APPLIED PHYSICS LETTERS 90, 031106 共2007兲

High-resolution mapping of the three-dimensional point spread function in the near-focus region of a confocal microscope Michael J. Nasse and Jörg C. Woehla兲 Laboratory for Surface Studies, Department of Chemistry and Biochemistry, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53211

Serge Huant Laboratoire de Spectrométrie Physique, Université Joseph Fourier Grenoble and CNRS, 38402 St. Martin d’Hères, France

共Received 2 October 2006; accepted 11 December 2006; published online 17 January 2007兲 Fluorescent nanobeads with a diameter of 20 nm were used to map the three-dimensional point spread function in the near-focus region of a confocal microscope at high spatial resolution. Fluorescence images were taken in 109 equidistant planes 共50 nm apart兲 parallel to the focal plane; postacquisition stacking of these images allows the reconstruction of the point spread function in the axial plane. The experimental distribution is compared to theoretical calculations based on an integral representation for the light intensity in the focus region that takes into account stratified media, polarization, the Gaussian illumination profile, and the finite exit pinhole size. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2431764兴 Many modern optical microscopy techniques such as fluorescence correlation spectroscopy1,2 or confocal scanning microscopy3 rely on the knowledge of the three-dimensional light distribution in the focal region of a microscope objective. Since the late 19th century it is known that the energy density in the vicinity of the focus, the illumination point spread function 共PSF兲, shows complicated features with side maxima and minima due to diffraction effects created by the lens aperture. Experimental verification of the near-focus PSF has been difficult because of the small dimensions involved, especially in systems with high numerical apertures 共NA兲.4–6 We have used commercial, fluorescent nanobeads 共F8888, Molecular Probes, Inc.兲 with a diameter of 20 nm as probes to map out the overall PSF in the focal region of a confocal microscope. Such nanobeads can be considered as pointlike emitters, since they are about an order of magnitude smaller than the diffraction limited spot size; they are probes of the electric field intensity averaged over the bead volume7 and have been used, for example, to develop a realistic model for the near field of an optical tip as used in aperture-type near-field scanning optical microscopy,7 and to perform in situ characterizations of these tips.8 In contrast to single molecules or quantum dots, fluorescent nanobeads are very photostable and show no blinking or abrupt photobleaching events. Also, a single nanobead is loaded with randomly oriented dye molecules; therefore, its fluorescence intensity is insensitive to the polarization state of the excitation light. This is crucial for the present case where the polarization of the focused light varies spatially 共especially for high NA objectives兲, even if the illumination is purely linearly polarized. The sample was prepared by drop depositing a very dilute, aqueous nanobead suspension on a standard glass cover slip with a nominal thickness of 0.17 mm. After drying, the sample was studied using an inverted microscope 共a modified Zeiss Axiovert 200M兲 equipped with a 40⫻ objective a兲

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共NA= 0.75兲. An argon ion laser beam 共488 nm line兲, emerging from a single mode optical fiber 共NA= 0.12兲 and collimated with a 200 mm lens, was injected into the microscope’s rear port. The filling parameter of the microscope objective was 0.128 共ratio of aperture radius to beam waist radius兲. The core of the single mode fiber 共mode field diameter of 3.2 ␮m兲 represented the input pinhole, while a 50 ␮m diameter multimode fiber served as the conjugated exit pinhole of the confocal setup. A closed-loop piezoelectric xy stage raster scanned the nanobead sample 共face up兲 across the illumination spot while the total fluorescence 共emission maximum: 520 nm兲 was collected through the same objective, focused by the detector/tube lens 共focal length: 164.5 mm兲 and detected by a single photon counting avalanche photodiode. Any excitation light was blocked by high quality dichroic and long-pass emission filters 共Chroma Technology兲. A total of 109 images have been acquired in 50 nm intervals by moving the microscope objective stage along the optical axis 共z axis兲, ranging from −4.50 ␮m 共focal plane in cover slip below nanobead兲 to +0.90 ␮m 共focal plane in air above nanobead兲. Figures 1共a兲–1共d兲 show a selection of these confocal images. We chose to image a fairly intense object, probably a small cluster 共less than three nanobeads across兲, in order to

FIG. 1. 共Color online兲 Top: Experimental confocal scanning images in the x-y plane of a small 20 nm fluorescent nanobead cluster with a 40⫻ objective of NA= 0.75 共128⫻ 128 pixels2, 6 ⫻ 6 ␮m2, integration time: 25 ms/ pixel, scale bar: 0.5 ␮m兲 at different objective z positions. Bottom: Calculated images at the corresponding distances 共focal plane at z = 0 ␮m兲. Colors of all images are linearly scaled from minimum to maximum.

0003-6951/2007/90共3兲/031106/3/$23.00 90, 031106-1 © 2007 American Institute of Physics Downloaded 18 Jan 2007 to 193.48.255.141. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

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Appl. Phys. Lett. 90, 031106 共2007兲

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acquire images with sufficient signal-to-noise ratio at greater z distances. We confirmed that the intense object shown here behaves exactly as single beads and is therefore clearly subdiffraction sized. Image 共a兲 represents the standard imaging situation where the nanobead is scanned in the objective’s focal plane 共z = 0 ␮m兲. As the objective is moved lower, a concentric intensity ring appears and becomes wider 关cf. Figs. 1共b兲 and 1共c兲兴. Finally, a second ring develops 关see Fig. 1共d兲兴. The small radial dissymmetry of the diffraction rings visible in the experimental images is due to residual aberrations in our optical system. Naturally, the question arises whether a theoretical description can be found that adequately reproduces these experimental results. For the higher NAs in modern microscopes, a theory should be used that takes the vector character of light 共polarization兲 into account.9,10 Furthermore, a Gaussian intensity profile is characteristic of the widely used fundamental laser mode or the light emitted by a single mode fiber, which can be accounted for by using a modified apodization function as in Ref. 11 but which properly satisfies Abbe’s sine condition.12 The comparison of results from such a theory with our experimental data suggested that the different media 共airglass-air兲 in the focal region need to be taken into account as presented in Ref. 13 共see also Refs. 14–17兲. For the present letter, we use an extended model that not only accounts for stratified media but also incorporates the correct apodization function and a more accurate calculation of the electric field focused by the detector lens.16 Furthermore, the model used in this letter does not assume a point detector, but a finite detector, and we consider a circularly polarized input beam in order to reflect our experimental situation. The nanobead is modeled as an ensemble of randomly oriented dipolar emitters; the integration over all dipole orientations can be written in analytical form.12 Using spherical coordinates 共r , 0 艋 ␾ ⬍ 2␲ , 0 艋 ␪ 艋 ␲兲 we can write the total detected intensity 共overall intensity PSF of the imaging system, which we will refer to as the imaging PSF兲 as18 Iimg共rs兲 ⬀

冕 冕冕 冕 2␲

0

R

0

2␲

0

␲

兩p · eill共rs兲兩2 · 兩edet共p,rs,rd兲兩2

0

⫻sin␪ p d␪ p d␾ p rd drd d␾d .

共1兲

Here, R denotes the finite detector radius, rd = 共rd , ␾d , ␲ / 2兲 is a point in the detector plane with the origin in the center of the detector surface, and rs is the location of the emitting source dipole with an orientation defined by the unit vector p = 共1 , ␾ p , ␪ p兲. The illumination light focused by the microscope objective creates an electric field eill in the near-focus region, the illumination field PSF; the illumination intensity PSF is given by 兩eill兩2. The complex square of the electric field emitted by the dipole p, collected by the objective and focused onto the detector surface by the detector/tube lens, is given by 兩edet兩2. It corresponds to the detection (intensity) PSF and describes the detection efficiency of the confocal setup for a given p. The scalar product of the dipole vector p with the electric field of the illumination light in the focal region eill in Eq. 共1兲 reflects the fact that a dipolar emitter is only excited by the field component parallel to its dipolar axis. The full details of the theory will be given elsewhere.12 All elements of the experimental setup are represented in the present model. It accounts explicitly for deviations from

FIG. 2. 共Color online兲 Comparison between 共a兲 postacquisition stacking of 109 experimental intensity profiles and 共b兲 theoretical calculation using Eq. 共1兲; see text for more details. Both images show the x-z plane, use the same logarithmic color scale, and have identical dimensions: 8 ␮m 共horizontal: x axis兲 by 5.4 ␮m 共vertical: z axis兲. The x axis corresponds to the lateral nanobead position, whereas z is the objective stage position. The objective stage position yielding a maximum fluorescence intensity is indicated by a dashed line. Inset: larger view of the theoretical PSF.

nominal design values, such as glass cover slip thickness 共design: 170 ␮m; measured: 180 ␮m兲 and refractive indices at the excitation and detection wavelengths 关design: 1.515; actual: 1.5297 at 488 nm and 1.5274 at 520 nm 共Ref. 19兲兴. We assumed a pointlike isotropic emitter with a monochromatic emission of 520 nm, positioned at the nanobead center 共10 nm from the glass-air interface兲. This position was held constant while the microscope objective was moved between each x-y image 关Figs. 1共e兲–1共h兲兴. We also considered that only light falling onto the multimode fiber core is detected 共flat sensitivity function兲. A rotational averaging algorithm was used in order to obtain an experimental plot in the x-z plane along the optical axis from the experimental scan images in the x-y plane. The radial intensity distribution profile was calculated for 109 experimental images 关ranging from z = + 0.90 to z = −4.50 ␮m; four of them are shown in Figs. 1共a兲–1共d兲兴. All of the profiles were then stacked and plotted with a logarithmic false color scale 关see Fig. 2共a兲兴. The obtained raster-scan resolution is 50 nm in the z direction and 47 nm in the x and y directions and thus around eight times higher than the diffraction limited optical resolution. The experimental x-z plot can be compared to theoretical calculations using Eq. 共1兲 as shown in Fig. 2共b兲. It can be seen that the theoretical image reproduces most of the features found in the experimental plot. Closer inspection reveals that the agreement is excellent up to a distance of ⬃2.5 ␮m from the focal plane. For example, the lateral extension 共full width half maximum obtained from a Gaussian fit兲 in the focal plane is 380± 1.0 nm which matches the theoretical value of 336± 1.5 nm when the size of the fluorescent probe is taken into account. Also, the depth of field in the experimental image is 1.39± 0.05 ␮m full width half maximum along the z axis; the value determined from image 2共b兲 is 1.22± 0.015 ␮m. This is a fairly small discrepancy

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Appl. Phys. Lett. 90, 031106 共2007兲

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given that the noise level along the z axis is inherently higher due to the time delay between each stacked image. At axial distances beyond 2.5 ␮m, however, the experimental image seems stretched; the on-axis minimum in the lower part of the experimental image 共see arrows in Fig. 2兲, for example, appears about 20% farther away from the focal plane than in the theoretical image. This discrepancy can also be seen in Figs. 1共d兲 and 1共h兲. Taking into account the spread of detection wavelengths 共and the associated refractive index changes兲 leads to a smoothing of the imaging PSF but cannot explain the stretching effect in Fig. 2共a兲, because image features are dominated by the 共unchanged兲 illumination PSF. It is possible, however, that sagging during the 12 h data acquisition or nonexecuted steps of the objective stage together with uncertainties in the experimental parameters contribute to it. In addition, the used model assumes that the emitting dipole is at least a few wavelengths away form any interface. In reality, the emission occurs from an ensemble of fluorescent molecules inside the nanobead that is located in direct vicinity of the air-glass interface; thus, refraction effects and self-interaction of the dipole with its back-reflection at the nanobead and air-glass interfaces alter the emission pattern. Incorporating approaches for modeling these effects20,21 may lead to a more quantitative agreement between experiment and theory at larger axial distances. An important feature of the experimental and theoretical images is the dissymmetry of the PSF relative to the focal plane. This has been observed before, albeit at much lower resolution,22,23 and significant discrepancies between experimental and theoretical PSFs have been reported 共see Ref. 24 and references therein兲. In part, this has been attributed to nonideal imaging conditions not reflected in the theoretical models that were used. The present model, however, allows to account for spherical aberrations due to a mismatch between design and actual imaging parameters, such as cover glass thickness, refractive indices, etc. These aberrations give rise to a dissymmetry with respect to the focal plane as shown in Fig. 2共b兲, correctly reproducing the experimental observations and underlining the importance of taking stratified media into account. It should be noted that the predicted PSF is indeed symmetric when calculated with nominal design values.

We have demonstrated that subdiffraction sized, fluorescent nanobeads can be used as scalar probes to experimentally map out the imaging point spread function of a confocal microscope on the nanometer scale with an unprecedented degree of detail. Furthermore, we have used an extended model, which is in excellent agreement with the experimental data in the near-focus region and which reproduces nicely the qualitative features at larger distances. The availability of high quality experimental data at a resolution limited only by bead size together with a refined theoretical model are key factors for enhanced data analysis and the development of innovative techniques in confocal scanning microscopy. E. L. Elson and D. Magde, Biopolymers 13, 1 共1974兲. D. Magde, E. L. Elson, and W. W. Webb, Biopolymers 13, 29 共1974兲. 3 J. Pawley, Handbook of Biological Confocal Microscopy, 3rd ed. 共Springer, New York, 2006兲. 4 F. S. Fay, W. Carrington, and K. E. Fogarty, J. Microsc. 153, 133 共1989兲. 5 S. F. Gibson and F. Lanni, J. Opt. Soc. Am. A 9, 154 共1992兲; 8, 1601 共1991兲. 6 R. Juškaitis and T. Wilson, J. Microsc. 189, 8 共1998兲. 7 A. Drezet, M. J. Nasse, S. Huant, and J. C. Woehl, Europhys. Lett. 66, 41 共2004兲. 8 A. Drezet, S. Huant, and J. C. Woehl, J. Lumin. 107, 176 共2004兲. 9 V. S. Ignatowsky, Transactions of the Optical Institute in Petrograd 1, 1 共1920兲. 10 B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 共1959兲. 11 A. Yoshida and T. Asakura, Optik 共Stuttgart兲 41, 281 共1974兲. 12 M. J. Nasse and J. C. Woehl, Opt. Lett. 共to be published兲. 13 O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, Opt. Express 11, 2964 共2003兲. 14 O. Haeberlé, Opt. Commun. 235, 1 共2004兲. 15 P. Török, Opt. Lett. 25, 1463 共2000兲. 16 P. Török and P. Varga, Appl. Opt. 36, 2305 共1997兲. 17 P. Török, P. Varga, Z. Laczik, and G. R. Booker, J. Opt. Soc. Am. A 12, 325 共1995兲; 12, 1605共E兲 共1995兲. 18 P. D. Higdon, P. Török, and T. Wilson, J. Microsc. 193, 127 共1999兲. 19 Sellmeier coefficients for Corning, Inc. borosilicate glass type 0211 共private communication兲. 20 L. Dai, I. Gregor, I. von der Hocht, T. Ruckstuhl, and J. Enderlein, Opt. Express 13, 9409 共2005兲. 21 H. Guo, J. Chen, and S. Zhuang, J. Opt. Soc. Am. A 23, 2756 共2006兲. 22 O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, Opt. Commun. 196, 109 共2001兲. 23 B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 共2004兲. 24 J. L. Beverage, R. V. Shack, and M. R. Descour, J. Microsc. 205, 61 共2002兲. 1 2

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