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380 J. Opt. Soc. Am. A/Vol. 17, No. 3 /March 2000 Indebetouw et al.scribed, and its feasibility has been demonstrated.3

More recently, the application of scanning holography tothree-dimensional microscopy has been contemplated.The holographic recording of three-dimensional fluores-cent specimens was shown possible,4 and the techniqueproved promising in locating fluorescent anomalies em-bedded in turbid media.5

The purpose of this paper is to present an analysis ofthe imaging properties of scanning holography, definetheoretically expected performances, and discuss somepractical issues associated with the technique. In Sec-tion 2 we briefly review the background of three-dimensional microscopy and holographic microscopy.The primary objective here is to identify the main draw-backs of conventional coherent holographic methods ap-plied to microscopy (i.e., speckle noise and a high-spatial-bandwidth requirement) and to show how the scanningheterodyne method may overcome some of the difficulties.In Section 3 we analyze the properties of the recon-structed image in detail. For this, the imaging process ofscanning holography is described in terms of its point-spread function as well as its transfer function. Numeri-cal examples are provided as an illustration of the ex-pected performance. Many of the formulas given arerestricted to the paraxial approximation, which often

cussed in Section 4. The coherence of the scanning holo-graphic process depends on the size of the detector andcan be varied from coherent to incoherent, in the sameway that the coherence in a conventional microscope de-pends on the size of the source. Thus scanning hologra-phy can in principle record phase objects, such as un-stained biological specimens, can record incoherentholograms, resulting in speckle-free reconstruction, andcan record holograms of fluorescent specimens. In Sec-tion 4 we outline some unique properties of scanning ho-lography, which include a posteriori compensation of ab-errations and a posteriori processing during recon-struction. Some practical issues related to the data ac-quisition time and the required bandwidth are also men-tioned in this section.

2. BACKGROUNDA. Three-Dimensional MicroscopyCommonly used methods for three-dimensional imagingin microscopy make use of optical sectioning, which gen-erally requires a three-dimensional sampling of the speci-mens volume. The two best-known examples are opticalsectioning microscopy and scanning confocal microscopy.Optical sectioning uses a conventional microscope to se-quentially record a series of images focused at differentImaging properholographic

Guy Indebetouw a

Department of Physics, Virginia Polytechn

Taegeun Kim an

Bradley Department of Electrical and Computer EnVirginia

Received May 7, 1999; revised manuscript rec

Scanning heterodyne holography is an alternative watering or fluorescent object. We analyze the propertiWe describe the possibility of varying the coherence ocess linear in intensity by changing the detection moddimensional point-spread function of the system andwith equal numerical aperture. We describe how it ialgorithm, to obtain an ideal transfer function equalof aberrations. Some practical implementation issue[S0740-3232(00)01703-8]

OCIS codes: 070.2580, 090.0090, 110.0180, 110.68

1. INTRODUCTIONScanning holography was invented as a clever applicationof the two-pupil interaction schemes, which are unique inextending the incoherent optical processing realm to op-erations requiring bipolar, or even complex, point-spread

1,2leads to convenient analytical solutions. It must bestressed, however, that the analysis presented here is

0740-3232/2000/030380-11$15.00 es of scanningicroscopy

Prapong Klysubun

nstitute, Blacksburg, Virginia 24061-0435

ing-Chung Poon

eering, Virginia Polytechnic Institute, Blacksburg,61-0111

d October 7, 1999; accepted October 15, 1999

f capturing three-dimensional information on a scat-f the images obtained by this novel imaging process.e system from a process linear in amplitude to a pro-We illustrate numerically the properties of the three-pare it with that of a conventional imaging system

ssible, by an appropriate choice of the reconstructionnity up to the cutoff frequency, even in the presencee also discussed. 2000 Optical Society of America

180.5810.

valid beyond these approximations. The main result ofthis section is that scanning holography produces recon-structed images with transverse and axial resolution com-parable with or better than those of a conventional micro-scope of equal numerical aperture and, in addition, offersunique opportunities for postprocessing, which are dis-depths.6 Suitable algorithms are then used to merge theimages into a three-dimensional representation and re-

2000 Optical Society of America

Indebetouw et al. Vol. 17, No. 3 /March 2000/J. Opt. Soc. Am. A 381duce the effects of out-of-focus blur. Scanning confocalimaging also requires a three-dimensional scan. Thismethod, however, can achieve true depth slicing. In con-focal imaging, the out-of-focus information in a selectedsection is rejected before detection by the use of conjugatepinholes.7 Both methods require precise three-dimensional positioning devices. This is particularlycritical for the confocal methods. For certain applica-tions in biology, one of the drawbacks of these instru-ments is that the data are acquired sequentially in a rela-tively slow three-dimensional scan of the specimen. Thislong data-acquisition time may be a drawback for in vivostudies. In addition, a long data-acquisition time mayexacerbate the photo-bleaching problem in fluorescencemicroscopy.8 Elimination of the need for a three-dimensional scan gave us the impetus to revisit holo-graphic methods for three-dimensional microscopy. Amost attractive quality of holography is its well-knownability to capture high-resolution images either in asingle shot, as in Gabors original idea,9 or, as in the caseof scanning holography, in a single two-dimensional scan.It should be mentioned that a direct comparison of holog-raphy and confocal imaging is inappropriate because thetwo methods lead to very different types of data. Confo-cal imaging efficiently extracts the information on asingle section, while a holographic reconstruction focusedat a particular depth remains corrupted by the out-of-focus data. The selection of a single slice and the rejec-tion of out-of-focus data have also been successfully dem-onstrated in white-light interferometric microscopy, alsocalled correlation microscopy.10 These methods, how-ever, also require the sequential recording of a large num-ber of transverse cross sections through the object.

B. Conventional HolographyIn conventional holography an object is illuminated co-herently, and the scattered light is made to interfere witha mutually coherent reference wave. This results in anencoding of each scatterer into a Fresnel zone patterncontaining the three-dimensional position information onthe scatterer. The interference pattern, or hologram, isthen recorded on a high-spatial-resolution medium.11

Because the recording medium is quadratic (phase insen-sitive), spatial heterodyning with the use of an off-axisreference wave at the recording stage and coherent spa-tial filtering at the reconstruction stage are needed to ex-tract the reconstructed image without twin-imageartifacts.12 The need for spatial heterodyning necessi-tates a coherent encoding and a recording medium withhigh spatial resolution. These are the sources of the twodrawbacks of conventional holographic microscopy.Namely, the ubiquitous speckle noise is unavoidable incoherent imaging, and the detection system requires ahigh spatial bandwidth.

The high-spatial-bandwidth requirement comes fromthe fact that for successful extraction of the reconstructedimage by using spatial heterodyning, the spatial carrierfrequency of the hologram must be at least one and a halftimes the spatial bandwidth of the object.12 Thus the to-tal hologram bandwidth is at least four times that of theobject. Storage and transmission of such a hologrammay become problematic. This difficulty may be easedsomewhat by using phase-shifting methods.13 Thespeckle noise, as is well known, is most severe in coherentimaging systems.14 The necessity for coherent encodingmeans that all spurious scattering from the specimen, itssupport, or the optics will interfere with the object wavethat is recorded and gives rise to speckles. In addition,reconstructing the hologram without artifacts requires co-herent spatial filtering, and the coherently reconstructedimage must then be magnified by some optical system forobservation. This coherent imaging process with magni-fication leads to an image covered with high-contrastspeckles having exactly the same size as that of the reso-lution limit of the instrument. Consequently, the fine de-tails of the image are irrevocably lost. A great deal of ef-fort has been devoted to reducing speckle noise incoherent imaging. Proposed methods include spatial av-eraging, statistical averaging, and other coherence-spoiling schemes.14 All these methods either result in areduction of spatial resolution or increase the systemscomplexity beyond reasonable limits. There are some ex-ceptions. In particle field analysis, for example, clevermethods have been described to minimize the specklenoise with the use of multiple beams and to avoid thetwin-image artifacts of on-line hologram reconstructionwith spatial filters.15

C. Scanning Heterodyne HolographyIn scanning holography1 a temporally modulated Fresnelzone pattern is created, for example, by the interferenceof a spherical wave and a plane wave shifted in frequency.This pattern is scanned in a two-dimensional raster overthe object, and the scattered, reflected, or fluorescent lightis collected on a spatially integrating detector. The pho-tocurrent is then heterodyned at the modulation fre-quency, or demodulated by other means, to produce a ho-lographic record in electronic form. As a consequence ofthe spatial scanning and the spatial integration on the de-tector, each scatterer is again encoded as a Fresnel zonepattern, as in conventional holography, but the processmay now be either coherent or incoherent. More impor-tant, the process occurs in the temporal rather than thespatial domain. With a spatially integrating detector,the imaging process is linear in intensity, and thus insen-sitive to spurious phase fluctuations, even if the scanningpattern is created by the interference of two coherent la-ser beams, as is most conveniently done in practice.With a pinhole detector, the imaging process is linear inamplitude and thus able to capture phase distributions.

The most important difference between conventionaland scanning holography is that in the latter the result-ing hologram is obtained in the form of a temporal ratherthan a spatial signal and that the extraction of the recon-structed image makes use of temporal heterodyningrather than spatial heterodyning. Consequently, the de-tection system need not be spatially resolving and can be,as may be needed for weakly scattering or weakly fluo-rescing specimens, a large spatially integrating detector.Furthermore, the holographic signal can be directlydownconverted at the recording stage, resulting in asingle-sideband holographic record with considerably re-duced bandwidth requirements. The reconstruction andthe subsequent magnification of an image focused at a

382 J. Opt. Soc. Am. A/Vol. 17, No. 3 /March 2000 Indebetouw et al.chosen depth within the specimen are performed digitallyby correlation of the hologram with a pattern matched tothe desired depth. Incoherent encoding and digital re-construction lead to reconstructed images that arespeckle free. In addition, a digital reconstruction schemepermits the straightforward implementation of variouspostprocessing operations to obtain, a posteriori, e.g.,dark field or gradient images, or to change the magnifica-tion, or to scan through the specimens depth, without anyoptics or mechanical motion.

D. Holographic MicroscopeThe principles of scanning holography have been experi-mentally demonstrated for simple, macroscopic objects,3

and the method has been extended to record holograms offluorescent specimens,4,5 thus demonstrating the incoher-ent nature of the process. A true holographic microscoperemains to be constructed, but the purpose of this paper isto discuss some of its expected properties. Certain limi-tations and unique properties can already be mentioned.

A holographic microscope, for example, will not performthe sharp optical sectioning characteristic of a confocalscanning microscope. This is simply because the holo-graphic information is acquired in a single two-dimensional scan, which prevents the possibility of reject-ing the out-of-focus information before detection.However, reconstruction from the holographic data canbenefit from the application of a number of algorithmsthat have been developed to process and improve imagesin the conventional optical sectioning methods.16,17 Res-toration and eventually ultraresolution methods18,19 canalso be used to advantage. A unique property of thescanning holographic method is that it offers the possibil-ity of correcting, during the reconstruction, the aberra-tions that may have affected the scanning pattern used inrecording the hologram. This may be of importance atwavelengths for which well-corrected, high-numerical-aperture optics are difficult or expensive to fabricate.

A conventional microscope has the capability of varyingthe degree of coherence by changing the size of the source.A broad source provides incoherent imaging, which mini-mizes speckle noise and artifacts but is blind to objectphase variations, whereas a point source provides spa-tially coherent illumination, making it possible to imagephase distributions such as unstained biological speci-mens. A holographic microscope presents an equivalentversatility because the imaging property of a scanning op-tical system can be varied from incoherent to coherentmode by changing the size of the detector.20,21 A large,spatially integrating detector leads to incoherent imag-ing, results in speckle-free images, and is capable of im-aging fluorescent samples, whereas a pinhole detector re-sults in coherent imaging capable of rendering phaseobjects visible and enabling the implementation of well-known microscopic techniques such as the Zernike phasecontrast and Nomarski interference contrast methods.

3. IMAGING PROPERTY OF SCANNINGHOLOGRAPHYA. Scanning FieldThe scanning pattern is formed by the superposition of aquasi-spherical wave and a plane wave interfering on theobject (as shown in Fig. 1, which will be described in Sub-section 3.C). The quasi-spherical wave emerges from apoint source created at the focal point of a well-correctedmicroscope objective of numerical aperture (NA) uni-formly illuminated by a plane wave. Within the domainof validity of the Debye integral representation of thefield,22,23 the amplitude distribution at a distance z fromthe geometrical focus is given by

U~r, z, t ! 5 2ikA exp~2ivt !E0

NA

exp@ik~1 2 s2!1/2z#

3 J0~ksr !~1 2 s2!21/2s ds

5 2ikAE~r, z !exp@i~kz 2 vt !#, (1)

where

E~r, z ! 5 E0

NA

exp~2i 12 ks2z !J0~ksr !~1 2 s

2!21/2s ds.

A is the uniform field amplitude in the aperture, k5 2p/l 5 v/c is the wave number of the radiation, v isits circular frequency, and c is the speed of light in vacuo.s 5 usu, where s is the transverse component of a unit vec-tor pointing from the geometrical focus to the point of ob-servation, so that n 5 s /l represents the transversespatial-frequency coordinate. J0 is a Bessel function ofthe first kind and zero order. r 5 uru is a transverse ra-dial coordinate.

In the paraxial approximation, this distribution can bewritten as, neglecting an unimportant factor,

Up~r, z, t ! 5 Ep~r, z !exp@i~kz 2 vt !#, (2)

with

Ep~r, z ! 5 E0

NA

exp~2i 12 ks2z !J0~ksr !s ds. (3)

The subscript p stands for paraxial approximation. Atsufficiently large distances from the geometrical focus,where the Fresnel number of the scanning aperture islarge, the amplitude distribution of Eq. (3) is correctly ap-proximated by a spherical wave truncated in space by acone of half-angle a 5 sin21(NA).23 In this case thescanning field takes the simple form

Et~r, z ! 5 exp@iF~r, z !#circ@r/a~z !#,

Fig. 1. Sketch of a scanning holographic microscope. AO1 andAO2 are acousto-optic modulators. P1 and P2 are point-sourceoutputs of single-mode fibers. The specimen is on a two-dimensional scanning stage.

Indebetouw et al. Vol. 17, No. 3 /March 2000/J. Opt. Soc. Am. A 383F~r, z ! 5 kr2/2 z. (4)

The subscript t stands for truncated wave approximation,circ(x) 5 1 for x , 1 and 0 otherwise, and

a~z ! 5 z 3 NA (5)

is the radius of the scanning pattern at a distance z fromthe point source. At this distance the scanning pattern ischaracterized by its Fresnel number

F~z ! 5 a2~z !/lz 5 ~NA!2z/l. (6)

This field is mixed on the object with a plane wave of am-plitude E0 shifted in frequency by V to produce a scan-ning pattern with amplitude

P~r, z, t ! 5 E~r, z ! 1 E0 exp~iVt !. (7)

The calculations in Ref. 23 show that the truncated-spherical-wave approximation is excellent for Fresnelnumbers F . 40 and is already quite good for F . 10.Noticeable discrepancies appear only near the boundaryof the pattern, which, in a practical setup such as thatshown in Fig. 1, could be clipped or tapered off by addi-tional apertures. It should be stressed, however, thatthis does not mean that a scanning pattern with a smallerFresnel number or one that does not satisfy the paraxialapproximation cannot be used. In this case, however, theamplitude distribution of the scanning field must be cal-culated exactly to achieve a correct reconstruction. Inthe following, but purely for convenience, we assume thatthe truncated-spherical-wave approximation is valid.

B. Reduced CoordinatesIn the following subsections, we consider a relatively thin,weakly scattering specimen located at a distance z0 fromthe point source. The depth variable in object space ismeasured from that distance, i.e.,

dz 5 z 2 z0 . (8)

If the object is thin compared with its average distancefrom the point source, i.e., dz ! z0 , the size of the scan-ning pattern and its Fresnel number are nearly constantwithin the object depth and are given by a 5 z0 3 NAand F0 5 (NA)

2z0 /l 5 a2/lz0 , respectively.

If we anticipate that the transverse resolution limit ofthe system will be on the order of l/NA and that the axialresolution limit will be on the order of l/(NA)2, it is natu-ral and useful to use dimensionless transverse and axialcoordinates scaled to these quantities. We thus definethe normalized transverse and axial coordinates

r 5 r 3 NA/l, (9)

j 5 dz 3 ~NA!2/l. (10)

Similarly, we define a dimensionless transverse spatial-frequency coordinate m scaled to the expected cutoff fre-quency nmax 5 NA/l. Thus

m 5 n/nmax 5 s/NA. (11)

With these notations the field amplitude of the quasi-spherical wave becomes, from Eq. (1),E~r, j; F0! 5 E0

1

exp(2i2p$1 2 @1 2 m2~NA!2#1/2%

3 ~F0 1 j!~NA!22)

3 J0~2pmr!@1 2 m2~NA!2#21/2m dm,

(12)

and becomes, in the paraxial approximation,

Ep~r, j; F0! 5 E0

1

exp@2ipm2~F0 1 j!#J0~2pmr!m dm.

(13)

When F0 is large enough and the object depth range issmall compared with the average object distance from thepoint source (i.e., j ! F0), the truncated-spherical-waveapproximation leads to a simplified expression for thefield amplitude:

Et~r, j; F0! 5 exp@ipr2~F0 1 j!

21#circ~r/F0!

. exp@ipr2~1 2 j/F0!/F0#circ~r/F0!. (14)

C. Holographic RecordWe now consider the recording of the hologram of a rela-tively thin, weakly scattering specimen that can be repre-sented by an amplitude transmittance T(r, j). Exten-sion of the following arguments to the case of three-dimensional reflecting surfaces is trivial, and theirextension to fluorescent specimens and rough surfaceswill be discussed below.

The scanning pattern with amplitude P(r, j, t) givenby Eq. (7) is projected through the specimen, which isscanned in a two-dimensional raster. If rs 5 rs(t) [orrs 5 rs(t) in reduced coordinates] represents the instan-taneous position of the object, the field amplitude behindthe object is approximately

Eobj~t ! 5 E d2rdjP~r, j, t !T~r 2 rs , j!. (15)As shown in Fig. 1, this amplitude is then Fourier trans-formed by a lens of focal length f and falls on a spatiallyintegrating quadratic detector through a mask with in-tensity transmittance M(r). The resulting detector cur-rent, for each instantaneous position rs 5 rs(t) of the ob-ject, is proportional to

i~rs! } E d2ruFr$Eobj%u2M~r!} E d2rd2r8d2r9E dj M~r!

3 exp@2i2p~r8 2 r9! a#P~r8, j!3 P*~r9, j!T~r8 2 rs , j!T*~r9 2 rs , j!, (16)

where * stands for complex conjugate. Here

Frs$Eobj% 5 E Eobj~rs!exp~2i2prs ar!d2rsis the Fourier transform of the field amplitude behind theobject, with a 5 @ f(NA)2/l#21 accounting for the scalingof the Fourier transform in the back focal plane of the

384 J. Opt. Soc. Am. A/Vol. 17, No. 3 /March 2000 Indebetouw et al.lens with focal length f. Expression (16) simplifies if wewrite it in terms of the Fourier transform

M~m! 5 E M~r!exp~2i2pm r!d2rof the mask intensity transmittance. We then have

i~rs! } E d2r8d2r9dj M~ar8 2 ar9!P~r8, j!3 P*~r9, j!T~r8 2 rs , j!T*~r9 2 rs , j!. (17)

From Eq. (7) the scanning amplitude P has a componentoscillating at the frequency V. The photodetector cur-rent can then be demodulated to extract the component atthe heterodyne frequency V. For example, this can bedone by mixing the photocurrent with reference signalscos(Vt) and sin(Vt) and low-pass filtering the result, as ina lock-in amplifier, to obtain two quadrature signalsC(rs) and S(rs), which are then digitized and combinedto form the single-sideband holographic record

H~rs! 5 C~rs! 1 iS~rs!. (18)

Equivalently, the signal i(rs) can be digitized directly, byusing a fast analog-to-digital converter (ADC), fast Fou-rier transformed, and filtered around the modulation fre-quency V. This of course assumes that the modulationfrequency is large enough, compared with the signal fluc-tuations resulting from scanning the object, for the de-modulation or the filtering to be performed without intro-ducing artifacts. In other words, the ShannonNyquistcriterion must be satisfied. This clearly imposes a limiton the scanning speed, as will be discussed below.

Using the definition of the scanning pattern from Eq.(7) in expression (16) and extracting the terms oscillatingat the temporal frequency V lead, to within some constantfactors, to the following holographic record:

H~rs! 5 E d2r8d2r9dj M~ar8 2 ar9!E~r8, j!3 T~r8 2 rs,j!T*~r9 2 rs,j!. (19)

In the truncated-wave approximation, E(r, j) is given byEq. (13), and in more general cases, it can be calculatedfrom Eq. (1).

Two extreme cases are of interest because they lead tolinear superposition integrals from which one can definepoint-spread functions and transfer functions. The firstcase is that of a coherent process. It results from using apinhole on the axis as a mask. Thus we have M(r). d (r), where d (r) is a Dirac delta function, and M(m). 1, leading to

Hcoh~rs! 5 H0E d2r8dj E~r8, j!T~r8 2 rs , j!, (20)where H0 5 *d

2r9 T*(r9 2 rs) is a constant complex fac-tor. In this case the amplitude T(r, j) of each objectpoint is encoded as a wave E(r, j). The process is linearin field amplitude and is thus coherent according to con-ventional wisdom. This hologram is sensitive to objectphase variations and thus is capable of recording phaseobjects such as thin unstained specimens, as encounteredin biomedical imaging, as well as the topography ofsmooth three-dimensional surfaces, as met in the micro-electronics industry. If the object is rough, however, weexpect the images to be corrupted by speckle noise. Thesecond extreme case is that of an incoherent process,which results from using an open mask and a large spa-tially integrating detector. Here M(r) . 1, and M(m). d (m). The holographic record is in this case

H inc~rs! 5 E d2r8dj E~r8, j!I~r8 2 rs , j!, (21)where I(r8, j) 5 uT(r8, j)u2. The process is linear in in-tensity and thus, according to conventional wisdom, inco-herent. This mode of operation is needed to record holo-grams of rough objects or rough surfaces without specklenoise and to record holograms of incoherent objects suchas fluorescent specimens. In both the coherent and theincoherent mode, a point object is encoded as the samewave E(r, j), which, for relatively large Fresnel numbers(F . 10) and relatively small numerical apertures (NA, 0.5), is well approximated by a truncated sphericalwave.

D. Hologram ReconstructionFor the reconstruction of an image focused at a distancezR from the point source used in the recording, that is, adistance jR into the object, in reduced coordinates, the ho-logram can be digitally correlated with the patternER(r, jR) matched to the desired depth. Thus the fo-cused reconstruction is, from Eq. (19),

R~r, jR! 5 E H~rs!ER*~rs 2 r, jR!d2rs5 E d2rsd2r8d2r9dj M~ar8 2 ar9!

3 ER*~rs 2 r, jR!E~r8, j!

3 T~r8 2 rs , j!T*~r9 2 rs , j!. (22)

In the coherent case, we obtain

Rcoh~r, jR! 5 E d2rsd2r8dj ER*~rs 2 r, jR!3 E~r8, j!T~r8 2 rs , j!, (23)

and in the incoherent case, we obtain

R inc~r, jR! 5 E d2rsd2r8dj ER*~rs 2 r, jR!3 E~r8, j!I~r8 2 rs , j!. (24)

It is remarkable that the reconstructed data have exactlythe same form whether the system operates in a coherentor an incoherent mode. The point-spread functions areidentical in both cases and in general are complex. Thisof course comes from the fact that the heterodyne detec-tion gives access to the phase of the photocurrent. Whenthe system operates in a coherent mode (with a pinholedetector), the current is proportional to the object ampli-tude and thus also carries information on the objectphase. When the system operates in an incoherent mode(with a spatially integrating detector), the photocurrent isproportional to the object intensity and is blind to its

Indebetouw et al. Vol. 17, No. 3 /March 2000/J. Opt. Soc. Am. A 385phase, but the phase of the photocurrent itself carries theencoded information on the object location. This is whatmakes it possible to record incoherent holograms, insen-sitive to object phases, but with a point-spread functionthat is not necessarily real positive. In fact, a complexpoint-spread function of arbitrary shape can in principlebe synthesized by choosing appropriate scanning and re-construction fields.

When the scanning field is a pure phase function, as itis, for example, in the truncated-wave approximation, theoptimum choice of reconstructing function is the scanningfield itself. The reconstruction operation, which is then acorrelation with a spherical wave of appropriate curva-ture, can be interpreted in two different ways. As isknown from the Huygens principle, the correlation of anoptical field with a spherical wave represents a free-spacepropagation of that field for a distance equal to the radiusof curvature of the wave. Thus the digital reconstructionis equivalent to propagating the field that would emergefrom the hologram for a distance zR , or F0 1 jR in re-duced coordinates, where the reconstructed image wouldbe observed. Correlation is also a pattern recognitionprocess. Consequently, the reconstruction operation canbe interpreted as a matched filtering of the hologram torecognize and extract from the hologram all the waveswith a curvature radius F0 1 jR . The distribution of theamplitude of these waves is of course identical with thedistribution of scatterers in a plane jR in the object, pos-sibly corrupted by out-of-focus images. The interpreta-tion in terms of pattern recognition may be helpful in de-signing reconstruction schemes based on nonlinearreconstruction processes rather than the linear process ofcorrelation. Such nonlinear operations, which can beperformed digitally, may lead to sharper depth discrimi-nation and sectioning than that provided by a linear im-aging process.

E. Point-Spread FunctionIn the two extreme cases of full coherence or incoherence,and when the reconstructing field is identical with thescanning field (a truncated spherical wave in common ap-proximation), the reconstructed data are either a linearsuperposition of object amplitudes or a linear superposi-tion of object intensities. With the change of variablesrs 2 r r8 2 12 r9, r8 r8 1 12 r9 in Eqs. (22) and (23),the reconstructed image can be written in the usual form:

Rcoh~r, j! 5 E d2r8dj PSF~r8; j, jR!T~r8 2 r, j! (25)in the coherent case and

R inc~r, j! 5 E d2r8dj PSF~r8; j, jR!I~r8 2 r, j! (26)in the incoherent case. In both cases the point-spreadfunction is

PSF~r; j, jR! 5 E d2r8ER*~r8 2 12 r, jR!E~r8 1 12 r, j!.(27)

The point-spread function for the reconstruction at adepth jR , as a function of the transverse coordinate r andthe axial coordinate j, is thus the correlation of the scan-ning field at j with the reconstructing field at jR . In par-ticular, the in-focus point-spread function is the autocor-relation of E(r, jR).

When the truncated-spherical-wave approximation isvalid, both the scanning wave and the reconstructingfunction are given by relation (14), and the point-spreadfunction can be calculated as

PSFt~r, j!

5 51

pF02 E

2~F02r/2!

1~F02r/2!

dxE2@F0

22~r/21x !2#1/2

1@F022~r/21x !2#1/2

dy

3 expH 2i2pF0 rx 1 ipjF02 @~r/2 1 x !2 1 y2#Jfor r , 2F0

0 for r . 2F0

.

(28)

For large enough Fresnel numbers, Eq. (28) is well repre-sented by empirical formulas.24 For the transverse dis-tribution in focus, these empirical formulas give, approxi-mately,

PSFe~r, 0; F0!

5 H P~r/2F0! J1@2pr~1 2 r/2F0!#pr~1 2 r/2F0! for r , 2F00 for r . 2F0

,

(29)

where P(x) 5 1 2 1.38x 1 0.031x2 1 0.344x3 and J1 isa Bessel function of the first kind and first order. For theaxial distribution, Eq. (28) gives

PSFe~0, j; F0! 5 sinc~j/2!, (30)

where the subscript e stands for empirical and sinc(x)5 sin(px)/px.

It is useful to compare the point-spread function ofscanning holography with that of a conventional imagingsystem having the same numerical aperture, which is,12

in the paraxial approximation, PSFcoh(r, 0) 5 J1(2pr)/pr for an aberration-free coherent system andPSFinc(r, 0) 5 @J1(2pr)/pr#

2 for an incoherent system.As already mentioned in Subsection 3.D, the first strikingdifference is that, although the imaging process in scan-ning holography with a spatially integrating detector isincoherent and linear in intensity, the point-spread func-tion is bipolar and even complex in general. It can beshown that for F0 . 5 the central lobe of the point-spreadfunction represented by Eq. (29) is nearly identical withthe central lobe of the conventional coherent point-spreadfunction (the Airy disk). Thus, in this case, the trans-verse resolution limit, defined as the radius of the centrallobe of the point-spread function, is, to a good approxima-tion, the same as that of a conventional coherent system.That is,

Dr 5 0.61 or Dr 5 0.61l/NA. (31)

From Eq. (30) the axial resolution, defined as the distancebetween the axial maximum at j 5 0 and the first axialzero, is found to be

386 J. Opt. Soc. Am. A/Vol. 17, No. 3 /March 2000 Indebetouw et al.Dj 5 2 or Dz 5 2l/~NA!2. (32)

Another interesting feature of the point-spread functionin the truncated-wave approximation is that it vanishesfor r . 2F0 (r . 2a), in contrast to the point-spreadfunction of a conventional imaging system, which hassidelobes extending over the entire image. This mayhave practical importance because the Fresnel number, inscanning holography, can be varied easily by changingthe distance between the object and the point source,without affecting the numerical aperture, and thus keep-ing the resolution constant. For certain types of objects,there may be some advantages in using a scanning pat-tern with a small Fresnel number to reduce the extent ofthe sidelobes of the point-spread function. At smallFresnel numbers, however, the truncated-wave approxi-mation is invalid, and the point-spread function must becalculated by using Eq. (1). Figures 24 illustrate thesepoints.

Figure 2 compares the profile of the in-focus point-spread function for different Fresnel numbers and nu-merical apertures with the Airy pattern of the coherentparaxial point-spread functions of a conventionalaberration-free imaging system. It is seen that the holo-graphic point-spread function does not vary much withthe Fresnel number, as one would expect theoretically.For modest numerical apertures, the point-spread func-tion is nearly identical with the Airy pattern, except for aslightly narrower main lobe and slightly larger sidelobes.For large numerical apertures, the main lobe is signifi-cantly narrower than that of the Airy pattern. As usual,this gain is obtained at the price of increased sidelobe am-plitudes. If the sidelobes are undesirable, it is fairlysimple to apply, a posteriori, an apodizing aperture to thereconstructing pattern to smooth them out, possibly atthe price of a reduced resolution. Figures 3 and 4 showother features of the point-spread function for modest(0.5) and high (0.95) numerical apertures, respectively.The topographical plots of Figs. 3(c) and 4(c) illustrate the

Fig. 2. Cross sections of the in-focus point-spread function am-plitude. For low numerical apertures, the point-spread functionis nearly independent of the Fresnel number and almost identi-cal with the point-spread function of a clear aperture of equal nu-merical aperture. For high numerical apertures, the point-spread function has a sharper central lobe than the Airy disk buthigher sidelobes.Fig. 3. Point-spread function (PSF) of a scanning holographicsystem with numerical aperture 0.5 and Fresnel number 5. ThePSF is nearly identical with that of a clear aperture of equal nu-merical aperture. (a) Axial sections through the PSF amplitude,where r is the radial transverse coordinate and z is the axial co-ordinate (j in the text), (b) three-dimensional representation ofthe in-focus PSF, (c) topographical plot of the PSF amplitude.

Indebetouw et al. Vol. 17, No. 3 /March 2000/J. Opt. Soc. Am. A 387relative advantage in depth resolution that is obtainedwith a high-numerical-aperture system.

F. Transfer FunctionAdditional information on the properties of the holo-graphic images is obtained by defining a transfer function

Fig. 4. Same as Fig. 3 but for a system with numerical aperture0.95 and Fresnel number 5. Compared with the PSF of Fig. 3and with J1(x)/2x, the PSF has a sharper central lobe and largersidelobes. The difference is also displayed in (c).as the FourierBessel transform of the point-spreadfunction.12 The results of Subsection 3.E establish thatthe point-spread function with a defocus j is the correla-tion of the scanning field at j with the reconstructingfunction at j 5 0. The transfer function with defocus j isthus

TF~m; j! 5 ER*~m; 0 !E~m; j!, (33)where ER(m; j) is the FourierBessel transform ofER(r, j):

ER~m; j! 5 2pE0

ER~r, j!J0~2pmr!r dr. (34)

Using the identity 2p*0m8J0(2pm8r)J0(2pmr)r dr

5 d (m 2 m8), which simply expresses the fact that theFourier transform of a J0 function is a d-ring distribution,and using the fact that if the domain of m8 is limited tothe range 0 , m8 , 1, so will be the domain of m, we findfrom Eq. (12) that

E~m; j! 5 exp(2i2p$1 2 @1 2 m2~NA!2#1/2%~F0 1 j!3 ~NA!22)@1 2 m2~NA!2#21/2circ~m!. (35)

The optimum choice for the reconstructing function isthat leading to a perfect in-focus transfer function; i.e., atj 5 0, the transfer function is equal to unity up to thecutoff frequency. From Eqs. (33) and (35), it is clear thatone must choose a reconstruction function that has aFourierBessel transform

ER~m; j! } E~m; j!@1 2 m2~NA!2#. (36)The resulting transfer function with defocus j is then,from Eq. (33),

TF~m; j! 5 exp(2i2p$1 2 @1 2 m2~NA!2#1/2%

3 j~NA!22)circ~m!. (37)

In particular, the in-focus transfer function is

TF~m; 0 ! 5 circ~m!, (38)

which is the ideal transfer function of a system with cut-off frequency m 5 1.

This result is valid when the Debye integral is a correctrepresentation of the scanning field and thus is valid be-yond the paraxial approximation. In the paraxial ap-proximation, the scanning field is approximated by Eq.(13), and we have the simplified expressions

Ep~m; j! 5 ERp~m; j! 5 exp@2ipm2~F0 1 j!#circ~m!,(39)

which leads to

TFp~m; j! 5 exp~2ipm2j!circ~m!. (40)

Comparing this result with relation (36) and Eq. (37),we see that the reconstructing function must be chosen soas to cancel out the phase of the FourierBessel trans-form of the scanning field and to level off its amplitudevariations. As is well-known, correcting the phases ismost important because phase variations in the transferfunction are akin to aberrations and always broaden the

388 J. Opt. Soc. Am. A/Vol. 17, No. 3 /March 2000 Indebetouw et al.size of the point-spread function. In practice, it is lessnecessary to equalize the amplitude, but, in fact, it is notdifficult to do so with a digital reconstruction. Problemswould arise only if the FourierBessel transform of thescanning field had zeros. This may occur if the scanningbeam is corrupted by large aberrations, but with a reason-ably well-corrected objective, the scanning field and itstransform have smooth amplitude and phase profiles withnearly spherical curvature, so that the reconstructingfunction also has a smooth amplitude profile with nearlyspherical curvature. Since the reconstructing function isgenerated digitally and then digitally correlated with thehologram to reconstruct an image, it is always possible tocalculate the reconstructing function that will correct theeventual aberrations of the scanning field and thus real-ize the ideal transfer function of Eq. (38). Consequently,the transfer function in scanning holography, whether itoperates in the coherent or the incoherent mode, may bemade flat up to the cutoff frequency mmax 5 1, or nmax5 NA/l, even if the scanning beam has some aberra-tions. This holds, of course, as long as these aberrationscan be duplicated in the reconstruction function. Theseattributes are to be contrasted with the transfer functionof a conventional imaging system. For a coherent sys-tem, the transfer function is the pupil distribution itself.12

Thus, for an aberration-free system with a numerical ap-erture NA, all the spatial frequencies lower than the cut-off frequency NA/l are transmitted integrally, and therest are blocked. Aberrations play a disastrous role inthis case, because they introduce spurious phase distor-tions in the pupil, which strongly affects the integrity ofthe image. In an incoherent system, the transfer func-tion is the autocorrelation of the pupil, which is alwaysmaximum at the origin and tapers off up to the cutoff fre-quency 2NA/l. Thus the low frequencies are always em-phasized, and the high frequencies are transmitted withattenuation. Aberrations always result in further at-tenuation of the high spatial frequencies. For severe ab-errations the transfer function may even change sign,leading to contrast inversion and severe image degrada-tions. In both cases a posteriori correction, or deblurring,is in general a nontrivial ill-posed inverse problem. Inholography, in contrast, conjugation of phase can readilybe obtained, enabling the application of a variety of aber-ration compensation schemes. For example, in electronholography, a posteriori compensation of spherical aber-rations has been demonstrated with the use of electroni-cally addressed phase masks.25

4. POSTPROCESSING AND PRACTICALCONSIDERATIONSA. Postprocessing PossibilitiesThe field function used to reconstruct the image digitallyin scanning holography can in principle be chosen at will.This degree of freedom can be used to accomplish a num-ber of processing operations while reconstructing the im-age. In other words, one can synthesize, a posteriori,various point-spread functions of the form

PSF 5 E ER*~r8 2 12 r, 0!E~r8 1 12 r, 0 !d2r8, (41)or, equivalently, one can synthesize in-focus transferfunctions of the form

TF~m; 0 ! 5 ER*~m; 0 !circ~m!. (42)

Most remarkable is that these synthesized point-spreadfunctions and transfer functions can be made to operateon the object amplitude, if a pinhole detector was used inrecording the hologram, or on the object intensity, if aspatially integrating detector was used. For example, itis easy to synthesize incoherent bipolar point-spreadfunctions or high-pass incoherent transfer functions.Such operations cannot be done directly in an incoherentimaging system. A few examples are discussed in whatfollows.

For simplicity, we assume in the following that thetruncated-wave approximation is valid and that the scan-ning field is given by relation (14) and its FourierBesseltransform is given by Eq. (39). Some examples of pos-sible postprocessing operations are briefly described inthe following paragraphs.

As already discussed in Subsection 3.F, if we choose areconstructing function ERt(r) 5 Et(r), the transferfunction is circ(m), and we obtain a perfect image with acutoff frequency m 5 1. It is now easy to add to the re-constructing function an amplitude factor that, for ex-ample, enhances the high frequencies for edge enhance-ment or tapers the high frequencies smoothly forapodization.

More can be done. If, for example, we choose a recon-structing function of the form ERt 5 Et(r)2 c circ(r/F0), where c is a real constant, the resultingtransfer function is TF(m) 5 Et(m)@E t*(m)2 cJ1(2pmF0)/m#. Thus, if c 5 (pF0)

21, the transferfunction is TF(m) . circ(m) 2 J1(2pmF0)/pmF0 (usewas made of the fact that the second term has a widthmuch smaller than unity). The frequency m 5 0 is en-tirely suppressed since TF(0) 5 0, and the low frequen-cies up to Dm 5 1.22/F0 are gradually attenuated from 0to 1. If F0 is large enough, this results in a dark field im-age.

As a final example, one may consider a reconstructingfunction of the form ERt 5 Et(r 1

12 ex) 2 Et(r 2

12 ex)

; 12 e(d/dx)Et(r), where e is smaller than a resolution el-ement and x is a unit vector in the x direction. This re-sults in a reconstruction revealing gradients in the x di-rection. The transfer function is TF } Et(m)sin(pemx),where mx is the spatial frequency in the x direction. Ife 5 12 , which corresponds to a reconstructing patternmade of two patterns Et(r) with opposite polarity andshifted by half a resolution element in the x direction, thetransfer function is TF 5 Et(m)sin(12pmx). This transferfunction is identical with that obtained with the Nomar-ski interference contrast method if Et(m) is interpretedas the spherical curvature in the pupil. When acting ona phase object recorded in the coherent mode, thisoperation reveals the phase gradients along x, as in theNomarski method. When acting on an object recorded inthe incoherent mode, it reveals the intensity gradientsalong x. Similarly, it is possible to extract axial gradi-ents by choosing a reconstructing function equal to the

Indebetouw et al. Vol. 17, No. 3 /March 2000/J. Opt. Soc. Am. A 389difference between two scanning patterns correspondingto a depth difference equal to half the axial resolution,e.g.,

ERt 5 Et~r 112 ej! 2 Et~r 2

12 ej! ;

12 e~d/dj!Et~r, j!.

B. Practical ConsiderationsSince we were unable to secure the funds necessary to ac-tually build a microscope, we will share our thoughts intrying to design one, in the hope that it may trigger some-ones interest. A possible design for a scanning hetero-dyne microscope was shown in Fig. 1. The illuminationmodule produces two quasi point sources of light, P1 andP2 , at the output of two single-mode fibers. An adjust-able frequency difference is provided by two acousto-opticmodulators driven in synchronism. The lights from thetwo point sources are combined at the beam splitter, andthe beams are shaped to produce, in the approximationdiscussed above, a spherical and a plane wave superposedon the object by means of a microscope objective of nu-merical aperture NA. An adjustable aperture limitingthe size of the scanning pattern is used to vary its Fresnelnumber. The objective images this aperture on the ob-ject. Light from P1 is focused by the objective near its fo-cal point, from which it travels toward the object, where itforms, in the truncated-spherical-wave approximation, aspherical wave truncated to a half-cone angle sin21(NA).Light from P2 is focused by an intermediate lens at thecenter of the objectives entrance pupil. The objective col-limates this light to project a plane wave of limited extenton the object. The Fresnel number of the scanning pat-tern can thus be changed without affecting the numericalaperture or the resolving power.

The possibility of using the instrument in transmissionmode, to obtain holograms of unstained specimens orphase objects, as well as in reflection mode, to form holo-grams of reflecting objects or of fluorescent specimens, isillustrated in Fig. 1. A mask in the pupil plane is used tovary the coherence between an imaging linear in inten-sity and an imaging linear in amplitude, as discussedabove. For fluorescence imaging the beam splitter is adichroic mirror transmitting the laser excitation wave-length and reflecting the fluorescence wavelength. Thereference detector uses a pinhole in a plane conjugate tothe object. The signal from this detector is used as a ref-erence signal for heterodyne detection. In this way,eventual shifts in signal frequency caused by mechanicalor thermal fluctuations in the illumination stage appearin both the signal and the reference and can be canceledout. The sample can be mounted on a computer-controlled, two-dimensional stage, or the scanning can beaccomplished by mirrors for faster scanning rates. Theresulting temporal signal from the detectors is sent to thedata acquisition stage, which can be either digital or ana-log. For digital acquisition the signal is converted by afast analog-to-digital converter (ADC) and filtered digi-tally to extract the holographic record. In this case weexpect the rate of the ADC to impose a limit on the ac-ceptable scanning rate. For analog acquisition the signalis mixed with the reference signal and filtered to obtaintwo downconverted quadrature signals, which are thenconverted from analog to digital and combined digitally toform the complex holographic record.The design of the data acquisition module involvessome trade-off between its bandwidth, which will eventu-ally limit the scanning rate, and its signal-to-noise ratio.For weak signals a lock-in amplifier as the phase-sensitive detection system may be best because it is spe-cially designed to extract weak signals from large, noisybackgrounds. But for that same reason, its bandwidth issmall, allowing only very slow scanning rates. Fasterscanning rates can be achieved with a detection systemhaving a larger bandwidth, but only at the price of a lowersignal-to-noise ratio. For example, let us consider a digi-tal acquisition system equipped with an ADC capable ofacquiring 250 3 106 samples per second at 8-bit resolu-tion. If the signal is sampled at twice the Nyquist rate,the maximum signal frequency must not exceed (2503 106/2)/2 5 62.5 MHz. Since, in a well-designed sys-tem, the smallest feature of the scanning pattern matchesin size the resolution limit, this frequency cutoff is thesum of the modulation frequency fm of the scanning pat-tern and the highest frequency fs resulting from scanningthe sample. From information theory we need fm . fs ,and for best bandwidth utilization, we want the largestpossible fs . We may, for example, choose, allowing somemargin for filtering without aliasing, fm 5 2 fs , leading tofm 5 41.7 MHz and fs 5 20.8 MHz. A frequency differ-ence of 41.7 MHz between the light beams interfering onthe sample can be obtained with a standard acousto-opticmodulator. Of course, the detector bandwidth must becompatible with this figure. If we specify to collect atleast four samples per resolution element, to exceed theNyquist criterion by a factor of 2, the maximum allowablescanning rate is fs/4 5 5.2 3 10

6 resolution elements persecond. The hologram of a sample with 512 3 512 reso-lution elements can thus be captured in (512 3 4 pixel perline) 3 (512 3 4 lines)/5.2 3 106 pixels per second;0.8 s, plus the return dead time of the scanning device.With NA ; 0.6 and l ; 0.5 mm, the resolution limit is;0.5 mm, so that the required scanning speed to reachthat rate is ;260 cm/s. Such a high speed requires a fastmirror scanning system. It should be stressed that thisacquisition time is to capture the holographic data, i.e.,the entire three-dimensional information. For example,if the sample is 100 focal depths thick (280 mm at NA; 0.6), the three-dimensional data captured in 0.8 s are512 3 512 pixels 3 100 slices 5 2.62 3 107 voxels. Butone must keep in mind that no real sectioning has beendone, as in confocal systems. The method adopted forscanning the specimen may itself limit the scanning rate.Mechanical stages, for example, are limited to speeds ,10cm/s. Thus scanning a 500-mm 3 500-mm specimen atfour samples per resolution element (0.5 mm) takes;(8 3 500 lines) 3 (5 ms/line) ; 20 s. With mirror scan-ners comparable with those used in confocal scanning mi-croscopes, the acquisition time of the same hologram is onthe order of 1 s. Another important factor that may limitthe signal-to-noise ratio is the dynamic range of the de-tector. With weakly scattering or fluorescent specimens,the modulation depth of the signal is expected to be small.In such cases the signal-to-noise ratio may be limited bythe detector dynamic range or the digitization noise. Theanalysis of these limitations requires case-by-casestudies.

390 J. Opt. Soc. Am. A/Vol. 17, No. 3 /March 2000 Indebetouw et al.5. SUMMARYWe have analyzed the imaging properties of a scanningholographic system and compared it with conventionalimaging. The salient points are the following. Varyingthe detection mode from pinhole detection to spatially in-tegrating detection allows one to vary the coherence prop-erty of the imaging process from linear in amplitude tolinear in intensity. The method is thus suitable for im-aging phase objects and relief surfaces, as well as for ob-taining incoherent (and thus speckle-free) holograms andfor imaging fluorescent specimens.

The three-dimensional point-spread function of the sys-tem was calculated as a function of two parameters,namely, the numerical aperture and the Fresnel numberof the scanning pattern. The results are valid beyond theparaxial approximation and are presented in terms of di-mensionless coordinates scaled to the theoretical resolu-tion limits of the system. This allows for direct compari-sons of systems with different numerical apertures. Asone might expect from theoretical considerations, it isfound that the amplitude distribution of the point-spreadfunction is nearly independent of the Fresnel number ofthe scanning pattern, within the domain of validity of thefield representation by a Debye integral. The Fresnelnumber can thus be used as a free design parameter, thevariation of which leaves the resolution unaffected.

For low and modest numerical apertures, the point-spread function is found to be nearly identical with that ofan aberration-free conventional imaging system of equalnumerical aperture. For higher numerical apertures, theholographic point-spread function, in reduced coordi-nates, exhibits improved transverse and axial resolutionlimits, compared with the Airy pattern.

An attractive feature of scanning holography is thatthe aberrations of the scanning pattern can easily be can-celed out by reconstructing the hologram digitally with anappropriate conjugate pattern. This process leads to anideal system transfer function equal to unity up to thecutoff frequency, independent of the aberrations of thescanning beam. This is valid for the system operating ineither the coherent or the incoherent mode. Scanningholography lends itself well to postprocessing operations,since the images are reconstructed digitally. A few suchpossibilities were mentioned, and some practical issueswere considered.

ACKNOWLEDGMENTSTing-Chung Poon and Taegeun Kim acknowledge thefinancial support of the National Science Foundation(grant ECS-9810158) for parts of this work.

Address correspondence to Guy Indebetouw at the loca-tion on the title page or by e-mail, gindebet@vt.edu.

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