Imaging and nulling properties of sparse-aperture Fizeau interferometers
Imaging and nulling properties of sparse-aperture Fizeau interferometers François Hénault Institut de Planétologie et d’Astrophysique de Grenoble, Université Joseph Fourier, CNRS, B.P. 53, 38041 Grenoble – France
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
General layout of a Fizeau interferometer B
O
P1
Telescope 1
P2
(P)
Telescope 2 FC
Relay optics 1
Relay optics 2
APS 1
Metrology beam 1
APS 2
Metrology beam 2
B’
Converging optics Diverging optics Fold mirror Beamsplitter Acromatic Phase Shifter
O’
Combining optics
(P’)
F’
Focal plane
• All telescopes assumed to be identical • All exit pupils optically conjugated with entrance pupils
Z
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Coordinates systems
On-sky angular coordinates V
Entrance pupil plane
Sky object
U Y
v u
s
sO
D P1
Exit pupil plane
X Y’
O P2
P’3
P4
X’
Detection plane
O’ P3
P’4
B
• Input pupils defined by vectors OPn • Output pupils defined by vectors O’P’n 1≤n≤N
P’1 D’
Conference 9146 - Optical and Infrared Interferometry IV
Y’’
P’2 sO ’
s’
X’’ F’ O’’
B’ M’
Z
MO’’ Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Image of a sky object projected onto the sky Most general Object-Image relationship : I(s) =
∫∫
2
N
O(s O ) PSFT (s - s O )
sO ∈ ΩO
∑a n =1
n
exp[i φ n ] exp[i k ξ (s O , s )] dΩ O
with ξ (s O , s ) = s O OPn − s O' P'n / m • PSFT(s) : PSF of an individual collecting telescope, projected back onto the sky • an : amplitude transmission factor of the nth telescope • ϕn : phase-shift along the nth interferometer arm for cophasing or nulling purpose • k = 2π/λ : wavenumber of monochromatic electro-magnetic field • m : optical compression factor between telescopes and their relay optics Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Golden rule of Fizeau interferometers • First-order approximation • Necessary condition : O’P’n = m OPn Pupil In
Pupil Out
• Object-Image relationship
I(s) = [PSFT (s) F(s)] ∗ O(s) • Invariant PSF over the Field of view (FoV)
F(s) = PSFT (s)
N
∑a n =1
Conference 9146 - Optical and Infrared Interferometry IV
n
[
exp[i φ n ] exp i k s OPn Montréal, 23 juin 2014
]
2
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Examples of application • Effect of pupil aberrations • Deviations from the “golden rule” : nonhomothetic exit pupil combiners • Simulation of images in nulling mode
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Effect of pupil aberrations (P)
n
(
n
O' Pn' = y 'n 1 + dz 'n F' s= cos u sin v ≈ v cos u cos v ≈ 1 − u 2 2 − v 2 2 dz 'n
)
dz2
B
Flat wavefront
• Develop all vectors at 2nd-order, e.g. : x ' (1 + dz ' F') sin u ≈ u
Input pupils P2 F
O
Z Beam compressors
P1 dz1
Collecting telescopes
• Insert into the OPD :
(
)(
)
Output pupils
ξ (sO , s ) ≈ dz n − dz 'n m + u 0 x n + v 0 y n − u x 'n + v y'n 1 + dz 'n F' m 2 0
)2
− dz n (u + v '
2
2
) 2m
F’ dz2’
• Use general Object-Image relation : I(s) =
∫∫
sO ∈ ΩO
2
N
O(sO ) PSFT (s - s O )
∑a n =1
n
exp[i φ n ] exp[i k ξ (s O , s )] dΩ O
• No longer a convolution operator !
X”
(P’)
B’
P’2
Flat wavefront
− dz n (u + v 2 0
O” O’
Z
P’1 dz1’
Multi-axial combiner
Focal plane
(exit pupil plane)
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
7
Imaging and nulling properties of sparse-aperture Fizeau interferometers
Pupil aberration: 2 telescopes PSF at 0.6 arcsec
0 2.5 arcsec
t PSF Invarian
Aberrated pupil Case 2
inges, r f d e v r Cu g PSF changin
No pupil aberration
PSF at 0.3 arcsec
Aberrated pupil Case 1
PSF at FoV centre
π
Achievable FoV
2.5 arcsec Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Pupil aberration: 8 telescopes PSF at FoV centre
PSF at 0.3 arcsec
PSF at 0.6 arcsec
π
Aberrated pupil Case 2
2.5 arcsec
Aberrated pupil Case 1
0
π
0
t PSF Invarian
No pupil aberration
0
π
π
0
Achievable FoV
2.5 arcsec Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Different combining schemes From collecting telescopes B
From collecting telescopes B APS 1
APS 2
Metrology beam 1 Metrology beam 2
Metrology beams 2-3
Metrology beams 1-4
Fringe tracker
Axial combiner
Crossedcubes nuller
(P’) O’ F’
B’ O’
(P’) F’
Focal plane Z
Focal plane Z
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
10
Imaging and nulling properties of sparse-aperture Fizeau interferometers
Axially Combined Interferometer (ACI) B
O
P1
Telescope 1
P2
(P)
Telescope 2
Relay optics 1
Relay optics 2
APS 1
APS 2
Metrology beam 1 Metrology beam 2
Axial beam combiner
Fringe tracker
• Co-axial combination by means of a balanced set of beamsplitters • Extinguishes all diffracted light originating from central star Only leaves planet photons • Specific Object-Image relationship
(P’) O’ F’
Focal plane
N Ι(s) = PSFT (s) * ∑ a n exp[i φ n ] exp i k s OPn n =1
[
]
2
O(s)
Z
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
11
Imaging and nulling properties of sparse-aperture Fizeau interferometers
Crossed-cubes nuller (CCN) • See the “Cheapest nuller in the World”: – One talk on Thursday afternoon, 2 posters on Wednesday afternoon
Cube 1
Focusing optics
Cube 2 Y’
X’ Z O"
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
12
Imaging and nulling properties of sparse-aperture Fizeau interferometers
Pseudo-images of a companion 2.5 arcsec
20 m
10 mm
Star
π
π
π
π
π
Planet
π π
π π π
π π
Fake sky objects
Telescope array
Phase-shifts maps
2.5 arcsec
2.5 arcsec
2.5 arcsec
Star
Planet photons only
Axial combiner
Planet
Maximal densification
Conference 9146 - Optical and Infrared Interferometry IV
Fizeau interferometer
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Conclusion • A simple Fourier optics formalism allows fast evaluation of nulling Fizeau interferometers performance – Including PSF, achievable Field of view (FoV), nulling maps and pseudo-images – Whatever is the telescope number N – In presence of optical defects (pupil aberrations) – Also applicable to non-Fizeau interferometers
• Simple formalism, no actual Fourier or Fresnel transforms required (gain in accuracy and computing time) • Simulations show that that the most efficient nulling schemes should use axial combiners or multi-axial combiners with maximal densification (e.g. Crossed-cubes nuller) Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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Imaging and nulling properties of sparse-aperture Fizeau interferometers
Previous publications • • • • • • •
“Fibered nulling telescope for extra-solar coronagraphy,” Optics Letters vol. 34, n° 7, p. 1096-1098 (2009) “Computing extinction maps of star nulling interferometers,” Optics Express vol. 16, 4537-4546 (2008) “Fine art of computing nulling interferometer maps,” Proceedings of the SPIE 7013, n°70131X (2008) “Simple Fourier optics formalism for high angular resolution systems and nulling interferometry,” JOSA A vol. 27, p. 435-449 (2010) “PSF and field of view characteristics of imaging and nulling interferometers,” Proceedings of the SPIE vol. 7734, n°773419 (2010) “Imaging power of multi-fibered nulling telescopes for extra-solar planet characterization,” Proceedings of the SPIE vol. 8151, n°81510A (2011) “Cheapest nuller in the world: crossed beamsplitter cubes,” Proceedings of the SPIE vol. 9146, this conference Perhaps one synthesis paper some day…
Conference 9146 - Optical and Infrared Interferometry IV
Montréal, 23 juin 2014
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