May 15, 2003 / Vol. 28, No. 10 / OPTICS LETTERS 801

Phase retrieval for high-numerical-aperture optical systems

Bridget M. Hanser

Graduate Group in Biophysics, University of California, San Francisco, San Francisco, California 94143

Mats G. L. Gustafsson, David A. Agard, and John W. Sedat

Department of Biochemistry and Biophysics, University of California, San Francisco, San Francisco, California 94143

Received November 15, 2002

We describe a phase retrieval approach for intensity point-spread functions of high-numerical-aperture opticalsystems such as light microscopes. The method calculates a generalized pupil function defined on a sphericalshell, using measured images at several defocus levels. The resultant pupil functions reproduce measuredpoint-source images significantly better than does an ideal imaging model. Availability of pupil functioninformation will facilitate new approaches to aberration correction in such systems. 2003 Optical Societyof America

OCIS codes: 100.5670, 070.2580, 180.0180, 100.0100.

Phase retrieval is a technique for estimating thewave-front properties of an imaging system fromone or more intensity images, typically of a pointsource.1 The retrieved wave-front information takesthe form of a complex pupil function, which permits amore direct and quantitative analysis of an imagingsystem than is possible from intensity informationalone. In astronomy, this technique can determinewave-front errors caused by aberrations within opticalsystems such as the uncorrected Hubble Space Tele-scope2 or by atmospheric turbulence.1 The retrievedwave-front information can be used to process imagedata3 or to correct alignment of segmented mirrortelescopes.4

Although they are often ignored, optical microscopyof biological specimens faces aberration problems thatare similar to those in astronomy. Analogously to at-mospheric turbulence, the biological sample itself cancause significant imaging aberrations that range fromthe depth-dependent spherical aberration caused byan overall refractive-index mismatch5 to complex, spa-tially varying aberrations that stem from refractive-index variations within the specimen.6 A method toextract wave-front information would fill a need inmicroscopy, as that information provides a recipe forremoving aberrations by use of wave-front correctionhardware7 or computational approaches and becausea pupil function description of the imaging system ismuch more easily modif ied to include further, knownaberrations than is the conventional intensity descrip-tion. Whereas phase-retrieval methods have been de-veloped for the zero numerical aperture (NA) conditionof telescopes,1 4,8 10 which subtend a negligible angleas seen from the distant source, these methods mustbe modif ied to apply to high-NA conditions, such asthose encountered in high-resolution microscopy. Inthis Letter we describe a modif ied phase-retrieval tech-nique that is suitable for high-NA optical systems.

Three-dimensional (3D) imaging properties canbe quantif ied in several ways (Fig. 1). It is com-mon to use the 3D intensity point-spread function11

(PSF)the image of a point source as a function

0146-9592/03/100801-03$15.00/0

of defocus, which can be directly measuredor,equivalently, the PSFs Fourier transform, the op-tical transfer function (OTF).12 However, thesedescriptions are incomplete: the intensity PSF is thesquared magnitude of the complex-valued amplitudePSFA and thus lacks the phase information of PSFA.

Let OTFA be the 3D Fourier transform of PSFA:

PSFAx, y, z Z Z Z

OTFAkx,ky ,kz

3 expikxx 1 kyy 1 kzzdkxdky dkz ,

(1)

where x, y, and z are real space coordinates andkx, ky , and kz are the corresponding Fourier space co-ordinates. For monochromatic light, OTFA is nonzeroonly on a two-dimensional (2D) surface: a sphericalshell of radius 2pnl (where n is the refractive indexof the immersion medium and l is the wavelength)limited by the aperture angle a of the objective lens(Fig. 1).13,14 For a low-NA system the 2D surface can

Fig. 1. Relations among four ways to characterize 3Dimaging properties: FT, Fourier transform; a, apertureangle; other abbreviations defined in text.

2003 Optical Society of America

802 OPTICS LETTERS / Vol. 28, No. 10 / May 15, 2003

be considered f lat, but in the high-NA case one musttake into account the surfaces spherical shape. Bymaking axial Fourier space coordinate kz a function oflateral coordinates kx and ky :

kzkx, ky 2pnl2 2 kx2 1 ky212, (2)

we can write Eq. (1) as a 2D integral over a 2D com-plex-valued pupil function P :

PSAAx, y, z Z Z

P kx,ky

3 expikzkx,kyzexpikxx 1 kyydkxdky . (3)

Pupil function P is nonzero on a support that corre-sponds to the systems aperture and effectively repre-sents the point sources in-focus wave front at the backfocal plane of the objective lens. P describes the imag-ing properties of the system compactly and completely,within the monochromatic scalar model of light usedhere.

Phase retrieval estimates the complex-valued pupilfunction P and thus the phase of the PSFAfrommeasured intensity PSF. It is possible to estimate theunknown phase information because of the redundancyprovided by the multiple focus levels in the measuredPSF and because of a priori knowledge of wavelengthand NA, which place geometric constraints on the pupilfunction.

Our method uses a modif ied version of the Ger-chbergSaxton algorithm15 as follows: From thecurrent estimate of the pupil function, estimatedPSFA sections were calculated with Eq. (3) (imple-mented as a 2D fast Fourier transform) for severalvalues of the defocus z; the PSFA amplitudes wereset pointwise equal to the square root of the corre-sponding sections of the measured intensity PSF;we then retransformed each section into a candidatepupil function by reversing Eq. (3) and discarded thevalues outside the known aperture. The average ofthese candidate pupil functions formed the updatedpupil function for use in the next iteration. Unlikein the original GerchbergSaxton algorithm, weallow the pupil functions magnitude to vary over theaperture, as one cannot generally assume that thepupil functions magnitude is constant over the pupilfor high-NA systems. For instance, high-angle raysgenerally suffer greater ref lection losses and are moresubject to polarization effects. The redundancy of themultiple focus levels compensates for the weakenedFourier space amplitude constraint.

We applied our modif ied phase retrieval approach towide-f ield f luorescence microscopy at high NA. Red0.12-mm-diameter f luorescent beads11 (Fluospheres,Molecular Probes) were diluted in ethanol, air driedonto a cover slip, and mounted in water. Thebeads were illuminated by a mercury arc lampand imaged with a 1003 1.3-NA Zeiss Axiomatplanoapochromatic objective lens onto a liquid-ni-trogen-cooled CCD. The passbands of the excitationand emission filters were 570590 and 610670 nm,respectively. The PSF data sets were preprocessedto correct for background, illumination intensity

variations, and photobleaching. Four sections, corre-sponding to 23-, 21-, 11-, and 13-mm defocus, wereselected from each PSF image stack for use in thephase retrieval. Each phase retrieval was run with awavelength of 615 nm for 25 iterations, with a Gauss-ian smoothing filter applied to the averaged pupilfunction at every f ifth iteration. The mean-squaredintensity error (MSE), normalized by the summedintensity of the measured data, was used as a measureof phase-retrieval quality. For 512 3 512 pixel sec-tions the phase retrieval takes approximately 40 s ona 600-MHz, EV67 Alpha processor. Figure 2 showsamplitude and phase of three phase-retrieved pupilfunctions for this microscope. Both the amplitudeand the phase components of the phase-retrievedpupil function are quite similar for different PSFscollected on the same optical system. The small dif-ferences, mostly in the amplitude, may be attributableto different signal-to-noise levels in the raw data,variations in the exact centering of the f luorescentbead with respect to the pixilated image, and the factthat the bead used to measure the PSF has a finiteand slightly variable size. The MSE value for eachof the pupils rapidly improved during the f irst fouriterations, then more slowly; final MSE values were0.030, 0.039, and 0.036 for the top, middle, and bottompupils, respectively. By comparison, phase retrievalon a noise-free, simulated PSF reached a final MSEof 7.3 3 1026, whereas simulated data that containedPoisson and readout noise, at levels comparable to

Fig. 2. Phase-retrieved pupil functions based on three dif-ferent measured PSFs from the same optical system. Thegray-scale range is zero to maximum amplitude, and 2p2to p2 rad for the phase.

May 15, 2003 / Vol. 28, No. 10 / OPTICS LETTERS 803

Fig. 3. Planar XY sections of (a) the measured PSF, (b) aPSF calculated from the central phase-retrieved pupil func-tion in Fig. 2, and (c) a theoretically calculated PSF. Thedefocus is 22 mm, 0 mm, and 12 mm for the top, cen-ter, and bottom rows, respectively. The gray scale rangesbetween the minimum and the maximum of each imageindividually.

Fig. 4. Axial XY sections comparing (top) measured,(middle) phase-retrieved, and (bottom) ideal PSFs. Theimage scaling is nonlinear to make the weak PSF featuresvisible.

those of the lowest-signal-to-noise measured data,produced a MSE of 0.0092.

For Figs. 3 and 4 the central pupil function inFig. 2 was used to reconstruct a phase-retrievedPSF. Compared to a theoretical, ideal PSF, whichcontains no lens-specific information, the phase-retrieved PSF reproduces the features of the measuredPSF significantly better. The measured PSF in

Fig. 3(a) shows the considerable radial asymmetrythat often occurs in microscope objective lenses; thesefeatures are well reproduced in the phase-retrievedPSF [Fig. 3(b)] but of course not in the ideal PSF[Fig. 3(c)]. Axial asymmetries are also well repro-duced (Fig. 4).

Although the simplified optical model used in thisLetter is able to reproduce most features of measuredPSFs, a number of refinements could be made. Forexample, a polychromatic model10 could be adopted totake into account the 2050-nm emission bandwidththat is typical of f luorescent specimens. Similarly,the scalar light model used could be modified to takeinto account the vectorial nature of the electromag-netic field, as these effects are signif icant at highaperture angles.16 Each of these corrections wouldtend to smooth the sharper features in the phase-retrieved, calculate PSF, improving the match withthe slightly more blurred features of the measuredPSF. To suppress noise, pupil functions could beexpressed with a parametric model, such as a finitebasis of Zernike polynomials, either after or duringthe iterative solution process.

This research was supported by National Institutesof Health grant GM-2510-24 to J. W. Sedat. B. M.Hanser acknowledges support from the Julius R. andPatricia A. Krevins, Lloyd M. Kozloff Fellowship, Uni-versity of California, San Francisco. D. A. Agard is aHoward Hughes Medical Institute Investigator. J. W.Sedats e-mail address is sedat@msg.ucsf.edu.

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