront errors. In comparison with the procedure described in Sect. 3.1 we deal with the same object now, and we must not care about any deviations in the position of the image pairs. This simplifies data aquisition for the measurement and diminishes the number of possible error sources.

4. Calibration of NAOS and CONICA static aberrations

4.1. Introduction An appropriate way to describe the shape of a wavefront in the telecope pupil is by using Zernike polynomials (Noll 1976). A set of Zernike coefficients indicates the linear combination describing the present wavefront. As a matter of course, we can regard this set of coefficients as a vector. We refer to the Noll notation (Noll 1976) which labels the focus with 4, the tangential and sagittal astigmatism with 5 and 6, coma with 7 and 8, and so on. Since the coefficients for piston (1), and tiptilt (2, 3) are extraneous to the image quality they are dropped. An extensive examination of the variety of error sources due to the practical and instrumental constraints was done in

the preceding Paper I. The induced aberrations due to defocusing by a shifted object in the CONICA stand-alone case have been simulated and proven to be negligible. The influence of the pupil shape and its numerization have been evaluated, errors taken in account with regard to the camera pixel scale and the defocus distance deviations have been simulated and the problem of different object structure was considered. Furthermore we focused in detail on the handling of data reduction, e.g. the influence of the different noise sources such as readout noise or pickup noise. In Paper I we state that all these error sources accumulate to ±35 nm rms for the focus coefficient (4). Since the presented calibration data of this paper are acquired with an optimized Zernike tool, the expected error should be well below this number. The accuracy of the higher order coefficients has not changed and amounts to about ±5 nm rms. In this section we describe how the overall wavefront error can be decomposed and assigned to its corresponding optical components. Then we present the experimental results for one camera objective and some selected filters of CONICA as well as the results for the dichroics of NAOS.

4.2. Disentanglement In the preceding sections we described how the static noncommon-path wavefront error can be measured, whether for CONICA stand-alone or for the entire instrument NAOSCONICA. However, each determination of the wavefront error is only valid for the particular instrument configuration in which it was measured. The tremendous number of instrument configurations6 makes it impractical to perform these calibrations for any possible instrument setup. For a practical application we need to split up the measured wavefront aberrations and assign the corresponding contributions to the divers optical components. This allows the construction of a configuration table7 with entries for each optical component of the instrument. When a special instrument configuration is selected, the corresponding wavefront error contributions can be read out, added together and delivered to the AO system. This enables the DM to pre-correct for the current static wavefront aberrations. In principal, we have to differentiate between three categories of optical components in the imaging path: the NAOS dichroics, the CONICA filters and the camera objectives. can be deterThe contribution of the NAOS dichroics adichro i mined by subtracting the overall NAOS-CONICA instrument from the total CONICA instrument aberraaberrations aNCtot i : tions aCtot i = aNCtot − aCtot . adichro i i i

(2)

The vector components are labeled by the Zernike number i running in our case from 4 to 15. Regarding the PD estimations of the different CONICA filters for one camera objective 6

Given the combinations of 5 NAOS dichroics, 40 narrow and broad band filters and 7 camera objectives! 7 In fact, a configuration file is generated. The instrument control software takes care of what coefficients have to be applied for the selected instrument setup. These processes are hidden and completely automatic.

M. Hartung et al.: Calibration of NAOS and CONICA static aberrations

Fig. 4. CONICA internal aberrations measured by 8 NB filters in Jand H-band with camera objective C50S and pinhole pair 0/2 mm. The thick line indicates the median representing the camera aberrations.

(see Fig. 4 and Sect. 4.3) we ascertain that generally the filter aberrations afil i are small and mainly the achromatic camera objective aberrations are seen. In any case the filter aberrations are not correlated with each other nor with the camera ones. This suggests that we can deduce the camera objective contribution by applying the median to the total CONICA internal . We prefer the median instead of the mean aberrations aCtot,fili i to avoid taking into account highly aberrant filters. The filter i which was used to determine the corresponding total CONICA is indicated by fili. The camera objecinternal aberration aCtot i tive and the residual filter contributions are obtained by these relations: = median(aCtot,fil1 , aCtot,fil2 , ..., aCtot,filn ) acam i i i i

(3)

Ctot − acam afil i = ai i .

(4)

The separation of the wavefront aberration into the contributions associated with the three categories of optical components (NAOS dichroics, CONICA filters and camera objectives) is only possible when we make use of both ways to introduce a focus shift, i.e. the DM to determine the NAOS-CONICA overall aberrations and the Zernike tool to determine the total CONICA aberrations. In addition, we note that even if we refer to these three categories by the notation dichroics, filters and objectives, the other components in the optical path are included, as well, even when they are not mentioned explicitly. E.g., the aberrations of the CONICA entrance window are included in the dichro aberrations and the aberrations of the CONICA collimator are an inextricable part of the camera aberrations.

4.3. Calibration of CONICA: Camera and filters Figure 4 shows the aberrations for all eight narrow band filters in J- and H-band of CONICA. The camera objective C50S and the pinhole pair (0/2 mm) is used to obtain the calibration data. expressing the defocus shows a The fourth coefficient aCtot,fili 4

5

Fig. 5. CONICA internal aberrations measured by 19 NB filters in K-band with camera objective C50S and pinhole pair 0/4 mm. The thick line indicates the median representing the camera aberrations. The dashed line highlights the aberrant filter NB2.09 which is picked out for the demonstration images in Fig. 6.

peak-to-peak variation of up to 60 nm. This implies a slight imprecision of coplanarity of the filters in the cold environment. The other measured coefficients associated with the different filters noticeably resemble each other. This is evidence that these narrow-band filters contribute little to the total aberration of the system and mainly the camera objective aberration is seen. Figure 5 displays the calibration results in the K-band. In total, 19 filters have been calibrated using the pinhole pair 0/4 mm. One of the strongly aberrant filters (NB2.09) is highlighted by a dashed line. A large defocus in comparison to the other ones is detected. This filter is expected to have a striking error of coplanarity. It is not surprising that the strong defocus comes along with a particularly high spherical aberration (i = 11). The other highly aberrant filters show the same behaviour in comparison with the common filters of minor aberrations. The spherical aberration expresses the next order of a radial symmetric Zernike mode. The probability that a strong default of coplanarity induces only a defocus and does not concern higher orders is small. The PD input images of this aberrant filter is depicted at the bottom of Fig. 6. The right image shows the PSF registered in focus, and the left image a PSF having introduced a defocus of 4 mm. Already the in-focus image reveals a strong degradation, but especially the phase inversion due to the high defocus can be clearly seen in the out of focus image. A bright spot emerges in the center of the “donut”. On the top of this couple of images another couple of images is depicted. These are the PD input data of a filter (NB2.06) with normal behaviour and without strong aberrations. As described in Sect. 4.2 the median of each Zernike number of the whole set of vectors yield the vector describing the camera contribution. The accuracy of separating the camera aberrations from the raw aberrations (filters including camera) by the method described above is striking. The median aberrations for the filters of the two different wavelength regions plotted in Figs. 4 and 5

6

M. Hartung et al.: Calibration of NAOS and CONICA static aberrations Filter NB2.06

In the following the properties of the five NAOS dichroics are itemized: – VIS: Visible light to WFS; J, H, K, L and M to CONICA; – N20C80: 20% of the incoming light to WFS; 80% to CONICA (J, H, K); – N90C80: 90% of the incoming light to WFS; 10% to CONICA (J, H, K); – K: K to NAOS; J and H to CONICA; – JHK: J, H an K to NAOS; L and M to CONICA.

Filter NB2.09

Four of these five dichroics have been calibrated. The dichroic JHK is omitted since only light in L and M band reaches CONICA. It is unreliable to sense the small wavefront errors of NAOS-CONICA at these wavelengths. Furthermore there is no need to, because the small static aberrations become completely negligible in L and M.

Fig. 6. Comparison of PD input images of a filter with small aberrations (NB2.06, on the top) and a filter with high aberrations (NB2.09 at the bottom). The in-focus images are placed on the left side, the out of focus images on the right side. The defocus distance is 4 mm for both filters (f/15).

are compiled in Table 2. The deviations of both median values are clearly below the expected error (see Sect. 4.1). Table 2 lists these median coefficients taken from all NB filters in J-, H- and K-band. Keeping in mind that the achievable precision is a few nm we state that the camera aberrations are very small. The highest contributions arise from the focus term (4) and the astigmatism (5, 6). Section 5.3 gives an idea of the impact on the image quality dealing with Zernike mode aberrations in this order of magnitude. The residual filter aberrations are obtained by Eq. (4). In general, besides the focus coefficient and a few deviating filters these values are close to zero, too.

4.4. Calibration of NAOS: Dichroics The calibration data are obtained with the fiber at the entrance focal plane of NAOS using the adaptive optic system itself for defocusing (see Sect. 3.2). Since the Zernike coefficients for the NAOS dichros are determined differentially, i.e. by subtraction of the total CONICA aberrations from the NAOS-CONICA overall aberrations, we can choose any reference camera and filter to perform the measurements as long as the components stay the same. A good choice is camera objective C50S and filter FeII1257. This objective oversamples even in the J-band and the filter has a small wavelength and therefore yields a higher accuracy in sensing wavefront errors. A suitable distance for the focus shift at this filter wavelength in the f/15beam is 2 mm. We can calculate suitable defocus distances using Eq. (1).

The calibration results are compiled in Table 3 and Table 4. The first table lists the direct PD results. Any correction performed by these coefficients would only apply to the instrument configuration that was used to obtain the calibration data. The second table lists the aberrations directly assigned to the dichroics. These were obtained by subtracting the total CONICA aberrations that have been measured with the same filter and camera objective using the Zernike tool. It is noteworthy that the sensed astigmatism (Zernike number 5, 6) in the separated case is higher than in the overall case. Obviously a part of the camera astigmatism is compensated by the dichroics. It is noteworthy that this tendency applies for all dichroics. Different reasons can cause this behaviour. First, the inclination of the dichroics artificially introduce an astigmatism. Even if the NAOS dichroics are designed for prism shape and do correct for this effect, a residual error cannot be excluded. Furthermore a certain amount of astigmatism can be introduced by components other than the dichroics lying in the same part of the light path, e.g. the parabolic output folding mirror or the CONICA entrance window (see Fig. 1). Nevertheless, it is not a limitation of the calibration method but only a question of assigning the contribution of the wavefront errors to the different optical components. In the end, only the sum of all aberrations has to be correct.

5. Image quality versus estimated aberrations

5.1. Strehl ratios by PD and focal plane image The PD calibration data can be used to investigate the available image quality in different ways. First, the knowledge of the wavefront allows us to calculate a SR. After the reduction of the calibration data the wavefront is described by a set of Zernike coefficients. Furthermore, we can just refer to the infocus image and calculate a SR with the measured point spread function. In the following we give a more detailed explanation of how these SRs are obtained.

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Table 2. Camera aberrations in nm RMS by the median over the filter + camera aberrations in the bands J, H (pinholes 0/2 mm), K (pinholes 0/4 mm) and all bands (J, H, K). Bands for median J, H K J, H, K

4 −12 −15 −15

5 −34 −39 −39

6 30 27 27

7 −6 −10 −9

8 4 7 6

9 2 3 2

10 13 15 13

11 −8 −9 −9

12 2 2 2

13 0 0 0

14 9 6 7

15 −1 −1 −1

Table 3. NAOS dichros, overall NAOS-CONICA aberrations in nm rms, reference filter: FeII1257. Dichro VIS N20C80 N90C10 K

4 15 2 −7 −8

5 −5 −1 −3 14

6 24 42 36 −17

7 −6 −2 −3 −4

8 23 30 19 18

9 5 5 6 3

10 −8 −4 −5 −5

11 −9 14 −28 −6

12 7 −1 1 7

13 −13 −19 −9 −14

14 −7 −8 −9 −10

15 3 4 1 2

Table 4. Separate NAOS dichro aberrations in nm rms. Dichro VIS N20C80 N90C10 K

4 −18 −32 −41 −42

5 37 41 38 56

6 −5 13 7 −47

7 3 8 7 5

8 16 23 12 11

Strehl by PD For small wavefront deviations the SR can be determined via the coherent energy referring to the wavefront variance σ2 in radian. The PD estimation yield the wavefront expanded in terms of Zernike coefficients ai . For small σ2 SR ' e−σ ' 1 − σ2 ' 1 − 2

m X

a2i

(5)

i=4

allows us to calculate the SR directly by the output of PD estimation. In principal the sum runs to infinity (m = ∞) but for our purpuse we stop at m = 15. We can compare these SR numbers to the ones that are directly determined by the in-focus images.

Strehl on image A straight-forward way to calculate a SR on the focus image (PSF) is to construct a theoretical diffraction-limited image PSFdiff taking into account the wavelength, the pixel scale, the aperture and the central obscuration8. Having normalized the total intensity of the PSF and PSFdiff to 1, the fraction of these values yields the SR (see Eq. (6)). In particular, in the case of the PSF sampling being close to the Nyquist criterion this approach has the disadvantage of being sensitive to the exact position of the PSF peak with respect to the pixel center. Furthermore, since the total intensity has to be determined by the integrated signal over a wider region around the PSF, the reliability of the SR value depends on a precise background correction. If the background is overestimated, then the SR will be overestimated, too, and vice versa. The reli8 The central obscuration which is caused by the secondary mirror will decrease slightly the central peak intensity and raise the side lobes of the Airy function.

9 2 1 3 −1

10 −21 −17 −18 −18

11 −44 −21 −64 −42

12 6 −3 −1 5

13 −13 −18 −9 −14

14 −14 −15 −16 -17

15 6 6 4 5

ability of the SR values can be enhanced when we switch from the image space to the Fourier space by R OTF( f ) PSF(α = 0) =R , (6) SR = PSFtheo (α = 0) OTFtheo ( f ) where OTF is the optical transfer function. Since in Fourier space only spatial frequencies are considered, a shift of the PSF is of no importance anymore. Aside from that, an elegant and reliable background correction can be performed using the zero spatial frequency. We calculate the SRs by the following procedure9 : – The image is corrected by its corresponding background; – The OTF is calculated. It is given by the real part of the Fourier transform of the image; – The residual background is corrected by the zero frequency. A fit of the very first spatial frequencies is used to extrapolate the true zero frequency value. The difference of the measured and the extrapolated value for the zero frequency yields the residual background; – The noise level is subtracted using the high frequencies beyond the diffraction limit; – A theoretical telescope OTF is constructed and multiplied by a Bessel function to account for the spatial spread due to the object size; – The detector response is taken into account. This is done by a further multiplication of the theoretical telescope OTF with the Fourier transform of the detector response; – The SR is obtained by the division of the normalized integrals from the measured and the theoretical OTF. All points with spatial frequencies higher than the diffraction limit are excluded. 9

This routine was developed by T. Fusco, ONERA.

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M. Hartung et al.: Calibration of NAOS and CONICA static aberrations

Fig. 7. Visualization of the theoretical and measured OTF at the example of filter FeII1644.

Figure 7 gives an example of how the described OTFs look for a PSF taken through the filter H2(1 0)S(7). To allow for a one dimensional representation, the circular mean of the two dimensional OTF is calculated. The raw, untreated OTF of the image is labeled “OTF image”. “OTF corr” displays the image OTF which is corrected for the residual background at zero frequency and for the noise level. The noise level of the uncorrected image OTF can be seen as plateau beyond the cutoff frequency D/λ and averages 1% of the maximum value. The theoretical diffraction-limited telescope OTF includes the reduction due to the object size10 and the correction for the detector modulation transfer function (MTF). It is labeled “MTF Det” and is located at the top of the plot. The detector response is constructed by the assumption that roughly 5% of the total intensity is contained at each of the adjacent pixels and 2.5% in the corner pixels. Here we refer to Finger et al. (2000). In this paper a measurement of the response of a comparable infrared array is described. Due to the lack of precise knowledge of the detector response it is constructed by a linear scaling in relation to the different pixel size11 . A linear scaling is implied by a linear behaviour of the diffusion length of the minority carriers in the detector material.

5.2. Comparison of Strehl ratios The resulting SRs for the narrow-band filters in J, H, and K are presented in Figs. 8 and 9. For each filter two SRs are given: the SR by PD and the SR on the image. A number of error sources contribute to the error of the SR values on image. Beside of small error contributions due to uncertainties of the pixel scale and the flatfield, the remaining uncertainty of the background correction and the detector response lead us to estimate an absolute error of ±4%. The exSince a 10 µm pinhole is used which is barely resolved, the influence is small and amounts for the smallest wavelength (1 µm) at most to 5%. 11 The pixel size of the CONICA detector (Aladdin 1K × 1K) is 27.0 µm. 10

Fig. 8. Comparison of SR versus wavelength in J and H band calculated directly and derived from PD results.

pected wavelength dependency of the MTF error is minor with respect to the remaining background error. Therefore it is neglected and we use the constant value given above derived from experience in reducing the experimental data. Recall that the SR by PD has a maximal wavelengthdependent error of ±5% at 1 µm and ±1% at 2 µm taking into account an error of ±35 nm RMS for the focus estimation (i = 4). The main contributor to this error is a systematic error in the precision of the pinhole positions in the Zernike tool (see Sect. 4.1). In general the PD SRs exceed the other SR values. This reflects the fact that the wavefront is expanded by a limited number of Zernike coefficients and the higher order aberrations are cut off. Note that it is not astonishing that in the case of very low SR values (worse than 50%) the PD SR value may lie below the image SRs (Fig. 9). Such strong wavefront errors violate the condition under which Eq. (5) is valid. Thus, we expect Eq. (5) to yield underestimated values. The comparison of the SR values determined by the different methods turns out to be consistent. The longer the wavelength, the more the image and PD SR values approach each other. This shows that the influence of aberrations scale with the wavelength. In other words, the fact that we cut off at a certain Zernike number (i = 15) has a greater impact at short wavelengths.

5.3. Focus adjustment Having in mind the small estimated wavefront errors that we presented in the previous sections we become conscious of the required precision of the most trivial aberration we regard: the focus. It is striking that even in the focus determination we depend on the precision of PD calibration. This becomes evident when we look at the conventional procedure of focus tuning and regard the loss of SR caused by the detected aberrations. To tune the focus of CONICA, the in-focus pinhole of the Zernike tool is imaged on the detector. Now, a focus curve is obtained by taking images at different axial position of the de-

M. Hartung et al.: Calibration of NAOS and CONICA static aberrations

9

to the AO system. The DM corrects for the residual focus deviations.

6. Closed loop compensation of NAOS-CONICA static aberrations

6.1. Rendering of aberrations

Fig. 9. Comparison of SR versus wavelength in K band calculated directly and derived from PD results.

tector stage (see Sect. 3.1). The maximum of the SRs indicate the proper focus position of the stage. The maximum of the obtained focus curves for the different cameras can be located with an accuracy of about 50 nm rms. For this wavefront error, Eq. (5) yields a loss of SR of 2.5% at a wavelength of 2 µm and almost 10% at 1 µm. Thus, in particular in the J- and Hband, the inaccuracy of determining the focus only by moving the detector stage gives reason for a significant loss of SR. Furthermore the whole effort of fine-tuning for the remaining static wavefront aberrations becomes irrelevant when the remaining focus error is in the regime of the highest higher-level aberrations (Zernike number i ≥ 5). Compare the aberrations for focus with astigmatism in Figs. 4 and 5. The only way to achieve a significant improvement of the wavefront error, and therefore of the SR after closed loop compensation, is to ensure that the residual focus deviation is corrected properly, too. This is guaranteed by following the procedure: – Determination of the rough nominal focus position of the CONICA detector for each camera with one reference filter. The in-focus pinhole of the Zernike tool serves as a reference; – Determination of the nominal focus for the whole instrument. The calibration point source in the NAOS entrance focal plane serves as a reference. The data points for the focus curve are obtained by moving the field selector in closed loop. This has to be done for every NAOS dichroic; – The corresponding data base entries are updated by the nominal focus positions (CONICA internal and NAOS). The nominal focus deviations are included in the data base. They are compensated for by moving the field selector in the case of switching the NAOS dichroics and by moving the CONICA detector stage in the case of switching the camera objectives; – Then the PD estimation reveals the residual focus error for each configuration, in particular for each filter. They are entered into the data base together with the higher order aberrations. For a certain instrument configuration the corresponding values are fetched automatically and delivered

Having explored in detail the application of phase diversity to calibrate NAOS and CONICA static aberrations in Paper I, we presented above the experimental results applying PD as a wavefront sensor. We described how the contributions of the different optical components in the light path are separated to create a complete calibration configuration table. For each possible configuration of the instrument the corresponding correction coefficients are rendered to NAOS and are used to adjust the AO system. In this manner the DM will take the shape needed for compensation of the static wavefront aberrations. To demonstrate the final gain in optical quality we compare the originally acquired images without correction for static aberrations with the images obtained after closed loop compensation. The gain will be quantified in terms of SR numbers.

6.2. Full AO correction The 10 µm calibration source in the entrance focal plane of NAOS simulates a star without turbulence. The visible WFS is used to correct for the common path aberrations. Therefore, the loop is closed on the 400 µm source as described in Sect. 3.2. The light is separated by the dichroic VIS, thus the WFS sees the visible part and the near-infrared is directed towards CONICA. Figure 10 shows two extreme cases of applying AO compensation. The upper pictures demonstrate the correction for a filter in J-band, the pictures below in K-band. In accordance with Figs. 4 and 5 the sensed aberrations in J and K band are very similar – recall that the main contribution arises from the achromatic camera objective and the NAOS dichroic. But even if similar correction coefficients are rendered to the AO system, the effect on the image is strongly wavelength dependent. This is due to the fact that the influence of the applied Zernike coefficients scales with the wavelength. Thus, we achieve a striking correction in J-band visible with the naked eye on the images before and after correction. The most important aberration, the astigmatism, vanishes and the PSF is contracted. In K-band the non-corrected image is already very close to the optimum and the improvement is hard to see directly on the image. But calculating the SRs shows that even in K-band the performed correction is still significant (Table 5). Note that the given error arises from a maximum estimate of all error sources as described in Sect. 5.2. The nature of the error is mainly systematic (e.g., caused by background correction) and affects the calculated SRs for the image pairs in the same way. SRs determined on experimental data are intrinsically afflicted by rather high error bars, but a direct inspection of the images (central intensity, shape of the diffraction rings) shows the relative gain of 2 to 3% in K-band to be true. Even this rather small appearing gain in K is of high importance. On the way to scientific goals such as

10

M. Hartung et al.: Calibration of NAOS and CONICA static aberrations Filter P gamma

cients have been used to quantify the image quality in terms of SR and be proven to be consistent with the SRs directly determined with the image data. Finally we gave a striking example of the acquired improvement of optical performance in comparing images with and without AO correction for static wavefront errors. It turned out that even starting with a very good image quality, we still could achieve a significant gain in terms of the SR ratio. The design of the instrument control software is harmonized with this calibration procedure and the AO loop parameters are automatically updated when the instrument setup changes. This ensures that the utmost optical performance is provided for all the configurations. Additionally, this implementation of wavefront sensing can be used to monitor the optical quality and to alert for small degradations of optical performance. In general, for future high performance AO systems, the presented technique is of great importance to achieve the challenging science goals of the astronomical community. It should be included in the instrumental design at a very early phase.

Filter Ks

Fig. 10. Comparison of PSFs before and after closed loop compensation. Above a couple of J-band images and at the bottom a couple of K-band images are shown. Left side: without AO correction. Right side: with AO correction. Especially in J-band, the sharpening of the PSF can be clearly seen.

Table 5. Comparison of SRs for two selected filters before and after AO compensation for static aberrations. The maximum mainly systematic error is given. Statistical errors are significantly smaller. Filter Pgamma Ks

SR no corr (%)

SR with corr (%)

60 ± 4 91 ± 4

70 ± 4 93 ± 4

e.g. planet detection, the total error budget must be tackled to eliminate every percentage point of loss in SR.

7. Conclusion In Paper I we presented a guideline for a PD approach to calibrate static wavefront aberrations. An extensive investigation of system limitations and error sources has been carried out. This approach was shown to be very flexible and powerful for precise wavefront sensing using experimental data of the first VLT AO system NAOS-CONICA. In Paper II we have given a detailed description of its implementation at the instrument and presented the experimental results of the calibration data for a variety of observing configurations. Especially, we turned our attention to the disentanglement of the measured overall wavefront errors which allows a convenient allocation to the divers optical components and makes the calibration procedure feasible for an instrument with a huge number of possible configurations. The sensed wavefront errors expanded in Zernike coeffi-

Acknowledgements. We would like to thank Eric Gendron and Wolfgang Brandner for their patience and assistance in the fine tuning of the instrument and interpretation of the measurements during the commissioning runs. Furthermore, we thank Gert Finger for the fruitful discussions about the detector characteristics. We are thankful to Norbert Hubin for the assistance to this work on the part of the European Southern Observatory. This research was partially supported through a European Southern Observatory contract and the Marie Curie Fellowship Association of the European Community.

References Blanc, A., Fusco, T., Hartung, M., Mugnier, L., Rousset, G., A&A, submitted (Paper I) Brandner, W., Rousset, G., Lenzen, R., et. al. 2002, The Messenger, 107, 1 Finger, G., Mehrgan, H., Meyer, M., et al. 1998, in Infrared Astronomical Instrumentation, ed. A. Fowler, SPIE, 3354, 87 Gonsalves, R. A. 1982, in Opt. Eng., 21-5, 829 Hartung, M., Bizenberger, P., Boehm, A., et al. 2000, in Optical and IR Telescope Instrumentation and Detectors, ed. M. Iye, & A. Moorwood, SPIE, 4008, 830 Jefferies, S. M., Lloyd-Hart, M., Keith Hege, E., & Georges, J. 2002, App. Opt., 41, 2095 Lenzen, R., Hofmann, R., Bizenberger, P., Tusche, A. 1998, in Infrared Astronomical Instrumentation, ed. A. Fowler, SPIE, 3354, 606 Noll, R. J. 1976, J. Opt. Soc. Am., 66-3, 207 Paxman, R. G., Schulz, T. J., & Fienup, J. R. 1992, J. Opt. Soc. Am. A, 9(7), 1072 Rousset, G., Lacombe, F., Puget, P., et al. 1998, in Adaptive Optical System Technologies, ed. D. Bonaccini, & R. K. Tyson, SPIE, 3353, 508 Rousset, G., Lacombe, F., Puget, P., et al. 2000, in Adaptive Optical Systems Technology, ed. P. L. Wizinowich, SPIE, 4007, 72 Spanoudakis, P., Zago, L., Ch´etelat, O., Gentsch, R., & Mato Mira, F., in Adaptive Optical Systems Technology, ed. P. L. Wizinowich, SPIE, 4007, 408

204ANNEXE H. CALIBRATION OF NAOS AND CONICA STATIC ABERRATIONS

Annexe I “Calibration and precompasenation of non-common path aberrations for extreme adaptive optics” J.-F. Sauvage et al. - JOSAA - 2007

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Calibration and precompensation of noncommon path aberrations for extreme adaptive optics Jean-François Sauvage,1,3,* Thierry Fusco,1,3 Gérard Rousset,2,3 and Cyril Petit1,3 1

Office National d’Etudes et de Recherches Aérospatiales, Départment d’Optique Théorique et Appliqué, BP 72, F-92322 Châtillon Cedex, France 2 Laboratoire d’Etudes et d’Instrumentation en Astrophysique, Université Paris 7, Observatoire de Paris, 5 place J. Janssen, 92195 Meudon Cedex, France 3 Groupement d’Intérêt Scientifique PHASE (Partenariat Haute Résolution Angulaire Sol Espace) between ONERA, Observatoire de Paris, CNRS, and Université Denis Diderot Paris 7, Paris, France *Corresponding authors: [email protected] Received October 25, 2006; revised February 2, 2007; accepted February 26, 2007; posted March 15, 2007 (Doc. ID 76411); published July 11, 2007 Noncommon path aberrations (NCPAs) are one of the main limitations of an extreme adaptive optics (AO) system. NCPAs prevent extreme AO systems from achieving their ultimate performance. These static aberrations are unseen by the wavefront sensor and therefore are not corrected in closed loop. We present experimental results validating what we believe to be new procedures of measurement and precompensation of the NCPAs on the AO bench at ONERA (Office National d’Etudes et de Recherches Aérospatiales). The measurement procedure is based on refined algorithms of phase diversity. The precompensation procedure makes use of a pseudo-closed-loop scheme to overcome the AO wavefront-sensor-model uncertainties. Strehl ratio obtained in the images reaches 98.7% at 632.8 nm. This result allows us to be confident of achieving the challenging performance required for direct observation of extrasolar planets. © 2007 Optical Society of America OCIS codes: 010.1080, 010.7350, 100.5070, 100.3190.

1. INTRODUCTION Exoplanet direct imaging is one of the leading goals of today’s astronomy. Such a challenge with a ground-based telescope can be tackled only by a very high performance adaptive optics (AO) system, so-called eXtreme AO (XAO), a coronagraph device, and a smart imaging process [1,2]. Most of the large telescopes nowadays are equipped with AO systems able to enhance their imaging performance up to the diffraction limit. One of the limitations of the existing AO system performance remains the unseen noncommon path aberrations (NCPAs). These static optical aberrations are located after the beam splitting, in the wavefront sensor (WFS) path and in the imaging path. The correction of NCPAs is one of the critical issues for achieving ultimate system performance [3] for XAO. These aberrations have to be measured by a dedicated WFS tool, judiciously placed in the imaging camera focal plane, and then directly precompensated in the closed loop process. An efficient way to obtain such a calibration is to use a phase diversity (PD) algorithm [4–6] for the NCPA measurement. For the correction, the wavefront references of the AO loop can be modified to account for these unseen aberrations in the AO compensation and to directly obtain the best possible wavefront quality at the scientific detector [7]. This type of approach has been successfully applied on the NAOS-CONICA [8] and Keck [9] telescopes and has led to a significant gain in global system performance [10]. Even if a real improvement can be seen on precompensated images, a significant amount of aberrations are 1084-7529/07/082334-13/$15.00

still not corrected. In the framework of the SpectroPolarimetric High-contrast Exoplanet REsearch (SPHERE) instrument development [1], we propose an optimized procedure to significantly improve the efficiency of the NCPA calibration and precompensation for high-contrast imaging. The goal is to achieve a residual wavefront error NCPA contribution of less than 10 nm rms after precompensation. The principle of the conventional procedure, as used in NAOS-CONICA [10,11] (hereafter shortened to NACO) is recalled and commented on in Section 2. The newly optimized algorithm for the NCPA measurements is described in Section 3 and is based on a maximum a posteriori approach (MAP) [12,13] for the phase estimation. The new approach for the NCPA precompensation is presented in Section 4. The application of the PD by the deformable mirror (DM) itself is discussed in Section 5. In Section 6 we detail the experimental results obtained with the ONERA AO bench for the validation of the key points of the proposed NCPA calibration and precompensation procedure.

2. PRINCIPLE OF THE NCPA CALIBRATION AND PRECOMPENSATION A. Phase Diversity for NCPA Calibration In order to directly measure the wavefront errors at the level of the scientific detector, a dedicated WFS has to be implemented. The idea is to avoid any additional optics. The WFS must therefore be based on the processing of the focal plane images recorded by the scientific camera itself. © 2007 Optical Society of America

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The PD approach [4–7] is a simple and efficient candidate to perform such a measurement. In this section we are going to briefly describe the PD concept and its interest with respect to our particular problematic. The principle of PD (as shown in Fig. 1) is to use two focal plane images differing by a known aberration (for instance, defocus), in order to estimate the aberrated phase. As shown in Eq. (2.1), the two images recorded on the imaging camera are nothing but the convolution of the object and the point-spread function (PSF) (the PSF being related to the pupil phase ␾) plus photon and detector additive noises: if = 兩FT−1关P exp共j␾兲兴兩2 ⴱ o + n, id = 兩FT−1兵P exp关j共␾ + ␾d兲兴其兩2 ⴱ o + n,

共2.1兲

where if is the conventional image, id is the PD image, j = 冑−1, P is the pupil function, ␾ is the unknown phase, ␾d is the known aberration, o is the observed object, n is the total noise, ⴱ stands for the convolution process, and FT stands for Fourier transform. The phase ␾共␳ជ 兲 is generally expanded on a set of basis modes (␳ជ being the position vector in the pupil plane). Using the Zernike basis to describe the aberrated phase, we can write

␾共␳ជ 兲 =

兺

k

ak · Zk共␳ជ 兲

共2.2兲

where ak are the Zernike coefficients of the phase expansion and Zk are the Zernike polynomials. The number of ak used to describe the phase will depend on the performance required and on the signal-to-noise ratio (SNR) characteristics. Nevertheless, since optical aberrations are considered, only first Zernike (typically between 10 and 100) are enough to well describe the NCPA. As shown in Eq. (2.1), there is a nonlinear relation between if and ␾ (and thus the ak coefficients). The estimation of ␾ has to be solved by the minimization of a given criterion [6,7]. We propose hereafter to define an optimal criterion adapted to the instrumental conditions (noise and wavefront aberrations) by using a MAP approach.

Fig. 1.

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The most simple known aberration to apply is defocus [4] (its amplitude is typically of the order of ␭), which can be introduced in several ways: • By translating the camera detector itself along the optical axis. The drawback of this approach is the range of displacement required for the detector, especially when high F ratios are considered. But this theoretically is the best option if it can be implemented. All the other options, presented below, may introduce not only defocus but other high-order aberrations of very low amplitudes, especially spherical aberration 共Z11兲. Even if they are easily quantified by using optical design software, they may, in the end, limit the accuracy of the calibration. • By translating a pinhole source in the entrance focal plane of the camera. This option has been used to calibrate the NCPAs on CONICA [10,11]. • By defocusing the image of a pinhole source in the entrance focal plane of the camera by translating the upstream collimator, for instance. • Last but not least, by using the DM directly. An adequate application of a set of voltages on the DM allows us to introduce the desired defocus with an accuracy related to the DM fitting capability. The two main advantages of the DM option are • No additional optical device is installed in the instrument; e.g., a software procedure may be developed to properly offset the voltages of the DM. • Other types of aberrations, such as astigmatism, can also be considered, leading to great flexibility in the procedure. Moreover, introducing other aberrations allows more accurate estimation of the focus itself, as presented in Section 5. The DM option was first used in NACO [10,11] and has also been applied to Keck [9]. A number of limitations have been identified in the PD [10,11]: photon and detector noise, detector defects (flat field stability), accuracy of the PD (amplitude and possibly additional high-order aberrations), and algorithm approximations [11] (including the number of Zernike polynomials). In the optimized pro-

(Color online) Principle of phase diversity: two images differing by a known aberration (here, defocus).

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Fig. 2.

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Principle of NCPA precompensation.

cedure that we propose below, the phase estimation is performed by minimizing a MAP criterion accounting for nonuniform noise model and phase a priori in a regularization term (see [13] for a detailed explanation of this approach). This optimized PD algorithm is presented in Section 3. B. NCPA Precompensation in the AO Loop The principle of NCPA precompensation is presented in Fig. 2. It consists of modifying the reference of the WFS to deliver a precompensated wavefront to the scientific path. A two-step process is therefore considered. Reference slopes are computed from PD data using a WFS model [14]. To accomplish that, a NCPA slope vector, as would be measured by the AO WFS (a Shack–Hartmann sensor, for instance), is first computed offline from the PD-measured set of Zernike coefficients by a matrix multiplication. The new reference vector is then added to the current WFS reference. Then closing the AO loop on the reference allows us to apply the opposite of the NCPAs to the DM. This leads to compensation for the scientific camera aberrations (in addition to the turbulence) and enhancement of the image quality at the level of the imaging camera’s detector. Any error in the WFS model directly affects the reference modifications, computed from the measured NCPA, and thus limits the performance of the precompensation process. Model errors have been identified as an important limitation of the approach in NAOS-CONICA [10]. As an example, an error of 10% on the pixel scale of the AO WFS detector directly translates into a 10% uncorrected amplitude of the NCPA. One way to reduce these model errors is to perform accurate calibrations of the WFS parameters. Nevertheless, uncertainties on calibrations (pixel scale of WFS, pupil alignment) will always degrade the ultimate performance of the precompensation process. In order to overcome this problem, a robust approach is proposed in Section 3. An important parameter in the NCPA precompensation is the selected number of Zernike polynomials to be compensated by the DM. This number is in fact limited by the finite number of actuators of the DM, i.e., the finite number of degrees of freedom of the AO system. The DM can

not compensate for all the spatial frequencies in the aberrant phase. In addition, the actuator geometry does not really fit properly the spatial behavior of the Zernike polynomials. All these problems are translated in fitting and aliasing errors on the compensated WF. These limitations have to be taken into account in the implementation of the precompensation procedure. It will be discussed later on in Section 6.

3. OPTIMIZATION OF THE NCPA MEASUREMENT A. Optimization of the PD algorithm As shown in Eq. (2.1), there is no linear relation between i and ␾. Therefore, the estimation of ␾ requires the iterative minimization of a given criterion. We propose here to define an optimal criterion adapted to our experimental conditions (noise and phase to estimate) by using a MAP approach [12,13]. The MAP criterion is based on a Bayesian scheme [see Eq. (3.1)] in which one wants to maximize the probability of having object o and phase ␾, knowing the images if and i d: P共o, ␾兩if,id兲 =

P共if,id兩o, ␾兲P共o兲P共␾兲 P共if兲P共id兲

.

共3.1兲

The decomposition of this probability makes different terms appear as discussed in detail in the following paragraphs. The denominator term P共if兲P共id兲 stands for the probability of obtaining the images if and id. As the images are already measured, this term is equal to 1. The term P共if , id 兩 o , ␾兲, called “likelihood term,” represents the probability of obtaining the measured data considering real object and phase. It is none other than the noise statistic in the image. The two main sources of noise are the detector noise and the photon noise: • For high flux pixels in the image, the dominant noise is the photon one. Hence, it follows a Poisson statistical law that can be approximated by a nonuniform Gaussian

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law with a variance ␴i2共rជ 兲 ⯝ i共rជ 兲, as long as i共rជ 兲 is greater than a few photons per pixel (rជ a position vector in the focal plane). • For low-flux pixels, the dominant noise is the detector noise, described by a spatially uniform distribution (same variance ␴2e for each pixel) and a Gaussian statistical law.

polishing defects, and misalignments. In a first approximation, their spatial power spectral density follows a 共n + 1兲−2 law, where n is the Zernike radial order (see Fig. 3). A general form for P共␾兲 is given by

Therefore, the global noise statistics can be approximated by a nonuniform Gaussian law of variance [13] ␴i2共rជ 兲 = ␴2e + i共rជ 兲. The second term, P共o兲, represents the a priori knowledge we have on the object. In our case, the object is marginally resolved (less than two pixels, for a diffraction FWHM of four pixels). Nevertheless, to account for its small extension as well as to account for pixel response, we have chosen to consider it as an unknown in the PD process. In the following, the only prior imposed on the object will be a positivity constraint (using a reparameterization: o = a2) leading to P共o兲 = P共a2兲 = 1. The probability P共o兲 does not have an impact on the criterion minimization. The third term, P共␾兲, is the regularization term for the phase estimation. It accounts for the knowledge we have on the NCPAs. Note that in our case we have chosen to parameterize the phase ␾ by the coefficients ak of its expansion on the Zernike basis. Then the term P共␾兲 will easily solve the problem of the choice of the number of Zernike polynomials to be accounted for in the estimation of the phase. Let us mention that in the conventional PD approach [5,6,11] the relatively arbitrary choice of a given number of Zernike (the N first) to be estimated is in fact an implicit regularization of the estimation problem by the truncation of the phase expansion in order to avoid the noise propagation on the Zernike high orders. This corresponds to a reduction in the dimension of the solution space. Here we will select in the algorithm a sufficiently large number of Zernike so as to not significantly reduce the solution space and to regularize the estimation by the term P共␾兲 in the criterion. Indeed, the NCPA are composed of static aberrations due to the optical design,

where R␾ is the phase covariance matrix and has on its diagonal the variance of the Zernike coefficients, of behavior similar to that given in Fig. 3, the other coefficients (covariances) being put equal to zero. The phase estimation is done by minimizing J共o , ␾兲 equal to −ln P共if , id , o , ␾兲:

P共␾兲 = exp共− ␾tR␾−1␾兲,

J共o, ␾兲 =

冐

if − hf ⴱ o

␴f共rជ 兲

冐冐 2

+

id − hd ⴱ o

␴d共rជ 兲

冐

共3.2兲

2

+ ␾tR␾−1␾ .

共3.3兲

This criterion makes the nonuniform noise statistics appear in the two images with standard deviations ␴f共rជ 兲 and ␴d共rជ 兲. The covariance matrix R␾ represents the prior knowledge of the phase. The minimization algorithm is based on an iterative conjugate gradient approach, allowing a fast convergence [7,13]. For the starting guess, all the Zernike coefficients are put to zero. Note that in the previous algorithm used for the NACO calibration [10,11], the PD algorithm did not include the nonuniform noise statistics and the phase regularization term. B. Simulation Results We present here only improvements brought by our new algorithm, the nonuniform noise model, and phase regularization, both in simulation in this section and experimentally in Section 6. Simulation is divided into two main parts: the generation of noisy aberrant images and the phase estimation by different PD algorithms. The simulated images are 128 ⫻ 128 pixels and are generated according to some realistic parameters of the ONERA AO bench (see Section 6): oversampling factor of 2.05 (this means 4.1 pixels in the Airy spot FWHM); the aberrant phase is modeled using the

Fig. 3. Typical aberration spectrum measured on existing optics. Diamonds, measured Zernike coefficient, integrated on radial order; dotted curve, the 共n + 1兲−2 approximation.

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into account an additive photon noise is useless. For high flux, photon noise is predominant and the algorithm with the nonuniform noise model allows us to increase the phase estimation accuracy of 15% to 20% with respect to the uniform noise model.

Fig. 4. Gain of the nonuniform model with respect to the conventional algorithm versus the maximum intensity value in the image. Randomly simulated images of 45 nm rms aberrant phase, 共n + 1兲−2 shaped spectrum. 75 Zernike modes from Z4 to Z78 are estimated by PD.

first 200 Zernike polynomials. The aberrations have a total rms error of 45 nm and a spectrum shape of 共n + 1兲−2. Finally, the images are noised with a uniform 1.6 electron noise per pixel and with photon noise. All ak values are given in nanometers. The maximum flux in the images is 100 photons in the case of the test of the regularized algorithm. We define the SNR in the images as the SNR of the focal plane image expressed by the ratio of the maximum of the image imax in photon electrons by the standard deviation of the sole detector noise in electrons: SNR = imax/␴e .

共3.4兲

1. Gain Brought by a Nonuniform Noise Model Let us first study the gain brought by accounting for a nonuniform noise model in the PD algorithm. For a given aberrant phase, the efficiency ⌺NU quantifies the gain in estimation accuracy for the nonuniform algorithm with respect to the uniform algorithm: ⌺NU = 100 ⫻

⑀U − ⑀NU ⑀U

,

2. Gain Brought by Phase Regularization The gain brought by the phase regularization term in Eq. (3.3) is quantified in this section. Figure 5 presents the results of the simulation using different algorithms. Table 1 shows the total estimation error for each algorithm and also the contribution of low orders (from a4 to a36) and the contribution of high orders (from a4 to a137) in the total error. The dashed curve represents the estimation error in nm2 for the 133 first Zernike coefficients (from a4 to a137), and a simple least-squares estimation (which means without regularization) is given. As comparison, the first 133 coefficients of the 200 Zernike polynomials simulated from the input spectrum (45 nm rms) to compute the images are plotted by the dotted curve. For this estimation, the error (noise propagation) is constant whatever the coefficient, as predicted by the theory (see [6]). For coefficients higher than a36, the estimation error becomes greater than the signal to be estimated. Since the SNR on these coefficients is lower than one, their estimation is not possible. The total error is 46 nm for the 133 coefficients, which is similar to the introduced WFE. In contrast, an estimation of only the first 33 coefficients a4 to a36 (solid curve) shows that the reconstruction error is still roughly constant whatever the mode but fainter than in the previous estimation. The total error is 29 nm for the 133 coefficients, considering that all the estimated coefficients from a37 to a137 are equal to zero. Here, reducing the number of parameters in the estimation (especially the high frequencies of the phase) leads to a better estimation on the low-order modes and a dramatic decrease in their estimation error. Nevertheless, as

共3.5兲

where ⑀U and ⑀NU are the reconstruction errors obtained, respectively, with uniform and nonuniform noise models (in nm2) defined by Nmax

⑀X =

兺 共a i=1

kmeas,X

− aktrue兲2

共3.6兲

where akmeasX are the estimated Zernike coefficients with the uniform noise model 共X = U兲 or with the nonuniform noise model 共X = NU兲 and aktrue are the true coefficients. Figure 4 shows the influence of the noise model on the estimation accuracy. ⌺NU is plotted with respect to the maximum intensity value in the image. Each point on the curve corresponds to only one occurrence of noise and aberrations, hence the relative instability found in computing ⌺NU. At low photon flux, the two estimation errors are identical. The image SNR is limited by detector noise (uniform noise in the full image), and therefore taking

Fig. 5. Noise propagation in the Zernike coefficient estimation for different cases of regularization. Dotted curve, average spectrum of the simulated aberrations [45 nm total WFE, 共n + 1兲−2 spectrum]. Dashed curves, phase estimation error for a 133 Zernike estimation, without any regularization term. Solid curve, the same estimation, with only 33 Zernike (regularization by truncation). Dashed–dotted curve, 133 Zernike estimated with the regularization term. SNR in the images is 103.

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Table 1. Estimation Error with Different Phase Regularizationsa Zernike Coefficients

Introduced WFE

Least Square 133 Coefficients

Truncated Least Square 33 Coefficients

Phase Reqularization 133 Coefficients

a4 to a36 a37 to a136 Total error

38 24 45

21 41 46

16 24 29

12 19 22

a

Introduced WFE is given in comparison. All values are given in nanometers.

explained before, this estimation is nothing but a first rough regularization and is not optimal. Because the previous regularization is arbitrary, we can refine the estimation by using prior knowledge on the phase to be estimated, that is, a 共n + 1兲−2 spectrum, as a regularization term. In that case, the estimated first 133 coefficients are given by the dashed–dotted curve. For all the coefficients, the error is lower than the input coefficients (dotted curve). More important, the total error 共22 nm兲 is smaller than the one for the estimation on 33 coefficients. The error computed with only the first coefficients a4 to a36 is also fainter with regularization, 12 nm compared to 16 nm. For high-order modes, the error tends to be equal to the phase itself, i.e., no noise propagation. MAP allows us to deal optimally with low SNR and avoids any noise amplification with the regularization term.

4. OPTIMIZATION OF THE NCPA PRECOMPENSATION A. Pseudo-Closed-Loop Process After PD measurement, the precompensation of NCPA has to be performed by a modification of the WFS reference. The compensation accuracy therefore depends greatly on AO loop model. In order to overcome this problem and reach much better performance, a new (to our knowledge) approach is proposed, “pseudo-closed loop” (PCL). The idea is to use a feedback loop for the NCPA precompensation, including the PD estimation (see the schematic of Fig. 2). Indeed after a first precompensation of the NCPAs, it is mandatory to have the capability to acquire a new set of two precompensated images in order to quantify the residual NCPAs due to model uncertainties. This can be done by closing the AO loop on the artificial source used for the PD image acquisition, accounting for the new WF reference. This ensures the stability of the precompensation by the DM during the image acquisition. By estimating a new set of Zernike coefficients, we have access to the residual phase after correction. We can take advantage of its measurement to offset the previously modified wavefront reference. The process can then be performed until convergence, resulting in quasi-null measurements of the Zernike coefficients (at least, free from any model error). Indeed, any error in the AO WFS model will result only in slowing the convergence of the process. In addition, after the first precompensation, the recorded images may exhibit a much better SNR due to the higher concentration of photons in the central core of the image, leading to a better estimation of the Zernike coefficients. The practical implementation of this PCL approach is summarized below. First, we perform a careful AO WFS

calibration: its detector pixel scale, the pupil image position, and the reference slope vector obtained using a dedicated calibration source at the AO WFS entrance focal plane. We are therefore able to adjust the initial AO WFS model. Second, an artificial quasi-punctual source is placed at the entrance of the AO bench and is used to calibrate the NCPAs. With this calibration source, the DM– WFS interaction matrix is calibrated, and, from the measurements, a new command matrix is computed. Then the multiloop measurement compensation process is as follows: 1. Measurement of NCPA with PD. This step can be summarized as follows: (i) Closing the AO loop using the calibrated AO WFS references and recording of a focused image on the science camera. (ii) Applying the defocus (using slope modification) and recording a defocused image with the AO loop, closed once again. (iii) Computation of NCPAs from this pair of images with the PD algorithm. 2. Computation of the incremental slope vector using the currently measured NCPAs. 3. Modification of the AO WFS references (to account for the latest measurements) and saving the new AO WFS references, 4. Measurement of residual NCPAs with PD and closed AO loop using the new references for the precompensation, similar to step 1. 5. Repeat steps 2 through 4 until convergence. A refinement could be to recalibrate at each step (3 to 4) the DM–WFS interaction matrix taking into account the influence of the reference offsets in the AO WFS response and recompute the command matrix to achieve the best possible efficiency with the AO system. An alternative approach, recently proposed, is to directly perform an interaction matrix linking the Zernike modes to be compensated and the PD estimation [15]. B. Number of Compensated Modes The number of Zernike modes that can be compensated for is determined by the number of actuators of the DM. The larger the number of actuators, the better the fit to Zernike polynomials by the DM. We performed the simulation of the capability of our DM (69 valid actuators) to compensate for the Zernike polynomials, using the DM influence functions as measured by a Zygo interferometer. The results, not presented here, show that considering Zernike polynomials of radial degree larger than 6 leads

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to significant fitting errors (larger than 45% of input standard deviation), reducing the overall performance of the NCPA precompensation. In most of the experimental results presented in this paper in Section 6, we have used the first 25 Zernike polynomials (from defocus Z4 up to Z28) for the NCPA compensation. It allows us to minimize the coupling effects between the compensated Zernike due to the limited number of actuators on the DM. The compensation of the first 25 Zernike polynomials brings already a significant reduction of the NCPA amplitudes. Because of the expected decrease of the amplitude of the NCPA with the order of Zernike (see Fig. 3), this choice is not an important limitation in the final performance.

5. DM APPLICATION OF A PHASE DIVERSITY To finalize the discussion on the procedure to measure and precompensate for the NCPA, let us now consider the application of the PD by the DM. As already stated, we consider that this approach is probably the best for a fully integrated AO system in an instrument if there is no science detector translation capability. For instance, defocus can be introduced by moving an optical element on the sole optical train of the AO WFS and closing the loop with this aberration, as first implemented in NACO (see [10,11]). But implementing a moving optical element is an issue with an instrument requiring high stability. Therefore, we propose to apply the PD by modifying the AO WFS references, the same as for the NCPA precompensation. This is a pure software procedure which uses the AO WFS model. Closing the AO loop with the modified references will apply the defocus to the science camera but also ensures the stability of this defocus. Application of the defocus directly on the DM voltages and not closing the loop will suffer from DM creeping. In fact, any other low-order aberration can be considered by this method as PD, allowing maximum flexibility. Due to the uncertainty of the AO WFS model, the introduced PD will not be perfectly known, which results in measurement errors. The main effect is the uncertainty in the amplitude of the PD. Considering defocus as the known aberration in the simulation, we observed a linear dependence of the NCPA defocus estimation error on defocus distance [11]. In other words, the error on the known defocus application translates directly into an error in defocus estimation. When introducing a defocus diversity, the NCPA a4 coefficient is the only polynomial affected by this bias. Considering an error of 10 nm on the known defocus, the error on the measured a4 is very close to 10 nm, whereas the total error on the other modes (mainly spherical aberration and astigmatism) is smaller than 1 nm. The same behavior was found when an astigmatism was used as known aberration. For 10 nm error on known astigmatism, 10 nm error is found on the astigmatism and only 1 nm for the other polynomials in total. Note that the PCL is not able to compensate for this systematic bias in the PD algorithm. The only way to determine the defocus is by using other approaches, e.g., trying different defocus values to optimize the image quality or using another diversity mode only for defocus

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measurement. In Section 6, we present experimental results of these two approaches.

6. LABORATORY RESULTS Both the PCL iterative compensation method and the various algorithm optimizations have been experimentally tested on the ONERA AO bench. It operates with a fibered laser diode source of 4 ␮m core size working at 633 nm and located at the entrance focal plane of the bench. The laser diode can be considered an incoherent source, since it is used at very low power and is therefore weakly coherent with a large number of modes. The wavefront corrector includes a tip–tilt mirror and a 9 ⫻ 9 actuator DM (69 valid actuators). The Shack–Hartmann WFS, working in the visible, is composed of an 8 ⫻ 8 lenslet array (52 in the pupil) and a 128⫻ 128 pixel DALSA camera. The WFS sampling frequency is set to 270 Hz. The imaging camera is a 512⫻ 512 Princeton camera with 4e-/ pixel/frame read-out noise (RON). The control law used for the AO closed loop is a classical integrator. An accurate estimation of image quality is mandatory to quantify the efficiency of the PCL and to compare the different modifications/improvements of the PD algorithm. The Strehl ratio (SR) is a good way to estimate image quality, but it is definitely not obvious how to compute it on a real image with high accuracy: This particular point is addressed in Appendix A with special care to the definition of error bars on SR estimation. A SNR of 104 in the focused image [see Eq. (3.4)] is sufficient to observe the first five Airy rings coming out of the RON. In this case, the PD estimation will therefore be highly accurate for the first Zernike modes. In order to take advantage of the regularization and to minimize the aliasing effect in the measurement, the phase estimation by PD is done on the first 75 Zernike polynomials starting at the defocus (from Z4 to Z78) and gives 75 Zernike coefficients (from a4 to a78). Nevertheless, we compensate for only the first Zernike polynomials (from Z4 to Z28) because of the limited number of actuators of the DM. These numbers of Zernike will always be used in the next subsection except where otherwise stated. A. Test of the Pseudo-Closed Loop Process For the test of this iterative method, the so-called PCL, the images used to perform PD are recorded with a very high SNR (SNR= 3 ⫻ 104 for the focused image) so as to be in a noise-free regime. This SNR level corresponds to an error on the first 25 Zernike polynomials of less than 0.5 nm as a result the noise. The measured SR before any compensation is 70% at 633 nm. A conventional PD algorithm (without regularization) is considered here (because of the high SNR). Figure 6 shows the 75 Zernike coefficients measured at different iterations of the PCL procedure. The first iteration corresponds to the measurement of the NCPAs without any precompensation. Only the first 25 modes are corrected, while 75 are measured at each iteration. The figure shows extremely good correction of the first 25, while the 50 higher-order modes remain quasi-identical.

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Fig. 6. Behavior of the measured Zernike coefficients with the number of iteration of the PCL. Zernike coefficients up to Z78 are estimated with conventional PD and SNR= 3 ⫻ 104. Coefficients up to Z28 are compensated by the PCL process using AO closedloop with integrator control law. Image wavelength is 632.8 nm. Dashed–dotted curve, coefficients before any compensation; dotted curve, coefficients after one iteration; solid curve, coefficients after ten iterations.

Fig. 7. Evolution of the residual error as a function of iteration number for the low order corrected (solid curve) and high-order uncorrected (dotted curve) Zernike modes and for all the measured modes (dashed curve). Conditions are the same as in Fig. 6.

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Figure 7 shows the evolution of the residual error ␴ k=N 2 = 冑共兺k=M ak兲 for the precompensated polynomials (M = 4 and N = 28), for the higher order noncorrected Zernike modes (M = 29 and N = 78) and for all the measured polynomials (M = 4 and N = 78). After four iterations, the global residual phase computed on the corrected Zernike modes (M = 4 and N = 28) is lower than 1 nm RMS (not limited by noise in the images), whereas the residual phase computed on the noncorrected Zernike (M = 29 and N = 78) modes remains quasi-identical passing from 22 to around 24 nm rms. After convergence, the total residual error on the first 78 Zernike polynomials is 24 nm. Finally, for each iteration, a SR value 共SRIm兲 can be measured on the focused image. In addition another SR value 共SRZern兲 can be computed using the coefficients estimated by the PD algorithm [see Eq. (A4) in Appendix A]. In Fig. 8 we compare the measured SRim and the estimated SRZern as a function of iteration number. Both SRim and SRZern have the same behavior. The maximum value achieved by SRim is 93.8% at 633 nm. After two iterations, SRIm reaches a convergence plateau. We plot on the same figure the ratio between SRim and SRZern. The difference between SRim and SRZern can be explained by the unestimated high-order coefficients (higher than a78) and the SR measurement bias due to uncertainty on the system (exact oversampling factor, background subtraction precision, exact fiber size and shape; see Appendix A for more details). This ratio SRim / SRZern is roughly constant after the first iteration, and its value at convergence can be estimated to 99.4%, which corresponds to an 8 nm rms phase error. The different values for the first iteration are explained by the approximation of SR by the coherent energy in SRZern, which is valid only for small phase variance. In Figure 9 we plot the focused images recorded on the camera without NCPA precompensation and after 1, 2, and 3 iterations of the PCL scheme. The correction of loworder aberrations allows for the cleaning of the center of the image, where two Airy rings are clearly visible, with the first one being complete. Around these rings, we observe residual speckles due to the uncompensated higherorder aberrations.

B. Defocus Determination As explained in Section 5, NCPA defocus estimation has to be considered with particular care since it is biased by uncertainty on the “known aberration” (actually, not perfectly known) introduced in the PD method. In order to overcome this bias, two approaches are proposed after the convergence of the precompensated scheme. The first one is based on a SR optimization, and the second one is a one-shot measurement with an “astigmatism” phasediversity.

Fig. 8. Evolution of SR with iteration number. SR is measured on the focal plane images 共SRim兲. Dotted curve, SR computed from the measured NCPAs 共SRZern兲. SR bias is uniform and estimated to 0.008. Dashed curve, ratio between SRim and SRZern. Conditions are the same as in Fig. 6.

1. SR Optimization After a few iterations of the PCL (enough to reach convergence), we modify the precompensated a4 coefficient in a given range around the estimated coefficient and measure the corresponding SR. Figure 10 shows the SR evolution with the value of a4. The maximum SR is obtained for

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Fig. 9. Focused images obtained on the ONERA AO bench (logarithmic scale) corresponding to the first four points of Fig. 8. The image on the left is the image obtained without any precompensation. The SR of the last image is 93.8%. Conditions are the same as in Fig. 6.

long as it is an even radial order (to solve the estimation phase indetermination) and feasible by the DM. At convergence of the PCL, we acquire a pair of images differing by astigmatism 共Z5兲. The PD measurement performed gives coefficients a4 to a78, with a biased estimation of a5 (the previous error due to model uncertainty is now done on a5 instead of a4), while the coefficient a4 is now correctly estimated. The value given for a4 by this method is a4 = 35 nm, which is fully compatible with the SR optimization of Subsection 6.B.1.

Fig. 10. Optimization of the coefficient a4 by changing its value before applying the correction slopes. a4 = 43 nm is the value estimated by PD and a4 = 34 nm is the value estimated by the maximum SR. Conditions are the same as in Fig. 6 except for the number of compensated Zernike modes, up to Z36.

Fig. 11. Evolution of SR with iteration number. Solid curve, conventional algorithm (uniform noise model); dotted curve, nonuniform noise model algorithm. The measurements were done at the time of each iteration. Conditions are the same as in Fig. 6 except for the use of the PD algorithm.

a4 = 34 nm, which is somewhat different from the value given by PD 共43 nm兲. The resulting gain in term of SR is around 1%. 2. Astigmatism Phase Diversity An alternative way to perform the NCPA defocus optimization is to use another known aberration between the two images. As explained in Section 1, defocalization is generally used because of its easy implementation. In our case the DM itself is used to generate the known aberration. Thus any Zernike polynomial can be considered, as

C. Test of Optimized Algorithms Let us now validate experimentally the various PD algorithm modifications proposed in Section 3 (that is nonuniform noise model and phase regularization). In order to test these improvements experimentally, two different regimes of SNR have been used: high-SNR regime as before, SNR= 3 ⫻ 104, and low-SNR regime, SNR= 102 obtained with the smallest exposure time while acquiring the pair of images. At high SNR, the correction remains extremely good whatever the algorithm configuration. The gain brought by the nonuniform noise model is rather small, and the limitation comes from other error terms, especially noncorrected modes. However, there is a slight gain of 0.1% of SR, shown in Fig. 11, which was predicted by the theory. At low SNR, the gain is much higher, 10% in SR, but this is the result of a one-shot test, not a mean gain obtained on a large number of trials. The gain brought by the use of the phase regularization term is shown in the low-SNR regime. At SNR= 102, the conventional algorithm without regularization barely estimates the phase. It leads to a poor result after precompensation: the saturation plateau remains around 72.1%, showing no real improvement on the image. When the regularization term is added, the SR value reaches 91.9%. The NCPA are estimated at almost the same accuracy as in high-SNR regime. As shown in Subsection 3.B, the use of regularization allows the minimization of the noise amplification on the high-order estimated modes and enhances the estimation accuracy of the lower orders. These properties will be particularly useful with an infrared camera where the SNR in the image could be limited and when a large number of modes have to be estimated. Note also the substantial gain brought by the use of the nonuniform noise model algorithm when compared with the conventional one and also when coupled to the phase regularization. Table 2 gathers the various SR values obtained after convergence of the PCL process for the different algorithm modification.

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Table 2. SR Obtained with the Different Optimized Alogorithms

Max SNR

Conventional Algorithm (%)

Nonuniform Noise Model (%)

Phase Regularization (%)

Regularization and Nonuniform Noise Model (%)

For High SNR For Low SNR

93.8 72.1

93.9 81.2

93.8 91.9

93.9 92.3

Fig. 12. PSF obtained after three iterations of NCPA precompensation. 42 modes are compensated (from Z4 to Z45), and the exact value of a4 has been optimized. SR is 98.7%, ␭ = 632.8 nm.

pared with the results of Subsection 6.A (only 1 nm). As explained in Subsection 4.B, the high-order Zernike modes of NCPA are not well fitted by the DM. That induces some coupling effects between the compensated highest-order modes (from Z29 to Z45) and the uncompensated ones (above Z45). They induced some aliasing effect on lower modes, slightly decreasing their precompensation efficiency. Nevertheless, the gain brought by the partial correction of highest-order modes is higher than the loss due to aliasing effects. Note that these performances were obtained using a Kalman filter in the AO loop as developed for optimized compensation of the turbulence [16] and not a simple integrator corrector as for the previous result. This Kalman filter uses the first 130 Zernike modes for the WFS phase regularized estimation, with the result that the limitations linked to the bad fitting of the high-order Zernike by the DM are partially overcome. It was not possible to obtain such a high SR (98.7%) with the integrator in the same conditions.

7. DISCUSSION

Fig. 13. Zernike coefficients measured before compensation (dotted curve) and after three iterations of PCL (dashed curve, corresponding to the PSF shown in Fig. 12). Conditions are the same as in Fig. 12.

D. Number of modes We performed an additional test with the PCL. In this test, the conditions are the same as previously, i.e., 75 Zernike modes are measured by PD. The SNR in the images used by PD is very high 共104兲. The PD algorithm used to perform NCPA measurement is the conventional one. The NCPA a4 estimation is unbiased (see Subsection 6.B). In this test 42 Zernike modes were compensated, instead of 25. The corresponding PSF is shown in Fig. 12, revealing up to four Airy rings. A SR of 98.7% is obtained. Figure 13 presents the measured Zernike coefficients before any compensation and after three iterations of PCL corresponding to Fig. 12. A 13 nm total residual error was estimated on the 75 Zernike coefficients, a substantial gain compared with the results shown in Fig. 7 共24.5 nm兲. Up to Z45, the residual error is 4.5 nm, while the higher-order contribution remains below 12.5 nm. We observe that on the first order, the residual error is slightly higher com-

We discuss here the gain brought by our new approach for NCPA measurement and compensation on extreme AO systems for extrasolar planet detection. This type of instrument requires very high AO performance in order to directly detect photons coming from very faint companions orbiting their parent star. An example of such an extreme AO system applied to direct exoplanet detection is Sphere Ao eXoplanets Observation (SAXO) [3], the extreme AO system of SPHERE [1]. SPHERE is a secondgeneration Very Large Telescope (VLT) instrument considered for first light in 2010. It will allow one to detect hot Jupiter-like planets with contrast up to 106. Characteristics of SAXO are the following [3]: a 41 by 41 actuator DM and a WFE budget of 80 nm after extreme AO correction (90% SR in H band). The specification of SAXO is to compensate for NCPA with at least 100 Zernike modes (goal 200) and to allocate a 8 nm WFE on these modes. The high number of actuators available on the DM allows one to fit the requested number of modes (no fitting problem). Therefore, the only error source is assumed to be a PD estimation error. According to Fig. 5 and to the fact that precision of PD is inversely proportional to the SNR in the image [6], a SNR of 104 is sufficient to have a precision of 0.4 nm2 on each of the 100 measured Zernike coefficients, i.e., a residual WFE of 6 nm. This SNR can be achieved by averaging the number of images recorded by the infrared camera. Moreover, this level of residual correction per Zernike mode is fully compatible with the results obtained on our AO bench (see Figs. 6 and 13). The goal of 100 Zernike

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modes for compensation and 8 nm of WFE after compensation is therefore fully achievable with the PCL and the optimized algorithms presented here.

8. CONCLUSION We have proposed and validated what we believe to be a new and efficient approach for the measurement and precompensation of the NCPAs. First, the measurement quality of the NCPAs has been improved via the optimization of the PD algorithm (accurate noise model, phase regularization). Moreover the limitation imposed by model uncertainties during the precompensation process has been overcome by the use of what we believe to be a new iterative approach, pseudo-closed loop (PCL). We have validated this new tool experimentally on the ONERA AO bench. Very high SR has been obtained (around 98.7% of SR at 633 nm, that is less than 14 nm of residual defects). The residual WFE on corrected modes is less than 4.5 nm rms. We have estimated the residual error on uncorrected aberrations to be less than 12.5 nm rms. Solutions have been proposed to deal with experimental issues (such as defocus uncertainty) of the PD implementation. The experimental results presented in this paper allow us to be confident in our capability of achieving the challenging performance required for direct detection of extrasolar planets. The residual errors obtained on our AO bench for NCPA compensation are fully compatible with the error budget of an extreme AO system like SAXO. Using 100 Zernike coefficients in the compensation, we should achieve an 8 nm residual error, corresponding to 99.9% SR at 1.6 ␮m.

APPENDIX A: STREHL RATIO ESTIMATION 1. Strehl Ratio Estimation in Focal Plane Images A widely used performance estimator in AO is the SR. Its experimental estimation is difficult and requires optimized algorithms able to deal with a number of experimental biases or noises. We propose here an efficient SR

Sauvage et al.

measurement procedure. The SR is defined as the ratio of the on-axis value (or tilt free) of the aberrated image iab on the on-axis value of the aberration-free image iAiry. Several parameters have to be taken into account in order to obtain accurate and unbiased values: the residual background and the noise in the image, the CCD pixel scale, and the size of the calibration source used to calibrate the NCPA. The SR value can be computed using the following equations:

SRim =

iab共0ជ 兲 iAiry共0ជ 兲

=

冕 冕

˜i 共fជ兲 · dfជ ab ,

共A1兲

˜i 共fជ兲 · dfជ Airy

where ˜iab and ˜iAiry are the optical transfer function (OTF) of the aberrant system and of the aberration-free system, respectively (i˜ standing for FT of i), and ជf is a position variable in Fourier space. We developed a procedure calculating the SR in the Fourier domain. Considering the OTF rather than the PSF for SR estimation presents several advantages that are summarized below and are illustrated in Fig. 14: • First, an analysis of the FT of the aberrated image (OTF) allows a fine subtraction of the residual background, which can be estimated from a parabolic fit at the lowest frequencies of the aberrated OTF excluding the zero frequency. • An important point is the adjustment of the cutoff frequency 共fc = D / ␭兲 in ˜iAiry (value in frequency pixel directly linked to the image pixel scale) to the experimental value in the aberrated OTF. The pixel scale is also used by PD to estimate the phase. • Actually, all the OTF values for frequencies greater than the cutoff frequency are only noise. An estimation of this level and then a subtraction to the aberrated OTF allow us to refine the SR estimation. In Fig. 14, for frequencies higher than fc the aberrated OTF presents a noise plateau at 1 . e − 3. • Finally, the procedure also takes into account the transfer function of the CCD and of the FT of the object when the latter is partially resolved by the optical system.

2. Errors in Strehl Ratio Estimation Practical instrumental limitations degrade the SR estimation accuracy even when an optimized algorithm is used as in Section 3. It is important to quantify their influence in order to give error bars on SR values.

Fig. 14. Measured OTF from the image (dashed curve), OTF corrected for noise and background contributions (solid curve), and adjusted Airy OTF as obtained by the SR measurement procedure (dotted–dashed curve. The cutoff frequencies are adjusted to be superimposed. The transfer function of the CCD is also given (dotted curve).

a. Influence of Residual Background Let us first study the influence of a residual uniform background ␦B per pixel on the estimation of the SR in an image if.

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Using Eq. (A1) we can express the background contribution in the SR computation as follows: 共if共0ជ 兲 + ␦B兲 SRim = iAiry共0ជ 兲

冉

冕

⯝ SR 1 +

冕

ជ 兲d␣ជ iAiry共␣

ជ 兲 + ␦B兲d␣ជ 共if共␣

␦B if共0ជ 兲

冊冢 冕 冣 1−

N 2␦ B

共A2兲

ជ 兲d␣ជ i f共 ␣

Note that Eq. (A2) takes into account the required norជ 兲 + ␦B兲d␣ជ 兲, malization by the total flux in the image 共兰共if共␣ ជ 兲 has N ⫻ N pixels and 兰iAiry共␣ជ 兲d␣ជ where the image if共␣ = 1. Considering that generally ␦B Ⰶ if共0ជ 兲, a residual background N2␦B equal to 1% of the total flux in the image modifies the SR value of 1% as well. The residual background in our images after subtraction of the calibrated background and after correction by fitting of the lowest frequencies of the OTF is estimated at ±0.1% of the total flux using images of 128⫻ 128 pixels. The SR estimation accuracy is therefore ±0.1%. b. Influence of an Uncertainty on the Pixel Scale The pixel scale is a parameter to be estimated experimentally since it depends greatly on component characterization and system implementation. It plays an important role in the NCPA estimation through the PD algorithm and also affects the quality of SR estimation. Its influence is on the whole procedure discussed in this paper is significant. In the following, we emphasize its influence on the SR measurement. Assuming that the OTF profile for an Airy pattern has a linear shape, a simple computation shows that the relative SR modification ␦SR/ SR is directly equal to twice the relative precision ␦e / e on the pixel scale e; that is,

␦SR/SR = − 2共␦e/e兲.

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c. SR Accuracy in Experimental Data It is now possible to estimate the global accuracy of the SR estimation using the results of Subsections A.2.a and A.2.b: ␴SR = 冑0.82 + 0.12 ⯝ 0.81%. The pixel scale bias is the main contribution to this value. 3. SR Estimated Using the Measured Zernike Coefficients Another way to estimate the SR is to use the residual 2 phase variance ␴␾2 to compute the coherent energy e共−␴␾兲. The residual phase variance can be obtained directly using the PD estimated Zernike coefficients. We therefore define the approximated SR SRZern by

冉

SRZern = exp −

Nmax

兺a

k=2

2 k

冊

,

共A4兲

where ak stands for the kth Zernike coefficient. On one hand, this expression of SRZern is a lower bound for the true SR for relatively large residual phase (low SR). On the other hand, SRZern is a good approximation of SR for small residual phases (high SR). However, it slightly over estimates it since SRZern only accounts for the Nmax first Zernike [Eq. (A4)]. Figure 8 shows the image measured SR 共SRim兲 and computed one 共SRZern兲. We verify on this figure the behavior described here above.

ACKNOWLEDGMENT The authors thank the Laboratoire d’Astrophysique de l’Observatoire de Grenoble for partial support of J.-F. Sauvage’s scholarship.

REFERENCES 1.

共A3兲

2.

It is clear that knowledge of e is essential to obtain an accurate estimation of SR, but the e value does not evolve with time. Therefore the estimation error on SR is constant for the whole test. The effect on SR is a bias. In other words, if the e value is critical for absolute SR computation, its influence is dramatically reduced when only the relative evolution of the SR is considered (the gain brought by a new approach of NCPA precompensation for example). Now the essential question is, “With what accuracy do we know the pixel scale?” In images taken on different days, the measured cutoff frequency is stable, with an uncertainty lower than half a frequency pixel. The random relative error on the pixel scale is 0.4% for 128 frequency pixels. Finally, we used the same measured pixel scale 共e = ␭ / 4.1D兲 in data processing of all the performed experiments (NCPA measurement and SR estimation). The relative error on SR estimation is therefore a bias of 0.8%. Because SR⯝ 100%, the absolute error on SR is 0.8%.

3.

4. 5. 6.

7. 8.

J.-L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, P. Puget, T. Fusco, and A. Boccaletti, “A planet finder instrument for the VLT,” in Proceedings of IAUC 200, Direct Imaging of Exoplanets: Science and Techniques, 2005 (International Astronomical Union, 2005); www.iau.org. C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, and D. Rouan, “Fundamental limitations on Earth-like planet detection with extremely large telescopes,” Astron. Astrophys. 447, 397–403 (2006). T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: Application to the SPHERE instrument,” Opt. Express 17, 7515–7534 (2006). R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982). R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992). L. Meynadier, V. Michau, M.-T. Velluet, J.-M. Conan, L. M. Mugnier, and G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38, 4967–4979 (1999). A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A 20, 1035–1045 (2003). G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, “NAOS, the first AO system of the

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J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007 VLT: on sky performance,” Proc. SPIE 4839, 140–149 (2002). M. A. van Dam, D. Le Mignant, and B. A. Macintosh, “Performance of the Keck Observatory adaptive-optics system,” Appl. Opt. 43, 5458–5467 (2004). M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. M. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations. Experimental results,” Astron. Astrophys. 399, 385–394 (2003). A. Blanc, T. Fusco, M. Hartung, L. M. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations. Application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003). J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, “Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra,” Appl. Opt. 37, 4614–4622 (1998).

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14. 15. 16.

L. M. Mugnier, T. Fusco, and J.-M. Conan, “MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected longexposure images,” J. Opt. Soc. Am. A 21, 1841–1854 (2004). G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. Alloin and J.-M. Mariotti, eds. (Kluwer, 1993), Vol. 243, pp. 115–137. J. Kolb, E. Marchetti, G. Rousset, and T. Fusco, “Calibration of the static aberrations in an MCAO system,” Proc. SPIE 5490, 299–308 (2004). C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Raboud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).

Annexe J “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics” T. Fusco et al. - JOSAA - 2007

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Optimal wave-front reconstruction strategies for multiconjugate adaptive optics Thierry Fusco, Jean-Marc Conan, Ge´rard Rousset, Laurent Marc Mugnier, and Vincent Michau Office National d’E´tudes et de Recherches Ae´rospatiales, De´partement d’Optique The´orique et Applique´e, BP 72, F-92322 Chaˆtillon cedex, France We propose an optimal approach for the phase reconstruction in a large field of view (FOV) for multiconjugate adaptive optics. This optimal approach is based on a minimum-mean-square-error estimator that minimizes the mean residual phase variance in the FOV of interest. It accounts for the C 2n profile in order to optimally estimate the correction wave front to be applied to each deformable mirror (DM). This optimal approach also accounts for the fact that the number of DMs will always be smaller than the number of turbulent layers, since the C 2n profile is a continuous function of the altitude h. Links between this optimal approach and a tomographic reconstruction of the turbulence volume are established. In particular, it is shown that the optimal approach consists of a full tomographic reconstruction of the turbulence volume followed by a projection onto the DMs accounting for the considered FOV of interest. The case where the turbulent layers are assumed to match the mirror positions [model-approximation (MA) approach], which might be a crude approximation, is also considered for comparison. This MA approach will rely on the notion of equivalent turbulent layers. A comparison between the optimal and MA approaches is proposed. It is shown that the optimal approach provides very good performance even with a small number of DMs (typically, one or two). For instance, good Strehl ratios (greater than 20%) are obtained for a 4-m telescope on a 150-arc sec ⫻ 150-arc sec FOV by using only three guide stars and two DMs. © 2001 Optical Society of America OCIS codes: 010.1080, 010.1330, 010.7350.

1. INTRODUCTION Atmospheric turbulence severely limits the angular resolution of ground-based telescopes. Adaptive optics (AO)1–3 is a powerful technique to overcome this limitation and to reach the diffraction limit of large telescopes. AO compensates, in real time, for the random fluctuations of wave fronts induced by the turbulent atmosphere. The turbulent wave front is measured by a wave-front sensor (WFS) using a guide star (GS) and optically corrected by a deformable mirror (DM) located in a pupil conjugate plane. This compensation allows the recording of long-exposure images with a resolution close to the diffraction limit. Because of anisoplanatism, the correction is efficient in only a limited field of view (FOV) (the socalled isoplanatic field) around the GS. This effect originates from the fact that turbulence is distributed in the volume above the telescope; then the wave fronts, coming from angularly separated points, are degraded differently. In the visible, the isoplanatic field is approximately a few arc seconds.4 Beyond this FOV, the correction degrades.5 Recently, a postprocessing method has been proposed to deal with the spatial variation of such an AO point-spread function.6 This method gives very good results, but it is limited by the decrease of the correction degree in the FOV, which leads to a decrease of the signal-to-noise ratio (SNR) in the corrected image. Classical AO therefore gives poor high-resolution performance in the case of large FOV. Improved performance is, however, expected with multiconjugate AO (MCAO).7,8 It consists in using several DMs conjugated at different heights in the atmosphere (see Fig. 1). With such a system, the turbulence effects are corrected not only on the telescope pupil but also in the turbulence volume; hence the increase of the correction field. Generally, several GSs are used to sense 0740-3232/2001/102527-12$15.00

the perturbation in different FOV positions and to control these mirrors. The choice of the number of GSs9–12 and DMs10,11,13,14 is crucial for the design of such systems. It is related to the turbulence profile C n2 (h), the telescope diameter, and the observation goals. Note that, in this paper, we consider only natural GSs, but all the theoretical development could be extended to the case of laser GSs provided that all their specificities are taken into account (analysis geometry, cone effect, tip–tilt measurement problems). We believe that one of the key issues is the phase reconstruction in MCAO. It is linked to the capability of the phase reconstruction algorithm to find the best deformation to apply on each DM from a set of WFS measurements, in order to obtain the best correction in a given FOV of interest. Since the minimization of the residual phase variance maximizes the image quality in the considered direction, we derive a minimum-mean-squareerror (MMSE) estimator that minimizes the mean residual phase variance in the FOV of interest.15 It accounts for the C n2 profile in order to optimally estimate the correction wave front to be applied to each DM. This optimal approach also accounts for the fact that the number of DMs will always be smaller than the number of turbulent layers, since the C n2 profile is a continuous function of the altitude h. Links between this MMSE approach and a tomographic reconstruction of the turbulence volume are established. In particular, it is shown that the MMSE approach consists of a full tomographic reconstruction of the turbulence volume followed by a projection onto the DMs accounting for the considered FOV of interest. The case where the turbulent layers are assumed to match the mirror positions, which might be a crude ap© 2001 Optical Society of America

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Fig. 1. Concept of a MCAO system. Several DMs are conjugated to different heights in the atmosphere. made on several GSs located in the FOV.

proximation, is also considered for comparison. This model-approximation (MA) approach will rely on the notion of equivalent turbulent layers.10 In both approaches, the regularization of the ill-posed problem for the phase reconstruction is studied carefully, and the prior knowledge available both on turbulence and on noise statistics has been incorporated into the reconstruction. Note that, in the paper, we consider only open-loop conditions; that is, all the wave-front statistics are derived from the Kolmogorov or the von Ka´rma´n theory. No temporal behavior is considered. This open-loop hypothesis is more restrictive than the study performed by Ellerbroek15 but allows us to obtain simple analytical formulas and to propose physical interpretations of the results. The theoretical development of the optimal approach for large-FOV phase reconstruction is proposed in Section 2. The crude model approximation where the turbulence is assumed to match the DM position, is considered in Section 3. A comparison of the performance of different phase reconstruction approaches [conventional truncated singular value decomposition (SVD), optimal approximation, and model approximation] is then proposed in Section 4. We study the influence of a well-chosen regularization (Kolmogorov statistics) on the phase reconstruction for a large FOV. The appeal of the optimal

The wave-front analysis is

phase estimation approach, and then the need of accurate C n2 measurements during the observing runs, is demonstrated.

2. OPTIMAL APPROACH FOR LARGEFIELD-OF-VIEW PHASE RECONSTRUCTION The concept of MCAO has been studied in the last ten years by many authors.7–20 The goal of MCAO is to compensate well for the turbulent wave fronts not only in one direction but also in a specified FOV of interest 兵 ␣其 FOV (larger than the classical isoplanatic patch4). Let us assume that the turbulent atmosphere is composed of a discrete sum of thin turbulent layers located at different heights.21 In the near-field approximation,21 the resulting phase ⌽(r, ␣) in the telescope pupil is given, for a sky direction ␣, by Nt

⌽ 共 r, ␣兲 ⫽

兺 ␾ 共 r ⫹ h ␣兲 , j

j

(1)

j⫽1

where r is the pupil coordinate and ␾ j ( ␳j ) are the phase perturbations in the jth atmospheric turbulent layer located at the altitude h j . N t is the number of turbulent layers.

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The wave front is measured in the telescope pupil for the discrete set of GS directions 兵 ␣i 其 GS . The correction is computed by using all these measurements 关 兵 ⌽ m (r, ␣i ) 其 GS兴 and considering several DMs located at different heights. Therefore the key points for the design of a MCAO system are the number and the position of DMs and GSs and, of course, the phase reconstruction method that gives the correction phase for the different DMs. In the present paper, we focus on this phase estimation algorithm, since the MCAO performance with respect to the DM and GS number and positions has already been studied.10–13 The main result of these previous works is that for telescope diameters of 4–8 m and for K-band (2.2-␮m) imaging, only a small number of DMs and GSs is needed to obtain quasi-uniform correction in a large FOV (typically larger than 1 arc min). A. Position of the Problem The goal of our approach is to minimize the residual phase variance in a specified FOV of interest, that is, to derive a MMSE estimator.15 This phase estimator is defined as the one that minimizes a quadratic distance averaged on the FOV of interest 兵 ␣其 FOV between the resulting true and correction phases:

⑀⫽

冓冕

兵 ␣其 FOV

ˆ (r, ␣) ⫺ ⌽(r, ␣) 储 2 d␣ 储⌽

冔

,

(2)

⌽,noise

where 具 • 典 ⌽,noise stands for a mathematical expectation on both turbulence and WFS noise outcomes, 储 • 储 2 denotes the spatial variance in the telescope pupil, ⌽(r, ␣) is the true phase in a given direction ␣ in 兵 ␣其 FOV , and ˆ (r, ␣) is the estimated correction phase in that direction ⌽ ˆ (r, ␣) under the con␣. The problem is to estimate ⌽ straint that it will be generated by a finite number of DMs, using not only the WFS measurements but also a priori information that we have on the turbulent wave front in the atmospheric volume. Let us consider that we have N GS GSs, i.e., N GS WFS measurements. For each WFS, we assume that the measured phase can be expressed as ⌽ m 共 r, ␣i 兲 ⫽ ⌽ 共 r, ␣i 兲 ⫹ n i 共 r兲 ,

Nt

⌽ 共 r, ␣i 兲 ⫽ m

兺 ␾ 共 r ⫹ h ␣ 兲 ⫹ n 共 r兲 . j

j

i

(4)

i

j⫽1

The unknowns of the problem are the correction phases ␾ˆ k to be estimated for each DM so as to minimize the criterion defined in Eq. (2). Of course, for practical reasons, the DM number (N DM) will always be smaller than the number of turbulent layers (N t ). In that case, we have, for a given direction ␣, N DM

ˆ 共 r, ␣兲 ⫽ ⌽

兺

␾ˆ k 共 r ⫹ h k ␣兲 .

(5)

k⫽1

The DM positions h k are, for instance, computed as presented in Refs. 10, 11, and 13 by using an average C n2 profile. Then Eq. (2) becomes

冓冕 冐兺

N DM

⑀⫽

兵 ␣其 FOV k⫽1

␾ˆ k 共 r ⫹ h k ␣兲

Nt

⫺

兺 ␾ 共 r ⫹ h ␣兲 j

j

j⫽1

冐 冔 2

d␣

.

(6)

␾ ,noise

For the sake of clarity, let us rewrite all the equations defined above in a matrix form. Equations (1), (4), and (5) become, respectively, N

(7)

N

(8)

⌽ 共 r, ␣兲 ⫽ M␣ t ␾, ⌽ m 共 r, ␣i 兲 ⫽ M␣ t ␾ ⫹ n i , i

ˆ 共 r, ␣兲 ⫽ MN DM␾ ˆ, ⌽ ␣ N M␣ t

(9)

N M␣ DM

where and are the matrices that perform the sum of the contributions of each wave front ␾ j ( ␳j ) and ␾ˆ k ( ␳k ) on the telescope pupil for a given direction ␣. ␾ ˆ are defined as and ␾

␾⫽

(3)

where ␣i is the angular position of the ith GS. For the sake of simplicity, Eq. (3) assumes that the WFS directly gives phase map measurements and that n i follows Gaussian statistics (central limit theorem). We suppose here that ⌽ m (r, ␣i ) is measured on a basis with an infinite number of modes. The measurements are limited only by the noise. Indeed, this noise on slope measurements given by a Shack–Hartmann (SH) WFS is given by the sum of several pixels in the calculation of the center of gravity and through the reconstruction; the noise on the phase results from a large number of subaperture contributions. To account for the noise propagation through the reconstruction from SH data, we color this Gaussian noise n i (r) with, in the Fourier domain, a power spectral density following an f ⫺2 law.22 The turbulent phase on the telescope pupil is given by the sum of all the turbulent layer contributions [see Eq. (1)]; then Eq. (3) can be rewritten as

2529

冉冊 冉 冊 ␾1 ] ␾j , ] ␾N t

ˆ ⫽ ␾

ˆ1 ␾ ] ˆk ␾ . ] ˆN ␾ DM

The criterion to be minimized is then

⑀⫽

冓冕

兵 ␣其 FOV

N

N

ˆ ⫺ M t ␾ 储 2 d␣ 储 M␣ DM␾ ␣

冔

(10)

.

(11)

␾ ,noise

B. Optimal Solution In general, the calculation of the MMSE estimator is not tractable unless the estimator is assumed to be linear with respect to the data (linear MMSE estimator). It is important to note that in the case of joint Gaussian statistics for the noise and the turbulence (which is the case in our problem), this linear estimator is identical to the true MMSE estimator.23 We can therefore seek the MMSE solution in the form ˆ ⫽ W⌽m ␾

with

N

⌽m ⫽ MN t ␾ ⫹ n, GS

(12)

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where the new unknowns are the elements of the matrix N W. MN t , ⌽m , and n are matrices and vectors defined as

WN DM⫽N t ⫽ C␾共 MN t 兲 T关 MN t C␾共 MN t 兲 T ⫹ Cn兴 ⫺1 . N

N

GS

N

GS

GS

(18)

GS

N MN t GS

⫽

N 共 M␣ t 1

,...,

N M␣ t i

,...,

N M␣ t N

兲,

冉 冊 冉 冊 冉 冊 ⌽ m 共 r, ␣1 兲 ] ⌽ m 共 r, ␣i 兲 , ] ⌽ m 共 r, ␣N GS

⌽m ⫽

(13)

n 1 共 r兲 ] n i 共 r兲 . ] n N GS 共 r兲

n⫽

⑀⫽ (14)

W is the reconstruction matrix

W⫽

W1 ] Wj , ] WN DM

ˆ j ⫽ Wj ⌽m . ␾

(15)

Putting Eq. (12) in Eq. (11) yields

冕

兵 ␣其 FOV

N Nt ␾ ⫹ Wn兲 具 储 M␣ DM共 WMN GS

N

⫺ M␣ t ␾ 储 2 典 ␾ ,noise d␣ .

(16)

This equation must be minimized with respect to W. The explicit minimization of Eq. (16) is presented in Appendix A. The final result is W⫽

冋冕

N

兵 ␣其 FOV

⫻

N

共 M␣ DM兲 TM␣ DM d␣

冋冕

N

兵 ␣其 FOV

N

册

⫹

共 M␣ DM兲 TM␣ t d␣

N

N

GS

N

GS

GS

兵 ␣其 FOV

N ˆ 储 M␣ t 共 ␾ ⫺ ␾ 兲储 2 d␣

冔

.

(17)

where C␾ and Cn , defined in Appendix A, are the generalization for several layers and several GSs of the classical turbulence and noise covariance matrices. The T and ⫹ superscript symbols denote the transpose and the generalized inverse matrix, respectively. Reintroducing Eq. (17) into Eq. (16), we can easily obtain an analytical expression of the MCAO error as a function of FOV angle.24 The true model derived above assumes an infinite number of modes to describe the turbulent phase and its correction. For practical reasons and for purposes of limiting the numerical calculations, the number of modes is, however, limited. This undermodeling in the direct problem induces correction errors. It is important to account for these errors, which is easily done by using a Monte Carlo simulation (that is, we simulate turbulent wave fronts, and we apply a correction derived by using W). Furthermore, such a simulation allows us to account for slight discrepancies from the true turbulence model. For instance, the wave fronts are simulated with von-Ka´rma´n statistics (finite outer scale), while the reconstruction matrix uses a Kolmogorov regularization. N N When M␣ DM ⫽ M␣ t , that is, when the DMs are exactly located on the turbulent layers, Eq. (17) simply reads as

(19)

␾ ,noise

In that case, it can be shown that the minimization of the residual phase variance in the telescope pupil ⑀ is equivalent to the minimization of (20)

that is, to the minimization of the residual phase variance in each layer (whatever the FOV of interest). Our estimator is therefore equivalent, in that case, to that of a tomographic approach.19 In particular, there is no dependence on the field angle. Such a DM correction minimizes the phase residual variance whatever the FOV position. But this case is only idealistic. In fact, the number of DMs will always be smaller than the number of turbulent layers. It is, however, interesting to note that in the general case, the solution given in Eq. (17) actually consists of this tomographic reconstruction on all turbulent layers corresponding to Eq. (18) followed by a projection onto the solution space (corresponding to the small number of altitudes where the DMs are located). The projection operator is therefore PN DM ,N t ⫽

冋冕

N

兵 ␣其 FOV

⫻

册

⫻ C␾共 MN t 兲 T关 MN t C␾共 MN t 兲 T ⫹ Cn兴 ⫺1 ,

冓冕

ˆ ⫺ ␾ 储 2 典 ␾ ,noise , ⑀⬘ ⫽ 具 储 ␾

so that

⑀⫽

Equation (11) then becomes

GS

冋冕

N

共 M␣ DM兲 TM␣ DM d␣

兵 ␣其 FOV

N

N

册

⫹

册

共 M␣ DM兲 TM␣ t d␣ .

(21)

This projection matrix is directly linked, through the integral in ␣, to the FOV of interest 兵 ␣其 FOV , where the correction is optimized. Indeed, when the mirror positions do not match the turbulent layers (N DM ⬍ N t ), an overall correction in the FOV is not possible. Optimizing for a particular FOV position may degrade the correction in other positions. Trade-offs have to be made for a specified set of FOV positions. The projection PN DM ,N t performs optimally these trade-offs. One can also show from Eq. (16) that, whatever the position in the FOV, the residual phase variance is minimal for one DM per layer (N DM ⫽ N t ). However, we will see in Section 4 that this ultimate performance is almost reached with a small number of DMs when considering a reasonable FOV of interest. Note that there is an analogy between the MCAO correction of the turbulence volume with a finite number of DMs, as presented here, and the correction of the turbulent phase in classical AO with a finite number of actuators, as proposed by Wallner.25 In both cases, the correction space is smaller than the unknown space (number of DMs smaller than the number of layers or the number of actuators smaller than the number of turbulent modes). This leads to a similar form of the reconstruction: first, a ‘‘full’’ reconstruction, followed by a projection onto the finite-space solution. ˆ ⫽ W ⌽m , which With the matrix W, one can compute ␾ gives the estimated correction phase on each DM that en-

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Vol. 18, No. 10 / October 2001 / J. Opt. Soc. Am. A

2531

sures a minimal residual phase variance for all the directions of the specified FOV 兵 ␣其 FOV . Of course, the computation of Eq. (17) requires the knowledge of the N turbulence profile for the computation of M␣ t and C␾ . A 2 real-time measurement of the C n profile can, for instance, be obtained with a generalized SCIDAR.26 In Section 3, a second approach, based on a cruder turbulence model, is presented.

3. MODEL-APPROXIMATION APPROACH In this approach, we assume that all the turbulence is located on the DMs. The C n2 profile is modeled only by a small number (N EL) of turbulent layers, called equivalent layers (ELs), in which are located the N DM ⫽ N EL DMs. The computation of the EL position and strength is done by a sampling of the C n2 profile into N EL slabs.10,11 Using this simplified turbulence model, one can estimate the correction phase with the approach proposed in Subsection 2B. All the equations remain valid, with N t and N DM replaced by N EL . Therefore the direct problem can be rewritten as N EL

⌽ ␣共 r兲 ⯝

兺 ␾ 共 r ⫹ h ␣兲 , j

j

j⫽1

N EL

⌽ ␣mi 共 r兲

⯝

兺 ␾ 共 r ⫹ h ␣ 兲 ⫹ n 共 r兲 . j

j

i

i

(22)

j⫽1

Consequently, the reconstruction matrix is deduced from Eq. (18): WMA ⫽ C␾共 MN EL 兲 T关 MN ELC␾共 MN EL 兲 T ⫹ CN兴 ⫺1 . N

N

GS

N

GS

GS

(23)

ˆ The reconstruction phases are therefore given by ␾ ⫽ WMA⌽m . This MA solution has already been derived in a previous paper10,11 following a maximum a posteriori approach. But MMSE and maximum a posteriori estimators are, in any event equivalent23 here on account of the Gaussian statistics of the noise and the turbulence. Now let us compare the two approaches and discuss their similarities and differences. They are both derived from the same theoretical development based on the MMSE criterion. The only (but important) difference is that the model approximation uses a simplified direct problem, which leads to a suboptimal solution.

4. RESULTS AND PERFORMANCE A. Simulation Tool Let us consider a modal decomposition of the wave fronts onto the Zernike basis. The phase screen on each turbulent layer j becomes ⬁

␾ j 共 ␳j 兲 ⫽

兺a

l, j Z l, j 共 ␳j 兲 ,

(24)

l⫽2

where Z l, j ( ␳) is the lth Zernike polynomial defined on a metapupil of diameter D j depending on the telescope diameter D, the layer altitude h j , and the maximal FOV angle ␣ max considered: D j ⫽ D ⫹ 2h j ␣ max .

(25)

Of course, all the equations presented above are still valid in this basis. In particular, one can note that in

Fig. 2. Decentered part of the metapupil associated with the altitude h j . The variable vector r is defined on the telescope pupil. The zone of interest is centered on h j ␣ i .

ˆ k are simply vectors of Eqs. (10) the the ␾j and the ␾ Zernike coefficients a l, j and a l,k ; C␾ is therefore a generalization of the Zernike covariance matrix given by Noll.27 The measured phase is also decomposed onto the Zernike polynomial basis. Noise is added on each Zernike coefficient by using a noise covariance matrix Cn . The SH WFS is not really simulated but its characteristic noise propagation is accounted for through the use of Cn , which considers that we measure Zernike polynomial derivatives.28 For a direction ␣ i , only a part of the metapupil associated with the layer j is viewed: a disk of diameter D cenN N tered on h j ␣i . In this particular basis, MN EL, MN t , N

GS

N

GS

M␣ t , and M␣ EL are computed as presented in Ref. 11. It consists of the decomposition of each decentered Zernike polynomial 关 Z l, j (r ⫹ ␣h j ) 兴 onto a Zernike basis defined on the telescope pupil (see Fig. 2). Ragazzoni et al.20 have shown that the number of modes required for such a decomposition is given by the number l of the metapupil Zernike polynomial. This result is important, since it ensures that the dimension of each matrix M will be linked only to the number of Zernike polynomials considered in each (turbulent or DM) layer. For example, in the case of a two-DM system, where 66 Zernike modes are sought on the first DM and 135 are sought on the second DM, the N dimension of the matrix M␣ DM will be only 135 ⫻ (66 ⫹ 135). Ideally, the number of Zernike modes must be infinite, but for practical reasons (computation time, matrix sizes), only a finite number of Zernike modes is considered both for the measured phases and for the DMs. B. Simulation Parameters Let us consider a four-layer profile defined as follows: No.

Position (km)

Strength (%)

1 2 3 4

0 2.5 5 7.5

25 25 25 25

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J. Opt. Soc. Am. A / Vol. 18, No. 10 / October 2001

The phase screens on each turbulent layer are simulated by McGlamery’s method.29 The simulated phase screens are large enough with respect to the telescope pupil to ensure that L 0 /D ⯝ 4, where L 0 is the outer scale of the

Fusco et al.

turbulence. The total r 0 is equal to 0.1 m at 0.5 ␮m. This leads to an isoplanatic angle ␪ 0 (defined with the Fried formula4) equal to 1.44 arc sec at 0.5 ␮m and 8.52 arc sec at 2.2 ␮m. We consider a 4-m telescope, and sev-

Fig. 3. (a) Turbulence (four layers) and DM (one, two, and four) repartition for the four systems presented in Subsection 4B. (b) Geometrical repartition of the GS pupil projection and FOV on the highest layer (h ⫽ 7.5 km). The physical size of the DM is equal to the physical size of the layers.

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Vol. 18, No. 10 / October 2001 / J. Opt. Soc. Am. A

eral MCAO systems are presented in Fig. 3. The several MCAO systems are constructed as follows: a. One GS on the optical axis and one DM (with 135 corrected modes) conjugated at 3.75 or 6.5 km (Subsection 4.D.1). b. One GS on the optical axis and four DMs (with, respectively, 66, 120, 135, and 230 corrected modes) conjugated on the four turbulent layers (tomographic reconstruction) (Subsection 4.D.1). c. Three GSs located at the vertices of an equilateral triangle with a separation equal to 70 arc sec and two DMs (with, respectively, 66 and 135 corrected modes) conjugated at 1.25 and 6.25 km (Subsections 4.C and 4.D.2). d. Three GSs located at the vertices of an equilateral triangle with a separation equal to 70 arc sec and four DMs (with, respectively, 66, 120, 135, and 230 corrected modes) conjugated on the four turbulent layers (tomographic reconstruction) (Subsection 4.D.2).

2533

compared with a more classical approach based on a least-squares minimization, is performed in Subsection 4.C.

C. Influence of the Kolmogorov Regularization Let us consider the three-GS and two-DM system (case c) presented in Subsection 4.B. The classical approach to inverting the ill-posed problem of the phase correction estimation in each DM is to use a least-squares minimization,18 that is, to consider a truncated singular value decomposition (SVD). With our notation, this wave-front estimator is therefore given by the following well-known relation: ˆ ⫽ 关共 MN EL 兲 TMN EL兴 ⫹ 共 MN EL 兲 T⌽ m , ␾ N N N GS

GS

GS

(27)

N

Ideally, a large number of the Zernike modes should be used, but for practical reasons the number of modes per DM considered here is quite reasonable. First, it is important to note that in cases b and d the number of DMs is equal to the number of true layers; therefore these two cases can be seen as the ultimate performance of cases a and c, as mentioned in Subsection 2B. The first two cases (a and b) correspond to the first step of a MCAO system, since they are composed of one or several conjugated DMs, but they still only use one GS. Therefore all the information on the off-axis phases is given only by the prior information that we have on the turbulence volume (the C n2 repartition and the Kolmogorov statistics of the phase). An example of such a system (case a) is under construction for the 8-m GeminiNorth telescope.30 This AO system (Altair) can be seen as the first order of a MCAO system. In our case, we have considered a 4-m telescope, but all the results can easily be extended to the 8-m case by a simple scaling of the FOV 兵 ␣其 FOV by the diameter ratio and the number of corrected modes by the square of this ratio. Cases c and d represent more complex systems, since they are composed of both several GS directions and several conjugated DMs. The Cn matrix is obtained by considering SH WFSs that measure the wave front in each GS direction. The SNR on each SH (defined as the ratio between the turbulence variance and the noise variance) is computed for a 7 ⫻ 7 subaperture SH and is equal to 10. It roughly corresponds to an 11th-magnitude GS. For each system, the maximal considered FOV (which defines the physical size of each DM; see Fig. 3) is equal to 150 arc sec. The performance of the different methods is evaluated in terms of a Strehl ratio (SR) approximated by 2 exp关⫺␴res ( ␣) 兴 , which is valid for good corrections. 2 ␴ res( ␣) is computed by 2 ˆ 共 r, ␣兲储 2 典 , ␴ res 共 ␣兲 ⫽ 具 储 ⌽ 共 r, ␣兲 ⫺ ⌽

(26)

where 具 • 典 is an average on 100 decorrelated simulated phases. First, a study of the gain brought by the regularization term (Kolmogorov regularization) in Eq. (17) and (18),

where MN EL is the interaction matrix between the DMs GS

N

N

and the WFSs. Because (MN EL) TMN EL is an illGS GS conditioned matrix (see Fig. 4), the inversion is made by using a SVD in which the lower-value modes are set to 0 in order to avoid the noise amplification. Of course, this truncation can be seen as a crude regularization, and it is easy to show that this approach is less optimal than the use of a well-chosen regularization term (the Kolmogorov statistics in our case), as shown for various turbulencerelated applications in Refs. 10, 23, 31, and 32. One can see in Fig. 5 that the use of a Kolmogorov regularization [model approximation, Eq. (23)], where C␾ is computed by assuming that all the turbulence is equally distributed on the two DMs, gives better results than the classical truncated SVD [Eq. (27)] whatever the chosen truncation threshold (note that the optimal choice of this threshold is one of the major problems of the SVD approach). For the optimal SVD threshold (␭ max/50 here), only 72 modes are corrected from the available 199 modes of the system (the piston is not considered). This optimal threshold is chosen as the one that gives the minimal mean residual variance in the whole FOV of interest (150 arc sec⫻150 arc sec). The great advantage of the well-chosen regularization (as derived from a MMSE approach) is that one does not have to adjust any parameters, since the optimal regularization is directly derived from the noise and turbulence (on each EL) statistics. It seems clear, in this example, that an adequate regularization of the inverse problem is required in a large FOV to obtain good performance. This Kolmogorov regularization avoids the noise amplification but also allows a good phase extrapolation where the phase is not measured or only partially measured. Now let us compare the optimal approach defined in Eq. (17) and the MA approach defined in Eq. (23). In Subsection 4.D.1, the one-GS and one-DM system (case a) is studied. In Subsection 4.D.2, the three-GS and twoDM system (case c) is considered. Note that for each GS configuration the best possible performance is obtained, as explained in Section 2, with one DM per turbulent layer, that is, four DMs here (cases b and d).

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J. Opt. Soc. Am. A / Vol. 18, No. 10 / October 2001

Fusco et al.

The DM is located at 3.75 km (center of gravity of the C n2 profile). If we use the MA method (all the turbulence supposed to be on the DM), then because the DM position is well chosen, a good extrapolation is possible. Indeed, as shown in Fig. 6, a quasi-optimal (close to the ultimate four-DM performance) SR is obtained in a 20-arc sec ⫻ 20-arc sec FOV. Beyond, if the science object is far from the optical axis (typically, 50 arc sec), the degradation of the extrapolation between one and four DMs becomes important (SR ⫽ 3% for one DM and 8% for four DMs). Now we take into account the knowledge of the true C n2 profile for the optimization of the DM correction in the science object direction. In our example, we consider a Fig. 4. Singular value of the systems versus mode number. The different considered thresholds are plotted. The optimal threshold (optimal result) is chosen as the one that gives the minimal residual variance in the whole FOV of interest (150 arc sec ⫻ 150 arc sec), as shown in Fig. 5.

Fig. 6. Comparison of the optimal phase estimation and MA approaches in the case of a one-GS (on the optical axis) and one-DM (conjugated at 3.75 km) system. In each case, an X cut of the FOV is presented. These simulation are made for a four-layer C 2n profile and a 4-m telescope. We plot the tomographic reconstruction (four DMs located on each turbulent layer) for comparison. Fig. 5. Comparison between the Kolmogorov regularization (solid curve) and a SVD using different thresholds. The SR versus the FOV position is plotted for the x axis defined in Fig. 3(b).

D. Comparison of the Optimal and Model-Approximation Approaches We have just shown that a well-chosen Kolmogorov regularization always gives better results than a simple truncated SVD approach. Let us now compare the two phase estimation methods presented in Sections 2 and 3 by using this regularization. More precisely, let us show the gain brought by the optimal phase estimation approach in which the prior information on the turbulence profile is more precise and for which an optimization in a given FOV of interest 兵 ␣其 FOV is performed. 1. Mono-Guide-Star and Mono-Deformable-Mirror System Let us first consider the simple but illustrative case of a system composed of only one GS and one conjugated DM (case a). In such a configuration, two different cases can be studied. • Case of a well-placed DM with respect to the turbulence profile (Fig. 6).

Fig. 7. Comparison of the optimal phase estimation and MA approaches in the case of a one-GS (on the optical axis) and one-DM (conjugated at 6.5 km) system. Note that the DM is misplaced with regard to the C 2n profile. In each case, an X cut of the FOV is presented. These simulations are made for a four-layer C 2n profile and a 4-m telescope. We plot the tomographic reconstruction (four DMs located on each turbulent layer) for comparison.

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Vol. 18, No. 10 / October 2001 / J. Opt. Soc. Am. A

2535

Fig. 8. Upper plots: comparison of iso-SR maps between the MA approach (two DMs) and the tomographic case (four DMs). Lower plots: iso-SR maps computed by using the optimal approach in the case of a two-DM and 3-GS system (case c). Three optimized FOVs are considered: 20, 60, and 120 arc sec. Note that, in all maps, only SR ⭓ 10% are plotted. Black corresponds to 10%, and the succeeding levels (dark to light) are 20%, 30%... .

Fig. 9. Comparison of the optimal and MA approaches is for two DMs and a four-layer atmospheric profile. The FOVs of interest are, for the optimal approach, 20, 60, and 120 arc sec. The MA approach (in which the results are independent of a given FOV) is plotted as a dotted curve. The tomographic phase estimation (four DMs in the four turbulent layers) is plotted (solid curve) for comparison. All these curves are, in fact, an X cut (at Y ⫽ 0) of each corresponding iso-SR map presented in Fig. 8.

5-arc sec ⫻ 5-arc sec FOV around ␣ ⫽ 50 arc sec. The optimal approach given by Eq. (17) is then used, and we obtain with only one DM nearly the same results as those for four DMs (the difference is only approximately 0.1% in SR) in this particular portion of the FOV.

• Case of a misplaced DM with respect to the turbulence profile (Fig. 7). The atmospheric conditions are the same as those in the case above; the only difference is in the DM position. Here the DM is misplaced with respect to the turbulence profile (DM located at 6.5 km). Of course, the model approximation gives a poor extrapolation, considering that to regard all the turbulence as being concentrated at 6.5 km is a bad approximation. In Fig. 7, the MA results (dashed curve) are strongly degraded compared with those in Fig. 6. But it is shown that the optimal reconstruction approach still gives good results, similar to the well-placed DM case, in the specified 5-arc sec ⫻ 5-arc sec FOV around ␣ ⫽ 50 arc sec. Since we have taken into account the true C n2 profile, the optimal approach is able to find the best DM deformation to optimize the correction in the direction of the science object, even if the DM position is far from the optimum. To summarize, even if we have only one GS, the best way to optimize the correction in a large FOV is to have the same number of DMs as the number of turbulent layers. Of course, this is impossible for practical reasons; we have then shown that the use of the information on the true C n2 profile in the reconstruction process yields impressive results even if we have only one DM. This approach could therefore be used in conventional AO systems (case of one DM at 0 km; that is, in the pupil plane) to increase their performance in the field when the C n2 measurements are available.

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J. Opt. Soc. Am. A / Vol. 18, No. 10 / October 2001

However, high and quasi-uniform correction quality in the whole FOV cannot be achieved with only one GS, and in Subsection 4.D.2 three-GS configurations are studied. 2. Multi-Guide-Star and Multi-Deformable-Mirror System Now let us consider a more complex MCAO system (case c of Subsection 4.B) composed of two DMs and three GSs located on the vertices of an equilateral triangle (GS separation ⫽ 70 arc sec). For the two-DM system, we plot in Figs. 8 and 9 a comparison between the optimal reconstruction method (for different FOVs of interest: 20, 60, and 120 arc sec) and the MA reconstruction method. The best-performance case, obtained with four DMs conjugated in the four turbulent layers (case d), is also plotted for comparison. Figures 8 and 9 show the appeal of the optimal phase reconstruction approach, which allows an optimal reconstruction in the FOV of interest 兵 ␣其 FOV . For example, let us consider a FOV of interest centered on the optical axis and having a size of 20, 60, or 120 arc sec. With only two DMs, the correction is nearly the same as the best performance obtained with the four-DM system. The correction at the center of the FOV, which is equal only to 31% (in terms of SR) with the MA approach, is equal to 49.3% for the 20-arc sec ⫻ 20-arc sec optimized area, 48.5% for the 60-arc sec ⫻ 60-arc sec optimized area, and 47% for the 120-arc sec ⫻ 120-arc sec optimized area. The SR for the limit case of four DMs is equal to 49.6%. We note the significant increase of the SR when using the true C n2 profile in the optimal approach. A very slow decrease of onaxis performance with increasing FOV of interest is observed. Note that in both the full tomographic and the MA case the best reconstruction is achieved in the GS directions. Of course, it is important that the FOV of interest be well specified, since outside this region of interest the SR decreases quickly. This is a consequence of the tight compromise performed in the optimal estimation.

Fusco et al.

Another example is presented in Fig. 10, where the optimized FOV consists of two areas located at ⫺60 and 60 arc sec (the size of each area is only 5 arc sec in diameter). In that case, for the ⫺60-arc sec position the SR goes from 7.2% (two-DM MA reconstruction) to 23% (two-DM optimal reconstruction), and for the 60-arc sec position the corresponding SR goes from 27% to 44%. Of course, we have globally lost in the 150-arc sec ⫻ 150-arc sec FOV, but we have optimized the correction in the two areas of interest.

5. CONCLUSION In this paper, we have presented an optimal phase reconstruction for MCAO systems. This optimal approach derives from a MMSE estimator that minimizes the mean residual phase variance in the FOV of interest. This optimal approach accounts for the fact that the number of DMs is always smaller than the number of atmospheric layers. It is shown to correspond to a full tomographic reconstruction of the turbulence volume followed by a projection on the DMs. This optimal approach requires a good knowledge of the C n2 profile, and therefore a generalized SCIDAR must be coupled to the MCAO system. Even if the mirror positions are not well adapted to the current C n2 profile, the knowledge of this profile is incorporated into our estimator, which therefore provides a quasi-ultimate performance in the FOV of interest. For comparison, we show the result obtained with a cruder approach. An equivalent C n2 profile composed of a small number of ELs (equal to the number of DMs) is computed, and we assume that each DM matches an EL. This can be an interesting alternative when only a crude C n2 profile knowledge is available. In both cases, great care has been taken to regularize the inverse problem. This well-chosen regularization brings a nonnegligible gain in the phase reconstruction for a large FOV, compared with that from a classical leastsquares estimation using a truncated SVD. To achieve a given performance, a system using an optimized approach will need fewer GSs and hence will gain in terms of sky coverage (if natural GSs are used) or system complexity (if laser GSs are used). Of course, high SRs in a large FOV require a larger number of GSs and a slight increase of DM number. However, even in such demanding conditions, the optimized reconstructor still limits the increase of the system complexity. In the present paper, we consider only an open-loop case (which allows us to use the turbulent phase covariance matrix), but future work should include a complete closed-loop modeling.

APPENDIX A: MINIMUM-MEAN-SQUAREERROR SOLUTION Fig. 10. Comparison of the optimal (dashed curve) and the MA (dotted curve) approach for two DMs and a four-layer atmospheric profile. The FOV of interest is, for the optimal approach, two areas of 5-arc sec diameter located at ⫺60 and 60 arc sec. The tomographic phase estimation (four DMs in the four turbulent layers) is plotted (solid curve) for comparison.

1. Matrix Differentiation Let us first recall some important results, used in the paper, on the theory of matrix differentiation.33 Let us consider two matrices A and B and define the first derivative of A with respect to B as follows:

Fusco et al.

冋 册 ⳵A

⳵ B11

⳵A

⫽

⳵B

⳵A

¯

⳵ Bm1

⑀⫽

⳵ B1n

]

⳵A

Vol. 18, No. 10 / October 2001 / J. Opt. Soc. Am. A

]

.

N EL

⳵ Bmn

⫻ 具 ␾␾

N

N

⫺

(A2)

具 ␾␾ T典 ( 具 nnT典 ) are denoted C␾ (Cn) and defined as

Cn ⫽

冋 冋

N

N

⫹ M␣ ELWCnW T共 M␣ EL兲 T兴 d␣. (A4) Using the formulas of the matrix differentiation described in Appendix A.1, we obtain ⳵⑀ N N N N ⫽ 关共 M␣ EL兲 TM␣ ELWMN t C␾共 MN t 兲 T GS GS ⳵W 兵 ␣其 FOV

冕

N

N

N

⫺ 共 M␣ EL兲 TM␣ t C␾共 MN t 兲 T GS

N N 共 M␣ EL兲 TM␣ t WCn兴 d␣

⫹ ⫽ 0, (A5) where 0 is the null matrix (matrix with all elements equal to 0). Equation (A5) leads to the final result: W⫽

冋冕

N

兵 ␣其 FOV

N

共 M␣ EL兲 TM␣ EL d␣

册 冋冕 ⫹

兵 ␣‰FOV

N

N

⫹

N

GS

N

GS

GS

N

共 M␣ EL兲 TM␣ t d␣

⫻ C␾共 MN t 兲 T关 MN t C␾共 MN t 兲 T ⫹ Cn兴 ⫺ 1 ,

册

(A6)

where A denotes the generalized inverse of the matrix A. Additional author information: All authors: phone, 33-1-46-73-40-40; fax, 33-1-46-73-41-71; e-mail, 兵last name其@onera.fr. URL: http://www.onera.fr/dota.

1.

N

具 ␾ 1 ␾ 1T典

0

0

0

0

0



0

0

0

0

0

␾ j ␾ jT典

0

0

0

0

0



0

0

0

0

0

T 具 ␾ Nt␾ N 典 t

具

N

REFERENCES

N M␣ t 兲 T

⫹ M␣ ELW具 nnT典 共 M␣ t W兲 T兴 d␣.

C␾ ⫽

N

GS

N

GS

N Nt 典 共 M␣ ELWMN GS

N

GS

N

WMN t ⫺ M␣ t 兲

T

N

GS

2. Optimal Minimum-Mean-Square-Error Solution The goal is to derive the MMSE criterion defined in Eq. (16) with respect to W. First, let us recall that for a given matrix A and a given vector v, we have the following relation: 储 Av储 2 ⫽ trace关 Av(Av) T兴 . Then, assuming that the noise and turbulent phase statistics are independent, Eq. (16) becomes

兵 ␣其 FOVtrace关共 M␣

N

N

• ⳵ A/ ⳵ BT ⫽ ( ⳵ AT/ ⳵ B) T. • (⳵/⳵B) 关 trace(BA) 兴 ⫽ ( ⳵ / ⳵ B) 关 trace(ATB) 兴 ⫽ ( ⳵ / ⳵ B) ⫻ 关 trace(AB) 兴 ⫽ AT. • ( ⳵ / ⳵ B) 关 trace(BTA) 兴 ⫽ ( ⳵ / ⳵ B) 关 trace(BAT兴 ⫽ A. • ( ⳵ / ⳵ B) 关 trace(ABC) 兴 ⫽ ATCT. • ( ⳵ / ⳵ B) 关 trace(BAB) T)] ⫽ 2BA. • ( ⳵ / ⳵ B)(xTBABTx) ⫽ 2xxTBA, which leads to ( ⳵ / ⳵ B) ⫻ 关 trace(CTBABTC) 兴 ⫽ 2CTCBA.

冕

N

trace关 M␣ t C␾共 M␣ t 兲 T

⫺ 2M␣ t C␾共 MN t 兲 TW T共 M␣ EL兲 T

Now let us suppose that A, B, and C are real matrices and that x is a vector. Then the following properties hold:

⑀⫽

N

兵 ␣其 FOV

⫹ M␣ ELWMN t C␾共 MN t 兲 TW T 共 M␣ EL兲 T

(A1)

⳵A

¯

冕

2537

册 册

具 n 1 n 1T典

0

0

0

0

0



0

0

0

0

0

具 n i n iT典

0

0

0

0

0



0

0

0

0

0

T 具 n N GSn N 典 GS

2.

3. 4. 5.

, 6.

7. 8.

,

(A3)

which can be seen as an N t -layer (N GS-GS) turbulence (noise) covariance matrix. Note that each 具 ␾ j ␾ jT典 is the Kolmogorov covariance matrix defined for the jth layer. Note that we have assumed that all the turbulent layers are statistically independent,21 as well as the noise on each GS measurement. Finally, the criterion to be minimized with respect to W is

9. 10. 11.

12.

J. W. Hardy, J. E. Lefevbre, and C. L. Koliopoulos, ‘‘Real time atmospheric compensation,’’ J. Opt. Soc. Am. 67, 360– 369 (1977). G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Le´na, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, ‘‘First diffraction-limited astronomical images with adaptive optics,’’ Astron. Astrophys. 230, 29–32 (1990). F. Roddier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, Cambridge, UK, 1999). D. L. Fried, ‘‘Anisoplanatism in adaptive optics,’’ J. Opt. Soc. Am. 72, 52–61 (1982). F. Chassat, ‘‘Calcul du domaine d’isoplane´tisme d’un syste`me d’optique adaptative fonctionnant a` travers la turbulence atmosphe´rique,’’ J. Opt. (Paris) 20, 13–23 (1989). T. Fusco, J.-M. Conan, L. Mugnier, V. Michau, and G. Rousset, ‘‘Characterisation of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,’’ Astron. Astrophys. Suppl. Ser. 142, 149– 156 (2000). R. H. Dicke, ‘‘Phase-contrast detection of telescope seeing and their correction,’’ Astron. J. 198, 605–615 (1975). J. M. Beckers, ‘‘Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,’’ in Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 693–703. R. Ragazzoni, ‘‘No laser guide stars for adaptive optics in giant telescopes?’’ Astron. Astrophys. Suppl. Ser. 136, 205– 209 (1999). T. Fusco, J.-M. Conan, V. Michau, L. Mugnier, and G. Rousset, ‘‘Efficient phase estimation for large field of view adaptive optics,’’ Opt. Lett. 24, 1472–1474 (1999). T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, and G. Rousset, ‘‘Phase estimation for large field of view: application to multiconjugate adaptive optics,’’ in Propagation through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., Proc. SPIE 3763, 125–133 (1999). T. Fusco, J.-M. Conan, V. Michau, G. Rousset, and L.

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13. 14. 15.

16.

17. 18.

19. 20. 21. 22.

J. Opt. Soc. Am. A / Vol. 18, No. 10 / October 2001 Mugnier, ‘‘Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,’’ in Adaptive Optical Systems Technology, P. Wizinowich, ed., Proc. SPIE 4007, 1044–1055 (2000). A. Tokovinin, M. Le Louarn, and M. Sarazin, ‘‘Isoplanatism in multiconjugate adaptive optics system,’’ J. Opt. Soc. Am. A 17, 1819–1827 (2000). M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, ‘‘New challenges for adaptive optics: extremely large telescopes,’’ Mon. Not. R. Astron. Soc. 317, 535–544 (2000). B. L. Ellerbroek, ‘‘First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,’’ J. Opt. Soc. Am. A 11, 783–805 (1994). M. Tallon, R. Foy, and J. Vernin, ‘‘3-D wavefront sensing for multiconjugate adaptive optics,’’ in Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 517–521. D. C. Johnston and B. M. Welsh, ‘‘Analysis of multiconjugate adaptive optics,’’ J. Opt. Soc. Am. A 11, 394–408 (1994). R. Flicker, F. Rigaut, and B. Ellerbroek, ‘‘Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8-m telescope,’’ in Adaptive Optical Systems Technology, P. Wizinovich, ed., Proc. SPIE 4007, 1032–1043 (2000). M. Tallon and R. Foy, ‘‘Adaptive telescope with laser probe: isoplanatism and cone effect,’’ Astron. Astrophys. 235, 549– 557 (1990). R. Ragazzoni, E. Marchetti, and F. Rigaut, ‘‘Modal tomography for adaptive optics,’’ Astron. Astrophys. 342, L53–L56 (1999). F. Roddier, ‘‘The effects of atmospherical turbulence in optical astronomy,’’ in Progress in Optics, E. Wolf, ed. (NorthHolland, Amsterdam, 1981), Vol. XIX, pp. 281–376. G. Rousset, ‘‘Wavefront sensing,’’ in Adaptive Optics for Astronomy, D. Alloin and J.-M. Mariotti, eds. (Kluwer Academic, Carge`se, France, 1993), pp. 115–137.

Fusco et al. 23. 24.

25. 26. 27. 28. 29. 30.

31.

32.

33.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968). T. Fusco, J.-M. Conan, V. Michau, G. Rousset, and F. Asse´mat, ‘‘Multiconjugate adaptive optics: comparison of phase reconstruction approaches for large field of view’’, in Atmospheric Propagation, Adaptive Systems, and Laser Radar Technology for Remote Sensing, J. D. Gonglewski, G. W. Kamerman, and A. Kohnle, eds., Proc. SPIE 4167, 168–179 (2000). E. P. Wallner, ‘‘Optimal wave-front correction using slope measurements,’’ J. Opt. Soc. Am. 73, 1771–1776 (1983). A. Fuchs, M. Tallon, and J. Vernin, ‘‘Focusing on a turbulent layer: principle of the ‘generalized SCIDAR’,’’ Publ. Astron. Soc. Pac. 110, 86–91 (1998). R. J. Noll, ‘‘Zernike polynomials and atmospheric turbulence,’’ J. Opt. Soc. Am. 66, 207–211 (1976). F. Rigaut and E. Gendron, ‘‘Laser guide star in adaptative optics: the tilt determination problem,’’ Astron. Astrophys. 261, 677–684 (1992). B. McGlamery, ‘‘Computer simulation studies of compensation of turbulence degraded images,’’ in Image Processing, J. C. Ulrich, ed., Proc. SPIE 74, 225–233 (1976). G. Herriot, S. Morris, S. Roberts, M. Fletcher, L. Saddlemyer, J.-P. Singh, G. Ve´ran, and E. Richardson, ‘‘Innovations in the Gemini adaptive optics system design,’’ in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3353, 488–499 (1998). J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, ‘‘Myopic deconvolution of adaptive optics images using object and point spread function power spectra,’’ Appl. Opt. 37, 4614–4622 (1998). L. M. Mugnier, C. Robert, J.-M. Conan, V. Michau, and S. Salem, ‘‘Regularized multiframe myopic deconvolution from wavefront sensing,’’ in Propagation through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., Proc. SPIE 3763, 134–144 (1999). W. J. Vetter, ‘‘Derivative operations on matrices,’’ IEEE Trans. Autom. Control AC-15, 241–244 (1970).

232ANNEXE J. OPTIMAL WAVE-FRONT RECONSTRUCTION STRATEGIES FOR MULTICONJ

Annexe K “Optimization of star-oriented and layer-oriented wavefront sensing concepts for ground layer adaptive optics” M. Nicolle et al. - JOSAA - 2006

233

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Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. A

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Optimization of star-oriented and layer-oriented wavefront sensing concepts for ground layer adaptive optics Magalie Nicolle, Thierry Fusco, and Vincent Michau Office National d’Etudes et de Recherches Aérospatiales, BP 72, 92322 Châtillon, France

Gérard Rousset Observatoire de Paris Meudon, LESIA, 5 place Jules Janssen, 92195 Meudon, France

Jean-Luc Beuzit Observatoire de Grenoble, BP 53, 38041 Grenoble, France Received February 3, 2006; accepted March 10, 2006; posted April 6, 2006 (Doc. ID 67687) Multiconjugate adaptive optics is one of the major challenges in adaptive optics. It requires the measurement of the volumic distribution of the turbulence. Two wavefront sensing (WFS) concepts have been proposed to perform the wavefront analysis for such systems: the star-oriented and layer-oriented approaches. We give a performance analysis and a comparison of these two concepts in the framework of the simplest of the multiguide-star adaptive optics systems, that is, ground layer adaptive optics. A phase-related criterion is proposed to assess analytically the performance of both concepts. This study highlights the main advantages and drawbacks of each WFS concept and shows how it is possible to optimize the concepts with respect to the signal to noise ratio on the phase measurement. A comparison of their optimized versions is provided and shows that one can expect very similar performance with the two optimized concepts. © 2006 Optical Society of America OCIS codes: 010.1080, 010.7350.

1. INTRODUCTION

Although adaptive optics (AO) systems1 enable groundbased telescopes to provide diffraction-limited images, their use remains restricted to a small fraction of the sky. Indeed, their need for a bright reference star and the narrowness of their corrected field of view2 (FOV) strongly reduce the number and variety of the accessible targets.3,4 At the same time, many scientific cases could take advantage of a wider corrected FOV. Thus one of the most challenging aims of next generation turbulence compensation systems is to increase the field where turbulence is compensated for. Such systems would have many applications, such as deep field mapping, astrometry, and spectroimaging of extended scenes.5 Hence multiconjugate adaptive optics (MCAO)6,7 has been proposed to achieve diffraction-limited images on fields typically ten times wider than that of a conventional AO system. Reaching such a performance requires knowing about the volumic distribution of turbulence, and thus a wavefront sensing (WFS) concept able to deal with a multidirectional phase measurement is needed. Two concepts have been proposed to perform this measurement. They are the star-oriented7 (SO) and layer oriented8,9 (LO) approaches. The purpose of this paper is to study the performance of both WFS concepts, to highlight their advantages and drawbacks, and to propose algorithms making possible their optimization. To simplify this study, we have chosen 1084-7529/06/092233-13/$15.00

to perform it in the context of the simplest of the systems that require a multidirectional phase measurement, that is, ground layer adaptive optics (GLAO). Moreover, it considers the case of natural guide stars (GSs). The study could be extended to laser GS asssisted systems, although the conclusions it could lead to may not be relevant for such systems. A presentation of the SO and LO phase measurement concepts adapted to GLAO is provided in Section 2. In Section 3 a phase-related criterion is proposed to characterize the effects of the noise propagation on the performance of both WFS concepts. Its expression is derived in each case and used to perform the study of the WFS concepts. In Section 4, the optimization of the concepts is proposed, and the gain in performance that it provides is analyzed.

2. MEASURING THE GROUND LAYER TURBULENCE The GLAO10 concept has been proposed as a solution for improving uniformly the quality of wide field images (typically 10 arc min). For this reason GLAO systems aim at compensating for the boundary layer of the atmosphere, which is at the same time the location of most of the atmospheric turbulence11–13 and the layer for which correction remains valid on a wide FOV. Hence a GLAO system has only one deformable mirror (DM), usually conjugated to the pupil of the telescope. As the turbulence © 2006 Optical Society of America

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Nicolle et al.

correction is only partial, GLAO is not supposed to achieve the diffraction limit. Rather, it has to be considered a seeing reducer. Compensating for the ground layer phase perturbations requires measurements of the distribution in altitude of the turbulence. Several phase estimation processes can give suitable information on the ground layer turbulence. The most commonly used is presented in Subsection 2.A. Then a brief description of the WFS concepts used for the phase estimation is provided in Subsection 2.B. A. Phase To Be Estimated The optimal way to get knowledge of the ground layer turbulence is to perform an optical tomography14–16 of the atmosphere from phase information obtained in various directions of the FOV. A simpler and widely approach is to take benefit of the angular decorrelation of the phase perturbations due to high altitude turbulence by simply averaging the phase perturbations over a given domain D* of the sky: ⌽to

be estim.共r兲

1 =

S D*

冕冕

␾共r, ␣兲d2␣ ,

共1兲

D*

where r stands for the pupil plane coordinates, SD* is the surface of the domain where the phase has to be averaged, and ␾共r , ␣兲 is the phase of the wavefront (WF) coming from the direction ␣ : ␾共r , ␣兲 = 兰0⬁␸共r − h␣ , h兲dh (with ␸ being the phase perturbation due to the turbulent layer located at the altitude h). Hence the resulting phase that is measured is due only to the correlated part of the turbulence, which is mainly located at low altitude and forms the ground layer. Nevertheless, only a finite number of directions of D* can be used for the WF analysis: the ones of the stars that are bright enough to contribute to the WF measurement process. If there are enough GSs in D*, then ⌽to be estim.共r兲 can be approximated by the following expression: ⌽to

be estim.共r兲

1 =

K*

兺 ␾共r, ␣ 兲,

K* k=1

k

共2兲

where K* is the number of GSs available in D* and ␾共r , ␣k兲 is the phase of the WF coming from the direction of the kth GS.

Fig. 1.

Note that the study of the shape of the domain D* where the GSs have to be found is beyond the scope of this paper (more information can be found on it in Fusco et al.17), as well as the way in which the finite average of Eq. (2) approximates the continuous average of Eq. (1). Here we are interested only in how WFS concepts make possible the use of the existing stars to measure the phase ⌽to be estim. [Eq. (2)]. Hence, in the following, D* is supposed to be superimposed onto the field of interest for the astronomical observations, and the goal of this paper is to study how accounting for more and more GSs of this FOV (with fainter and fainter photon fluxes) modifies the performance of the WFS concepts. Measuring a phase average over various angular directions requires specific WFS concepts. They are presented in Subsection 2.B. B. Wavefront Sensing Concepts for Ground Layer Adaptive Optics The two WFS concepts proposed to perform the measurement of the phase perturbations due to the ground layer are derived from MCAO. They are the star-oriented (SO) and the layer-oriented (LO) approaches. Their GLAO versions are represented in Fig. 1 and discussed in Subsections 2.B.1 and 2.B.2, respectively. 1. Star-Oriented Wavefront Sensing Concept In the SO concept [Fig. 1(a)] the WF is measured independently in each GS direction by a dedicated WFS device, so that we get as many measurements as there are GSs in the FOV, each being corrupted by noise: 兵␾meas.,k共r兲 = ␾k共r, ␣k兲 + Noise共r,Nk兲其k苸关1,K*兴 ,

共3兲

where ␾共r , ␣k兲 is the phase coming from the direction of the kth GS and Noise共r , Nk兲 is the noise term related to the kth WFS device and is dependent on the flux of the kth GS 共Nk兲. Note that here and in the following the 兵␾meas.,k其k苸关1,K*兴 are reconstructed phases. It is assumed that the measurements delivered by the WFS devices allow reconstruction of the first Pmax Zernike modes, Pmax being a function of the size of the WFS subapertures (that is to say, of the number of subapertures for a given telescope diameter). The reconstruction model used hereafter accounts only for noise effects on the reconstructed phase. Aliasing and temporal errors are not considered here,

(Color online) SO and LO WFS concepts for GLAO.

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first because their effects are negligible when compared with the noise effects when the signal to noise ratio (SNR) of the WFS devices is low, and second because in the following all the WFS will be considered identical, leading to the same order of magnitude of these errors for both the SO and LO approaches. Hence the 兵␾meas.,k其k苸关1,K*兴 have the following expression: Pmax

"k,

␾meas.,k共r兲 =

兺a

meas.,k Zp共r兲, p

apmeas.,k = apk + npk ,

p=2

共4兲 is the projection of ␾共r , ␣k兲 onto the pth where Zernike mode and npk is the noise of the kth measurement propagated on the same mode.18 The K* reconstructed phases are then recombined to drive the DM.14,16,19 A simple way to recombine the 兵␾meas.,k其k苸关1,K*兴 is to make an average of the individual phases:

2235

detection increases the SNR of the WFS device,20 but the WFs coming from different directions in the FOV are mixed and weighted by the GS flux, leading to a loss of information and possibly a reconstruction problem.20,21 In the SO case, WFs are measured separately in different directions without any information loss, but with a lower sensitivity when noisy detectors are considered.20 It is rather hard to predict the performance of these two WFS concepts in conditions close to the ones of on-sky use. We propose an analytical criterion in Section 3 that makes possible some study of the performance of the SO and LO concepts and some comparison between them.

apk = ap共␣k兲

⌽SO共r兲 =

1

K*

兺 关␾ 共r兲 + Noise共r,N 兲兴. k

K* k=1

k

共5兲

This average is not optimal with regard to the propagation of the noise, first because Eq. (5) makes no distinction between the measurements coming from WFS devices with low SNR and the ones obtained with good SNR, and second because there is no modal optimization considered in the reconstruction of the 兵␾meas.,k其k苸关1,K*兴 with respect to the SNR. This makes the SO approach very sensitive to noise, especially when faint GSs are considered. To work around this problem, one can apply two strategies: • Optimize the phase reconstruction with respect to the SNR on the WFS; this solution will be explored in Section 4. • Use fewer WFS devices; this is the point of the LO concept, presented in Subsection 2.B.2. 2. Layer-Oriented Wavefront Sensing Concept In the LO concept [Fig. 1(b)] the photon fluxes of several stars are coadded and sensed simultaneously with one unique WFS device conjugated to the ground layer.9 Hence only one resultant phase is measured. In the WFS process the phase information is coded in terms of intensity, so that the measured phase is the average of the phases in each GS direction weighted by the corresponding GS flux:

3. CRITERION FOR EVALUATION OF THE PERFORMANCE OF WAVEFRONT SENSING CONCEPTS The performance of an AO system can be assessed by using criteria based either on the corrected phase or on the corrected point spread function. When more specifically considering the WF analysis performance, the use of a phase-related criterion is more comfortable. This is one of the reasons that make the residual phase variance a very commonly used criterion for characterizing AO loop performance. In the case of a wide field AO (MCAO or GLAO), a suitable criterion for assessing only the effects of the noise propagation on the WFS concept performance is the variance of the difference between the phase that should be reconstructed from the WFS data and the phase that is actually reconstructed. In the GLAO case, the phase to be reconstructed is the projection of ⌽to be estim. [Eq. (2)] onto the first Pmax Zernike modes:

⌽to

兺N ␾ k

⌽LO =

Ntot

k

+ Noise共Ntot兲,

共6兲

where ␾k and Nk are the same as those for Eq. (3) and Ntot K* is the total flux provided by all the GSs: Ntot = 兺k=1 Nk. As only one WFS device is used for the WF perturbation measurements, only one noise term exists, dependent on the total flux in the FOV. Finally, just as in the case of the SO concept, ⌽LO is reconstructed onto the first Pmax Zernike modes. The main difference between the two concepts is the way in which they use the photons coming from the GS. In the LO approach the flux coaddition made before the

=

K* Pmax

兺 兺 a Z 共r兲,

K* k=1

k p

共7兲

p

p=1

where apk = ap共␣k兲 is the same as that in Eq. (4). The phase actually reconstructed is ⌽SO [Eq. (5)] or ⌽LO [Eq. (6)], according to the considered WFS concept. Basing this performance study on only the Pmax Zernike modes corrected by the DM excludes the fitting error from the performance analysis; this error is assumed to be the same for both concepts. Then the quality criterion (QC) that is proposed here for the evaluation of both SO and LO performance is

K*

k=1

be rec.共r兲

1

QC冉SO = LO冊

冓冕冕

P

共r兲兴2d2r 关⌽to be rec.共r兲 − ⌽冉SO LO冊

冔

. turb.

共8兲 The lower this criterion is, the better the considered WFS concept approximates ⌽to be rec.. This expression is derived for both the SO and LO approaches in the following subsections. A. Quality Criterion Expression for the Star-Oriented Concept Introducing the expressions for ⌽to be rec. [Eq. (7)] and ⌽SO [Eq. (5)] into Eq. (8), we get

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J. Opt. Soc. Am. A / Vol. 23, No. 9 / September 2006

冓冕冕 再兺 冎 冔 K*

1

QCSO =

K*

␾k共r兲 −

2 P K*

Nicolle et al.

k=1

B. Quality Criterion Expression for the Layer-Oriented Concept Introducing the expressions for ⌽to be rec. [Eq. (2)] and ⌽LO [Eq. (6)] into Eq. (8), we get

兺 关␾ 共r兲 k

k=1

2

+ Noisek共r兲兴

d 2r

. QCLO =

Then the difference between ⌽SO and ⌽to be estim. is simply the noise introduced by each WFS device into the reconstruction process, and QCSO becomes 1 QCSO =

兺␴

K2* k=1

2 Noisek ,

k

=

␨k Nk

P

k=1

+

2 ␰k␴RON k

Nk2

1 −

K*

Nk Ntot

册 冔

冊

␾k共r兲

2

− Noise共r,Ntot兲

K*

2 is the variance of the noise in the kth meawhere ␴Noise k surement. This average is not optimal because it does not properly manage a WFS device with a bad SNR, but it makes possible a first evaluation of the SO concept. Whatever the type of considered WFS device, each noise term 2 ␴Noise can be expressed22,23 as

2 ␴Noise k

冓冕冕 冋兺 冉 K*

turb.

d 2r

. turb.

Here it is clear that the linear coefficients affecting the individual phases ␾k introduce an additional error source into the WF measurement process. Besides, as only one WFS device is used for the WF perturbation measurement, only one term of noise exists, depending on the total flux in the FOV. Denoting 具Nᐉ典GS = Ntot / K* as the average flux per GS and ⌬k = 具Nᐉ典GS − Nk as the gap between the flux of the kth GS and the average flux, and using the same hypothesis as that for deriving Eq. (9), we get

, 共10兲

where Nk is the statistical average of the flux in one subaperture of the kth WFS device (corresponding to the kth 2 is the variance of the CCD readout noise GS), ␴RON k (RON) of the kth WFS device, and ␨k 共␰k兲 is a numerical coefficient for the photon (detector) noise that takes into account the WFS device characteristics (CCD readout mode and sampling, spot position estimator, etc.), the propagation of the noise through the reconstruction process (number Pmax of reconstructed modes), and the filtering of the AO loop. Assuming that all the WFS parameters are identical (that is to say, "k 苸 关1 , K*兴, ␨k = ␨, ␰k = ␰, 2 2 = ␴RON ), QCSO can simply be written as and ␴RON k

共9兲 where 具·典GS is the arithmetical average over the GS directions of the included quantity. The first term of Eq. (9) represents the photon noise error, and the second term represents the CCD RON error. This model, where the WFS parameters are all considered identical, is pessimistic with regard to the SO performance. Indeed, on a real AO system the spatial and temporal sampling frequencies of the detector of the WFS can be adapted in order to improve the SNR when the corresponding GS is faint. Moreover, as already said in Subsection 2.B, an optimization could be performed. But this expression clearly shows that in the SO concept, the quality of the ground layer phase estimation is highly dependent on the flux on each WFS device. Indeed, according to Eq. (9) GSs with low flux make the SO performance decrease at least as the inverse of their flux, and even as the inverse of the square of their flux when noisy detectors are considered. This great sensitivity to noise is the main drawback of the SO concept. The LO concept has been proposed precisely to deal with this problem.

Pmax 具ap共␣j兲ap共␣k兲典turb. Cjk = 兺p=0

is the covariance of the where turbulent phase on the first Pmax Zernike modes between 2 are the the directions ␣j and ␣k and where ␨, ␰, and ␴RON same as those for Eq. (9). Chassat24 showed that by knowing the turbulence profile, it is possible to estimate analytically 具ap共␣j兲ap共␣k兲典turb. and thus Cjk. Three terms can be identified in the expression for QCLO. We recognize in the last two terms of Eq. (10) the photon and detector noise effects on the phase estimation performance. In the LO case these terms depend on the total flux available in the FOV. This means that adding one GS, even faint, in the WFS process makes this term decrease. The first term of Eq. (10) is related to the turbulent phase covariance from one GS direction to another. This error term is due to the inhomogeneities in the repartition of the total flux between the GSs, which prevents the average of the uncorrelated part of the turbulence from being null, as expected in Subsection 2.A. If all the GSs have the same flux 共"k , Nk = 具Nᐉ典GS兲, or if all the turbulence is in the pupil plane, then QCLO is sensitive only to the noise on the WFS and Eq. (10) gets simplified: QCLO,no

turb. term =

␨ Ntot

+

2 ␰␴RON 2 Ntot

.

共11兲

In addition, it can be shown that when all the GSs have the same flux, and if only photon noise is considered, then the LO and SO WFS concepts are equivalent. In all the other cases, comparing SO and LO performance requires a careful study. Indeed, on the one hand, the LO approach is much less sensitive to noise than the SO approach because of the coaddition of the light coming from all the GSs on its single WFS device. But, on the other hand, the mismeasurement of the phase due to the flux differences between the GSs degrades its performance, so that it is impossible to predict which WFS concept performs better without a careful study.

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Otherwise, some attempts have been made to render the SO performance less sensitive to noise by reconstructing the weighted phase average measured by the LO method with the individual measurements of the SO method. This is the numerical layer-oriented20 (NLO) WFS concept, which is presented next. C. Quality Criterion Expression for the Numerical Layer-Oriented Concept The NLO concept has already been studied.20 Here only a few results are recalled. The point of the NLO concept is to treat the independent measurements obtained in a SO concept so as to reconstruct a phase similar to the one measured by a LO concept. This introduces into the phase average some weighting coefficients that are proportional to the fluxes of the GSs, and thus a discrimination is made between the phase information coming from bright GSs and the ones coming from fainter GSs. Hence the NLO phase can be written as ⌽NLO =

1

K*

兺 N 共␾

Ntot k=1

k

k

+ Noisek兲.

共12兲

This expression for the reconstructed phase leads to the following expression for QCNLO: 1 QCNLO =

K* K*

兺兺

2 Ntot j=1

k=1

⌬j⌬kCjk +

␨ Ntot

+ K*

冉 冊 2 ␰␴RON 2 Ntot

. 共13兲

The first two terms of this equation are identical to the ones for QCLO. This means that in the presence of photon noise only, it is possible to get the same performance with both the SO and LO methods, by reconstructing a NLO phase. Unfortunately, the term in Eq. (13) related to the detector noise corresponds to the LO term multiplied by the number of GSs. Subsection 3.D compares the performance of the SO, LO, and NLO concepts. D. Comparison of the Star-Oriented, Layer-Oriented, and Numerical Layer-Oriented Performance The performance of the WFS concepts described above can be studied thanks to the mean square error (MSE) criterion. In Subsection 3.D.1 the conditions of the study are described. The performance of the WFS concepts is then analyzed in Subsection 3.D.2. 1. Conditions of the Study The present study can be described by the following conditions: • The system is characterized by an 8 m pupil, an 8 arc min FOV, and an imaging wavelength ␭ima = 2.2 ␮m. • The turbulence profile is a test profile, characterized by 1. Seeing of 1 arc sec at 0.5 ␮m. 2. Outer scale L0 = 25 m. 3. 80% of the turbulence in the pupil plane and 20% in an upper layer. 4. Full decorrelation of the phase perturbations in the upper layer. This makes the covariance Cjk become Cjk = ␴␸2 + ␴␸2 ␦jk, where ␦jk is the Kronecker symbol grd alt (␦jk = 1 if j = k, ␦jk = 0 otherwise). With the turbulence profile considered here, we have ␴␸2 = 25.2 rad2 and grd

2237

␴␸2 = 6.3 rad2. Moreover, this assumption also makes the alt equations for QCLO and QCNLO simpler:

QCLO = K*␴␸2 alt

QCNLO = K*␴␸2 alt

具⌬2ᐉ 典GS 2 Ntot

具⌬2ᐉ 典GS 2 Ntot

+

+

␨ Ntot

␨ Ntot

+

+

2 ␰␴RON 2 Ntot

,

2 K*␰␴RON 2 Ntot

.

• The WFS devices are all identical Shack–Hartmann type, characterized by the following: 1. There are 14⫻ 14 subapertures (which means 190 reconstructed modes). 2. The photon and/or detector noises are simulated, with ␴RON = 1 e− per pixel. 3. The focal plane is sampled so that the full width at half-maximum of a diffraction-limited subaperture spot corresponds to 2 pixels of the camera (Shannon criterion). 4. Phase slopes in the subaperture are measured by computing the center of gravity of the subimages on windows twice as wide as the turbulent spot. In this case, the linear coefficients ␨ and ␰ can be determined with expressions derived by Rousset22 and Rigaut and Gendron.18 • Star fields are generated numerically by using a statistical model of the galactic population.25,26 Various pairs (latitude, longitude) of galactic coordinates are considered: (30°,0), (60°,0), and (90°, 90°). Figure 2(a) shows the evolution of the mean number of GSs with respect to the limiting magnitude for these three sets of galactic coordinates. These curves clearly show that when the working latitude is high, the number of available GSs is very low. • For each set of galactic coordinates, 1000 star fields are generated. • For each star field, the criteria QCSO [Eq. (9)], QCLO [Eq. (10)], QCLO, no turb. term [Eq. (11)], and QCNLO [Eq. (13)] are computed versus the limiting magnitude of the GSs; increasing the limiting magnitude means considering in the WFS process more and more GSs, but with fainter and fainter photon flux. Hence increasing the limiting magnitude means the following: 1. Increasing the number of GSs. 2. In the SO and NLO cases, measuring more individual phases, but with lower SNR. 3. In the LO case, increasing the total flux available for the WFS (integrated flux). Figure 2(b) shows the evolution of the integrated flux in the FOV with respect to the limiting magnitude. This curve clearly shows that when the working latitude is high, the integrated flux saturates earlier than at low galactic latitudes, due to the fact that faint GSs are scarcer [cf. Fig. 2(a)]. This saturation means that there is not much information to get by increasing the limiting magnitude of the system. • For each star field and limiting magnitude we impose the requirement that there must be at least 4 GSs to perform a relevant measurement of the ground layer turbulence; no upper limit is set. • The results presented here are the averages of the criteria over the star field occurrences, fulfilling the condition on the minimum number of GSs.

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Fig. 2. (a) Mean number of GSs in the simulated star fields with respect to the considered limiting magnitude for various galactic latitudes, (b) integrated flux with respect to the limiting magnitude.

Fig. 3.

Comparison of SO, LO, and NLO performance.

2. Performance Analysis of the Star-Oriented, Layer-Oriented, and Numerical Layer-Oriented Wavefront Sensing Concepts Figure 3(a) shows the comparison of the performance of the LO, SO, and NLO WFS concepts when increasing the limiting magnitude of the considered GS and in the presence of photon noise only. In the LO case two curves are plotted: one corresponding to the total error [Eq. (10)] and one for which the term related to the GS flux dispersion is ignored [Eq. (11)]. This case mimics a case where all the turbulence would be located in the pupil plane or where all the available flux would be equitably distributed among the GSs. Thus it represents the best performance achievable with the LO system. The difference between this case and the true LO performance is the turbulence-related term of Eq. (10) [the first term of Eq. (10)]. Concerning this point, Fig. 3(a) shows that the contribution of the turbulence-related term prevails in the LO measurement error and even makes the LO performance worse than the SO one, whereas the single noise-related term of QCLO

共QCLO, no turb. term兲 is significantly better than QCSO. Besides, as expected, increasing the GS limiting magnitude improves the LO performance because it increases the integrated flux on the single WFS device. On the contrary, for the SO concept, increasing the limiting magnitude worsens the performance. This is due to the fact that increasing the limiting magnitude of the GS implies that more WFS devices with bad SNR are incorporated into the WF estimation. Nevertheless, according to Eq. (9), one can note that an optimal limiting magnitude should exist for the SO case. Indeed, the SO performance is proportional to the total photon noise on all the WFS, but also inversely proportional to the square of the number of GSs. This means that increasing the limiting magnitude of the GS should be fruitful if the induced increase of the photon noise remains low with respect to the increase of the number of GSs. But Fig. 3(a) shows that such a situation may occur only with very bright GSs (limiting magnitude lower than 15), which is a marginal situation when considering natural GSs. Finally, one can note that, as expected, the NLO perfor-

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mance is exactly the same as the LO one when only photon noise is considered. But because of the turbulencerelated term [first term of Eq. (13)], combining the individual SO measurements into a NLO measurement degrades the WF analysis performance. When some additional detector noise is considered [Fig. 3(b)], the SO performance is dramatically degraded, especially at high limiting magnitude, that is to say, when measurements obtained with very faint GSs are included in the WF estimation process. On the contrary, with a 1 e− per pixel rms detector noise, the LO performance is still dominated by the turbulence effects, so that its performance hardly varies with regard to the case where only photon noise is considered. But note that with a 3 e− per pixel rms detector noise, the error due to the detector noise and the one due to the turbulence would have the same order of magnitude. Finally, one can note that combining the individual SO measurements in the NLO way improves its performance, but not sufficiently to be competitive with the true LO concept, which remains the best because of the integration of the whole flux on its single WFS device. These results depend on the GS characteristics and thus on the working galactic latitude. Figure 4 shows the

Fig. 4.

2239

influence of the GS statistics on SO and LO behavior. First, when only photon noise is considered [Fig. 4(a)], one can note that only the SO performance is sensitive to an increase of the galactic latitude. Indeed, when increasing the galactic latitude for which the star fields are simulated, the number of GSs decreases, as shown in Fig. 2, and bright GSs become scarce. This degrades the SO performance because of its sensitivity to the noise, whereas the LO performance is still dominated by the turbulence effects and hardly evolves. This makes the SO performance closer to the LO performance. When some detector noise is considered [Fig. 4(b)], the SO measurement is still completely hopeless. In the LO case one can note that increasing the galactic latitude makes the detector noise effects progressively dominate its performance. Indeed, the global error (curves with crosses) becomes almost equal to the error due only to the noise (curves with diamonds). All of this analysis makes it possible to highlight key points in the SO and LO performance: • The two WFS concepts are not equivalent when only photon noise is considered. The LO performance is dominated by the turbulence effects, which makes it worse

Comparison of SO, LO, and NLO performance for high galactic latitudes.

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than the SO one. Nevertheless, this advantage of the SO approach shrinks when the working galactic latitude increases, because of the bright GS scarcity. Moreover, the LO performance could be improved by an optimization aiming at reducing its turbulence-related term. • When some detector noise is considered, the SO performance is dramatically degraded, which is unacceptable. The combination of the SO measurements into a NLO scheme brings better results, but not good enough to make it competitive with the true LO approach. Hence optimizing the way in which the SO measurements are combined is mandatory. Concerning the LO method, the detector noise effects and the turbulence effects have almost the same influence on its performance for a low RON and a low galactic latitude. In this case the optimization proposed just above (reduction of its turbulence-related term) should still provide a significant gain. But when considering detectors with higher RON or regions of the sky where bright GSs are scarcer, the detector noise effects becomes dominant, so that a mode-by-mode optimization of the reconstruction should be planned also.

1 QCOSO =

K2*

冋兺 兺 兺 兺

K* K* Pmax Pmax

共1 − ␩pk兲共1 − ␩qj兲

j=1 k=1 p=1 q=1

⫻具apk aqj典

冕冕

K* Pmax

Zp共r兲Zq共r兲d2r +

P

⌽OSO共r兲 =

1

K* Pmax

兺兺␩a

K* k=1

k meas.,k Zp共r兲, p p

共14兲

p=1

where the apmeas.,k = apk + npk are defined in Eq. (4) and 兵兵␩pk其k苸关1,KA兴其p苸关1,Pmax兴 are the coefficients to optimize. This is the best way to perform the optimization because each mode measured from each GS direction is optimally used in the reconstruction. Hence Eq. (8) becomes

册

Pmax

QCOSO =

兺 QC

OSO,p ,

共15兲

p=1

with

冉 冊 冋兺 兺 1

QCOSO,p =

K*

2

K* K*

j=1 k=1

兺 共␩ 兲 共␴ 兲 k 2 p

k=1

A. Analytical Developments The SO optimization can be done rather simply by decomposing ⌽SO, given by Eq. (5), onto the Zernike polynomials, accounting for the propagation of the noise on the reconstructed modes18 and introducing linear coefficients:

k 2 p

where 共␴pk兲2 is the variance of the noise in the kth WFS device and propagated onto the pth Zernike polynomial.18,22 In this expression, the 兵Zp其p苸关1,Pmax兴 are Zernike polynomials and thus are orthogonal: 兰兰PZp共r兲Zq共r兲d2r = ␦pq, where ␦pq stands for the Kronecker symbol. Hence

+

As stated in Subsection 3.D.2, the great sensitivity of the SO method with regard to the noise on the WFS devices makes its optimization mandatory. This optimization can be performed rather easily because each phase is independently estimated. It is therefore possible to weight independently each WF measurement in the average process, attributing to each a coefficient that accounts for SNR in the corresponding WFS. But this optimization can be improved again by accounting for the propagation of the noise on the reconstructed modes. Hence a mode-bymode optimization can be performed, where the coefficients of the optimization are computed independently for each mode reconstructed from each WF measurement. After exposing the analytical layout of the optimization, a comparison of the optimized SO approach with the “classical” one and the NLO approach is performed.

k 2 p

k=1 p=1

K*

4. OPTIMIZATION OF THE STAR-ORIENTED PERFORMANCE

兺 兺 共␩ 兲 共␴ 兲

k 2 p

共1 − ␩pj兲共1 − ␩pk兲具apj apk典

册

.

共16兲

The minimization of QCOSO,p with respect to the 兵␩pk其 provides their optimal values. This optimization is easy to do as it corresponds to inverting a linear matrix equation. Indeed, Eq. (16) can be expressed as QCOSO,p = 共1 − ␩p兲tCp共1 − ␩p兲 + ␩pt Np␩p ,

共17兲

where ␩p is the vector of the linear coefficients 兵␩pk其 for the Zernike mode Zp, Cp = 关具apj apk典兴jk is the covariance matrix of the pth Zernike coefficient between the different GS directions, and Np is a diagonal matrix gathering the noise variances of the WFS devices, still for Zp. One can differentiate this equation with respect to ␩p and look for the zeros of the expression of the derivative. Practically speaking, the inversion of the derivative equation was easy to realize in the cases that we studied because of the good conditioning of the problem. Otherwise, it is possible to invert the derivative equation with a truncated singular value decomposition algorithm. In the performance study in Subsection 4.B this optimization is compared with the “classical” SO and NLO methods. B. Comparison of Optimal Star-Oriented Performance with Star-Oriented and Numerical Layer-Oriented Performance Figure 5 shows the comparison of the optimized SO approach with the classical one and the NLO method in the same conditions as those for Fig. 3. One can note that the optimized SO performance is not degraded any longer by increasing the limiting magnitude of the GSs in the presence of photon noise as well as when some detector noise is considered. For high limiting magnitudes, this means that the faintest GSs have very low weighting coefficients and are hardly accounted for in the phase reconstruction. Here again it is important to clarify that this optimization method does not pretend to manage a global optimization of a GLAO system but is only a comparison of sev-

,

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Fig. 5.

2241

Comparison of the OSO performance with the “classical” and NLO performance.

Fig. 6.

Gain brought by dimming the brightest GS for the LO approach.

eral WFS concepts, all involving the same set of star fields. The analysis of the way in which the ground layer is actually approximated by an average over the selected and weighted GS directions is beyond the scope of this paper. With regard to the noise propagation, the gain of the SO optimization is very fruitful. After exposing in Section 5 the LO optimization, both WFS concepts will be compared in Section 6.

5. OPTIMIZATION OF THE LAYER-ORIENTED PERFORMANCE A. Effect of Dimming the Brightest Guide Stars As we underlined when commenting on Fig. 3, the LO performance is degraded by the fact that when there are some differences of flux among the GSs, the phase measurement is sensitive to the high altitude turbulence [cf. the first term of Eq. (10)]. Figure 3 showed that this phenomenon can even make the LO performance worse than the SO performance. This problem could be solved by nulling the flux differences among the GSs. Hence the first term of Eq. (10)

would be null, and only the photon and detector noises would degrade the performance. Doing this means dimming all the GSs to bring their flux to that of one of the faintest GSs. This is necessarily an optical process, which has to be done before the light coaddition and the detection by the WFS device. Therefore it makes the total flux coming into the WFS device decrease and the term due to photon and detector noises increase. Hence nulling the GS flux dispersion is clearly not optimal. It can even be shown that this always makes the LO performance worse than the SO performance when considering only the photon noise. Indeed, if we denote as Nmin the flux of the faintest GS, then by definition "k 苸 关1 , K*兴, Nmin 艋 Nk, and we have 1 "k

Nk

艋

K*

1 Nmin

Û

1

兺N

k=1

k

艋

K* Nmin

Û

␨ K*

冓冔 1

Nk

艋 GS

␨ K*Nmin

.

Therefore it is necessary to make a compromise between the performance degradation due to the turbulencerelated term and the one due to photon and detector noises. This means optimizing the dimming of the GSs. Figure 6 illustrates the effect of such an optimization.

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This figure has been obtained by simulating 1000 sky fields corresponding to the galactic latitude 30 ° N, with a limiting magnitude set to 17, computing QCLO for each of them for different dimmings, and making an average over the different scenes. Here the dimming of the GSs is not rigorously optimized. Only the gap in magnitude between the brightest GS and the faintest one is, with a bin of 1 mag. Hence, when the tolerated gap in magnitude is null, all the GSs are dimmed, so that they all have the same flux as that of the faintest one. If a gap of 1 bin of magnitude is tolerated, all the GSs of magnitude lower than the limiting magnitude minus 1 are dimmed, so that their magnitude is brought back to the limiting magnitude minus 1B. Figure 6(a) shows a diagram representing this dimming operation. Figure 6(b) shows the evolution of QCLO with respect to the gap in magnitude tolerated among the GSs. This figure shows that as explained in the proceeding paragraph, it is possible to make a compromise between the photon and detector noises and the GS flux dispersion effects by adapting the dimming of the GSs. Here optimizing the gap in magnitude leads to QCLO divided by 30 in the undimmed case (high tolerated gap) and divided by 2 in the case where all the GSs have the same flux (null tolerated gap). The optimal gap in magnitude varies from 0 to 4 when the limiting magnitude varies from 14 to 20. The higher the limiting magnitude is, the higher the optimal gap in magnitude is too, due to the high number of faint GSs. Indeed, the GS flux variance decreases when increasing the limiting magnitude because faint GSs are much more numerous than bright ones, as shown by Fig. 7. Finding the optimal dimming for the GSs is made possible by using the same analytical formalism as that for the SO case. B. Analytical Developments The optimization of the LO performance by dimming the brightest stars can be introduced into Eq. (6) thanks to new coefficients, one per GS, called 兵␭k其k苸关1,K*兴. But another optimization can be considered too. Indeed, just as in the SO case, a mode-by-mode optimization can be performed when using ⌽LO to approximate ⌽to be estim. Hence each mode of ⌽LO can be weighted by a coefficient ␩pOLO

Fig. 7. Dispersion of the flux of the GSs available in a star field around the mean flux per GS.

Nicolle et al.

depending on the SNR in the WFS device and on the propagation of the noise onto the Zernike modes.18 Then, accounting for these two optimizations, we get Pmax

兺␩

⌽OLO共r兲 =

OLO p

p=1

冤冢

K*

兺

k=1

␭ kN k

apk

K*

兺␭ N ᐉ

ᐉ=1

冉 兺 冊冥

ᐉ

冣

K*

+ np

ᐉ=1

␭ ᐉN ᐉ

Zp共r兲.

共18兲

Here the 兵␭k其k苸关1,K*兴 coefficients correspond to the optical attenuation of each GS 共"k , ␭k 苸 关0 , 1兴兲. Accounting for this new expression, Eq. (10) becomes Pmax

QCOLO =

兺 QC

共19兲

OLO,p ,

p=1

with K* K*

QCOLO,p =

兺兺

j=1 k=1

⫻具apj

冢

1 K*

apk典

−

␩pOLO␭jNj K*

兺

ᐉ=1

+

␭ ᐉN ᐉ

共␩pOLO兲2

冤兺

冣冢

1 K*

␨p

K*

ᐉ=1

␭ ᐉN ᐉ

−

K*

兺␭ N ᐉ=1

+

冣 冊冥

␩pOLO␭kNk ᐉ

ᐉ

2 ␰p␴RON

冉兺 K*

ᐉ=1

2

␭ ᐉN ᐉ

.

共20兲 As already said, the attenuations are done before the detection process and thus degrade the WFS SNR. As the optimization of the 兵␩pOLO其p苸关1,Pmax兴 depends on the WFS SNR, a global optimization would be required for the 兵␩pOLO其p苸关1,Pmax兴 and the 兵␭k其k苸关1,K*兴. Moreover, no analytical solution can be found for the 兵␭k其k苸关1,K*兴 by trying to minimize QCOLO. A numerical multivariable minimization algorithm would be mandatory, leading to a substantial increase in the computation time. In addition, the practical realization of the optimal dimming requires a precision and a system complexity that may be very difficult to reach for real systems. For these two reasons we propose a “rough” but more realistic LO optimization: First, the numerical method described in Subsection 5.A is used to determine the 兵␭k其k苸关1,K*兴; once these coefficients are known the 兵␩pOLO其p苸关1,Pmax兴 are determined by deriving the analytical solution of the minimization of Eq. (20). The LO approach optimized in this way is compared with the “classical” LO approach in Subsection 5.C. C. Comparison of Optimized Layer-Oriented and Layer-Oriented Performance Figure 8 shows the comparison of the optimized LO method with the classical LO method in the same conditions as those for the previous figures. Note that a “rough” optimization of the LO concept has been performed, as described in the previous paragraph. For this reason, the optimized LO (OLO) performance may not be as good as it would be if we could set the GS optical attenuations as precisely as required. Nevertheless, one can note that,

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Fig. 8.

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Comparison of LO and OLO performance.

with or without detector noise, the optimization significantly improves the LO performance. Note that at the considered galactic latitude (30°N), the number and the brightness of the GSs are important, so that the WFS SNR is very good, and the optimization that brings the highest gain is the dimming of the brightest GS.

6. COMPARISON OF OPTIMIZED LAYER-ORIENTED AND OPTIMIZED STAR-ORIENTED PERFORMANCE Figure 9 shows the comparison of the optimized SO (OSO) and OLO performance at different galactic latitudes. The optimization of the optical dimming of the OLO concept is the same as that for Fig. 8. When only photon noise is considered, the two optimized WFS concepts have almost the same performance. As said previously, the improvement of the LO performance is mainly due to the dimming of the brightest GS, which made possible the reduction of the turbulence effects. The OLO performance remains slightly worse than the OSO performances but could be improved a little if the OLO optimization were rigorously performed, as explained in Subsection 5.A. When considering high galactic latitudes, both WFS concepts have their performance degraded in the same way, and the residual phase variance remains lower than 1 rad2. When detector noise is considered, the OLO method is better than the OSO method, due to its smaller number of wavefront sensors. But when increasing the galactic latitude, this advantage of the LO approach is reduced. This is due to the fact that the GSs become scarcer and fainter. Hence only one or two bright GSs gather almost all the flux available in the FOV, and the way in which the phase average is performed has not much importance. Hence the higher the galactic latitude is, the closer the two WFS concepts are to each other. This result should appear at lower galactic latitudes if noisier detectors are considered because the detection limit of the WFS devices is reduced. Hence the optimization of the SO and LO method brought a significant gain for both WFS concepts, what-

ever the galactic latitude and the considered limiting magnitude. Moreover, it made it possible for their performances to be very close to each other. Nevertheless, optimizing these two concepts does not have the same cost in terms of system complexity. Hence the SO optimization is a software matter, whereas the LO optimization mainly relies on the optical attenuation of the bright GSs (except for very high latitudes), the attenuations being a function of the turbulence conditions and of the GS position and magnitude. This requires additional optical elements along the optical path of the GS and could introduce an optical design upset.

7. CONCLUSION The analytical modeling of the two WFS concepts for phase estimation in the framework of a GLAO system has highlighted key points in the evolution of their performance. Concerning the “classical” SO and LO performance, it has been underlined that the LO performance is mainly limited by the error introduced by the mixing and the weighting of the phase information before detection whereas the SO performance is significantly limited by the noise propagation in the reconstruction process, above all in the presence of detector noise. For these reasons, optimizing both the SO and LO approaches is very fruitful. A mode-by-mode optimization has been proposed for the SO method. It consists in a numerical operation, the proper coefficients being computed by simple inversion of a linear matrix equation. LO optimization consists in the optical attenuation of the brightest GS (to reduce the GS flux differences) and in the mode-by-mode optimization of the phase reconstruction thanks to numerical coefficients. The GS attenuations and the numerical coefficients of the optimization are interdependent and have to be determined together by using a multivariable minimization algorithm. Nevertheless, it has been shown that pretty good results can be obtained with a crude but more realistic GS attenuation algorithm, which consists in a simple limitation of the gap

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in magnitude between the brightest GS and the faintest one. The OSO and OLO methods have very similar performance in the presence of photon noise; when considering additional detector noise, the OLO approach remains bet-

Fig. 9.

Nicolle et al.

ter than the OSO approach, but the difference is much lower than in the unoptimized cases and decreases when considering growing galactic latitudes or a noisier WFS device. In both optimized cases the dependency of the measurement error with respect to the limiting magni-

Comparison of OSO and OLO performance.

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tude of the GS is very faint, because both optimizations manage very well the WFS SNR. Nevertheless, optimizing these two concepts does not have the same cost in terms of system complexity. That is, the SO optimization is just a software matter, whereas the LO optimization requires some additional optical elements along the optical path of the GS, these elements needing to be adaptable to outer parameters such as turbulence conditions and GS properties (magnitude and position). Once again, it is necessary to keep in mind that this study does not mean to present an optimization of the GLAO system. This work studies only one point of a total error budget that includes all the other facets of the GLAO system, i.e., the reconstruction algorithm (tomography, for instance), the shape of the domain D* where GSs are to be found, and the sky coverage problem. Finally, this study should be fruitful for the performance analysis of the first stage of a multiple-FOV WFS concept,9 with the results presented here probably needing modification with respect to the characteristics of this concept.

view layer-oriented adaptive optics. Nearly whole sky coverage on 8 m class and beyond,” Astron. Astrophys. 396, 731–744 (2002). F. Rigaut, “Ground-conjugate wide field adaptive optics for the ELTs,” in Beyond Conventional Adaptive Optics, R. Ragazzoni and S. Esposito, eds. (European Southern Observatory, 2001). J. Vernin and C. Munoz-Tunon, “Optical seeing at La Palma Observatory. 2: Intensive site testing campaign at the Nordic Optical Telescope,” Astron. Astrophys. 284, 311–318 (1994). R. Avila, J. Vernin, and S. Cuevas, “Turbulence profiles with generalized SCIDAR at San Pedro Mártir Observatory and isoplanatism studies,” Publ. Astron. Soc. Pac. 110, 1106–1116 (1998). A. Tokovinin, S. Baumont, and J. Vasquez, “Statistics of turbulence profile at Cerro Tololo,” Mon. Not. R. Astron. Soc. 340, 52–58 (2003). M. Tallon, R. Foy, and J. Vernin, “3-d wavefront sensing for multiconjugate adaptive optics,” in Progress in Telescope and Instrumentation Technologies, ESO Conference and Workshop Proceedings, M. H. Ulric, ed. (European Southern Observatory, 1992), pp. 517–521. R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999). T. Fusco, J.-M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wavefront reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001). T. Fusco, M. Nicolle, G. Rousset, V. Michau, A. Blanc, J.-L. Beuzit, and J.-M. Conan, “Wavefront sensing issues in MCAO,” C. R. Phys. 6, 1049–1058 (2006). F. Rigaut and E. Gendron, “Laser guide star in adaptative optics: the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992). B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994). D. Bello, J.-M. Conan, G. Rousset, and R. Ragazzoni, “Signal to noise ratio of layer oriented measurements for multiconjugate adaptive optics,” Astron. Astrophys. 410, 1101–1106 (2003). M. Nicolle, T. Fusco, V. Michau, G. Rousset, A. Blanc, and J.-L. Beuzit, “Ground layer adaptive optics: analysis of the wavefront sensing issue,” in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., in Proc. SPIE 5490, 858–869 (2004). G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, 1999), Chap. 5, pp. 91–130. M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack–Hartmann wavefront sensor measurement for extreme adaptive optics,” Opt. Lett. 29, 2743–2745 (2004). F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. (Paris) 20, 13–23 (1989). A. Robin, C. Reylé, S. Derrière, and S. Picaud, “A synthetic view on structure and evolution of the Milky Way,” Astron. Astrophys. 409, 523–540 (2003). T. Fusco, A. Blanc, M. Nicolle, V. Michau, G. Rousset, J.-L. Beuzit, and N. Hubin, “Improvement of sky coverage estimation for MCAO systems: strategies and algorithms,” Mon. Not. R. Astron. Soc. (to be published).

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ACKNOWLEDGMENTS The authors thank A. Blanc, J.-M. Conan, N. Hubin, and L. Mugnier for very fruitful discussions. This work was partly supported under EC contract RII3-CT-2004001566. Corresponding author: [email protected].

Magalie

Nicolle,

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F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999). D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982). J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993). R. W. Wilson and C. R. Jenkins, “Adaptive optics for astronomy: theoretical performance and limitations,” Mon. Not. R. Astron. Soc. 278, 39–61 (1996). M. Le Louarn and N. Hubin, “Wide-field adaptive optics for deep-field spectroscopy in the visible,” Mon. Not. R. Astron. Soc. 349, 1009–1018 (2004). R. H. Dicke, “Phase-contrast detection of telescope seeing and their correction,” Astron. J. 198, 605–615 (1975). J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Very Large Telescopes and Their Instrumentation, ESO Conference and Workshop Proceedings (European Southern Observatory, 1988), pp. 693–703. R. Ragazzoni, “No laser guide stars for adaptive optics in giant telescopes,” Astron. Astrophys., Suppl. Ser. 136, 205–209 (1999). R. Raggazzoni, E. Dialaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of

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Annexe L “Experimental comparaison of optimized star oriented and numerical layer oriented using the MAD test bench” T. Fusco et al. - A&A - 2007

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Experimental validation of Star oriented optimisation for ground layer adaptive optics on the MAD test bench T. Fusco, M. Nicolle, C. Petit, J. Kolb, S. Oberti, E. Marchetti, N. Hubin, E. Fedrigo and V. Michau 10 septembre 2007

To be submitted in AetAss

1 Introduction Multi-conjugate adaptive optics (MCAO) [Dicke 1975, Beckers 1988] has been proposed to achieve diffraction limited images on fields typically ten times wider than the one of a conventional adaptive optics (AO) system [Roddier 1999]. Such a performance is made possible only by overcoming the anisoplanatism effects due to the turbulent layers located in the higher part of the atmosphere [Fried 1982] (typically above the first kilometer). In order to get some information on the volumic distribution of the turbulence, one needs a wave-front sensing (WFS) concept able to deal with a multi-directional phase measurement. Two concepts have been proposed to perform this measurement. They are the star oriented [Beckers 1988] and the layer oriented [Ragazzoni 1999, Raggazzoni et al. 2002] approaches. A comparative study of both WFS concepts performance has been proposed in a previous paper [Nicolle et al. 2006] (see appendix ??) in the framework of a GLAO system. This paper highlighted that the SO concept is mainly limited by the propagation of the detector noise in the reconstruction process, whereas the LO performance is mainly limited by the sight of the high altitude turbulence when the guides stars have different magnitudes. This paper also proposed a way to optimize each WFS concept in order to make the average performance over the field of view (FOV) as good as possible. Then it has been shown that both optimized concepts could reach very close performance, even in presence of detector noise. In this last case, the optimized LO concept was prooved to be slightly better than the optimized SO, but it has been underlined that this gain in performance is made possible only by introducing some optical attenuation elements in the optical paths corresponding to the bright guide stars (GS), possibly leading to some optical design upset. All the results presented in [Nicolle et al. 2006] where found by modeling analytically and numerically the SO and LO WFS concepts. Moreover, the quality criterion used to perform their study and their optimization was a phase-related criterion, which is unusual to characterize a GLAO system performance. We propose here some experimental results on the SO (optimized or not) and LO behavior. These results have been obtained on the Multi-conjugate Adaptive optics Demonstrator (MAD) of ESO. In section 2 we briefly describe the MAD test bench and its main characteristics. At the time we made our tests, the layer oriented unit of MAD was not settled yet. Only the star oriented unit 1

was operational. Nevertheless this was enough to enable some tests on three reconstruction modes for the GLAO : the conventionnal star oriented, the numerical layer oriented and the optimized star oriented. We present these three modes in section 3, and describe also the different guide stars configurations we used for the tests. Finally, we present the experimental performance obtained with the three reconstruction modes in section 4.

2 Presentation of MAD

F IG . 1 – Optical layout of MAD, the multi-conjugate adaptive optics demonstrator of ESO. (credits : ESO, www.eso.org)

MAD (fig. 1) is the ESO’s Multi-conjugate Adaptive optics Demonstrator [Marchetti et al. 2003]. Its objective is to show feasibility and performance of a wide field AO system on the sky, in presence of natural guide stars. Planned to be integrated in the current of the year 2007 on the telescope unit telescope Melipal of the VLT (UT3), it will be then the first MCAO system for astronomy operating on the sky. The total field of MAD covers 2 arc-minutes and is imaged by one infrared camera (CAMCAO [Amorim et al. 2004]) made of 2048 × 2048 square pixels 28 mas wide. The total field of the camera being approximately of one arcmin, the camera moves into the focal plane of the instrument in order to cover the totality of the 2 arcmin. MAD can correct for the turbulence thanks to two bimorph deformable mirrors (DM) of 60 actuators each. One of these DMs is conjugated to the pupil of the telescope and the other one to 8.5 km of altitude. The mirror that is conjugated to the pupil of the instrument is assembled on a tip/tilt mount that allows to preserve the whole dynamic range of the DMs for the correction of the high order modes of the turbulence. Let us add that it is possible to use the DMs individually or together, hence exploring almost all wide field AO concepts : MCAO, GLAO or LTAO [Tokovinin et al. 2004].

2

Concerning the wave-front analysis, MAD will be equiped with two wave-front sensing units : a star oriented one, made of three identical Shack-Hartmann (SH) wave-front sensors (WFS) [Rousset 1999], and a layer oriented one, including eight pyramid WFS [Ragazzoni 1996]. However, when the data presented here were recorded, only the SO unit was available [Marchetti et al. 2006]. The characteristics of its SH WFS are summarized in table (1). A calibration unit allows to pre-compensate for the static aberrations of the system by correcting the reference slopes [Kolb 2005].

wavelength 0.7 ± 0.25 µm Pixel size 300 mas

Number of sub-apertures 8×8 Sampling of diffraction limited PSF Shannon /4

Detector noise

usefull sub-apertures 52

Pixels per sub-aperture 8×8

Reading modes

maximal frequency

Binnings 1 × 1 et 2 × 2

400 Hz (binning 1 × 1) 636 Hz (binning 2 × 2)

Treatments before CoG computation

(independent of reading frequency)

background substraction, Thresholding of negative pixels and windowing 6 × 6 pix.

9.6 photo-electrons/pixel/frame

TAB . 1 – Characteristics of the SH wavefront sensors of the SO unit of MAD.

The measurements of the WFS are then recovered by the real time computer (RTC), responsible for the wave-front reconstruction. This one is carried out by one integrator/proportional algorithm. MAD makes it possible to adapt the gain coefficients (integrator and proportional) of the loop, but also to allocate different gains to each WFS. These gains are in fact simple multiplicative coefficients applied to the slopes measured by the various WFS before the wave-front reconstruction. This possibility of adapting the gain of each WFS was crucial for the work which is presented here. Indeed it made it possible to simulate the behavior of the LO by generating a Numerical Layer Oriented mode (NLO) and to test the optimization of the SO for the GLAO. When being integrated on the VLT telescope, MAD will run on the sky, with natural guide stars. In laboratory, it is associated with a turbulence generator (Multi-Atmospheric Phase screens and Stars, MAPS (fig. 2 a) [Kolb et al. 2004]. MAPS simulates the turbulence using three phase screens reproducing an atmospheric profile typical of the turbulence in Paranal (see table 2). MAPS also includes a constellation of variable intensity light sources simulating the sky (fig. 2 b). It is possible to vary collectively the flux of all the sources, or to introduce flux variations between them by the mean of calibrated optical densities. These sources are used at the same time as guides for the WFSs and as reference point like sources to estimate the quality of the correction in the field on the imaging camera.

3 Configurations used for the tests In order to bring an experimental validation to the performance study presented in Nicolle et al. [2006], we have mainly used MAD in GLAO mode. As in Nicolle et al. [2006] and in accordance with an idea broadly spread in the community, we choosed to estimate the ground layer phase by 3

(a) MAPS

(b) MAPS guide sources

F IG . 2 – MAPS is both the turbulence generator and the sky simulator for MAD when it is in the lab.(Credits : ESO)

Altitude (km) Wind speed (m · s−1 ) Screen 1 0. 7. Screen 2 6. 13. Screen 3 8.5 30. Global characteristics seeing (arcsec) r0 (cm) τ0 (ms) θ0 (arcsec) 0.84 11.9 3.2 2

% of total profile 64. 22. 14. L0 (m) 22

TAB . 2 – Characteristics of the turbulence generated by MAPS.

4

averaging the phases measured in several directions. With the three SH WFSs of the SO unit, the phase to estimate is then : 3 1X φGL φk (1) c = 3 k=1

Actually, as any AO RTC, the one of MAD does not reconstruct the phase φGL c but directly command the DMs actuators from thePSH WFS slopes measurements. In the GLAO case, the RTC is fed with 3 1 the slopes vector pGL c = 3 k=1 pk and commands only the pupil DM. We exploited three different ways of estimating the slopes vector pGL c : – Star Oriented P (SO) : the gain of all the WFSs is set to the unity, so that we compute : pSO = 13 3k=1 (pk + bk ), where bk is the noise vector.

– Numerical Layer Oriented (NLO) : the gain of each WFS corresponds to the fraction of the P3 Nk total available flux that it collects. Hence we compute : pNLO = k=1 Ntot (pk + bk ), where P Nk is the flux of the kth guide source and Ntot = 3k=1 Nk . This mode mimics the LO mode. It can be shown [Bello et al. 2003] that when only photon noise exists, the NLO and the LO have exactly the same performance. Meanwhile, it has to be kept in mind that the effects of the detector noise is multiplied the number of guide sources in the NLO case with compared to the true LO case (here, multiplied by 3). – Optimized Star Oriented (OSO) : the gain of each WFS is here optimized as described in Nicolle et al. [2006]. The meaning of this optimisation is to make the best compromise between the information available from the kthPWFS and the way its measurement is degraded by the noise. Hence we compute : pOSO = 3k=1 ηk (pk + bk ), where the ηk are the optimal coeffi2 cients computed in order to minimise kφGL c − φOSO k . For this study we used the guide sources number 5, 10 and 15 of MAPS (see fig. 2), that are regularly distributed along a circle of diameter 108 arcsec. As the GLAO is characterized by a low correction, some parameters of the AO loop have been chosen in order to guaranty its robustness more than its best performance : 1. We chose to set the SH WFS reference slopes in order to work around the center of the SH sub-pupils . This is justified by considering two characteristics of the system : – To estimate and take into account the static aberrations on the analysis paths one needs the presence of a focal plane in the close vicinity of the beam splitter, where the reference source is to be set. However this possibility does not exist on MAD : the calibration unit is located rather far before the beam splitter (almost at the entrance of the bench), which introduces errors in the measurement of the reference slopes. – Moreover, table (1) shows that the PSF of one SH sub-aperture is strongly under-sampled by the pixels of the detector (4 times less than Shannon). This under-sampling induces a strong non-linearity in the measurement of the displacement of the Hartmann spots when they are not centered between four pixels. In addition it was established by the MAD team at ESO that not taking in account the on axis static aberrations does not degrade significantly the performance of the system in GLAO mode. 2. As the pixels strongly under-sample the PSF of the sub-apertures, we decided not to use the 5

fast reading mode of the WFSs CCD (binning 2 × 2). Moreover, in order to be less limited by the detector noise, we worked at a frequency of 200 Hz, that is to say half of the maximum frequency. 3. Finally we used a pure integrator command law to control the DM. The delay of the AO loop correction being of two frames, we chose to work with an integrator gain of 0.3, lower than the optimal gain but allowing to work at low flux. Once defined all these confuguration parameters, one has to choose a suitable quality criterion to estimate the system performance. In Nicolle et al. [2006], the system performance and its optimization have been studied considering a phase-related criterion. On the bench, we do not have acces to such a variable. Meanwhile the MAPS module makes possible to record long exposure PSF in 34 directions roughly uniformly spread in the 2 arcmin FOV. From these PSF, one can extract reduced quality criteria such as the Strehl Ratio, the full width at half maximum of the PSF or the Ensquarred Energy (EE). As GLAO provides only a partial correction, it has been chosen to present the following results using the ensquarred energy in a 200 × 200 mas area. It has been checked that the results are the same whatever the size of the integration area for computing the EE, and also if considering the Strehl Ratio criterion.

4 Performance study for Star Oriented, Numerical Layer Oriented and optimized Star Oriented The main objectives of this study were : – To quantify the degradation of the SO performance due to the detector noise, – To quantify the impact of high altitude turbulence on the NLO performance when the guide sources have different magnitudes, – To quantify the gain brought by the SO optimisation proposed in Nicolle et al. [2006]. In order to reach these objectives we have worked with the three guide sources mentionned above (sources number 5, 10 and 15 on the MAPS representation in fig. 2). We tested the GLAO performance in four conditions of flux. For each condition the table 3 gives the flux measured in one fully enlighed sub-aperture of each SH WFS and the equivalent visible magnitude of the source.

Configuration M 000 M 102 M 120 M 320

GS 5 V flux 10.7 1400 10.0 2675 12.6 220 13.8 70

GS 10 V flux 10.6 1540 8.75 8460 13.3 120 13.2 140

GS 15 V flux 10.5 1690 11.67 575 11.7 600 11.4 740

TAB . 3 – Visible magnitude of the guide sources and flux detected in one fully enlighted SH subaperture (in photo-electrons / sous-aperture / frame).

6

4.1 The Star Oriented performance One can first analyse the GLAO performance when computing the phase average simply by averagingg the noisy measurements obtained in the three WFS directions. Figure (3) shows the GLAO performance with the SO reconstruction, in the four flux conditions described in the table (3), when the full turbulence profile is considered (the three phase screens of MAPS are on). The first map (M 000) was realized at very high flux and with roughly the same flux on each WFS. This flux condition makes possible to achieve the best performance of the GLAO in SO mode. Hence the performance map of the configuration (M 000) can be considered as a reference for the GLAO performance. One can notice that the average across the FOV of the EE in 200 mas is 16.8 %. In open loop, the avegaged ensquarred energy is 6.7 % only. The gain brought by the GLAO is then a 2.5 factor in average over the FOV. One can notice in addition that the performance is rather uniform across the FOV, with an root mean sqarre (RMS) variation of 1.4 %. The three other maps of figure (3) show the effects of some differences in the fluxes of the guide sources. These maps clearly show how detector nois affects the correction in the directions of the faint sources. This is the case for instance of the star number 15 in the (M 102) case and of the star number 5 in the (M 320) case. This degradation acts on the two aspects of the correction : between the configurations (M 000) and (M 320), the average EE looses 4 %, whereas its RMS doubles. As an alternative to the classical SO, the numerical LO is studied in the following paragraph.

4.2 The Numerical Layer Oriented The study of the numerical LO is particularly interesting because, as it has been explained in section 3, its behavior under strong flux conditions is exactly identical to the one of the true LO concept. Nevertheless, one has to keep in mind that the contribution of the detector noise is multiplied by the number of WFS in the NLO case when compared to the LO (here, multiplied by 3). Figure (4) compares the maps of EE in 200 mas for the SO and the NLO in a configuration where the stars have different fluxes (configuration M 120) and where only the ground layer turbulence is considered. As only the phase screen conjugated to the pupil plane is present, the global seeing of the turbulence is lower than in the cases shown by figure 3 (see table 2), and the global performance of the GLAO is improved. One can see also that the RMS fluctuations of performance in one map are increased too. This is due to the fact that the lower strength of the turbulence makes the GLAO performance more sensitive to the static aberrations of the bench. Appendix A shows this phenomenon in its study of the static aberrations of the bench. Nevertheless, one can see that in Fig. 4 the fluctuations of performance are more important than they should be accroding to app. A. This is due to the fact that the measurements recorded for fig. 4 and for app. A have been taken at two different times, and that a few time after the data of fig. 4 have been recorded, we realized that some frost was on formation on the imaging camera, inducing some additionnal non common path aberrations. From the maps of Fig. 4, one can suppose that some frost was already present on the imaging camera, creating additionnal fluctuations of performance in the FOV.

7

F IG . 3 – Maps of ensquarred energy in 200 mas for GLAO in SO mode. The three phase screens of MAPS are on. (s = 0.84 arcsec)

8

SO

NLO

F IG . 4 – Maps of ensqarred energy in 200 mas for the GLAO in NLO mode and in SO mode. The sources configuration (M 120) is used, and only the turbulent phase screen located in the pupil plane is running on MAD (s = 0.64 arcsec).

9

Concerning the comparison of SO and NLO, it is noted that in accordance with the forecasts made in Nicolle et al. [2006] when all the turbulence is located in the pupil plane, the two reconstruction modes offer almost the same performance : their average performance differ from less than 1 % in EE, that is to say almost three times less than the RMS fluctutation of performance in one map. Moreover the RMS fluctuations within the maps are identical. In particular, one notes the same degradation of the performance in the direction of the weakest star (number 10, in bottom on the right). Nevertheless things change radically when the high altitude phases screens are incorporated in MAPS. Let us study this point in the case of the configuration (M 102). This configuration is interesting here because fluxes on the sensors are very strong (cf table 3), so that the detector noise is not an issue for WFSs. Moreover the flux variations between the three WFSs are important, with a gap of 3 magnitudes between the brightest GS and the faintest one. In this situation, as we have just said it, the NLO behaves exactly as an optical LO would do. The figure (5) shows the EE maps obtained for the SO and the NLO.

SO

NLO

F IG . 5 – Maps of EE in 200 mas for the GLAO in NLO mode and in SO mode. The sources configuration (M 102) is used, and the three phase screens of MAPS are on. (s = 0.84 arcsec).

One can notice that the performance of the NLO is clearly distorted in the direction of the brightest star (here, star 10, in bottom on the right). One also observes a loss of performance in the direction of star 15 (on the left) more important than in the case of the SO : the weighting of the measurement taken in this direction is too important, some information is lost. The signal to noise ratio of the WFSs being identical in both SO and NLO cases, the degradation observed for the NLO in the presence of the three phase screens is only due to the combined effects of the high altitude turbulence and the flux variations between the guide stars. The loss of performance of the NLO with compared to the SO is slightly higher than 1 % on the averaged ensquarred energy, which remains acceptable. But in the 10

same time the RMS fluctuations of performance in the FOV increase by 60 %.

M320

M120

F IG . 6 – Maps of ensqarred energy in 200 mas for the GLAO in NLO mode, in presence of three turbulent layers.

This degradation of the NLO performance can also be observed in the case of the other flux configurations considered here. This is what figure (6) shows. One can notice that in the flux configurations (M120) et (M 320) the performance is magnified in the direction of the brightest star, that is to says the star number 15 (on the left). In order to be more precise, the table 4 compares the SO and the NLO average performance across the FOV of MAD. hEE200 mas iF OV ± RM SF OV (EE200 mas ) (%) Configuration SO NLO

M 000 16.8 ± 1.4

M102 15.6 ± 3.4 14.5 ± 5.4

M120 15.7 ± 2.1 14 ± 2.6

M320 12.9 ± 3.1 13.9 ± 3.6

TAB . 4 – Comparison of SO and NLO performance with high altitude turbulence (average performance ± RMS, in % EE200 mas ). This table clearly shows that when the detector noise is not an issue for wavefront sensing (configurations M 102 and M 120) the SO concept gives a better performance for the GLAO than the NLO. This is true when considering the average performance as well as its RMS across the FOV. In particular, one notices that for the configuration (M 120), where the gap of magnitude between the brightest GS and the faintest one in only 1.6 magnitude, the loss of performance of the NLO is more than 1.5 %.

11

Finally, The NLO concept is better than the SO only when one of the WFS is limited by the detector noise (configuration M 320). One can notice that the true optical LO being less sensitive to noise than the NLO, the gain brought by a true LO should be even better for this configuration. Nevertheless, one can notice also that the gain brought by NLO in the average performance of the GLAO is paid by a slight loss in the uniformity of the correction (the RMS fluctuations increase by 17 %).

4.3 Optimization of SO Thanks to the analytical model developped in Nicolle et al. [2006], we have computed for each GS configuration the optimal gains of the WFS. Table 5 shows the computed gains and compare them with the gains coefficients corresponding to the NLO concept (that is the fraction of the total flux available per WFS). Configuration Coefficients NLO

Coefficients OSO

M102 η5 = 0.30 η10 = 0.57 η15 = 0.13 η5 = 1.03 η10 = 1.05 η15 = 0.91

M120 η5 = 0.23 η10 = 0.13 η15 = 0.64 η5 = 1.03 η10 = 0.65 η15 = 1.25

M320 η5 = 0.07 η10 = 0.15 η15 = 0.78 η5 = 0.45 η10 = 0.86 η15 = 1.55

TAB . 5 – Gain coefficients per WFS in the case of the NLO and of the optimized SO.

The main difference between the two concepts is the criterion used to determine the gains coefficients : in the case of the NLO, gains coefficients correspond to the fraction of total flux illuminating each WFS. In the optimized SO (OSO), it is the signal to noise ratio on WFS that is the interest parameter. The difference between the two approaches is obvious in the case of the configuration (M 102) : whereas the gains assigned to the three WFS are very different in the case of the NLO, they are very close in the case of the OSO : the three coefficients are almost equal to the unit, so that its performance is identical to that of the SO. The maps presented by figure 7 show the results obtained with the OSO in the two configurations with weaker stars that we have tested. Table 6 presents a comparison of the average performance of SO, NLO and OSO. Configuration SO NLO OSO

M 320 hEEi (%) ± 12.9 ± 13.9 ± 16. ±

RMS 3.1 3.6 2.8

M 120 hEEi (%) ± 15.7 ± 14. ± 15. ±

RMS 2.1 2.6 2.2

TAB . 6 – Comparison of SO, NLO and OSO performance with high altitude turbulence (average performance ± RMS, in % EE200 mas ). One can notice that the results of the optimization of the SO are very convincing in the case of 12

F IG . 7 – Maps of ensqarred energy in 200 mas for the GLAO in OSO mode, with high altitude turbulence.

the configuration (M 320) : the optimization of the SO makes it possible to exceed the performance of both the SO and the NLO, whether it is in average performance or in uniformity. Let us notice however that one cannot completely extrapolate these results with the optical LO : one of the WFS being dominated by the detector noise, the performance of the NLO is necessarily worse here than in the case of a true LO. However all analytical studies previously undertaken show that one reaches better performance with an optimized SO than with a LO [Nicolle et al. 2006]. This is precisely for this reason that the LO optimization should be seriously considered. The results of the SO optimization are on the other hand less convincing in configuration (M 120) : If the OSO gets better performance than the NLO, it remains slightly lower than the SO. The variation is however lower than 1 % of ensquarred energy. The fact that the OSO is in an intermediate situation between the NLO and the SO shows that the coefficients computed for the OSO are in fact too much dispersed. The problem doubtless comes from a bad modeling of the noise on the WFS measurements in the simulation that was used to compute the gains of the optimization. Indeed, as shown by figure (8), two points make the SH model used for the optimal gain computation unrealistic : – The noise modeling did not take into account the thresholing of the negative pixels of the WFS cameras. The point is that thresholding improves the slope variance at low flux (this is precisely how negative pixels are not accounted for in the slope estimation process). This is what we can see at low flux on figure (8) : the SH error variance is over-estimated in the model that is used for the computation of the gains with respect to the error variance actually measured on the test

13

F IG . 8 – Error in the modeling of MAD’s SH WFS behavior in the optimization algorithm.

bench. Hence, one can deduce that the gain coefficients computed for the very faint sources should be a little bit higher than what they are in table (5). – Much more annoying, the saturation effects due to the turbulent displacements of the Hartmann spots was not accounted for in the optimization algorithm. This point has a strong effect when the flux of the source is important, and is magnified by the strong under-sampling of the Hartmann spots by the pixels of the CCD. We can see this effect on the figure (8), by considering the experimental points obtained in presence of turbulence (⋄). As one can see, the consequence of this effect is that when the flux in one sub-aperture is higher than a few hundreds of photoelectrons per sub-aperture and per frame, the SH model is completely under-estimating the real slope error variance. Hence, in the case of the configuration (M 120), for which the source number 15 provides approximatively 600 photo-electrons per sub-aperture and per frame, one can see that the slope error variance is three times weaker in the optimization algorithm than what it is acutally. Then the gain coefficients computed for this star is much higher than it should be. Unfortunately, we have ackowledged the real SH behavior with respect to the flux after the experimental session on MAD. At the date we made the tests of the OSO performance, we did not nkow our SH model was wrong. Hence, for this configuration, the gain coefficients we computed (cf. table 5) are more dispersed than they should be. Finally, it has to be said that the error in the modeling of the SH WFS of MAD has also an impact on the performance of the optimized SO in the configuration (M 320). This means that the OSO performance described by figure 7 could have been even better than they are.

5 Conclusion This experimental validation made possible to confirm several important points highlighted by the analytical and numerical study presented in Nicolle et al. [2006] : – The study of the Numerical Layer Oriented enabled us to confirm the important degradation

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of the performance of a Layer Oriented concept by high altitude turbulence when this one is confronted with stars guides of different magnitudes. This degradation of the performance with compared to that of the SO affects the average performance in the corrected field at strong flux (configurations M 102 and M 120). But it is especially sensitive on the uniformity of the performance in the field : the correction is clearly exaggerated in the direction of the brightest guide star, even for weak variations of magnitude : it is the case for example in the (M 320) configuration, where the gap of magnitude between the three guide stars is only 1.5 magnitude in the visible. In this case the NLO brings a better performance in average value, but worse in uniformity than the performance of the SO. – The study of the optimized Star Oriented enabled us to confirm the profit brought by an optimal management of the directions of analysis in the rebuilding of the phase to apply to the deformable mirror of the GLAO. But it also allowed us to stress the importance of a good modeling of the wave-front sensors for the optimization of the SO. One thus saw with the example of the configuration M 320 that the ignorance of the behavior of the sensors with respect to the detector noise can make the wave-front reconstruction slightly sub-optimal. It is also important to integrate into the Shack-Hartmann model a modeling of the saturation behavior of the measurement error variance at very strong flux. If not, one exposes oneself to the risk of degrading the uniformity of the correction of the GLAO without any gain in the average performance.

R´ef´erences A. Amorim, A. Melo, J. Alves, J. Rebordao, J. Pinhao, G. Bonfait, J. Lima, R. Barros, R. Fernandes, I. Catarino, M. Carvalho, R. Marques, J.-M. Poncet, F. Duarte Santos, G. Finger, N. Hubin, G. Huster, F. Koch, J.-L. Lizon, and E. Marchetti. The CAMCAO infrared camera. In A. F. M. Moorwood and M. Iye, editors, Ground-based Instrumentation for Astronomy., volume 5492, pages 1699– 1709. Proc. Soc. Photo-Opt. Instrum. Eng., September 2004. J. M. Beckers. Increasing the size of the isoplanatic patch with multiconjugate adaptive optics. In Very Large Telescopes and their Instrumentation, ESO Conference and Workshop Proceedings, pages 693–703, Garching Germany, March 1988. ESO. D. Bello, J.-M. Conan, G. Rousset, and R. Ragazzoni. Signal to noise ratio of layer oriented measurements for multiconjugate adaptive optics. Astron. Astrophys., 410 :1101–1106, November 2003. R. H. Dicke. Phase-contrast detection of telescope seeing and their correction. Astron. J., 198(4) : 605–615, 1975. D. L. Fried. Anisoplanatism in adaptive optics. J. Opt. Soc. Am., 72(1) :52–61, January 1982. J. Kolb. Outils d’´etalonnage et de test pour les syst`emes d’Optique Adaptative Multi Conjugu´ee : Application au d´emonstrateur MAD de l’ESO. PhD thesis, Universit´e de Paris VII, 2005. th`ese. J. Kolb, E. Marchetti, S. Tisserand, F. Franza, B. Delabre, F. Gonte, R. Brast, S. Jacob, and F. Reversat. MAPS : a turbulence simulator for MCAO. In D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, editors, Advancements in Adaptive Optics., volume 5490, pages 794–804. Proc. Soc. Photo-Opt. Instrum. Eng., October 2004.

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E. Marchetti, N. Hubin, E. Fedrigo, J. Brynnel, B. Delabre, R. Donaldson, F. Franza, R. Conan, M. Le Louarn, C. Cavadore, A. Balestra, D. Baade, J.-L. Lizon, R. Gilmozzi, G. J. Monnet, R. Ragazzoni, C. Arcidiacono, A. Baruffolo, E. Diolaiti, J. Farinato, E. Vernet-Viard, D. J. Butler, S. Hippler, and A. Amorin. MAD the ESO multi-conjugate adaptive optics demonstrator. In P. L. Wizinowich and D. Bonaccini, editors, Adaptive Optical System Technologies II., volume 4839, pages 317–328. Proc. Soc. Photo-Opt. Instrum. Eng., SPIE, February 2003. E. Marchetti, R. Brast, B. Delabre, R. Donaldson, E. Fedrigo, F. Frank, N. Hubin, J. Kolb, M. Le Louarn, J.-L. Lizon, S. Oberti, F. Quir´os-Pacheco, R. Reiss, J. Santos, S. Tordo, A. Baruffolo, P. Bagnara, A. Amorim, and J. Lima. MAD Star Oriented : laboratory results for Ground Layer and Multi- Conjugate Adaptive Optics. In B.L. Brent L. Ellerbroek and D. Bonaccini, editors, Advances in Adaptive Optics II, volume 6272. Proc. Soc. Photo-Opt. Instrum. Eng., 2006. to be published. M. Nicolle, T. Fusco, V. Michau, G. Rousset, and J.-L. Beuzit. Optimization of star oriented and layer oriented wave-front sensing concepts for ground layer adaptive optics. J. Opt. Soc. Am. A, 23 : 2233–2245, September 2006. R. Ragazzoni. No laser guide stars for adaptive optics in giant telescopes ? Astron. Astrophys. Suppl. Ser., 136 :205–209, April 1999. R. Ragazzoni. Pupil plane wavefront sensing with an oscillating prism. Journal of Modern Optics, 43 :289–293, February 1996. R. Raggazzoni, E. Dialaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman. Multiple field of view layer-oriented adaptive optics. nearly whole sky coverage on 8 m class and beyond. Astron. Astrophys., 396 :731–744, December(III) 2002. F. Roddier, editor. Adaptive Optics in Astronomy. Cambridge University Press, Cambridge, 1999. G. Rousset. Wave-front sensors. In Roddier [1999], chapter 5, pages 91–130. J.-F. Sauvage, T. Fusco, G. Rousset, C. Petit, B. Neichel, A. Blanc, and J.-L. Beuzit. Fine calibration and pre-compensation of non-common path aberrations for high performance AO system. In R. K. Tyson and M. Lloyd-Hart, editors, Chromaticity effects in adaptive optics ; wavelength dependence of amplitude compensation., volume 5903, pages 88–95. Proc. Soc. Photo-Opt. Instrum. Eng., January 2005. A. Tokovinin, S. Thomas, B. Gregory, N. van der Bliek, P. Schurter, R. Cantarutti, and E. Mondaca. Design of ground-layer turbulence compensation with a Rayleigh beacon. In D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, editors, Advancements in Adaptive Optics., volume 5490, pages 870–878. Proc. Soc. Photo-Opt. Instrum. Eng., October 2004.

A

Characterization of MAD static aberrations

A.1 Characterization without turbulence In the absence of any turbulence, the main limitation of any AO system lies in its static aberrations. For a multi-WFS system, one can distinguish three categories of field aberrations :

16

– Common field aberrations : those aberrations are the ones that are located before the beam splitter on the optical path. They affect both the imaging and the wave-front sensing optical paths. But the way in which they are perceived depends on the guide star direction. Hence, the correction of these aberrations is best in the GS direction, but is degraded in the other directions of the corrected FOV. – Non common path aberrations : these are the abberrations affecting only the optical path of the imaging camera, after the beam splitter. Unless a specific process has been implemented to correct for them [Sauvage et al. 2005], they are not seen, and neither corrected by the AO loop. Hence, their impact on the system performance is the same whatever the AO system is or wherever its GS are. – Wave-front sensing paths aberrations : these aberrations are located after the beam splitter, on the optical path of each WFS. They are seen only by the corresponding WFS, that tries to correct for them whereas they do not affect the imaging path. When the AO system is calibrated (by the acquisition of reference slopes), these aberrations are automatically compensated by the AO loop. As we have seen in section 3, here it is not the case. The MAD SO unit static aberrations are visible on the maps of the figure (9). This figure shows EE maps obtained in closed loop without any turbulence and in four different cases : figures 9 - a, b and c give the performance obtained with a single conjugate AO loop closed respectively on the GS number 5, 10 and 15 . The fourth map is for the performance in GLAO mode, the SO unit being looking simultaneously at the three same GS and the estimate of the ground layer phase being the simple average of the WFSs measurements. In these maps the quality criterion is the percentage of ensquarred energy in 200 mas. Finally, we worked at high and roughly uniform flux (configuration M 000, see table 3). First we can see on figure 9 that EE in 200 mas is fairly high, with an average level around 65 %. This shows how weak are the static aberration of MAD. We can see also that figs. 9 a, b and c are rather similar. By realizing the average of the measurements in the three GS directions, GLAO (fig. 9 d) highlights the aberrations that are common to figs. 9 a, b and c. This is especially the case of the performance degradation that one can see in the lower left corner of the FOV on all the maps. These are the non common path aberrations. Although the effect of such aberrations on EE is not linear, it is possible to partially clear them by substracting two by two the maps of fig. 9. Hence, common field aberrations and wave-front sensing paths aberrations are more perceptible, as shown by figure (10), representing the point-to-point difference between the maps (9 a), (9 b) et (9 c). Nevertheless, as here each GS direction is allways used by the same wave-front sensor, we won’t be able to discriminate those two sorts of aberrations. One can see in particular on figure 10 that the order of magnitude of the common field aberrations and wave-front sensing paths aberrations is approximatively the same as the fluctuations in the FOV for one given AO loop. This means that these aberrations are not an issue for the understanding of MAD behavior. Finally, one can see that the GLAO mode also partially clear these aberrations. This is traduced in its performance map (9 d) by an improvement of both the average performance over the corrected FOV and the correction uniformity.

17

Guide stars positions.

(a) AO, GS 5

(b) AO, GS 10

(c) AO, GS 15

(d) GLAO, GSs 5, 10 et 15

F IG . 9 – Maps of EE in 200 × 200 mas, without any turbulence. Maps (a), (b) et (c) are for single conjugate AO loop closed on GS number 5, 10 and 15 respectively. Map (d) is for GLAO performance, using simultaneously GSs number 5, 10 and 15.

18

(a) AO, GS 5 - GS 10

(b) AO, GS 5 - GS 15

(c) AO, GS 10 - GS 15

F IG . 10 – Differences between the single conjugate AO performance maps of figure 9.

19

A.2 Characterization with turbulence A.2.1 Open loop Figure 11 shows the system performance in open loop, for the two turbulence profile we have used in the tests presented in this paper, that is a profile where only the phase sreen conjugated to the pupil was considered (11 a), and a total turbulence profile, where all the phases screen were used (11 b). One can notice that the two performance maps are very flat and almost identical. Nevertheless, here again the non common path aberration noticed in figure 9 (lower left corenr) is perceptible when only the ground layer turbulence is simulated. Indeed, the global seeing of the turbulence si weaker when only one phase sreen is considered (see table 2). (a) Ground layer phase screen only (s = 0.64 arcsec)

(b) 3 phase screens (s = 0.84 arcsec)

F IG . 11 – Open loop, EE in 200 mas.

A.2.2 Ground layer turbulence only Another interesting point is the performance of the system when only the ground layer turbulence is simulated by MAPS. Inded, if the turbulence phase screen is well conjugate to the pupil plane, then all the WFS must see exactly the same turbulence, and provide the same performance. Figure 12 shows the same maps as the ones of figure 9 and 10, this time with ground layer turbulence. First can can notice that in all the AO maps (12 - a, b et c), the performance is more homogeneous from one point to another of the FOV. The performance is also more homogeneous in one given point of the FOV, when corrected thanks to different AO loops (GS + WFS) This point means that an important part of the static abberrations of the bench are negligible with respect to the turbulence residuals after correction. This is confirmed by the maps (12 - e, f et g), representing the common field aberrations and the wave-front sensing paths aberrations : indeed the gaps of performance in one point of the FOV are lower thant in figure 10.

20

(a) AO, GS 5

(b) AO, GS 10

(c) AO, GS 15

(d) GLAO, GSs 5, 10 et 15

(e) a - b

(f) a - c

(g) b - c

F IG . 12 – Cartes d’´energie encadr´ee dans 200 mas en pr´esence uniquement de la turbulence proche du sol (seeing : 0.64 arcsec). Les cartes (a), (b) et (c) pr´esentent les r´esultats en AO classique pour chacune des 3 e´ toiles guides. La carte (d) pr´esente le r´esultat du GLAO. Les cartes (e) a` (g) repr´esentent les diff´erences points a` points entre les cartes (a), (b) et (c).

21

A.3 Summary Table (7) summarizes EE performance obtained on MAD in GLAO mode, at very strong and uniform flux, with respect to the turbulenc profile and to the applied correction. Configuration

hEE200 mas iF OV (%)

σF OV (EE200 mas ) (%)

No turbulence, closed loop Ground layer turbulence, closed loop Ground layer turbulence, open loop Full turbulence (3 layers), open loop

68.4 47.0 8.4 6.7

3.92 2.67 0.79 0.40

TAB . 7 – GLAO performance at high and uniform flux, with respect to the turbulenc profile and to the applied correction.

22

270ANNEXE L. EXPERIMENTAL COMPARAISON OF OPTIMIZED STAR ORIENTED AND NU

Annexe M “Sky coverage estimation for multiconjugate adaptive optics systems : strategies and results” T. Fusco et al. - MNRAS - 2006

271

Sky coverage estimation for MCAO systems: strategies and results

Journal: Manuscript ID:

Monthly Notices of the Royal Astronomical Society MN-06-0193-MJ.R1

Manuscript Type:

Main Journal

Date Submitted by the Author:

12-Apr-2006

Complete List of Authors:

Keywords:

Fusco, Thierry; ONERA, DOTA Blanc, Amandine; ONERA, DOTA Nicolle, Magalie; ONERA, DOTA Beuzit, Jean-Luc; Observatoire de Grenoble, LAOG Michau, vincent; ONERA, DOTA Rousset, Gerard; Observatoire de Paris, LESIA Hubin, Norbert; ESO, AO atmospheric effects < Astronomical instrumentation, methods, and techniques, instrumentation: adaptive optics < Astronomical instrumentation, methods, and techniques, instrumentation: high angular resolution < Astronomical instrumentation, methods, and techniques

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Mon. Not. R. Astron. Soc. 000, 000–000 (0000)

Printed 12 April 2006

(MN LATEX style file v2.2)

Sky coverage estimation for MCAO systems: strategies and results T. Fusco1⋆ , A. Blanc1,2 , M. Nicolle1, J.-L. Beuzit2 , V. Michau1 and G. Rousset1 and N. Hubin3 1 ONERA

/ D´epartement d’Optique Th´eorique et Appliqu´ee BP 72, F-92322 Chˆatillon cedex, France

2 Laboratoire 3

d’Astrophysique de l’Observatoire de Grenoble, BP 53, F-38041 Grenoble Cedex 9, France

ESO-European Southern Observatory, Karl-Schwarzschild-Straβe 2, Garching, D-85748 Germany

12 April 2006

ABSTRACT We propose an improvement of the sky coverage estimation for Multiconjugate AO (MCAO) systems. A new algorithm is presented which allows to account for the real corrected FoV surface corrected by an MCAO system (depending on GS positions and system characteristics) as well as the type of strategy (Star Oriented or Layer Oriented) considered for the wavefront sensing. An application to the ESO MCAO demonstrator (MAD) system is considered. In the context of this particular application, the importance of parameters such as the GS geometry, the generalized isoplanatic angle, the magnitude difference between GS is highlighted). Key words: techniques: high angular resolution, instrumentation: adaptive optics, anisoplanatism, sky coverage, wavefront sensing.

1 INTRODUCTION Adaptive Optics (AO) is a powerful technique to correct for the degradation induced by the atmospheric turbulence and to reach the diffraction limit of large ground-based telescopes. Due to anisoplanatic effects, the correction is efficient only in a limited field of view (FoV) around the wavefront sensor (WFS) guide star (GS). These effects originate from the fact that the turbulence is distributed in the volume above the telescope. Then, wavefronts from different directions in the sky have different aberrations. For typical atmospheric conditions and at near infrared wavelengths, the isoplanatic domain is only of a few tens of arcseconds (Fried (1982)). In addition, ⋆

E-mail: [email protected]

c 0000 RAS

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T. Fusco, A. Blanc, J.-L. Beuzit, M. Nicolle, V. Michau, G. Rousset and N. Hubin

AO systems need to have relatively bright GS (typically lower than magnitude 16-17 (see Rousset et al. (2002)) to produce a significant correction. The combination of these two points dramatically limits the portion of the sky accessible with classical AO. In order to extend the field of view (FoV), a modification of the classical AO is required leading to the concept of Multiconjugate AO (MCAO) (Dicke (1975); Beckers (1988); Ellerbroek & Rigaut (2000)). Using a GS asterism to measure the wavefront in several directions in the FoV and several deformable mirrors (DM) conjugated at different selected altitudes allows the MCAO system to correct for turbulence over a field larger than the isoplanatic patch. When the guide stars are natural ones, a key point of AO is related to the part of the sky accessible for a given system performance. This is provided by a global sky coverage study. Classically, sky coverage provides the fraction of the sky which contains stars verifying given conditions on their fluxes in order to achieve a given system performance. The generalization to MCAO implies being able to account for the number, magnitude, and magnitude difference of the Natural GS (NGS). Several algorithms have been proposed to deal with all these parameters (Le Louarn et al. (1998); Marchetti et al. (2002)). Nevertheless, they do not take into account the relative positions of the stars in the FoV. And yet, this parameter is essential to quantify the final performance of a MCAO system (Fusco et al. (2000)). We propose in this article an extension of the “classical” notion of sky coverage (Le Louarn et al. (1998)) for MCAO by introducing additional parameters in the sky coverage algorithm. These parameters are related to the observing conditions (GS geometry and magnitude difference between GS (Raggazzoni et al. (2002)), isoplanatic angle) and to the system itself (wavefront sensor concept). The article is structured as follows. We first recall, in Section 2 the different approaches already proposed for sky coverage in MCAO. In Section 3, the wavefront sensing and the reconstruction concepts in MCAO are briefly described. We particularly highlight the difference between star and layer oriented concepts. In section 4 definitions and descriptions are proposed for the various types of FoV which must be considered in a MCAO system and which are mandatory for an accurate sky coverage estimation. In Section 6 a description of our new approach for sky coverage computation in MCAO is proposed. Improvements with respect to existing algorithms are highlighted. All these points are gathered in a new sky coverage algorithm called ”‘Surface Sky Coverage”’ (SSC). This new algorithm uses, as a basic input, a statistical model of stellar population, called Besanc¸on model (see Robin et al. (2003) for more details) which is described in Section 6.4. Finally, examples of application are proposed in Section 7 for the MCAO demonstrator (MAD) developed at c 0000 RAS, MNRAS 000, 000–000

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Sky coverage estimation for MCAO systems: strategies and results

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ESO. It allows us to compare Classical Sky coverage estimation with our new approach and to highlight all the possibilities of the new sky coverage algorithm.

2 SKY COVERAGE IN MCAO The notion of sky coverage is rather simple for classical adaptive optics. Indeed, in this case, the system performance can be directly related to the GS magnitude and its separation from the object of interest (for given atmospheric conditions). The AO sky coverage is therefore nothing but a simple count of GS’s in a given sky region. Its value is given by the number of GS’s (of magnitude lower than a given limit) multiplied by the isoplanatic angle over the global size of the sky region. Unfortunately, the generalisation to MCAO system is not obvious. Instead of one single parameter, the final system performance is given by a combination of multiple factors in interaction. Two different kinds of studies can be performed to derive sky coverage information from MCAO design and performances requirements. The first one, which is considered in this article, consists in a high level study and, in fact, in a generalisation of the classical AO sky coverage estimation. The goal of our approach is to rapidly derive a first estimation of the sky coverage related to a given MCAO concept and WFS scheme. The idea here is to link the MCAO global performance to physical information on guide stars (number, magnitudes and positions) and on system (diameter, FoV, wavefront sensor approaches and WFS devices). A first system analysis allows observational constraints to be determined (magnitude ranges, number of GS, maximum distance between GS, maximum magnitude difference between GS) as a function of system characteristics (FoV, telescope diameter, system throughput, reconstruction approaches, WFS device, number of corrected modes ...) and expected performance (correction level and uniformity in the FoV). It is clear that such a kind of approach will only give tendencies. But it is essential to explore a large domain of parameters and to rapidly obtain tendencies on general behaviors as well as order of magnitudes for sky coverage (as a function of galactic position for instance) for various kind of MCAO systems. It allows us to obtain the first trade-offs concerning the system design and to adjust the scientific requirements. In particular it may help us to select the systems for which laser guide stars are mandatory. It would also help us to provide first inputs for MCAO systems on future Extremely Large Telescopes. The second one consists in a complete simulation of a given MCAO system for a large number of GS configurations obtained from the real star fields or statistical models (Arcidiacono et al. (2004); Assemat (2004)). This approach is complex and time consuming (since a lot of simulations c 0000 RAS, MNRAS 000, 000–000

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T. Fusco, A. Blanc, J.-L. Beuzit, M. Nicolle, V. Michau, G. Rousset and N. Hubin

have to be performed to reach a good statistical convergence). Nevertheless, it allows to account for complex system characteristics and provide accurate and exhaustive results. It has to be performed at a final stage of a system design in order to confirm and refine the global system performance and the system design. These two approaches complement each other and have to be considered at different stages of the study of a MCAO system. In any case, an information is required on the star repartition in the sky (for various spectral bandwidths). This information can be provided by: • a direct use of real star fields given by cross-correlation of catalogues. • a statistical approach which provides synthesis star fields. In the following, we have used the statistical approach using the so-called Besanc¸on model (Robin et al. (2003)) which will be briefly presented in Section 6.4.

3 WAVEFRONT SENSING AND RECONSTRUCTION CONCEPTS IN MCAO Wavefront sensing and reconstruction processes are two of the main issues of any MCAO system. Indeed, an accurate wavefront sensing concept is essential to obtain a pertinent information on the turbulent volume the MCAO has to correct for. In addition, these wavefront sensor measurements have to be coupled with an optimal reconstruction process (see Ellerbroek (1994); Fusco et al. (2001) and Le Roux et al. (2004)) in order to find the best command to apply on each deformable mirror. Two approaches have been proposed so far to measure and correct for the turbulent volume in MCAO: the Star oriented (SO) and the Layer oriented (LO) schemes. they are briefly described hereafter. 3.1 The star oriented concept k (r)) is measured in each GS direction (αk ) by a In the SO case (see Figure 1) the wavefront (ΦαSO

dedicated WFS as shown in Equation 1. nl oK⋆ nX n
oK⋆ αk = ϕltrue (r − αk hl ) + Noise N k ΦSO (r) k=1

(1)

k=1

l=1

where

ϕltrue

represents the true turbulent wavefront in the layer l, N k is the flux per GS direction,

K⋆ the number of GS directions, nl the number of turbulent layers used to estimate the turbulent volume and hl the altitude of the lth layer. r stands for the pupil coordinates. The Noise(.) function depends on the WFS characteristics. c 0000 RAS, MNRAS 000, 000–000

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Figure 1. Star Oriented concept in MCAO. In this WFS concept, one WFS per GS is used and the measurements form all the sensors are combined to control several DMs.

The reconstruction process consists in the computation of the turbulent volume from the wavefronts measured in the GS directions followed by a projection onto the deformable mirrors in order to obtain the best correction for a specified field of interest (see Fusco et al. (2001) for a complete description). Hence, the quality of this process is directly related to the noisy data of each WFS, i.e. to the magnitude of each GS. From a photometric point of view, the only constraint for the SO concept consists in a limiting magnitude (mlim per GS). mlim is defined as follow : considering a GS distribution in the technical FoV and a given set of system characteristics (sampling frequency, sub-aperture number, detector noise, transition ...), the MCAO performance is fulfilled within the corrected FoV when all the GS have a magnitude lower or equal to mlim .

3.2 The layer oriented concept In the LO case (see Figure 2), the phase measurement is performed using one WFS per DM. The WFS device is optically conjugated to a given altitude hj . In that case, the measured phase c 0000 RAS, MNRAS 000, 000–000

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T. Fusco, A. Blanc, J.-L. Beuzit, M. Nicolle, V. Michau, G. Rousset and N. Hubin W F Slayerj

(ϕLO

(r)) is much more complex when compared to the SO case as shown in Equation 2. ( )nW F S W F Slayerj

ϕLO

(r)

=

j=1

( PK  ⋆

 (r − αk [hj − hl ])  k=1 ηk N k P up (r − αk hj )  P  NW F S K⋆ 1  Noise γ η N [P up (r − α h )] j k j k=1 k k γj +  PK⋆   k=1 ηk N k P up (r − αk hj ) k=1

P nl

ηk N k · PK⋆ 

l l=1 ϕtrue

(2)

j=1

Where P up stands for the pupil function and γj represents the flux separation between the WFS PNwf s ( j=1 γj = 1) and ηk the optical attenuation in each GS direction before the flux coaddition

onto the WFS. The main interest of such an approach is the light co-addition before the detection

,which increases the signal to noise ratio per WFS specially when noisy CCD are considered (Bello et al. (2003)). There is no reconstruction process needed in that case since the measured W F Slayerj

phase ϕLO

(r) can be used to directly control the DM’s. Nevertheless, this co-addition has a

drawback, the wavefronts coming from different directions in the FoV being mixed and weighted by the GS flux (leading to an information loss and a possible phase estimation problem Bello et al. (2003); Nicolle et al. (2006)). As shown in Equation 2, the wavefront coming from the brighter stars are dominant in the phase measurement process. A way to deal with GS flux differences is to optically attenuate the flux of the brighter stars before the detection (ηk coefficients in Equation 2, see Nicolle et al. (2006)). From a photometric point of view the relevant parameters in the LO case are the integrated flux over all the GS, the magnitude difference between GS and the values of the attenuation coefficients ηk (0 6 ηk 6 1). Using these coefficients, one reduces the effects of magnitude difference on the LO measurement (Nicolle et al. (2005)) but it leads to a reduction of the global available flux and then to an increase of the noise measurements. To summarize, if a SO approach is considered, the critical parameter would be the GS magnitude, while two parameters would be used in the LO case: the integrated flux per WFS (or layer) and the magnitude difference between GS (including a possible attenuation coefficient per GS direction).

4 VARIETIES OF FIELD OF VIEW IN MCAO Whatever the chosen WFS concept, the use of several bright guide stars is mandatory in order to sense the turbulence volume. These GS have to be found in a FoV large enough to ensure a good measurement of the turbulence volume and thus a good correction in the directions of interest. c 0000 RAS, MNRAS 000, 000–000

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Figure 2. Layer Oriented concept in MCAO. In this case, there is one WFS per DM (conjugate at the same altitude). The flux coming from all the GS are co-added before the detection on each WFS.

According to the scientific needs, the FoV in which GS have to be found may be different than the one in which the MCAO performance has to be fulfilled. More generally, in MCAO, four notions of FoV can be defined to fully describe the system requirements and final performance. This clearly increases the complexity of the sky coverage estimation. These four FoV are described bellow and illustrated in Figure 3. 4.1 The technical FoV The technical FoV (Tech-FoV) is the field in which GS have to be found in order to sense the turbulent volume. Several constraints govern the final choice of this field: • technical constraints related to the telescope and system design. • scientific constraints, that is the average value and the evolution of the system performance in the field (Strehl ratio, encircled energy ...). These constraints are linked to the telescope and system design as well as to the atmospheric parameters (seeing, Cn2 profile ...). In particular, the size of the technical field directly impacts on the performance of a MCAO system since an increase of the technical FoV implies a larger volume of turbulence to be sensed and thus a possible increase of tomographic reconstruction error. It is interesting to mention here that larger telescope diameters may allow to deal with larger technical FoV Ragazzoni (1999); Fusco et al. (2000) which should be potentially interesting for MCAO system based on natural GS for extremely large telescopes. c 0000 RAS, MNRAS 000, 000–000

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Technical FOV

Scientific FOV

Observable FOV

Corrected FOV Figure 3. Definition of the various notions of FoV in MCAO : The technical FoV (larger one) in which the GS have to be found, the scientific FoV (6 the technical FoV) in which the required MCAO performance should be fulfilled, the corrected FoV in which, for a given GS configuration the MCAO performance are actually achieved and the observable FoV which represents the part of the scientific FoV actually corrected for a given GS configuration.

4.2 The scientific FoV The scientific FoV (Sci-FoV) corresponds to the field in which the scientific targets have to be located and thus in which the system performance has to be fulfilled. Scientific FoV can be smaller than the technical one depending on the telescope design. The performance to be achieved in the scientific FoV depends on the GS separation and magnitudes. For instance, in term of Strehl ratio, the performance SRα (α ⊆ Sci-FoV) can be expressed as follows: 2

SRα ≃ e−σα

(3)

 2 σα2 = Mα C σnoise,k k

(4)

with (see ?)

• Mα is a projection matrix from the reconstructed phase in the DM onto a given direction α of the scientific FoV. Mα depends on the scientific FoV size, the Cn2 and its sampling by the deformable mirrors. c 0000 RAS, MNRAS 000, 000–000

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 2 • C σnoise,k which is the noise propagation from the WFS measurements to the tomographic k  2 reconstruction of the volumic phase. C σnoise,k depends on the GS flux, the WFS characteristics k

and the GS relative positions. k stands for the k th GS direction.

From Equation 4, it is clear that the MCAO performance in the scientific FoV will depend on the following parameters: • the GS relative positions in the technical FoV but also with respect to the scientific one (see Sections 4.3 and 4.4) which allow to compute Mα • the GS limiting magnitudes and GS magnitude difference in the case of the LO WFS concept. These parameters should be taken into account in the MCAO sky coverage in order to provide accurate results. They will be used to provide conditions on GS which allow to ensure that σα2 is small (or uniform) enough to fulfill the performance requirement in the whole scientific FoV.

4.3 The corrected FoV The corrected FoV (Corr-FoV) represents, for a given GS configuration, the part of the technical FoV actually corrected by the MCAO system. The corrected FoV (see Figure 3) is equal to the convolution of the surface defined by the stars within the technical FoV with the isoplanatic field. Note that we have assumed here that the MCAO system is able to interpolate the wavefront inside the surface defined by the GS. Such an approximation is directly linked to the telescope diameter, the number of corrected modes and the C2n profile.

4.4 The observable FoV The observable FoV (Obs-FoV) is the last, but the really important FoV (from an astronomer point of view). It corresponds to the part of the scientific FoV actually corrected by the MCAO system for a given GS configuration. It is nothing but the intersection of the corrected FoV with the scientific FoV. Obs-FoV = Corr-FoV ∩ Sci-FoV

(5)

It is interesting to note here that when the technical FoV is equal to the scientific FoV, it automatically implies an equality between corrected and observable FoV. c 0000 RAS, MNRAS 000, 000–000

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5 ISOPLANATIC ANGLE In classical AO, the isoplanatic angle is defined as a distance from the GS position where the AO performance is better than a minimum requirement, this limit value beeing usually expressed as a residual variance or a Strehl ratio. It seems clear that, generally speaking, this isoplanatic angle should depend on the required performance (limit value), the system characteristics (actuator number, temporal sampling frequency ...), GS characteristics (flux) and of course the atmospheric conditions (Cn2 profile). A particular case is the θ0 angle defined by Fried (Fried (1982)) which represents the isoplanatic angle obtained in the case of a perfect AO system (infinite number of actuator, infinite sampling frequency, no noise) and for a residual phase smaller than 1 rad2 in the FoV. Under this assumption, Fried gives a simple expression of θ0 which only depends on atmospheric conditions but remains pessimistic for realistic AO systems. The generalization of the classical AO isoplanatic angle to MCAO systems is not straightforward and two points have to be distinguished :

• the system capability to interpolate and correct for the turbulent volume within the Field defined by the GS positions. This is characterized by the number and the positions of the GSs for the wavefront interpolation (Fusco et al. (2000)) and by the number of DM for the turbulent volume correction (?). In this paper, we will assume that the MCAO system is well dimensionned, i.e. the required performance is achieved in the field surface delimited by the GS positions. In other words, the choice of the technical FoV is done to ensure an efficient interpolation between GS. • the degradation of the correction outside the corrected area due to angular decorrelation of the wavefront with respect to the DM correction. In a first approximation, this decorrelation should be the same in MCAO and classical AO and barely depends on the size of the corrected area (large for MCAO system and equal to a single point for classical AO). In order to validate this assumption, an estimation of the isoplanatic angle is obtained on simulation (using the 100 GS realisations) as a function of a minimum SR specification for a classical AO system and a MCAO one (MAD like system as defined in Section 7.1). The results are presented in Figure 4 .The isoplanatic angle on classical AO and MCAO are nearly identical. For this particular system and for a typical Paranal profile, the evolution of the isoplanatic angle with the minimum SR can be fitted by a linear law. Depending on the chosen criterion, the isoplanatic angle goes from 50 arcseconds down to 10 arcseconds. This has to be compared with the Fried’s θ0 of 14.4 arcseconds (which is known to be a pessimistic value) for the considered atmospheric conditions and imaging wavelenghts (2.2µm). c 0000 RAS, MNRAS 000, 000–000

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Figure 4. Evolution of the generalized isoplanatic angle with respect to the specification in terms of minimum performance to achieve in the FoV. Average and rms values [fill circle] obtained on 100 GS configurations (randomly determined) are considered. The same results [open circle] are plotted in the case of a classical AO system (using the same parameters for the turbulence profile). The dashed line represents a linar fit of the measured isoplanatic values.

These results allow us to consider simple AO data to estimate the generalized isoplanatic angle to be used in the surface sky coverage algorithm and thus avoid a complex MCAO simulation.

6 SKY COVERAGE IN MCAO 6.1 The conventional definition The performance of a MCAO system depends on the quantity and quality of the wavefront measurements that can be obtained in the technical FoV. Thus, the knowledge of star number and magnitude within a given region is a key point to study: the conventional MCAO sky coverage (Le Louarn et al. (1998); Marchetti et al. (2002)) provides this information. It is based on the computation of the probability to find stars that check a set of conditions within one technical FoV, in the considered region of the sky. The classical sky coverage corresponds to the average number (in percent) of technical FoV in which the conditions on stars (number and/or magnitude) are verified (Figure 5) without any information on GS repartitions in the FoV. Assuming that the stars distribution follows Poisson statistics, one can compute the probability c 0000 RAS, MNRAS 000, 000–000

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technical FOV

Figure 5. Conventional MCAO Sky coverage. The solid line represents the technical FoV, the dashed line the actually corrected FoV given the GS configuration. In the classical MCAO sky coverage definition, the technical FoV is assumed to be corrected whatever the GS configuration.

P to find at least one star within a given radius r (Le Louarn et al. (1998)): πr 2 ν(m) (6) PN stars>0 (m, r) = 1 − exp − 36002 where ν(m) is the density of stars brighter than the magnitude m (per square degree) in a given galactic position (provided by the Besanc¸on model). The probability to find more than X stars is obtained from Equation 6: PN stars>X (m, r) = 1 −

X X

PN stars=X−i (m, r)

(7)

i=0

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This relation does not allow to introduce specific conditions on stars as limiting magnitude difference, etc. For this reason, a simulated process proposed by Marchetti et al. (2002) is considered. To compute the sky coverage for different conditions (Galactic coordinates, limiting magnitude, FoV, etc.), and in order to account for GS positions in the FoV, star fields are generated using the following process. We first define conditions on stars and on the technical FoV. • a 1 square degree field is simulated on 512×512 pixels, • the number of stars is generated using a Poisson distribution with the mean equal to the star density provided by the Galactic model at this magnitude (Besanc¸on model), • for each magnitude, the positions of the stars are defined by random deviate drawn from a uniform distribution. The sky coverage is computed using these simulated stellar fields. We search all the technical FoV (defined as proposed in Section 4.1) within the 1 square degree star field in which the searched conditions on stars are verified. The process is repeated typically 500 times and the sky coverage is obtained by averaging these results. In the classical sky coverage approach, it is implicitly assumed that all the scientific FoV is corrected if the right number of GS is found in the technical FoV. In other words, it does not account for the relative position between GS within the technical FoV. Figure 5 illustrates this approximation: the two grey technical FoV (equal here to the scientific one) contain the searched GS. In the classical sky coverage, they have the same weight in the computation whereas the part of the technical FoV really sensed by these two star configurations is very different. The results of the classical sky coverage in MCAO are thus optimistic. Hence, this sky coverage definition needs to be improved. In particular, accounting for the GS geometry is essential to well describe the system performance in a given FoV. This leads us to introduce some new parameters in the Sky coverage computation to be as close as possible to the real description of the MCAO performance in the FoV and in particular the four FoV definitions (Technical, Scientific, Corrected and Observable) presented in Section 4.

6.2 The “Surface” sky coverage definition In order to refine the sky coverage estimation, the classical approach defined in the previous section has to be corrected from the relative position of GS within the technical FoV with respect to the scientific FoV. This process is detailed below. c 0000 RAS, MNRAS 000, 000–000

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6.2.1 Refinement of the observable FoV computation The computation of the observable FoV is obtained using the GS distribution in each technical FoV, the turbulence parameters (definition of an isoplanatic angle which accounts for turbulence profile and system expected performance) and the system characteristics (see Section 5): • limiting magnitude per GS • integrated magnitude • ∆m between GS • minimum number of WFS (NW F S,min ) required to obtain a correct wavefront reconstruction (minimum number of GS) • maximum number of WFS (NW F S,max ) From these data, one can compute the corrected surface from the available GS fulfilling flux conditions (limiting GS magnitude per WFS and/or limit difference of magnitude between GS) and the scientific FoV. Three cases have to be considered for each technical FoV: • the number of GS is smaller than NW F S,min , in that case the surface value is set to 0; • the number of GS is larger than NW F S,min but smaller than NW F S,max , in that case we compute largest possible surface including all the GS; • the number of GS is larger than NW F S,max . In that case we compute all the possible surfaces including the maximum number of GS and we define the global surface as the union of all the computed surfaces. For each case, the surface is obtained by the convolution of the polygon formed by the GS position with a disk of a diameter equal to the isoplanatic angle (the higher the performance, the smaller the isoplanatic angle as illustrated in Figure 4). The surface optimisation is rather simple when only conditions on individual GS magnitudes are considered (limiting magnitude per GS). But when a condition on the magnitude difference between GS is added (including the possibility to dim the brightest star as presented in Section 3.2), the optimisation process becomes more complex. Indeed, the maximisation of the observable FoV has to be performed under the constraint of a maximum flux difference between GS in addition to the constraint on GS flux. The surface maximisation algorithm accounts for this kind of attenuation and allows to find the optimal observable surface for a given maximum flux difference between GS (with the possibility of an attenuation of the brightest star) and a given integrated flux. c 0000 RAS, MNRAS 000, 000–000

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6.3 The surface sky coverage computation The observable surface is introduced in the sky coverage computation by introducing the ratio between the surface of the observable FoV over the scientific FoV in the sky coverage computation. This new sky coverage definition is called “Surface” Sky coverage. For each GS geometry, the fraction of the scientific FoV really sensed by the GS is defined by: Sobs−f ov µs = Ssci−f ov µs is called the weighted coefficient ∈ [0,1].

(8)

The smaller the scientifics FoV in comparison to the technical FoV is, the closer to 1 µs should be. The “Surface” sky coverage defines the percentage of the sky which can be observed at a given galactic coordinate for: • a given technical FoV • a given scientific FoV • a given system performance. PNSurface stars>X (m, r) = hη . µs i

(9)

where η is a boolean number which is set to 0 when conditions on GS in the FoV are not fulfilled and set to 1 otherwise. h.i stands for a statistical average on random realisations obtained using a statistical model of stellar population (see Section 6.4). It is interesting to notice here that if µs is always set to 1 when , PNSurface stars>X (m, r) is nothing but the classical sky coverage for MCAO.

6.4 Model of stellar population The sky coverage algorithm requires a statistical knowledge of the star repartition in the Galaxy. In the following, we have considered the “Besanc¸on model” (Robin et al. (2003)). This model reproduces the stellar content of the Galaxy using some physical assumptions and a scenario of star formation and evolution.

7 SKY COVERAGE ESTIMATIONS 7.1 System assumptions In order to demonstrate the interest of our approach for sky coverage estimation in MCAO, and to highlight the importance of the new parameters integrated in the algorithm (surface defined by the c 0000 RAS, MNRAS 000, 000–000

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GS repartitions, isoplanatic angle, magnitude difference between GS in case of LO WFS) we have considered two MCAO system configurations. The first one based on a Star oriented scheme based on the MAD design (see Section 7.2) with 3 wavefront sensors and 2 DM. In that case, the important parameter is the limiting magnitude by WFS, that is the minimum number of photons required to obtain an accurate wavefront reconstruction per direction. The second one is based on a LO scheme (see Section 7.3) with two measured layers (i.e two WFS) and 2 DM. Between 3 and 8 GS are co-added on each wavefront sensor and a 50/50 beam splitting between the two measured plans. The upper limit (8 GS) is fixed by the number of devices used to collect the light from each star and to co-add this light onto a single detector. The lower limit (3 GS) is fixed by the need of an average of the wavefronts coming from several sky direction in order to isolate the contribution of the layer to be sensed by the WFS. In addition to a minimum number of GS, a limiting magnitude difference ∆m ) between GS is also mandatory to ensure a good estimation of the phase in the turbulent layers (Raggazzoni et al. (2002); Nicolle et al. (2005)) as presented in Section 3.2. For the LO WFS, the important parameter is the limiting integrated magnitude (that is the equivalent magnitude of all the GS co-added on each WFS) defined as magint,lim = 2.5log10

0.5

Ngs X

10

−0.4magi

i=1

!

(10)

where the 0.5 factor come from the beam splitting between layers (that is between WFS). It is interesting to note that the larger the number of measured layers is the smaller the flux separation coefficient shall be. Both SO and LO WFS performance depend on a limiting magnitude (or integrated magnitude in case of LO). It is clear that such a value depends both on the required performance of the MCAO system and the on telescope and system characteristics: • Global throughput, imaging and analysis wavelengths, sampling frequency • WFS type: Shack-Hartmann, Pyramid, Curvature ... • Detector characteristics: read-out noise, quantum efficiency. In the following we will lean on the MAD (Multiconjugate AO Demonstrator, currently developed at ESO) to set the main system characteristics even if several parameters will be modified in order to study their impact on the system sky coverage (in particular the CCD characteristics). The MAD bench is composed of two deformable mirrors of 66 actuators each (bimorph technology) c 0000 RAS, MNRAS 000, 000–000

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respectively conjugated at 0 and 7 km from the telescope pupil. We have chosen a MAD-like configuration because this system already includes a SO (based on shack-hartmann devices) and a LO (based on Pyramid devices) WFS mode.

7.2 The Star Oriented mode Let us first consider the case of SO measurements. As presented before, 3 WFS (that is 3 GS) are considered. The technical FoV is a disk of 2 arcmin diameter and scientific FoV equal to the technical one. For each WFS a Shack-Hartmann device is considered with 8x8 sub-apertures (8x8 pixels per sub-aperture) with a 2.4 FoV diaphragm.Each Shack-Hartmann device can pick up a GS in the entire FoV. Considering first that the GS magnitude is faint enough to ensure a good wavefront measurement per GS direction, we plot in Figure 6 the SR evolution in the FoV for four various GS configurations (randomly positioned in the FoV). It is clear that the definition of a corrected surface depends on a criterion on the system performance. In the following we have chosen to consider a minimum SR (equal to 30 %) as a limit value defining the corrected FoV. This figure highlights the importance of the introduction of a surface parameter in the sky coverage computation. As explained before, the surface sky coverage algorithm uses an isoplanatic angle (see Section 5) to account for the different corrected surfaces associated to various performance criteria (the only assumption is that the required performance is fulfilled within the GS configuration). The surface obtained from the GS positions is convolved with a isoplanatic angle to provide the estimated corrected surface for the given GS positions. An estimation of the isoplanatic angle is obtained as a function of minimum SR specifications using the Figure 4. The last, but essential parameter to consider is of course the GS limit magnitude. This limiting magnitude depends on a lot of parameters going from expected system performances to wavefront sensor and detector characteristics, system throughput and atmopsheric conditions. The first point to study is the behaviour of the system performance with respect to WFS noise for typical atmospheric conditions. In order to limit the number of variables, we will consider a reduced parameter on each WFS to characterize its performance. This parameter is a Signal to Noise ratio term which can be defined as the ratio of the signal to be measured by the WFS (angle of arrival in the case of a SH WFS) onto the measurement noises (photon and detector noises): 2 σaoa SNR = 2 2 σph + σdetec

(11)

2 SNR depends on atmospheric conditions through σaoa and system and GS characteristics through c 0000 RAS, MNRAS 000, 000–000

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Figure 6. SR evolution in the technical fov (2 arcmin diameter field) for four various GS configurations. Iso-SR are plotted.The diamond symbols represent the GS positions.

2 2 σph + σdetec . As an example, we plot in Figure 7 2D map performance obtained for various SNR

(5, 10, 25, 50) and assuming an equilateral triangular GS geometry. The 3 WFS are assumed to have the same SNR. A more complete description of the SNR influence is proposed in Figure 8 where the averaged SR (and its RMS fluctuation in the technical FoV) are plotted as a function of the SNR (same for each GS). The GS geometry is the same than the one considered in Figure 7. It shows that an average SR greater than 40 % is reached for a SNR greater than 10. A plateau is achieved for SNR larger than 50. In the following we will consider that a SNR of 10 is a limit to ensure a pertinent WF reconstruction. Figure 9 connects the WFS SNR to the GS magnitudes c 0000 RAS, MNRAS 000, 000–000

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SNR = 5

SNR = 10

SNR = 25

SNR = 50

19

Figure 7. SR evolution in the technical FoV (2 arcmin diameter field) for four SNR on GS (same SNR per GS). Iso-SR are plotted.

for various system sampling frequencies and assuming a given system throughput (here 2.1010 photons / m2 / s for a 0 magnitude GS) and detector characteristics (EMCCD with 0 read-out noise) From Figures 4, 8 and 9, it is now possible to use our surface sky coverage algorithm in order to determine the real sky coverage of a MCAO system as a function of the required performance in the field (that is the generalized isoplanatic angle derived from Figure 4) and as a function of the galactic latitude. Because the GS limiting magnitude is extremely dependant on the system c 0000 RAS, MNRAS 000, 000–000

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Figure 8. Average and RMS values of the SR in the technical FoV for various SNR on the three WFSs.

Figure 9. Evolution of the WFS SNR as a function of the GS magnitude (R band) and for various sampling frequency (from 50 to 500 Hz). Photon noise limited detector is considered.

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characteristics, we have performed the sky coverage for two limiting GS magnitudes (16 and 18). The different plots in Figure 10 show the over-estimation of a classical MCAO Sky Coverage estimation with respect to our new Surface Sky Coverage algorithm. Of course this over-estimation depends on the generalized isoplanatic angle or, in other words, on the tolerated performance in the FoV. The lower the limit performance is the smaller the over-estimation should be.

7.3 The Layer Oriented mode In addition to its SO mode, the MAD MCAO system has a LO WFS developped by INAF. In this mode, 8 star-enlarger devices (see Raggazzoni et al. (2002)) are used to pick up the flux coming from 8 GSs and to recombine the beams of these 8 GSs onto 2 pyramid WFS respectively conjugated to 0 and 7 km. As explained in Section 3.2 the main difference between SO and LO from a photometric point of view is that the limiting GS magnitude per WFS of the SO approach is replaced by the integrated magnitude per WFS (sum of the flux coming from all the GS after separation between the two WFS) and a magnitude difference per GS. This limiting magnitude difference per GS (essential to ensure a good WFS measurement and reconstruction in a LO scheme as shown in Nicolle et al. (2006)) can be ensured using optical densities for the brighter GS direction before the beam coaddition. A complete study of the GS magnitude difference on MAD performance is beyond the scope of this article but it is interesting to study the behaviour of the surface sky coverage with respect to this parameter. As explained in Section 6.3, our algorithm is able to find the set of GS maximising the observable surface under constraints on • minimum integrated magnitude, • maximum magnitude difference. Another constraint is to consider at least 3 GS (minimum value) up to 8 GS (maximum number of star enlarger devices). The optimisation of the surface defined by the GS position under the photometric constraints and GS number (3 to 8) leads to a complex multi-variable iterative process. It is nevertheless possible to find the optimal GS configuration which fulfils all the photometric requirements and ensures the largest corrected surface. The figure 11 presents the sky coverage estimations obtained with our algorithm in the LO case and for various maximum difference of magnitude between GS. The case of 30 (upper plot) and 60 (lower plot) degres of galactic latitude are considered. For each galactic latitude 4 integrated magnitudes per WFS are considered: 15, 16, 17 and 18. c 0000 RAS, MNRAS 000, 000–000

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Figure 10. Comparison between classical and surface sky coverage estimations for various galactic latitudes and various generalized isoplanatic angles. Two cases have been considered in terms of limiting GS magnitude per WFS (16 [up] and 18 [down]). The SO MAD system has been considered.

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Figure 11. Surface sky coverage as a function of the magnitude gap between GS for the LO concept. [up] Galactic latitude = 30, [down] Galactic latitude = 60. For each plot, four different integrated magnitude per WFS are considered.

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T. Fusco, A. Blanc, J.-L. Beuzit, M. Nicolle, V. Michau, G. Rousset and N. Hubin As expected, increasing the magnitude gap between GS leads to significant improvements in

term of sky coverage. Indeed, it allows to consider a large number of GS for a given integrated magnitude, or in other words to increase the number of photons for a given GS configuration, because the brightest stars are less (or even no more) optically attenuated before the detection. Figure 11 highlights the importance of the authorized gap of magnitude between GS in terms of sky coverage. A compromise has to be found between requirement in terms of wavefront measurement accuracy (which is degraded by the magnitude difference between GS in a layer oriented scheme) and the expected sky coverage of the MCAO system. Such kind of study will be proposed in a future paper.

8 CONCLUSION We have proposed an improvement of the conventional sky coverage for MCAO. The new algorithm accounts for different conditions on GS (number and flux conditions) as well as (which is new) the observed surface defined by the GS. To well represent the diversity of the MCAO both in terms of concept and expected performance, four different notions of FoV have been introduced. The new sky coverage algorithm proposed in this article allows to deal with SO and LO WFS concepts, with GS geometry in the FoV, with photometric issues (GS magnitude, integrated magnitude and magnitude difference between GS in the case of LO concept), turbulence characteristics and system expected performance (through the use of a generalized isoplanatic angle). An application to the MAD system developed at ESO has been considered. In the frame of this particular application, the importance of parameters such as the GS geometry, the generalized isoplanatic angle, the difference magnitude between GS has been highlighted.

ACKNOWLEDGMENTS The authors thank J.M. Conan for fruitfull discussion. This work was partly supported by the EC contract RII3-CT-2004-001566.

REFERENCES Arcidiacono C., Diolaiti E., Ragazzoni R., Farinato J., Vernet-Viard E., 2004, in Bonaccini Calia D., Ellerbreok B., Ragazzoni R., eds, Advancements in Adaptive Optics Vol. 5490, Sky coverage c 0000 RAS, MNRAS 000, 000–000

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for layer-oriented mcao: a detailed analytical and numerical study. Bellingham, Washington, pp 563–574 Assemat F., 2004, PhD thesis, Universit de PARIS VII, Observatoire de Meudon Beckers J. M., 1988, in Very Large Telescopes and their Instrumentation ESO Conference and Workshop Proceedings, Increasing the size of the isoplanatic patch with multiconjugate adaptive optics. Garching Germany, pp 693–703 Bello D., Conan J.-M., Rousset G., Ragazzoni R., 2003, Astron. Astrophys., 410, 1101 Dicke R. H., 1975, Astron. J., 198, 605 Ellerbroek B. L., 1994, J. Opt. Soc. Am. A, 11, 2 Ellerbroek B. L., Rigaut F. , 2000, Nature, 403 Fried D. L., 1982, J. Opt. Soc. Am., 72, 52 Fusco T., Conan J.-M., Michau V., Rousset G., Mugnier L., 2000, in Wizinowich P., ed., Adaptive Optical Systems Technology Vol. 4007, Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics. Bellingham, Washington, pp 1044–1055 Fusco T., Conan J.-M., Rousset G., Mugnier L. M., Michau V., 2001, J. Opt. Soc. Am. A, 18, 2527 Le Louarn M., Foy R., Hubin N., Tallon M., 1998, MNRAS, 295, 756 Le Roux B., Conan J.-M., Kulcs´ar C., Raynaud H.-F., Mugnier L. M., Fusco T., 2004, J. Opt. Soc. Am. A, 21 Marchetti E., Ragazzoni R., Diolaiti E., 2002, in Wizinowich P. L., Bonaccini D., eds, Adaptive optical system technologies II Vol. 4839, Which range of magnitudes for layer oriented MCAO ?. SPIE, pp 566–577 Nicolle M., Fusco T., Michau V., Rousset G., Beuzit J.-L., 2005, in Astronomical Adaptive Optics Systems and Applications II Vol. 5903, Performance analysis of multi-object wave-front sensing concepts for GLAO. SPIE Nicolle M., Fusco T., Michau V., Rousset G., Beuzit J.-L., J. Opt. Soc. Am. A, accepted Ragazzoni R., 1999, Astron. Astrophys. Suppl. Ser., 136, 205 Raggazzoni R., Dialaiti E., Farinato J., Fedrigo E., Marchetti E., Tordi M., Kirkman D., 2002, Astron. Astrophys., 396, 731 Robin A., Crz M., 1986, Astron. Astrophys., 157, 71 Robin A., Reyl C., Derrire S., Picaud S., 2003, Astron. Astrophys., 409, 523 Rousset G., Lacombe F., Puget P., Hubin N., Gendron E., Fusco T., Arsenault R., Charton J., Gigan P., Kern P., Lagrange A.-M., Madec P.-Y., Mouillet D., Rabaud D., Rabou P., Stadler E., c 0000 RAS, MNRAS 000, 000–000

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Zins G., 2002, in Wizinowich P. L., Bonaccini D., eds, Adaptive Optical System Technology II Vol. 4839, NAOS, the first AO system of the VLT: on sky performance. SPIE, Bellingham, Washington, pp 140–149

c 0000 RAS, MNRAS 000, 000–000

Annexe N “MISTRAL : a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images.” L. Mugnier, T. Fusco et J.-M. Conan - JOSAA - 2004

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MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images Laurent M. Mugnier, Thierry Fusco, and Jean-Marc Conan Office National d’E´tudes et de Recherches Ae´rospatiales, Optics Department, B.P. 72, F-92322 Chaˆtillon cedex, France Received August 8, 2003; revised manuscript received April 14, 2004; accepted April 15, 2004 Deconvolution is a necessary tool for the exploitation of a number of imaging instruments. We describe a deconvolution method developed in a Bayesian framework in the context of imaging through turbulence with adaptive optics. This method uses a noise model that accounts for both photonic and detector noises. It additionally contains a regularization term that is appropriate for objects that are a mix of sharp edges and smooth areas. Finally, it reckons with an imperfect knowledge of the point-spread function (PSF) by estimating the PSF jointly with the object under soft constraints rather than blindly (i.e., without constraints). These constraints are designed to embody our knowledge of the PSF. The implementation of this method is called MISTRAL. It is validated by simulations, and its effectiveness is illustrated by deconvolution results on experimental data taken on various adaptive optics systems and telescopes. Some of these deconvolutions have already been used to derive published astrophysical interpretations. © 2004 Optical Society of America OCIS codes: 100.1830, 100.3020, 100.3190, 010.1080, 010.1330, 110.6770.

1. INTRODUCTION The performance of high-resolution imaging with large astronomical telescopes is severely limited by atmospheric turbulence. Adaptive optics1–3 (AO) offers realtime compensation of the turbulence. The correction is, however, only partial,2,4–7 and the long-exposure images must be deconvolved to restore the fine details of the object. Because of the inevitable noise in the images, great care must be taken in the deconvolution process to obtain a reliable restoration with good photometric precision. A key point is to recognize that noise makes it necessary to add some prior knowledge on the observed object into the deconvolution method; failure to do so usually results in unacceptable amplification of the noise.8,9 Additionally, fine modeling of the noise statistics contributes to the accurate restoration of objects with a high dynamic range. Finally, the fact that the residual point-spread function (PSF) is usually not perfectly known10,11 adds to the difficulty. This paper presents a deconvolution method that falls within the maximum a posteriori (MAP) framework, or, equivalently, the penalized-likelihood framework, and that addresses these three points. It uses a prior that is well adapted to astronomical objects that are a mix of sharp structures and smooth areas, such as planets and asteroids; for pointlike objects such as binary stars, an alternative and more appropriate prior can be used. This method takes into account the presence of a mixture of photon and electronic noises. It also estimates the PSF given some prior information on the average PSF and its variability. The implementation of this method is called 1084-7529/2004/101841-14$15.00

MISTRAL (for Myopic Iterative STep-preserving Restoration ALgorithm). Although it is presented in the context of long-exposure images recorded on AO-corrected telescopes, this method can be used in other contexts as well. In particular, it has already been successfully used for Hubble Space Telescope data.12

2. IMAGING MODEL AND PROBLEM STATEMENT Within the isoplanatic angle, defined as the size of the angular patch in which the PSF due to turbulence can be considered constant, the image i of the observed object o at the focal plane of the system consisting of the atmosphere, the telescope, the AO system and the detector is given by i ⫽ 关 h * o 兴 〫n,

(1)

where * denotes the convolution operator and [•] the sampling operator, h is the PSF of the system, n is a corruptive noise process (often predominantly photon noise), and the symbol 〫 represents a pixel-by-pixel operation.8 If the noise is additive and independent of the noiseless image 关 h * o 兴 , then the symbol 〫 simply represents addition. In the following sections we shall consider that the object and the image are sampled on a regular grid, yielding a vectorial formulation for Eq. (1): i ⫽ 共 h* o兲 〫n,

(2)

where o, i, and n are the vectors corresponding to the lexicographically ordered object, image, and noise, respectively. © 2004 Optical Society of America

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Note that the raw image must be carefully preprocessed to yield an image that closely follows this imaging model. The preprocessing includes at least the correction of the background and of the flat field, the correction of the camera’s bad pixels and possibly of its correlated noise, and the scaling of the image in photons. In the case of background-dominated (e.g., thermal infrared) images, one must often record tens of images to avoid detector saturation. The preprocessing then includes the (relative) recentering of these images13 and their addition prior to deconvolution. One could contemplate processing these images jointly rather than co-adding them. Yet for AO-corrected images this would be costly in computation time while bringing very little additional information, because all PSFs are essentially the same as soon as the exposure time is long with respect to the evolution time scale of turbulence. The deconvolution procedure needs a measurement of the PSF. The usual procedure consists in recording the corrected image of a nearby unresolved star shortly before and/or after observing the object of interest. Since the correction quality depends on the observing conditions (turbulence strength, wind speed, magnitude, and spatial extent of the source used for wave-front sensing), the unresolved star image is not a perfect measurement of the PSF associated with the image to be deconvolved.10 A more precise estimate of the PSF can be obtained via control-loop data accumulated during the acquisition of the object of interest.14 The PSF estimated in this fashion still has a limited precision owing to noise and corresponds to an infinite integration time, so it is also imperfect. Furthermore, it does not intrinsically include the effect of static aberrations, even though these can be calibrated either by using the image of a star and its controlloop data14 or by a dedicated setup of the instrument.15,16 The problem is to obtain an estimate oˆ of the observed object o given the image i, a more-or-less precise knowledge of the PSF h, and some prior information on the noise statistics and on the object.

3. DECONVOLUTION APPROACH A. Deconvolution with Known Point-Spread Function We first address the classical case in which the PSF is assumed to be known. Most deconvolution techniques boil down to the minimization (or maximization) of a criterion. The first issue is the definition of a suitable criterion for the given inverse problem. The second issue is then to find the position of the criterion’s global minimum, which is defined as the solution. This second issue will be addressed in Section 4. Following the probabilistic (Bayesian) MAP approach, the deconvolution problem can be stated as follows: We look for the most likely object oˆ given the observed image i and our prior information on o, which is summarized by a probability density p(o). This reads as oˆ ⫽ arg max p 共 o兩 i兲 ⫽ arg max p 共 i兩 o兲 ⫻ p 共 o兲 . o

(3)

o

Equivalently, oˆ can be defined as the object that minimizes a compound criterion J(o) defined as follows:

J 共 o兲 ⫽ J i 共 o兲 ⫹ J o 共 o兲 ,

(4)

where the negative log-likelihood J i ⫽ ⫺ln p(i兩 o) is a measure of fidelity to the data and J o ⫽ ⫺ln p(o) is a regularization or penalty term, so the MAP solution can equivalently be called a penalized-likelihood solution. Note that the Bayesian approach does not require that o truly be the outcome of a stochastic process; rather, p(o) should be designed to embody the available prior information on o, which means that J o should have higher values for objects that are less compatible with our prior knowledge,8 e.g., that are very oscillating. When o is not the outcome of a stochastic process, J o usually includes a scaling factor or global hyperparameter, denoted by ␮ in the following, which adjusts the balance between fidelity to the data and fidelity to the prior information. If no prior knowledge is used, which corresponds to setting p(o) ⫽ constant in Eq. (3), one then maximizes p(i兩 o) (likelihood of the data) so that the solution is a maximum-likelihood solution. In this case the criterion of Eq. (4) is constituted only of the term J i . The Richardson–Lucy algorithm17 is an example of an iterative algorithm that converges toward the minimum of J i when the noise follows Poisson statistics. 1. Noise Model If the noise statistics are additive, stationary, white Gaussian, then the data fidelity term is J iswG , a simple least-squares difference between the actual data i and our model of the data for a given object, h* o. In astronomical imaging, the noise is often predominantly photon noise, which follows Poisson statistics and has the following negative log-likelihood: J iP 共 o兲 ⫽

兺 共 h* o兲共 l, m 兲 ⫺ i共 l, m 兲 ln关共 h* o兲共 l, m 兲兴 , l,m

(5) where the sum extends over all pixels (l, m) of the image. This model can perform notably better than simple leastsquares for objects with high dynamic range on a dark background, because the noise variance (h* o)(l, m) varies strongly between pixels for images of such objects. Yet in dark regions of the image the noise is usually predominantly detector noise, which follows Gaussian, and approximately stationary, statistics. A fine noise model should thus take into account both components of the noise.18 This is why we adopt a nonstationary white Gaussian model for the noise, which is a good approximation of a mix of photon and detector noise:

J imix共 o兲 ⫽

兺 2␴ l,m

1 2

共 l, m 兲

关 i共 l, m 兲 ⫺ 共 o* h兲共 l, m 兲兴 2 ,

(6) 2 2 where ␴ 2 (l, m) ⫽ ␴ ph ⫹ ␴ det is the sum of the photon noise and the detector noise variance. In the absence of detector noise, J imix from Eq. (6) is actually a second-order expansion of J iP defined in Eq. (5). Additionally, from our experience, the use of J imix rather than J iP makes the com-

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putation of the solution with gradient-based techniques faster. We believe that this can be attributed to the fact that J imix is quadratic. 2 2 The variance ␴ det and the variance map ␴ ph (l, m) can both be estimated from the image. Currently we esti2 mate ␴ det by fitting the histogram of negative-valued pixels with the left half of a centered Gaussian, which leads to

2 ␴ det

⫽

␲ 2

estimation, used by Brette and Idier26 for image restoration, and recently applied to imaging through turbulence27,28: J o 共 o兲 ⫽ ␮ ␦ 2

兺 ␾ 共 ⵜo共 l, m 兲 / ␦ 兲 ,

(7)

Of course this estimate can be accurate only if the camera offset has been subtracted carefully. The photon noise variance map is also estimated before the deconvolution, as 2 ␴ ph 共 l, m 兲 ⫽ max关 i共 l, m 兲 , 0兴 .

(8)

This estimate is quite precise for the bright regions in the image, where photon noise dominates. Its poor accuracy in the dark regions of the image is unimportant because in these regions the main contribution to the noise variance is given by the detector. Note that if the image to be processed is actually obtained by subtraction of two images (e.g., in the infrared, an image of interest and a sky background), then the noise variance map of the difference image can be obtained by adding the individual variance maps estimated by means of Eq. (8) on each of the two images. Finally, one could also contemplate estimating the pho2 ton noise variance ␴ ph (l, m) at each iteration from the current object estimate. This may make J i a nonconvex function of o, so we have discarded this possibility. 2. Object Prior This section aims at deriving an object prior for objects that are either smooth or piecewise smooth, such as planets and natural or artificial satellites. The choice of a Gaussian prior probability distribution for the object can be justified from an information theory standpoint as being the least informative, given the first two moments of the distribution. In this case, a reasonable model of the object’s power spectral density (PSD) can be found10 and used to derive the regularization criterion J o , which is then quadratic. The model parameters can even be estimated from the image itself, as shown recently in the context of phase diversity.19–21 The disadvantage of a Gaussian prior (or equivalently of a quadratic regularization term), especially for objects with sharp edges such as asteroids or artificial satellites, is that it tends to oversmooth edges. A possible remedy is to use an edge-preserving prior that is quadratic for small gradients and linear for large ones.22 The quadratic part ensures a good smoothing of the small gradients (i.e., of noise), and the linear behavior cancels the penalization of large gradients (i.e., of edges), as explained by Bouman and Sauer.23 Such priors are called quadratic–linear, or L 2 – L 1 for short.24 Here we use a function that is an isotropic version of the expression suggested by Rey25 in the context of robust

(9)

l,m

where

␾ 共 x 兲 ⫽ 兩 x 兩 ⫺ ln共 1 ⫹ 兩 x 兩 兲 关 具 i共 l, m 兲 典 共共 l,m 兲 ;i共 l,m 兲 ⭐0 兲 兴 2 .

1843

(10)

and where ⵜo(l, m) ⫽ 关 ⵜx o(l, m) 2 ⫹ ⵜy o(l, m) 2 兴 1/2, and ⵜx o and ⵜy o are the object finite-difference gradients along x and y, respectively. This functional J o is indeed L 2 – L 1 because ␾ (x) ⬇ x 2 /2 for x close to 0 and ␾ (x)/ 兩 x 兩 → 1 for x → ⫾⬁. Thus parameter ␦ is a (soft) threshold, in the sense that J o switches, at each pixel (l, m), between the quadratic and the linear behaviors depending on whether ⵜo(l, m) is smaller than or greater than ␦. The global factor ␮ and the threshold ␦ have to be adjusted according to the noise level and the structure of the object. These two hyperparameters currently have to be adjusted by hand. As a rule of thumb, a reasonable set of hyperparameters for L 2 – L 1 regularization is to take ␮ ⬇ 1 and ␦ to be on the order of the image gradient’s norm, i.e., generally several times smaller than the object gradient’s norm 储 ⵜo储 ⫽ 关 兺 l,m 兩 ⵜo(l, m) 兩 2 兴 1/2 (because for high signal-to-noise ratios, the image is essentially a smoothed version of the object). This is supported by Ku¨nsch’s findings in the context of noise removal with L 2 – L 1 regularization.29 When adjusting these hyperparameters, one should bear in mind that their sensitivity is logarithmic; i.e., one must increase or decrease them by a factor of 10 to see a notable difference. Additionally, we have noticed that convergence is somewhat faster when the inversion is more regularized, i.e., when ␮ or ␦ or both are large. This is not surprising, as the inverse problem is then better conditioned, and it suggests that one should begin with large values of the hyperparameters and make them decrease rather than the other way around. See, e.g., Ref. 12 for additional information on a working strategy of hyperparameter adjustment. The functional J o is strictly convex because ␾ ⬙ (x) ⫽ 1/(1 ⫹ 兩 x 兩 ) 2 ⬎ 0, and J imix of Eq. (6) is convex because it is quadratic, so the global criterion J ⫽ J imix ⫹ J o is strictly convex. This ensures uniqueness and stability of the solution with respect to noise and also justifies the use of a gradient-based method for the minimization. The photometric quality of the restoration is an everpresent concern for astronomers, as the interpretation of the restored image may heavily depend on it. An appealing property of the prior of Eq. (9) is that it does not bias the global photometry of the restoration. Indeed, because it is a function of the local pixel value differences, this prior is insensitive to a global offset of the object. This is notably different from other priors such as the several variants of the entropy, which incorporate prior knowledge on the pixel values and thus bias the photometry. If the object is a stellar field, then a stronger prior can be used, namely, the fact that the unknown object is a collection of Dirac delta functions. In this case, the unknown parameters are no longer a pixel map but the po-

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sitions and magnitudes of each star.11 The implementation of this alternative object prior in our restoration method, assuming that the number of stars is known, is hereafter called ␦-MISTRAL. The case of a crowded stellar field with an unknown number of stars is a difficult problem of its own often called ‘‘deconvolution of spike trains.’’ It has been addressed in many papers, including Refs. 30–33, and is outside the scope of this paper. See, e.g., Ref. 34 for a review on this subject. B. Myopic Deconvolution As mentioned in Section 2, the true PSF is seldom available. The deconvolution of turbulence-degraded images with an unknown PSF is a difficult subject. It has been tackled by many authors (see, e.g. Ref. 10 or Ref. 28 for a short review) and was first cast into a probabilistic framework, as a joint maximum-likelihood problem, by Holmes.35 Considering the PSF to be completely unknown usually leads to unacceptable results, because estimating both the object and the PSF from a single image is a highly underdetermined problem. Typically, even when the positivity36,37 of the object and the PSF is strictly enforced, the estimated PSF (or the estimated object) can be reduced to a Dirac delta function. It is thus necessary to regularize the problem by adding more constraints, both on the object (see Subsection 3.A.2) and on the PSF. In particular, the band-limitedness of the PSF has been found to be an effective one.35,36,38 For shortexposure images, the fact that the optical PSF is completely described by a phase function over the aperture is a very effective constraint.36,39 Unfortunately, for longexposure images, this constraint is not fulfilled; for such images, Conan et al. have proposed a scheme called myopic deconvolution10 that constrains the transfer function softly at all frequencies, which is more informative than the sole band-limitedness of the PSF. This scheme is adopted here; it consists in jointly estimating the PSF and the object in the same MAP framework. This joint MAP estimator is

˜ ⫽ E关h ˜ 兴 is the mean transfer function and S where h m h ˜ ˜ ⫽ E 关 兩 h( f ) ⫺ hm( f ) 兩 2 兴 is the associated spatial PSD. J h is interpretable as the energy of a set of springs (one per spatial frequency) that draw the transfer function toward its mean with a stiffness given by the PSD of the PSF, which characterizes the variability of the transfer function at each frequency. Such a regularization obviously ensures that the estimated transfer function is close to the mean transfer function with respect to error bars given by the PSD. The regularization in particular en˜ ( f ) above the cutforces a zero value for the estimate of h off of the telescope, because S h ( f ) is zero above the cutoff. This ensures the aforementioned band-limitedness of the PSF. In practice, the mean PSF and its PSD are estimated by replacing, in their definitions, the expected values (E 关 • 兴 ) by an average on the different images recorded on an unresolved star. This star must be in a region where the seeing is the same as when observing the object. Additionally, the star flux should be chosen so that the wavefront sensing noise on the star is similar to the wave-front sensing noise on the object, the latter being a function of the object’s spatial extent and flux.41 It is possible to estimate the PSF from the statistics of the wave-front sensing data,14,42 which avoids the errors due to seeing fluctuations. The quantity estimated by this technique is actually the part of the mean PSF that is due to turbulence; the myopic approach is thus interesting in that it accounts for the PSF uncertainties due to constant aberration calibration errors,11 residual speckle patterns because of the finite exposure time, and the wave-front sensing noise for faint stars. Note that the new criterion is convex in o for a given h and convex in h for a given o, but it is not necessarily convex on the whole parameter space. However, using reasonable starting points (see Section 4), we did not encounter the minimization problems associated with nonconvexity.

ˆ 兴 ⫽ arg max p 共 o, h兩 i兲 关 oˆ, h o,h

4. IMPLEMENTATION ISSUES

⫽ arg max p 共 i兩 o, h兲 ⫻ p 共 o兲 ⫻ p 共 h兲 o,h

⫽ arg min关 J i 共 o, h兲 ⫹ J o 共 o兲 ⫹ J h 共 h兲兴 .

(11)

o,h

The myopic criterion contains the two terms of Eq. (4), the first one now being a function of o and of h, plus an additional term J h ⫽ ⫺ln p(h) that accounts for the knowledge, although partial, available on the PSF. Such a three-term criterion has also been obtained in a deterministic approach.40 The regularization term for the PSF can quite naturally be derived from our probabilistic approach. The PSF is considered to be a Gaussian stochastic process since it is the temporal average of a large number of short-exposure PSFs. Additionally assuming that the difference between the PSF and its mean is stationary, J h is given by10: J h 共 h兲 ⫽

1

兺 2 f

˜共 f 兲 ⫺ h ˜ 共 f 兲兩 2 兩h m S h共 f 兲

,

(12)

A. Minimization Method The criterion of Eq. (11) is minimized numerically to obtain the joint MAP estimate for the object o and the PSF h. The minimization is performed by a conjugategradient method,43 which is usually recognized to be faster than expectation-maximization-based algorithms. The convergence of the conjugate-gradient method to a stationary point (in practice, to a local minimum) is guaranteed44 because the criterion is continuously differentiable. We have found that the convergence is faster if the descent direction is re-initialized regularly; this can be attributed to the fact that the criterion to be minimized is not quadratic. This modified version of the conjugate gradient is known as the partial conjugate-gradient method and has similar convergence properties (see Sec. 8.5 of Ref. 45). The simplest way to organize the unknowns for the minimization is to stack the object and the PSF together into a vector and to run the conjugate-gradient routine on this variable. Yet this can be slow, as the gradients of the

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criterion with respect to the object and to the PSF may have different orders of magnitude. We have found that the minimization is speeded up by splitting it into two blocks and alternating between minimizations on the object for the current PSF estimate and minimizations on the PSF for the current object estimate. Additionally, for a given (current) object, the criterion of Eq. (11) is quadratic with respect to the PSF, so its minimum in h, without the positivity constraint, is analytical. This analytical expression can still be used as a starting point for the estimation of h if one wants to enforce positivity of the PSF. The minimization starts by estimating the object for a fixed PSF taken as the mean PSF. The initial guess for the object is either the image itself or a Wiener-filtered version of the image. The minimization is not stopped by hand but rather when the estimated object and PSF no longer evolve (i.e., when their evolution from one iteration to the next is close to machine precision).

B. Positivity Constraint The object intensity map is a set of positive values, which is important a priori information. One should therefore enforce a positivity constraint on the object. This constraint can be implemented in various ways,46 such as criterion minimization under the positivity constraint, reparameterization of the object, or explicit modification of the a priori probability distribution (e.g., addition of an entropic term). The first two methods can actually be interpreted as an implicit modification of the a priori distribution that gives a zero probability to objects having negative pixel values. Note that with some expectationmaximization-based algorithms such as the Richardson– Lucy algorithm, the positivity constraint is automatically satisfied provided that the initial guess is positive. The addition of an entropic term notably slows down the minimization; additionally, it degrades the photometric quality of the restored image as mentioned above. The reparameterization is easy to implement, but the only one we found that does not notably slow down the minimization is47 o(l, m) ⫽ a(l, m) 2 , which induces local minima because it is not monotonic. We have found that the best way to ensure positivity, with respect both to speed and to not introducing local minima, is to directly minimize the criterion under this constraint. We do so by means of the projected gradient, as proposed in image processing by Commenges48 and by Nakamura et al.49

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5. VALIDATION ON SIMULATED DATA This section successively validates the contribution of the various components of our restoration method to the quality of the restored image, by means of simulations. These components are the noise model, the positivity, the edge-preserving object prior, and the myopic capability. To have a realistic PSF and PSF variability, we use as PSFs five experimental NAOS –CONICA50 images of an unresolved bright star recorded in the Br␥ narrow band (wavelength ␭ ⫽ 2.166 ␮ m). They are shown in Fig. 1. Their Strehl ratios range from 52% to 66%, and the Strehl ratio fluctuation is due to seeing variations. In all subsections but Subsection 5.D, the PSF is assumed to be known. Two types of objects are used: a planetary-type object (synthetic asteroid of uniform level with a 10% brighter broad feature and three 30% brighter spots, surrounded by a few stars), and a galaxy (scaled image of M51). The images are obtained by convolution of these objects with a PSF (the fourth of the PSFs shown in Fig. 1) and contamination by noise. The noise is the combination of photon noise (Poisson statistics with a total flux of 107 photons) and detector noise [stationary Gaussian statistics with a standard deviation of 10 photoelectrons/pixel (ph/pix)]. Figure 2 shows the objects and the simulated noisy images. The ideal, i.e., noiseless diffraction-limited, images are also shown for comparison.

A. Effect of the Noise Model The restored object minimizes Eq. (4) with J o being the quadratic regularization derived from the Gaussian prior used in Ref. 10. Figure 3 shows the best restorations obtained for each of two white-noise models when the global hyperparameter ␮ is varied. These models are the simple stationary model, which leads to the least-squares criterion J iswG mentioned in Subsection 3.A.1, and the nonstationary model J imix of Eq. (6), which accounts for both photon noise and detector noise. The root-meansquare error is slightly smaller for the finer noise model (2052 ph/pix) than for the simple one (2115 ph/pix). This error is computed as the mean square difference between the restoration and the true object on all pixels where the object’s value is nonzero, so it is indeed indicative of the restoration quality of the object rather than of the background. Visually, the presence of the star on the right of

Fig. 1. Five experimental VLT-NAOS-CONICA images of an unresolved bright star recorded on Sep. 29, 2002, at 20:58 UT in the Br␥ narrow band and used in the simulations (wavelength ␭ ⫽ 2.166 ␮ m, exposure time 2 s). The fourth one is used as the true PSF. The estimated Strehl ratios are 57%, 52%, 64%, 66%, and 58%. The corresponding Fried parameters are r 0 ⫽ 17.5, 15.6, 20.7, 21.0 and 17.0 cm.

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Fig. 2. From left to right: original object used for the simulation, noiseless image by a perfect diffraction-limited telescope (for comparison), and simulated image obtained by adding noise to the convolution of the object by the PSF (the fourth of the PSFs shown in Fig. 1). Top, asteroid; bottom, galaxy.

Fig. 3. Best restorations of the asteroid obtained with quadratic regularization for the two noise models, without a positivity constraint. Left, stationary model; right, nonstationary model, which accounts for both photon noise and detector noise. Restoration errors are 2115 and 2052 ph/pix, respectively. These restored images are zoomed by a factor of two for better visualization.

Fig. 4. Restorations of the asteroid obtained with quadratic regularization and the nonstationary noise model, without (left) and with (right) the positivity constraint. Restoration errors are 2052 and 1764 ph/pix, respectively. These restored images are zoomed by a factor of 2 for better visualization.

the asteroid is more obvious with the finer model, which is the one used from now on. B. Effect of the Positivity Constraint The influence of the positivity constraint is illustrated on the asteroid case in Fig. 4. It is implemented through projection, as discussed in Subsection 4.B. This constraint helps reduce noise and ringing in the dark regions of the image, i.e., where it is actually enforced. Indeed, the root-mean-square error drops from 2052 to 1764 ph/ pix; this constraint will thus be used in all the following restorations. Yet, some ringing remains inside the object because of its sharp edges and the use of a quadratic regularization; indeed, quadratic regularization precludes spectral extrapolation and thus can produce a restored object with a sharp cutoff in Fourier space. In the object space, this sharp cutoff takes the form of ringing, akin to Gibbs oscillations.51 Such artifacts prevent any astrophysical interpretation that would rely on precise photometry.

Fig. 5. Restorations obtained with the edge-preserving prior, the nonstationary noise model, and the positivity constraint for the best values of the hyperparameters. Restoration errors are 1201 ph/pix for the asteroid (left) and 985 ph/pix for the galaxy (right). These restored images are zoomed by a factor of 2 for better visualization.

C. Effect of the Edge-Preserving Object Prior Figure 5 shows the restoration of the asteroid and of the galaxy with the edge-preserving prior of Eq. (9). They are obtained with ( ␦ , ␮ ) ⫽ (0.03, 30) and (30, 0.03), respectively. The restoration errors are, respectively, 1201

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Keck-AO) and different kinds of astronomical objects (planetary objects, pointlike objects, etc). The average PSF and the PSD of the PSF is computed from a set of images of a nearby star recorded after (and also before, whenever possible) the object of interest. For each of these examples, astrophysical results have been derived from the deconvolved images and have lead to scientific publications. A. Planetary Objects

Fig. 6. Classical deconvolution using the mean PSF as the true one (left); myopic deconvolution (right). Restoration errors are 5139 and 2333 ph/pix, respectively, for the asteroid (top) and 1671 and 1365 ph/pix for the galaxy (bottom).

and 985 ph/pix. The enhancement of the photometric quality of the L 2 – L 1 restoration of the asteroid, with respect to the quadratic restoration of Fig. 4, is obvious: The 10% brighter broad feature and the largest of the three 30% brighter spots are visible, and the piecewise uniformity of the asteroid is respected. D. Effect of the Myopic Capability This subsection assumes that the true PSF is unknown and that we instead have five noisy PSFs taken shortly before or after the object of interest. These are the five images of a nearby star shown in Fig. 1. We thus use the myopic scheme described in Subsection 3.B, which is capable of estimating the PSF and the object simultaneously. We use the edge-preserving prior of Eq. (9) and the fine noise model of Eq. (6). The mean PSF and the PSD of the PSF are estimated by replacing, in their definitions, the expected values (E 关 • 兴 ) by an average on the five images. The quality of the restoration can be favorably compared with that of a ‘‘classical’’ (i.e., with known PSF) deconvolution by using the mean PSF: for the asteroid, the latter deconvolution gives an error of 5139 ph/ pix, whereas the myopic deconvolution gives an error of 2333 ph/pix. For the galaxy, these restoration errors are 1671 and 1365 ph/pix, respectively. Figure 6 shows the restored asteroid and galaxy for the two restoration methods. As expected, the myopic deconvolution performs better than the classical deconvolution with mean PSF because the former does not assume erroneously that the PSF is known, but not as well as the classical deconvolution with the true PSF (see Fig. 5).

6. DECONVOLUTION OF EXPERIMENTAL DATA This section presents results obtained on different types of AO systems (BOA, NAOS, HOKUPA’A, ADONIS, PUEO,

1. Ganymede Observed with BOA Ganymede is the biggest moon of Jupiter (visual magnitude M v ⫽ 4.6, diameter approximately 1.7 arcsec). It was observed by the 1.52-m telescope of Observatoire de Haute Provence by ONERA’s AO bench called BOA (for Banc d’Optique Adaptative) on September 28, 1997.52,53 The seeing conditions were particularly severe: D/r o ⬇ 23 at the imaging wavelength 0.85 ␮m. The object itself is used for the wave-front sensing. The corrected image (100 s exposure time) is shown in Fig. 7(a) and does not exhibit any detail (the Strehl ratio is approximately 5%). The field of view is 3.80 arcsec. The estimated total flux is approximately 8 ⫻ 107 photons. The star ␪ Cap (M v ⫽ 4.1), located 1.5 deg away from Ganymede, was then observed to provide a PSF calibration. A neutral density, i.e., a light attenuator that is not chromatic, was used to have approximately the same wave-front sensing conditions. The PSD of the PSF and the mean PSF were estimated from a series of 50 images recorded with an exposure time of 1 s. The difference in exposure times between the reference star and Ganymede was accounted for in the PSD computation.10 The MISTRAL de-

Fig. 7. (a) image of Ganymede with ONERA’s AO bench at the 1.52-m telescope of Observatoire de Haute Provence on 1997/09/ 28, 20:18 UT. (b) MISTRAL deconvolution. (c) Synthesized image obtained by mapping broadband probe images of NASA’s data base into a view of Ganymede seen from Earth at the time (a) was taken (courtesy NASA/JPL/Caltech, see http:// space.jpl.nasa.gov/). (d) Same synthesized image convolved with the Airy disk of a 1.52-m telescope, for comparison with the deconvolved image.

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convolution is shown on Fig. 7(b). Although a probe passing by Jupiter obviously exhibits a better resolution than a 1.52-m telescope on the ground, many features of Ganymede (various dark areas, bright spot) are visible in the deconvolved AO-corrected image and can be compared with the NASA/JPL high-resolution image [Fig. 7(c)]. The latter image was synthesized by mapping broadband probe images of NASA/JPL’s database into a view of Ganymede seen from Earth at the time Fig. 7(a) was taken (see the solar system simulator at http:// space.jpl.nasa.gov/). A fairer comparison can be done between the MISTRAL deconvolution and the image of Fig. 7(c) convolved with the Airy pattern of a 1.52-m telescope; the latter image is shown on Fig. 7(d). 2. Io Observed with Keck-AO Io, the innermost Galilean satellite of Jupiter (with angular size of 1.2 arcsec) was observed in near IR in February 2001 from the 10-m Keck II telescope with use of its AO system. After deconvolution with MISTRAL, the resolution, approximately 100 km on Io’s disk, is comparable with the best Galileo/NIMS (Near Infrared Mapping Spectrometer) resolution for global imaging and allows, for the first time, investigation of the very nature of individual eruptions54 (see Figs. 8 and 9). On February 19, two volcanos, Amirani and Tvshatar, with temperatures

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differing from the Galileo observations, were observed. On February 20, a slight brightening near the Surt volcano was detected that turned into an extremely bright volcanic outburst two days later. Thanks to the quality of the photometric restitution obtained with MISTRAL,55 these outburst data have been fitted with a silicate cooling model, which indicates that this is a highly vigorous eruption with a highly dynamic emplacement mechanism, akin to fire fountaining. Its integrated thermal output was close to the total estimated output of Io, making this the largest Ionian thermal outburst yet witnessed. 3. Io Observed with NAOS Io was observed on December 5, 2001, with the recently commissioned VLT/NAOS AO system50,56 and its infrared camera CONICA.57 The Strehl ratio on this Bracket-␥ (2.166 ␮m) observation is estimated to be 35%. This wavelength range mainly gives information about reflected sunlight modulated by various surface features. The image deconvolved with MISTRAL is shown in Fig. 10. Dark caldera, such as Pele and Pillan patera, are visible in the southeast area of the disk. The low-albedo area that is at the North of the center of the disk corresponds to Lei–Kung fluctus, a lava flow field. No hightemperature volcanic hot spot was detected during this observing night. Ground-based monitoring programs us-

Fig. 8. Jupiter-facing hemisphere of Io observed with the Keck AO system in J, H, and K band (from left to right). The basic preprocessed images from February 20, 2001, are displayed on the first row. The second row corresponds to the same images after deconvolution. Albedo features, comparable with the 20-km resolution reconstructed Galileo/SSI image (right column) are easily detected. The last row shows the February 22, 2001, images, which are dominated by the presence of the Surt outburst.

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Fig. 9. Io observed by the Keck from February 19, 2001, and deconvolved with MISTRAL. Two hot spots, corresponding to Tvashtar (north) and Amirani, are clearly detected in the H and K bands. The bad seeing conditions of this observation induced very poor AO correction in the J band, which explains the poor restitution quality in the J band.

ing AO systems on 8-m–10-m-class telescopes will help astronomers characterize the time evolution of Io’s volcanic activity, including the frequency, spatial distribution, and temperature of hot spots and surface changes.58,59 Indeed, with the Galileo mission coming to an end, the future monitoring of Io’s volcanism lies in the hands of terrestrial observers. 4. Neptune Observed with HOKUPA’A The 36-actuator curvature AO system called HOKUPA’A was used to observe Neptune at the Canada–France– Hawaii Telescope both in November 1997 and in July 1998. It produced the first sharp infrared images of Neptune. These images show the fine structures of its cloud bands with high contrast, allowing the details of Neptune’s atmospheric activity to be observed from the ground for the first time.60 Figure 11 shows images of Neptune obtained on July 6, 1998 in a methane absorption band (1.72 ␮m) and deconvolved with MISTRAL. At this wavelength Neptune’s atmosphere is very dark, and high altitude clouds appear with a high contrast. The top three images are individual 600-s exposures taken at the time indicated above each image. Note how the fine structure in the cloud bands can be followed from one frame to another as the planet rotates. The upper left and right images have been numerically rotated about Neptune’s rotation axis to match the central image and added to it to form the bottom left image, thus improving the signal-to-noise ratio. The bottom right image is the same as the bottom left one except for its color table, which better shows the low-light levels. The periodic pattern of bright dots seen just above the southernmost cloud band (at the bottom of each image) is particularly remarkable. Such a regular pattern of small clouds has never before been observed on Neptune and may be indicative of gravity waves in Neptune’s atmosphere. 5. Uranian System Observed with ADONIS MISTRAL has been applied to infrared images of Uranus acquired on May 2, 1999 with the ADONIS AO system. The deconvolved images in the J and H bands exhibit structures on the planet (bright polar haze).61 When looking at low intensity levels (see Fig. 12), one can also see the structure of the Epsilon ring and of the innermost rings, as well as very faint satellites discovered by Voyager 2 in 1986 and never reobserved since.

Fig. 10. Left: Io observed with NAOS –CONICA on December 5th, 2001 (7:14:59 UTC), in the Br␥ band (2.166 ␮m); north is up and east is left. The object itself is used for the wave-front sensing. The camera pixel scale is 13.25 mas. The seeing was 0.9 arcsec, and the estimated Strehl ratio is 35%. Right, deconvolution using MISTRAL. The two images are given in the same linear color scale. The white square represents the telescope diffraction limit at the observing wavelength.

B. Disklike Objects: MBM 12 Association Observed with Pueo In the younger association MBM 12, seven binaries and a quadruple system including a protoplanetary disk have recently been detected and deconvolved.62 For the young protoplanetary disk LkH␣ 263 C seen almost perfectly edge-on, MISTRAL was applied to recover the maximum spatial information possible (see Fig. 13). The deconvolved images were then compared with synthetic images of a disk model so as to extract structural parameters such as outer radius, dust mass, and inclination.62

C. Pointlike Objects 1. Capricornus Association Observed with ADONIS The young, nearby stellar associations are ideal laboratories to study the formation and evolution of circumstellar disks, brown dwarfs, and planets around solar-type stars. Owing to their proximity (closer than 100 pc), small separations can be reached to explore the faint circumstellar environment of such associations. In the Capricornus association (distance 48 pc; age less than 30 Myr), the source HD 199143 has been previously resolved as a binary system.63 The strong IR color (J ⫺ K ⫽ 1.37) of

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Fig. 11. False-color images of Neptune obtain at the Canada–France–Hawaii Telescope with the HOKUPA’A AO system on July 6, 1998, in a methane absorption band (1.72 ␮m) and deconvolved by MISTRAL. The object itself is used for the wave-front sensing. The top three images are individual 600-s exposures taken at 11:54:10, 12:29:38, and 13:03:29 UT time. The left and right images have been numerically rotated about Neptune’s rotation axis to match the central image and co-added to it to form the bottom images, thus improving the signal-to-noise ratio. The color scale in the bottom right image shows the fainter details.

Fig. 12. Logarithmic display of the images restored by MISTRAL; north is up and east is left. The chosen color scale saturates the high levels so as to make visible the faint details. Innermost rings and faintest satellites, first observed with Voyager in 1986, are also detected.

the companion had been estimated from an image deconvolved with IDAC (Iterative Deconvolution Algorithm in C)38 and attributed to the presence of a ‘‘circumsecondary’’ disk. New observations of Chauvin et al., after deconvolution by ␦-MISTRAL (see an example of deconvolution

with MISTRAL in the J band in Fig. 14), did not confirm the photometry in the J band of HD 199143 B. Instead, these new results showed that no disk is needed to explain the re-estimated IR color (J ⫺ K ⫽ 0.81) of HD 199143 B, now interpreted as a late M2 dwarf.64

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Fig. 13. Top row, raw images of the circumstellar disk surrounding LkH␣ 263 C in J, H, and Ks (from left to right). corresponding deconvolved images. On all images the field of view is 2.2 arcsec, with north up and east to the left.

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Fig. 14. Left, ADONIS image in the J band of HD 199143 A and B. The object itself is used for the wave-front sensing. Right, resulting image obtained with MISTRAL.

2. GJ263 Observed with NAOS Figure 15 shows how diffraction-limited imaging with NAOS –CONICA at a wavelength of 1.257 ␮m shows the individual components of the close binary star GJ 263. The angular distance between the two stars is only 0.040 arcsec. Spatially resolved observations combined with precise photometric deconvolution of binary stars like this one will allow the determination of orbital parameters and ultimately of the masses of the individual binary star components.

D. Other Objects and Other Adaptive Optics Systems A nonexhaustive list of astronomical images taken on various systems and deconvolved with MISTRAL is presented below with their corresponding publications. • Titan observed with PUEO65 and with Keck-AO66: detection of albedo surface features.

Fig. 15. NAOS –CONICA image of the double star GJ 263; the angular distance between the two components is 0.040 arcsec. The raw image, as directly recorded by CONICA, is shown in the middle, with a MISTRAL deconvolved version to the right. The recorded PSF is shown to the left. The object itself was used for the wave-front sensing. The C50S camera (0.01325 arcsec/pixel) was used with an FeII filter at the near-infrared wavelength 1.257 ␮m. The exposure time was 10 s. (See http:// www.eso.org/outreach/press-rel/pr-2001/pr-25-01.html for more details).

• Tucana–Horologium association observed with ADONIS67: precise photometry on several binaries. • The main belt asteroid 216-Kleopatra observed with ADONIS68: bifurcated shapes, density, and origin. • The main belt asteroid Vesta observed with KeckAO: shape, mineralogy, etc.

7. CONCLUSION AND PERSPECTIVES A deconvolution method has been derived in a Bayesian framework. Its three main components are a fine noise

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model, a PSF estimation capability, and a carefully designed object regularization term. The nonstationary noise model accounts for both photonic and detector noise and yields a large dynamic range in the restored images. Additionally, this method reckons with the usually imperfect knowledge of the PSF by estimating the PSF simultaneously with the object under soft constraints that embody our uncertainty about the PSF. Finally, this method comprises a regularization term that is appropriate for a wide class of objects, namely, objects that are a mix of sharp edges and smooth areas. This regularization does not bias the photometry and can restore sharp edges without ringing effects. The implementation of this method, called MISTRAL, allows a positivity constraint to be enforced both on the object and on the PSF. The contributions of the different components of this method to the overall quality of the restoration have been validated by simulations. The effectiveness of MISTRAL has been illustrated by several results on experimental and scientific data taken on various AO systems and telescopes. Additionally, this method has already been successfully employed in a number of astronomical publications to derive astrophysical results. Future work should include the automatic tuning of the hyperparameters. The deconvolution of images larger than the isoplanatic patch also deserves further study. At least when the PSF is known, the result of the use of MISTRAL ‘‘as is’’ on such data can be predicted. Indeed, it has been shown that for AO-corrected images, the PSF within an angular patch away from the guide star is the on-axis PSF convolved with an anisoplanatism PSF,69 which is a delta function on axis and widens as the considered patch direction gets farther from the guide star. Hence MISTRAL should be able to deconvolve the image from the on-axis PSF and restore an object smoothed only by this anisoplanatism PSF. Finally, integrating the PSF out of the problem rather than estimating it jointly with the object should also be studied. Indeed, the reverse operation, i.e., integrating the object out of the problem when estimating the PSF of an instrument, has already been proven to be a successful approach in the context of phase diversity, since it offers robustness to noise and should asymptotically remove the local minima associated with the joint estimation.20,21

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ACKNOWLEDGMENTS We are grateful to our colleagues of the High-Resolution Imaging team at ONERA for making the Banc d’Optique Adaptative happen as well as for many fruitful discussions. We warmly thank all the astronomers who participated in the data reduction and used the deconvolved images for their astrophysical interpretations, in particular Franck Marchis, Gae¨l Chauvin and the NAOS –CONICA team, Franc¸ois and Claude Roddier, Christophe Dumas, and Athena Coustenis. We acknowledge the referees for their most careful reviews and constructive comments. Corresponding author Laurent Mugnier can be reached at [email protected].

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314ANNEXE N. MISTRAL : A MYOPIC EDGE-PRESERVING IMAGE RESTORATION METHOD

Annexe O “Post processing fo differential image for direct extrasolar planet detection from the ground.” J.-F. Sauvage - SPIE “Astronomical Telescopes” - 2006

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Post processing of differential images for direct extrasolar planet detection from the ground J.-F. Sauvagea , L. Mugniera , T. Fuscoa and G. Rousseta, b a ONERA,

b LESIA,

BP-72, 92322 Chˆatillon Cedex, France; Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon, France ABSTRACT

The direct imaging from the ground of extrasolar planets has become today a major astronomical and biological focus. This kind of imaging requires simultaneously the use of a dedicated high performance Adaptive Optics [AO] system and a differential imaging camera in order to cancel out the flux coming from the star. In addition, the use of sophisticated post-processing techniques is mandatory to achieve the ultimate detection performance required. In the framework of the SPHERE project, we present here the development of a new technique, based on Maximum A Posteriori [MAP] approach, able to estimate parameters of a faint companion in the vicinity of a bright star, using the multi-wavelength images, the AO closed-loop data as well as some knowledge on non-common path and differential aberrations. Simulation results show a 10−5 detectivity at 5σ for angular λ separation around 15 D with only two images. Keywords: Image processing, exoplanet detection, differential imaging, inverse problem, regularisation

1. INTRODUCTION Today more than 150 exoplanets have been detected. But a great number among them are known by indirect gravitational effects on their parent star. This indirect detection and study allows one to estimate physical parameters of the companion, like its orbital period or mass, but does not indicate its atmosphere composition or its temperature. Exoplanet direct detection from the ground represents today a great scientific gain on our knowledge of exoplanet, since it allows one to perform spectroscopy of the planet. But such a detection needs a major improvement of technologies in use, since the star and its companion are separated by a fraction of arcsecond, and the flux ratio between them is extremely high (106 ). The SPHERE instrument,1 a VLT Planet Finder, will allow to detect photons coming from hot Jupiter planets and will be installed on VLT in 2010. This instrument is composed of a high performance extreme AO system,2 an optimised coronagraphic device3 and a dual band imager.4 But a dedicated post-processing method is mandatory in order to achieve the ultimate detection level of SPHERE. In this paper, we will consider the case of extreme AO coupled to differential imaging. The common use of spectral differential images is to perform differences between images at different wavelength in order to calibrate the residuals of aberration not corrected by AO and the residuals of diffraction not canceled by the coronagraph. The main limitation of differential imaging comes from differential aberrations between the two images, or between object images and reference images obtained at different times. The principle of differential imaging is detailed in the next section. We propose in the third section an optimised method dedicated to our specific issue, based on maximum a posteriori approach and able to estimate the turbulence parameters and the object in a pair of images. We present in the fourth section simulation results for the estimation of the turbulent phase structure function and the object.

2. SPECTRAL DIFFERENTIAL IMAGING Spectral differential imaging is an instrumental method that aims at “attenuating” the flux of the central star with respect to the flux of the potential companion. This method was first initiated by Racine5 and Marois.4 Thus, differential imaging plays a role slightly similar to a coronagraph. The difference between differential imaging and a coronagraph is that a coronagraph subtracts only the coherent light to the signal, but before email : [email protected]

detection. Therefore photon noise in coronagraphic images is also attenuated. The differential imaging is able to subtract also the star light, but after detection. The photon noise variance in differential imaging is therefore doubled in the combined images. Spectral differential imaging consists in acquiring two simultaneous images of a system star-companion at different wavelengths. These two images are rescaled spatially and in intensity and combined in a subtraction that reduces the flux of central star and of the residual speckle. Here we will only treat differential imaging without coronagraph, for simplicity. The subtraction should reduce the star light, but not the companion contribution in the image. This is possible if there are strong features in the companion spectrum. In the case of the giant gaseous exoplanets searched by the SPHERE project, a strong absorption band due to methane exists at 1.62µm and can be used in such a subtraction: the imaging wavelengths have to be chosen inside and outside the methane band (for example 1.575 and 1.625 µm), so that the companion emits at one wavelength, but is drastically less visible at the other one. On the other hand, the wavelengths have to be close enough to ensure the speckle pattern of the central star only differ in the two images by a spatial and intensity scaling. Let us study the image formation theory and the limitation of differential imaging. The expression of the two images iλ1 and iλ2 can be written as a convolution of the observed object by the Point Spread Function [PSF] of the instrument plus additive noises due to photon statistics and electronics:

iλ1 (~ α) = hλ1 (~ α) ∗ oλ1 (~ α ) + n1

iλ2 (~ α) = hλ2 (~ α) ∗ oλ2 (~ α ) + n2

(1)

with hλ1 and hλ2 the PSF’s in the two imaging channels which depends on turbulence parameters and static aberrations in the imaging path, α ~ the angular position in the image field or the object field, o the observed object, * stands for the convolution process, n1 and n2 stand for the noise in the images. For unresolved planets and star, the observed object is the sum of Dirac functions weighted by the total flux of the star and the companions at their respective position. The images are centred on the star, whose flux is supposed to be the same at the two wavelengths.

oλ1 (~ α) = F0 δ(~ α0 ) +

X

F1,i δ(~ αi )

i

α) = F0 δ(~ α0 ) + oλ2 (~

X

F2,i δ(~ αi )

(2)

i

with F0 and α ~ 0 the total flux and position of the central star, F1,i , F2,i and α ~ i the flux of the different companions at wavelengths λ1 and λ2 and their positions. There are two different ways to perform image subtraction: the Single Difference [SD] cancels the effect of the common static aberrations, the Double Difference [DD] cancels the effect of both common and differential aberrations and is therefore photon noise limited. - The SD consist in directly subtracting the two images and allows therefore to cancel the effect of the common aberrations in the two imaging channels. The images have to be spatially rescaled at the same wavelength by a λ2 λ1 dilation in the focal plane of the second image. iSD (~ α) , iλ1 (~ α) − iλ2 (

λ2 α ~) λ1

(3)

If we consider the case where there is no differential aberration, and if λ1 and λ2 are sufficiently close then 4 the PSF hλ1 can be well approximated by hλ2 rescaled at λ1 . The limitation of this rescaling is ∆λ Therefore λ . the SD gives a good approximation of the difference of the companions convolved by first PSF hλ1 , as the star light has been totally reduced:

iSD (~ α) =

X i

F1,i δ(~ αi ) −

X i

! λ2 F2,i δ( α ~ i ) ∗ hλ1 + n1 − n2 λ1

(4)

But in a more realistic case, the static differential aberrations are not null and the difference between the image iλ1 and the image iλ2 rescaled at λ1 makes appear the effect of differential aberrations. The two images have to be acquired simultaneously in order to see the same acquisition conditions (turbulence parameters, guide star magnitude, AO performance...). Two imaging channels are therefore used, each of them acquiring an image centred on the imaging wavelength. The efficiency of this subtraction depends on differential aberration amplitude between the two optical imaging channels, since these aberrations are the main difference between the two combined images.6 - The DD aims at solving the SD limitation by using two reference images obtained on a reference star with the same imaging tool but at another time, the DD therefore cancels the effect of the differential aberrations (assuming that they have not evolved between the two observations): iDD (~ α) =

      λ2 λ2 α ~ − iref,λ1 (~ α ~ iλ1 (~ α) − iref,λ2 α) − iλ2 λ1 λ1

(5)

The reference images are acquired at a different time, and on a different position on sky. This method is therefore sensitive to the evolution of the observing conditions between the acquisition of the two pairs of scientific and reference images. The evolution of turbulence parameters, AO performance, and most of all the evolution of quasi-static aberrations are the main limitations of the DD method.

3. POST PROCESSING FOR DIFFERENTIAL IMAGING As explained before, the detection of low flux companions (contrast around 106 between central star and companion) requires the perfect calibration of both differential static aberrations and system parameters (AO performance). In a first approximation, we assume that the static aberrations in each imaging channel are perfectly known. This is well achieved by using a phase diversity calibration, as described by Sauvage et al..7 In this framework, we present here a new post-processing deconvolution method based on a MAP approach that estimates the turbulence-induced PSF and the observed object.

3.1. Separation static / turbulent aberrations in long exposure images The image formation from the ground of stellar objects is perturbed by two factors: the atmospheric turbulence and the static aberrations of the telescope. The aberrant pupil phase is therefore the sum of two terms: φ = φt +φs with φt the turbulent part and φs the static part of the phase. The turbulent phase φt is a random variable of time and position in pupil plane and is therefore characterised by its structure function Dφ , whereas the static phase φs does not depend on time and is deterministically known. If the turbulent phase is stationary8 (as for uncorrected turbulence) then it has been shown by Roddier9 that the OTF is the product of the long exposure turbulence-induced OTF and of the static OTF:   ZZ     1 1 ˜ ~ ~ h(f ) = exp − Dφ (λf ) P (~r + λf~) exp i.φs (~r + λf~) .P (~r)∗ . exp −i.φs (λf~) d2~r 2 Spup Spup

(6)

with • P (~r) the pupil function • Dφ (λf~) the atmospheric phase structure function after AO correction at wavelength λ: ρ) , h|φt (~r + ρ~) − φt (~r)|2 i Dφt (~

(7)

The phase structure function Dφt (λf~) is a statistical term that quantifies the turbulent phase variations for two points separated by ρ ~ = λf~ in the pupil plane and its shape depends on turbulence parameters and on AO performance. If the turbulence is corrected, Dφ depends both on ~r and ρ ~ ~) = h|φt (~r + ρ ~) − φt (~r)|2 i Dφt (~r, ρ

(8)

The average h·i in the expression of Dφt (~r, ρ ~) is theoretically an average on phase occurrences (and thus on time), but may be approximated by an average h·i~r on ~r (stationarity approximation). This simplified expression Dφt (~ ρ) = h|φt (~r + ρ ~) − φt (~r)|2 i~r is therefore independent of ~r. This stationarity approximation is justifiable in the case of a Kolmogorov turbulence statistic, and often used also in the case of AO-corrected turbulent phases.8 Equation (6) shows that the global OTF is the product of a turbulence-induced OTF and a static OTF: ˜ f~) = h ˜ t (f~) · h ˜ s (f~) h(

(9)

˜ s (f~) the OTF due to telescope and the with ˜ht (f~) the long exposure OTF due to turbulence only, and h aberrations. The structure function at a wavelength λ2 can be rescaled at another wavelength λ1 by the operation described in Equation 10, in order to compute the turbulence-induced OTF in the first image. Thus, the turbulent OTF ˜ t,λ in the first image can be computed thanks to this structure function at λ2 by using relation 11. The image h 1 formation described in Equation 1 can now be rewritten as in Equation 12 taking explicitly into account the turbulent and static components of the phase. Dφt ,λ1 (~ ρ) = (

λ2 λ2 2 ) Dφt ,λ2 ( ρ ~) λ1 λ1

  1 λ2 2 λ2 ˜ ht,λ1 = exp − ( ) Dφt ,λ2 ( ρ ~) 2 λ1 λ1 iλ1 (~ α) iλ2 (~ α)

= ht,λ1 (~ α) ∗ hs,λ1 (~ α) ∗ oλ1 (~ α)

= ht,λ2 (~ α) ∗ hs,λ2 (~ α) ∗ oλ2 (~ α)

(10)

(11)

(12)

ρ), with ht,λ1 and ht,λ2 the turbulent long exposure PSF depending on turbulent phase structure function Dφ,λ (~ hs,λ1 and hs,λ2 the PSF depending on static aberrations φs,1 and φs,2 in the two imaging channels.

3.2. The post-processing framework The main limitation of differential imaging comes from differential aberrations in the two spectral channels which creates different static pattern in the images in the case of SD, and the evolution of these patterns due to system state modification in the case of DD. Therefore in our new approach we propose to estimate the PSF and the observed object o in the two images iλ1 and iλ2 . The estimation of the PSF reduces to that of its residual turbulent component as the static aberrations are supposed to be measured separately. The estimation is done thanks to the minimisation of an adequate MAP criterium J(Dφ , o). The MAP approach is based on writing the probability P = P (o, Dφ |iλ1 , iλ2 ) of a given object and Dφ knowing the images using Bayes’ theorem (see Equation 13). Finding the best object and structure function means maximising the probability P with respect to o and Dφ . P(o, Dφ ) = P (o, Dφ |iλ1 , iλ2 ) ∝ P (iλ1 , iλ2 |o, Dφ ) · P (o) · P (Dφ )

(13)

The first factor P (iλ1 , iλ2 |o, Dφ ) is called “likelihood term” and embodies the relationship between data and the sought parameters. Its statistics is given by the noise statistics in the image (stationary white Gaussian

noise in a first approximation). The other probabilities are the a priori knowledge we have on the parameters to estimate. These regularisation terms allow to smooth the criterium and to accelerate its minimisation. For instance, the turbulent phase structure function has a particular shape depending on turbulence parameters and may therefore be taken into account in this regularisation term. The criterium to minimise is J = − ln(P) with P written in Fourier space and may be rewritten as (Equation 14).

J (Dφ , o)

˜ t,λ (Dφ , f~) · h ˜ s,λ (f~) · o˜λ (f~)||2 = ||˜iλ1 (f~) − h 1 1 1 ~ ˜ ~ ~ ˜ ˜ + ||iλ2 (f ) − ht,λ2 (Dφ , f ) · hs,λ2 (f ) · o˜λ2 (f~)||2 + JR,Dφ (Dφ ) + JR,o (o)

(14)

where ˜ denotes the Fourier transform, JR,Dφ and JR,o denote regularisation terms accounting for a priori knowledge we may have on the parameters to estimate.

3.3. Assumption and subsequent simplified method In the framework of this deconvolution process, we make the following assumption in order to simplify the minimisation and demonstrate the feasibility of such a global technique: We assume that the companion presents particular spectral signature : it emits light at the first wavelength and is totally undetectable at the second wavelength, it means the object is an ideal hot Jupiter and presents strong absorption line around 1.6µm. The second image iλ2 can therefore be seen as a calibration PSF, and the global minimisation may be approximated by the three following steps : 1) Estimation of the structure function Dφ (λ2 · f~) in the image iλ2 without the object, knowing the static aberration. This corresponds to the minimisation of the two middle term with respect to Dφ in Equation 14: likelihood term on iλ2 , and regularisation term on Dφ . 2) Rescaling of the structure function (estimated at λ2 ) at wavelength λ1 according to Equation 10 and computation of global PSF h1 of the first image iλ1 , knowing the static aberration of the first channel according to Equation 11. 3) Deconvolution of the first image with the previously computed PSF h1 , and estimation of the object in iλ1 . This corresponds to the minimisation of the two terms depending on o only (first and last) in the criterium J with respect to o .

3.4. Estimation of phase structure function Dφ The turbulent phase structure function gives a statistical knowledge on a turbulent phase. For a turbulent phase 5 following a Kolmogorov profile, the structure function is given by the relation Dφ (ρ) = ( rρ0 ) 3 , with r0 the Fried parameter. But for a turbulent phase corrected by an AO system, this relation takes into account the AO system parameters and is here numerically estimated. The Figure 1 shows typical profiles of Dφ for the turbulence and AO conditions explained in section 4, with variations of seeing. Let us study the first step of the method : the estimation of structure function Dφ in a calibration long exposure PSF. A criterium (see Equation 15) is used for this minimisation, based on the likelihood term and a regularisation term on Dφ .

J (Dφ ) = +

1 ˜ s,λ ||2 ||˜iλ2 − F · exp(− Dφ ) · h 2 2 JDφ (Dφ )

(15)

Figure 1. Profiles of phase structure function for a turbulent phase corrected by AO. Condition of simulation : Paranal + SAXO, with different values of seeing.

with F the flux of the observed star, Dφ the structure function to estimate and ˜hs,λ2 the static PSF due to static aberration. Dφ is the only estimated parameter, since the star flux as well as the static aberrations (and therefore the static PSF) are assumed to be known. The regularisation term JDφ (Dφ ) is an adaptive smoothness term on estimated Dφ designed to avoid noise amplification during estimation and to allow the extrapolation of Dφ to regions where the static OTF is very small (or even null). The regularisation is done using the gradient of structure function ∇Dφ and penalises deviations between two adjacent pixels according to a typical adaptive variance, depending on pixel position. This term is computed as explained in Equation 16. JDφ (Dφ ) =

1 t −1 (∇Dφ ) C∇D (∇Dφ ) φ 2

(16)

with C∇Dφ the covariance matrix of the gradient of phase structure function ∇Dφ . This covariance matrix has been estimated on different occurences of ∇Dφ , these occurences have been generated with different value for r0 , wind speed or star magnitude. C∇Dφ quantifies the typical variability of ∇Dφ and allows one to correctly weigth the regularisation term. A common issue in regularised inversion methods and criterium minimisation is how to choose the hyperparameter, that balances the two terms of the criterium. With the Bayesian apporach adapted here and with the use of a C∇Dφ estimated by simulations, there is no such hyper-parameter to be tuned and the estimation of Dφ is completely unsupervised.

3.5. Object estimation In our procedure, the structure function and the object estimation are done sequentially. This simplified approach gives a good idea of the global approach performance, even though a global minimisation should be even more precise and therefore lead to a slightly better object estimation (which is the final goal).

The object estimation is done using MISTRAL10 algorithm developed at ONERA. This algorithm is based on the minimisation of the following criterium, and gives the best object given an image, its PSF and a priori knowledge: ˆ t,λ ∗ hs,λ ∗ o||2 + JR (o) J (o) = ||iλ1 − h 1 1

(17)

ˆ t,λ the turbulent PSF at λ1 computed with the estimated Dφ , JR (o) a regularisation term accounting with h 1 for a priori knowledge on the object. This regularisation term may contains different terms. In our particular problematic, we used a positivity constraint and a quadratic linear-quadratic regularisation.10

4. RESULTS In this section, we validate our post-processing method on simulated data. The simulation conditions are detailed in the following list, and correspond to a 8m class Telescope with an Adaptive Optics system of high performance like SPHERE, and a turbulence profile corresponding to a typical Paranal sky. The goal of this simulation is to compare the detectivity of Single Difference, Double Difference and our approach. Conditions : • λ1 = 1.60µm, λ2 = 1.58µm (corresponding to two wavelengths inside and outside the methane absorption line) • Turbulence parameter : a typical Cn2 profile for Paranal is being used with an average wind speed of 12.5 m/s, and seeing of 0.8 arcsecondes. • Adaptive Optics parameters : as extreme-AO. 41×41 actuators with spatially filtered Shack Hartmann WFS, a L3CCD working at 1.2kHz sampling frequency. The guide star has a V-magnitude of 8. • The static aberration component is randomly generated according to a n12 spectrum (n being the radial Zernike order) with 1300 Zernike coefficients and a differential wavefront error of 10nm RMS in each channel. i.e., the total differential wavefront error is 14nm RMS. • Imaging parameters : 256×256 images, with a 8m telescope. The different PSF’s hλi are generated at λi arcsecond on sky) and are therefore already spatially rescaled. The images Shannon (i.e., one pixel is 2D iλ1 and iλ2 are generated by convolution of the object and the PSF of each channel. • The object at the first wavelength is a star and three companions with a flux ratio of 10−3 for the first image, and the star alone for the object at the second wavelength. The star flux is set to a total of 107 photons for each wavelength. The companions are located close to the star, respectively at 2.5, 5.0 and 7.5 λ D. Such conditions allow us to generate quite realistic images (see example on Figure 2) that are processed by our method.

4.1. Dφ estimation : simulation results The image iλ2 is processed in order to estimate the phase structure function via the minimisation of the criterium presented in previous section. Figure 3 shows results of Dφ estimation. The true Dφ (used to generate the images) on the left shows the plateau value and central features characteristic of AO system. In the middle, the estimated structure function without regularisation (only the likelihood term is used in the criterium). Noise on the edge of the circular support of Dφ is amplified. The use of adaptive regularisation (on the right) allows us to reduce this noise amplification and gives a far better estimation of Dφ . The error profiles are plotted on Figure 4. Without regularisation, the error is lower than 0.03 rad2 . Figure 4 shows the gain brought by regularisation on Dφ estimation : the maximum error for high frequencies is one order of magnitude fainter when regularisation is used during Dφ estimation. The adaptive aspect of the regularisation allows a powerful smoothing of the estimated Dφ at the edges, and simultaneously a data-driven precise estimation of the quite oscillating Dφ near the center.

Figure 2. Example of spectral images, logarithmic scale, two of the three companions around the central star are visible on the left image (λ1 = 1.6µm), and only the star in the right image (λ2 = 1.58µm).

4.2. Computation of h1 The Dφ estimated at λ2 is now used to compute h1 , the PSF of the first image. This computed PSF will then be used in the deconvolution of the first image iλ1 . Once again we assume to know perfectly the static aberrations of first imaging channel. The OTF ˜ h1 is thus computed as product of turbulent OTF and static OTF by mean of Equations (9), (10) and (11).

4.3. Object estimation One can estimate different objects with the different Dφ estimated previously : the rough Dφ without regularisation, regularised Dφ or by using the true Dφ , used for images simulation. The different objects estimated are gathered in Figure 5, and compared with results of differential imaging in different cases. [Top left] The observed object. Central star has a total flux of 107 photon, the three companions have a ratio λ inside the AO halo. of 10−3 compared to central star and are situated at 2.5, 5.0 and 7.5 D [Top middle] result of single differential imaging. The two images at λ1 and λ2 are spatially rescaled before subtraction. The effect of differential aberrations on central star reduces contrast around it, the first companion is unseeable and the second one is visible. [Top right] result of double differential imaging. A difference of reference images has been subtracted to the single difference of images. This combination of images plays the role of a calibration of differential aberrations and allows to reduce their effect. This result is ideal since it does not account for slow variations of static aberrations between the two images, of evolution of turbulence parameters between the acquisition of object images and reference images. It is therefore a perfect DD, only limited by photon noise. [Bottom left] object estimated after deconvolution by the PSF h1,Dφ . This PSF has been computed with estimated Dφ with regularisation. The two farest companions are clearly visible, the flux ratio is almost respected. The closest companion is quite visible, but a bit hidden by residual flux coming from the central star.

Figure 3. True Dφ , Dφ estimated without regularisation, Dφ estimated with regularisation.

Figure 4. X cross-section for [left] true and estimated structure function with and without regularisation, and [right] absolute error on estimated structure function with and withoutu regularisation.

[Bottom right] object estimated after deconvolution with the PSF obtained with regularised Dφ . The noise in this object is slightly fainter than in the previous one, and the central star and the first object are better defined.

5. PERFORMANCE EVALUATION The method performance is evaluated here by the 5σ detectivity profiles. Detectivity profile is obtained as 5 times the standard deviation computed azimutally on the result of Single Difference, Double Difference, and object estimation by our approach normalised to peak flux in image.

Figure 5. logarithmic scale of [Top left] observed object, and result of [Top middle] SD and [Top right] DD, [Bottom left] deconvolved object with the PSF computed with non-regularised structure functions, [Bottom right] deconvolved object with the PSF computed with regularised structure function.

These detectivity profiles are shown on Figure 6. The simulation conditions are the one listed on section 4. As the static differential aberrations are weak (10nm RMS on each imaging channel), the DD is better than SD λ ) when differential aberrations effects dominates. Far from the optical only close to optical axis (closer than 5 D axis, the two differences are photon noise limited and the SD gives therefore slightly better detectivity. We found √ the expected gain in 2. Whatever the angular separation, our method gives better detectivity. The gain is more than 5 in the whole field of view. It is due both to the concentration of the object light in one pixel and to photon noise reduction due to our regularised deconvolution.

Figure 6. Averaged detection profiles at 5σ in the case of rough image, Single difference, double difference and our simplified object estimation.

6. CONCLUSION We propose a method which allows to solve SD limitations (differential aberrations) and gives better results than λ DD, without reference images. In a perfect case, detectivity at 5σ reaches less than 10−5 at 15 D . This result has still to be tested in more complex cases (slowly evolving static aberrations or mis-calibration, residual background) but gives a rough idea of the potentiallity of the method. The perspectives for this method are to perform a global estimation of the parameters (structure function, object in the two images and static aberrations), to process real images obtained on a differential imager like NACO SDI, and to generalise our approach to coronagraphy.

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