Page 1

Scanning holographic microscopy with resolution

exceeding the Rayleigh limit of the objective by

superposition of off-axis holograms

Guy Indebetouw, Yoshitaka Tada, Joseph Rosen, and Gary Brooker

We present what we believe to be a new application of scanning holographic microscopy to superresolu-

tion. Spatial resolution exceeding the Rayleigh limit of the objective is obtained by digital coherent

addition of the reconstructions of several off-axis Fresnel holograms. Superresolution by holographic

superposition and synthetic aperture has a long history, which is briefly reviewed. The method is

demonstrated experimentally by combining three off-axis holograms of fluorescent beads showing a

transverse resolution gain of nearly a factor of 2.© 2007 Optical Society of America

OCIS codes:

090.0090, 180.0180, 180.2520, 110.0180, 110.6880.

1.

Extending the resolution beyond the theoretical Ray-

leigh limit of the objective has been, and still is, an

important quest in high-resolution microscopy. In

the recent past, many different approaches toward

achieving this goal have been proposed and demon-

strated. A number of superresolution methods are

based on the rearrangement and smart use of the

degrees of freedom of the image and of the optical

system.1–6The general idea is to utilize degrees of

freedom that are deemed unnecessary.7–11The de-

grees of freedom that can be sacrificed to improve the

spatial resolution are numerous. For example, they

can be dimensional,12temporal,13,14or polarization.15

Different approaches to superresolution are based on

analytic extrapolation, iterative use of a priori knowl-

edge,16,17or deconvolution.18Closest to the method

described in this paper, structured illumination, spa-

Introduction

tial encoding, and methods based on aperture syn-

thesis have also been successfully demonstrated for

superresolution. These are based on the early idea

that an object’s spatial frequencies exceeding the

limit of the pupil of the imaging system can be ac-

cessed by illuminating the object with gratings or

interference patterns that shift the otherwise inac-

cessible spatial frequencies back into the pupil’s

transmission disk.19,20This idea has been imple-

mented in a large number of ways.21–27

Many superresolution methods deal with the

improvement of a single planar image. For 3D spec-

imens (as is often the case in biological studies, for

example), it is thus necessary to process each axial

section separately, and sequentially, which can be

time consuming. Holographic methods alleviate this

limitation by offering the possibility of improving the

resolution of the entire 3D information recorded on

the hologram. The idea of improving the resolution by

adding holograms dates from the early developments

of electronic holography.28,29Recent advances in de-

tector and computer power have triggered renewed

interest in applying these ideas with modern digital

holography.30–37

Digitalholographyhasbeenshowncapableofhigh-

resolution imaging of 3D biological specimens as well

as of quantitative measurement of their phase infor-

mation.38,39However, one limitation of digital holog-

raphy is that it requires a minimum degree of spatial

and spectral coherence in order to encode the holo-

gram phase in the form of interference fringes be-

tween the light scattered by the specimen and a

reference wave. This necessity precludes the use of

G. Indebetouw (gindebet@vt.edu) and Y. Tada are with the

DepartmentofPhysics,VirginiaTech,Blacksburg,Virginia24061-

0435, USA. J. Rosen and G. Brooker are with the Department of

Biology, Integrated Imaging Center, Johns Hopkins University,

Montgomery County Campus, 9605 Medical Center Drive, Rock-

ville, Maryland 20850, USA. J. Rosen is also with the Department

of Electrical and Computer Engineering, Ben-Gurion University of

the Negev, P.O. Box 653, Beer-Sheva 84105, Israel, from which he

is presently on leave.

Received 30 August 2006; revised 13 October 2006; accepted 16

October 2006; posted 16 October 2006 (Doc. ID 74610); published 2

February 2007.

0003-6935/07/060993-08$15.00/0

© 2007 Optical Society of America

20 February 2007 ? Vol. 46, No. 6 ? APPLIED OPTICS993

Page 2

digital holography with fluorescent specimens. Yet

modem biological studies rely heavily on specific flu-

orescence excitation and the emission of specific flu-

orescent probes. In scanning holography a phase

encoding excitation pattern is scanned over the spec-

imen in a 2D raster, and the hologram phase is en-

coded in the temporal domain.40The method has

been shown capable of reconstructing high-resolution

holographic images of 3D fluorescent specimens with-

out requiring the spatial coherence of the specimen’s

emission,41as well as of measuring the phase of a

specimen quantitatively.42The method has also been

shown to offer the possibility of synthesizing different

point spread functions (PSF)43and of locating small

fluorescent features in 3D with submicrometer accu-

racy.44

In this paper we demonstrate experimentally that

synthesizing a pupil exceeding the objective’s pupil

disk is easily implemented in scanning holographic

microscopy and leads to images of 3D fluorescent

specimens reconstructed with a spatial resolution ex-

ceeding the Rayleigh limit of the objective. The re-

sults shown here are illustrative in demonstrating

the principle, but do not attempt to test the limits of

capability of the method. In Section 2, the basic ideas

of scanning holographic microscopy are briefly re-

viewed and how superresolution is achieved by im-

plementing the idea of pupil synthesis is described.

Section 3 specifies the experimental setup, and an

experimental demonstration is presented and dis-

cussed in Section 4. Final remarks and a summary

are given in Section 5.

2.

In conventional scanning holography, two pupils are

combined by a beam splitter and superposed in the

entrance pupil of the objective. One pupil is a spherical

wave filling the objective’s pupil disk, and having a

curvature chosen to produce, in the back focal plane of

the objective (which is also the object’s space), a spher-

ical wave with a radius of curvature z0. The other is a

point at the center of the entrance pupil of the objec-

tive, producing a plane wave in the object’s space.

The interference of these two waves results in a 3D

Fresnel zone pattern (FZP) that is scanned over the

specimen in a 2D raster. The scattered or fluorescent

light is collected by a nonimaging detector (photo-

diode, or photomultiplier). A single-sideband holo-

gram can be obtained from this data in two ways:

either by heterodyne detection using a frequency dif-

ference between the two pupils,45or by homodyne

detection46using three scans with three fixed phase

differences between the two pupils.47,48Other meth-

ods of data acquisition have also been proposed.49

The hologram can be reconstructed numerically by

Fresnel back propagation or by correlation with an

experimental PSF. The spatial resolution of the re-

constructed image is determined by the numerical

aperture of the FZP scanned over the object. With

an on-line hologram, the spatial frequency spec-

trum of the reconstruction is confined to the disk of

the objective’s pupil, which has a cutoff frequency

Theoretical Background

?MAX? NAOBJ??, where NAOBJis the numerical aper-

ture of the objective, and ? is the wavelength of the

radiation.

To extend the spectrum beyond the objective’s cut-

off in a particular direction specified by a unit vector

n ˆ, one can record an off-axis hologram by scanning an

off-axis FZP on the specimen. If the spatial frequency

offset of the FZP is ?0n ˆ, the spatial frequency spec-

trum of the reconstruction is extended up to a value

?MAX? ?0in the direction of n ˆ. By combining several

holograms with shifts in different angular directions,

it is thus possible to tile an object’s spatial frequency

spectrum that extends, in principle, far beyond the

objective’s pupil disk.

The two pupils needed to create an off-axis FZP are

P˜1?? ???exp?i??z0?2?Disk????MAX?,

P˜2j?? ?????? ? ??0n ˆj?. (1)

Disk?x? ? 1 for x ? 1, and ?0 for x ? 1. In these

expressions, ? ? is the transverse spatial frequency vec-

tor in the pupil plane, proportional to the transverse

spatial coordinate vector r ?Pin the pupil: ? ? ? r ?P?

?fOBJ, where fOBJis the focal length of the objective.50

In the object’s space, P1?r ?? is a converging spherical

wave with radius of curvature z0, and P2j?r ?? is a plane

wave with a transverse spatial frequency ?0n ˆj. After

heterodyne detection, or phase-shifted homodyne de-

tection, each object point is encoded as an off-axis

spherical wave Sj?r ?, z?, where z is the axial coordinate

in object space measured from the focal plane of the

objective. In Fourier space, we have

S˜j?? ?, z??P˜1?? ?, z??P˜2j?? ?, z?

?exp?i???z0?0

?Disk??? ? ??0n ˆj???MAX?,

2??z0?z???2?2? ? ·?0n ˆj???

(2)

where Q symbolizes a correlation integral, and

P˜1,2?? ?, z? are the generalized (defocused) pupils51de-

fined as

P˜1,2?? ?, z??P˜1,2?? ??exp?i2?z???2??2?,(3a)

or in paraxial approximation,

P˜1?? ?, z??exp?i2?z???exp?i???z0?z??2?

?Disk????MAX?,

P˜2j?? ?, z??exp?i2?z???exp??i??z?0

2???? ? ??0n ˆj?.

(3b)

The Fourier transform of the specimen hologram is

given by

H˜

Oj??dzI˜?? ?, z?S˜j?? ?, z?, (4)

994APPLIED OPTICS ? Vol. 46, No. 6 ? 20 February 2007

Page 3

where I˜?? ?, z? is the 2D transverse Fourier transform

of the 3D object intensity distribution I?r ?, z?. The re-

construction of the hologram in a chosen axial plane

of focus z ? zRcan be obtained by digital Fresnel

back propagation from the hologram for a distance

z0? zR. In the experiment discussed in Section 4, we

chose to reconstruct the hologram by correlation with

the experimental hologram of a subresolution point

source. As has been shown previously,41this method

offers a way of reducing the phase aberrations of the

objective.41With high NA objectives, these aberra-

tions are attributable to the fact that the objective is

not (and cannot be) used in the geometry for which it

was designed, and can be severe. In the experiment

described in Section 3, this scheme is implemented by

recording a reference hologram HRj?r ?? of a point

source ??r ?, z?, and propagating it to the desired re-

construction plane by using a propagation factor

Pj?r ?, zR?. In Fourier space, we have

H˜

Rj?? ???exp?i??z0??? ? ??0n ˆj?

?Disk??? ? ??0n ˆj???MAX?,

2??

(5)

P˜j?? ?, zR??exp??i??zR??2?2? ? ·?0n ˆj??. (6)

The Fourier transform of the reconstructed image is

then given by

R˜j?? ?, zR??H˜

Oj?? ???H˜

Rj?? ??P˜j?? ?, zR??*

??dzI˜?? ?, z?exp?i???zR?z???2?2? ? ·?0n ˆj??

?Disk??? ? ??0n ˆj???MAX?, (7)

where the asterisk represents a complex conjugation.

In the plane of focus z ? zR, the reconstruction has a

spatial frequency spectrum consisting of the object’s

spatial frequencies located within a disk of radius

?MAX, centered at ?0n ˆj. By adding coherently the re-

construction amplitudes of a number of holograms

with different spatial frequency shifts, one can, in

principle, tile a synthetic pupil with an area far ex-

ceeding the pupil disk of the objective.

It must be emphasized that although the pupil rep-

resentation in Eq. (1) is, strictly speaking, valid only

in the paraxial regime,52the method of scanning ho-

lographic microscopy is not limited to small NAs. In

fact, since we actually measure the PSF, and recon-

struct the holograms by this experimental PSF, our

method is valid regardless of the system’s NA, and it

is independent of the theoretical representation of

the PSF.

3.

The experimental setup of a scanning holographic

microscope has been described in detail in previous

publications.40,41For completeness, a sketch is given

in Fig. 1. The only addition to the previous arrange-

ments is the introduction of a wedge prism in a con-

jugate object plane to shift the spatial frequency of

the plane wave creating the FZP by a chosen amount

in a chosen direction. The wedge is then rotated in

discrete angular steps to cover a desired area of the

object’s spatial frequency spectrum. For experimen-

tal simplicity we chose to introduce both the spherical

wave and the off-axis plane wave forming the FZP

through the pupil of the objective. With this method,

the largest spatial frequency shift achievable is the

objective’s frequency cutoff, namely, ?0? ?MAX. Opt-

ing for this geometry is not a necessary requirement.

Experimental Arrangement

Fig. 1.

electro-optic phase modulator. L1,2are achromat doublet lenses, 16 cm focal length, L3is a collecting lens, 1 cm focal length. The wedge

on a rotating stage is used to create off-axis Fresnel patterns on the specimen.

Schematic of an off-axis scanning holographic microscope. M, mirrors; BS, beam splitters; DBS, dichroic beam splitter; EOM,

20 February 2007 ? Vol. 46, No. 6 ? APPLIED OPTICS 995

Page 4

In principle, it is possible to introduce the off-axis

plane waves externally at any angle, although this

practice may be difficult to implement at high NA.

Nevertheless, the transfer function of such system

could in principle be extended to its theoretical fre-

quency limit of 2?? and cover the entire spectral re-

gion inside this limit. Additionally, it may also be

possible to introduce several plane waves simulta-

neously by using an array of sources.35–37

The experiment uses a Mitutoyo Plan Apo objective

20x, 0.42 NA, chosen for its long working distance.

The detector is a commercial Hamamatsu photomul-

tiplier tube (R1166). The specimen is scanned in a 2D

raster using a piezo scanning stage (PI Hera 625).

The FZP modulation is provided by an electro-optic

phase modulator (Linos LM0202) driven by a saw

tooth waveform at 25 kHz. The signal is sampled at

100 kSample?s by a GageScope 1602. The hologram

is constructed line by line after bandpass filtering

at the modulation frequency, as previously de-

scribed, and reconstructed digitally using MATLAB

codes (MATLAB 7.1.0.144?R14 SP3) on a standard

personal computer (PC).

It was found that at high NA, the fact that the

off-axis FZP has to cross the cover glass of the spec-

imen before illuminating it introduced a fair amount

of aberrations and distortions. Furthermore it also

changed the spatial frequency offset by an amount

difficult to predict. In principle, if the optical proper-

ties of the specimen and its mounting environment

are known, this effect can be compensated for numer-

ically. We chose to eliminate these difficulties by us-

ing a specimen without cover glass. The specimen

used in the demonstration in Section 4 is a cluster of

1 ?m diameter fluorescent beads (R0100 Polymer

Microspheres, Duke Scientific, Palo Alto, Calif., USA)

smeared on a microscope slide without a cover slip.

4.

To assess the gain in the resolution of the system,

we used the reconstruction of a subresolution pin-

hole (0.5 ?m diameter)toestimatethesizeofthePSF.

With a NA of 0.42, the theoretical Rayleigh resolution

limit of the objective is 1.22?EM?2NA ? 0.9 ?m. ?EM

? 600 nm is the emission wavelength of the beads.

The wrapped phase of the on-line hologram of the

Experimental Results

Fig. 2.

of a 0.5 ?m diameter pinhole. The scale bar is 10 ?m. The phase

distribution has a radius ?18 ?m, a Fresnel number ?12, and a

radius of curvature ?50 ?m. (b) Amplitude of the reconstruction of

the 0.5 ?m pinhole using the on-line hologram. FWHM ?1.0 ?m.

(Color online) (a) Wrapped phase of the on-line hologram

Fig. 3.

grams of the 0.5 ?m pinhole illustrating the idea of pupil synthe-

sis. The scale bar is 10 ?m. (b) Amplitude of the reconstruction of

the 0.5 ?m pinhole using the composite off-axis holograms. FWHM

?0.7 ?m.

(Color online) (a) Wrapped phase of three off-axis holo-

996APPLIED OPTICS ? Vol. 46, No. 6 ? 20 February 2007

Page 5

0.5 ?m diameter pinhole is shown in Fig. 2(a), and

its reconstruction is shown in Fig. 2(b). The holo-

gram has a Fresnel number F ? 12, and a radius

a ? 18 ?m. Using the relation F ? a2??EXz0with

an excitation wavelength ?EX? 532 nm, the radius

of curvature of the hologram is found to be z0?

50 ?m, and its equivalent NA is a?z0? 0.35. This

leads to an expected theoretical Rayleigh limit

?0.9 ?m, which is close to the Rayleigh resolution of

the objective. The observation that the smaller equiv-

alent NA of the hologram leads to the same resolution

limit as that of the objective is attributable to the fact

that the objective forms an image at the emission

wavelength while the hologram is formed at the

shorter excitation wavelength. A dye with a larger

Stokes shift would lead to an even larger gain. The

reconstruction shown in Fig. 2(b) has a FWHM equal

to ?1.0 ?m. Since this is the convolution of the ac-

tual PSF with the 0.5 ?m pinhole, the size of the

experimental PSF is estimated to be ?0.9 ?m, or

smaller. Three off-axis holograms of the pinhole

were recorded with a spatial frequency offset ?0?

?MAX ? 0.66 ?m?1in directions 120° apart. The

wrapped phases of the three holograms are combined

in Fig. 3(a) to illustrate the wider spatial frequency

coverage of the composite hologram. The reconstruc-

tion of the pinhole from this hologram is shown in

Fig. 3(b). Its FWHM is ?0.7 ?m. The actual size of

the PSF of the composite hologram is thus estimated

Fig.4.

a collection of ?1.0 ?m fluorescent beads (excitation?emission

wavelengths ? 532 nm?600 nm) at the best focus distance of

47.5 ?m from the focal plane of the objective. The scale bar is 5 ?m.

Bead clusters are just barely resolved. (b) Same reconstruction at

a focus distance of 49 ?m. The two planes are within the Rayleigh

range of the on-axis scanning FZP.

(Coloronline)(a)Reconstructionoftheon-axishologramof

Fig. 5.

the reconstructions of three off-axis holograms recorded with off

sets 120° apart. The scale bar is 5 ?m. (a) Best focus at 47.5 ?m

from the focal plane of the objective. (b) Same reconstruction at a

focus distance of 49 ?m. The distance between the two planes is

close to the Rayleigh range of the synthesized FZP, and different

bead clusters are focused in different planes.

(Color online) Coherent sum of the complex amplitudes of

20 February 2007 ? Vol. 46, No. 6 ? APPLIED OPTICS997

Page 6

to be smaller than ?0.6 ?m, which corresponds to an

equivalent NA larger than ?0.54. The reduction of

the resolution limit by a factor ?0.6 or smaller could

be further improved by combining more than three

holograms. If the off-axis FZP are introduced on the

specimen through the objective, as done in the exper-

iment, we can expect a resolution limit down to a

factor ?0.5 that of the objective. Further improve-

ment is possible in principle down to the theoretical

limit ??2, by introducing off-axis FZP externally (al-

beit this may involve some technical difficulties).

To demonstrate the reality of the achieved resolu-

tion improvement, we chose a sample consisting of

fluorescent beads with a diameter slightly larger

than the ?0.9 ?m resolution limit of the objective at

the emission wavelength, and slightly larger than the

?0.9 ?m resolution limit of the on-axis hologram at

the excitation wavelength. The expectation is that

the reconstruction of the on-axis hologram will show,

at best, barely unresolved beads, while the composite

reconstruction should reveal beads that are resolved.

The three holograms of the fluorescent beads were

recorded in reflection, reconstructed individually,

and the amplitudes of their reconstructions were

added coherently. Figures 4(a) and 4(b) show the re-

constructions of the on-axis hologram in two different

planes. As expected, individual beads are detected in

the best plane of focus [z ? 47.5 ?m in Fig. 4(a)], but

the clusters are not resolved. Shifting the focal plane

to z ? 49 ?m does not reveal anything different be-

cause the two planes with an axial separation of

1.5 ?m are well within the Rayleigh focal distance of

?3.5 ?m. The composite reconstructions of the three

off-axis holograms in the same two focal planes are

Fig. 6.

struction of a 0.5 ?m diameter pinhole from the on-axis hologram.

The phase profile is typical of an Airy pattern with a central lobe

diameter ?1.5 ?m. The scale bar is 1 ?m.

(a) Absolute value and (b) wrapped phase of the recon-

Fig. 7.

coherent sum of the reconstructions of a 0.5 ?m diameter pinhole

from the three off-axis holograms. The threefold symmetry of the

destructive interference results in a narrower amplitude distribu-

tion (a), and a central lobe diameter ?1.0 ?m. The scale bar is

1 ?m.

(a) Absolute value and (b) wrapped phase of the

998APPLIED OPTICS ? Vol. 46, No. 6 ? 20 February 2007

Page 7

shown in Figs. 5(a) and 5(b). The clusters of beads are

now well resolved. Furthermore Figs. 5(a) and 5(b)

reveal that different clusters are in best focus in dif-

ferent planes. This is to be expected since the two

planes are 1.5 ?m apart, while the axial Rayleigh

distance of the composite hologram is ?1.8 ?m.

It is interesting to observe how the gain in trans-

verse and axial resolution is a result of the mutual

interference of the complex amplitudes of the recon-

structions of the off-axis holograms. This is illus-

trated in Figs. 6 and 7, which show the absolute

values and the wrapped phases of the reconstructions

of the 0.5 ?m pinhole for the on-axis hologram (Fig.

6), and for the three combined off-axis holograms

(Fig. 7). The middle gray in the phase figures repre-

sents a phase zero, while the black and the white

represent a phase ?? and ??, respectively. Figure 6

is that of a typical Airy pattern. Combining the com-

plex amplitudes of the three off-axis holograms leads

to destructive interferences at the rim of the main

lobe of the Airy disc, leading to a narrowing of its size.

The threefold symmetry of the destructive interfer-

ences is clearly identified in Fig. 7.

5.

We have shown experimentally that it is possible to

overcome the spatial resolution limit of the objective

in holographic microscopy by combining a number of

off-axis scanning holograms to synthesize a pupil

area larger than that of the objective. The principle

of superresolution by holographic superposition and

synthetic aperture has a long history, which is briefly

reviewed. The implementation of this principle using

incoherent holographic scanning microscopy is dem-

onstrated by using holograms of fluorescent beads.

This research was funded in part by a grant from

the National Institute of Health, Office of Extra-

muralResearch,grantR21RR018440,andbyagrant

from the National Science Foundation, grant DBI-

0420382. J. Rosen’s research was supported in part

by the Israel Sciences Foundation, grant 119-03.

Summary

References

1. G. Toraldo di Francia, “Super-gain antennas and optical re-

solving power,” Nuovo Cimento, Suppl. 9, 426–438 (1952).

2. G. Toraldo di Francia, “Resolving power and information,” J.

Opt. Soc. Am. 45, 497–501 (1955).

3. G.ToraldodiFrancia,“Degreesoffreedomofanimage,”J.Opt.

Soc. Am. 59, 799–804 (1969).

4. W. Lukosz, “Optical systems with resolving power exceeding

the classical limits,” J. Opt. Soc. Am. 56, 1463–1472 (1966).

5. W. Lukosz, “Optical systems with resolving power exceeding

the classical limits, II,” J. Opt. Soc. Am. 57, 932–941 (1967).

6. I. J. Cox and J. R. Sheppard, “Information capacity and reso-

lution in an optical system,” J. Opt. Soc. Am. A 3, 1152–1158

(1986).

7. D. Mendlovic and A. W. Lohmann, “Space-bandwidth product ad-

aptation and its application to superresolution: fundamentals,”

J. Opt. Soc. Am. A 4, 558–562 (1997).

8. D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space-

bandwidth product adaptation and its application to super-

resolution: examples,” J. Opt. Soc. Am. A 4, 563–567 (1997).

9. Z. Zalevsky, D. Mendlovics, and A. W. Lohmann, “Optical

systems with improved resolving power,” in Progress in Op-

tics, E. Wolf, ed. (Elsevier, 2000), Vol. 40, pp. 271–341.

10. T. Leizerson, S. G. Lipson, and V. Sarafi, “Superresolution in

far-field imaging,” J. Opt. Soc. Am. A 19, 436–443 (2002).

11. M. A. Grimm and A. W. Lohmann, “Superresolution image for

one-dimensionalobjects,”J.Opt.Soc.Am.56,1151–1156(1966).

12. D.Mendlovics,A.W.Lohmann,N.Konforti,I.Kiryuschev,and

Z. Zalevsky, “One-dimensional superresolution optical system

for temporally restricted objects,” Appl. Opt. 36, 2353–2359

(1997).

13. A. Shemer, D. Mendlovics, Z. Zalevsky, J. Garcia, and P.

Garcia-Martinez, “Superresolving optical system with time

multiplexing and computer decoding,” Appl. Opt. 38, 7245–

7251 (1999).

14. A. W. Lohmann and D. P. Paris, “Superresolution for nonbi-

refringent objects,” Appl. Opt. 3, 1037–1043 (1964).

15. E. N. Leith, D. Angell, and C. P. Kuei, “Superresolution by

incoherent to coherent conversion,” J. Opt. Soc. Am. A4, 1050–

1054 (1987).

16. R. W. Gerchberg, “Super-resolution through error energy re-

duction,” Opt. Acta 21, 709–720 (1974).

17. M.BerteroandC.DeMol,“Superresolutionbydatainversion,”

in Progress in Optics, E. Wolf, ed. (Elsevier, 1996), Vol. 36,

pp. 129–178.

18. B. Colicchio, O. Haeberle, C. Xu, A. Dieterlen, and G. Jung,

“Improvement of the LLS and MAP deconvolution algorithms

by automatic determination of optimal regularization param-

eters and prefiltering of original data,” Opt. Commun. 244,

37–49 (2005).

19. P. C. Sun and E. N. Leith, “Superresolution by spatial–

temporal encoding methods,” Appl. Opt. 31, 4857–4862 (1992).

20. M. G. L. Gustafsson, “Surpassing the lateral resolution limit

by a factor of two using structured illumination microscopy,” J.

Microsc. 198, 82–87 (2000).

21. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining

optical sectioning by using structured light in a conventional

microscope,” Opt. Lett. 22, 1905–1907 (1997).

22. J.T.Frohn,H.F.Knapp,andA.Stemmer,“Three-dimensional

resolution enhancement in fluorescence microscopy by har-

monic excitation,” Opt. Lett. 26, 828–830 (2001).

23. J. Garcia, Z. Zalevsky, and D. Fixler, “Synthetic aperture su-

perresolution by speckle pattern projection,” Opt. Express 13,

6073–6078 (2005).

24. O. Haeberle and B. Simon, “Improving the lateral resolution in

confocal fluorescence microscopy using laterally interfering ex-

citation beams,” Opt. Commun. 259, 400–408 (2006).

25. M.Martinez-Corral,P.Andres,C.J.Zapata-Rodriguez,andM.

Kowalczyk, “Three-dimensional superresolution by annular

binary filters,” Opt. Commun 165, 267–278 (1999).

26. M. Gu, T. Tannous, and C. R. J. Sheppard, “Improved axial

resolution in focal fluorescence microscopy with annular pu-

pils,” Opt. Commun. 110, 533–539 (1994).

27. M. Martinez-Corral, M. T. Caballero, E. H. K. Stelzer, and J.

Swoger, “Tailoring the axial shape of the point spread function

using the Toraldo concept,” Opt. Express 10, 98–103 (2002).

28. J. W. Goodman and R. W. Lawrence, “Digital image informa-

tion from electronically detected holograms,” Appl. Phys. Lett.

11, 77–79 (1967).

29. T. Sato, M. Ueda, and G. Yamagishi, “Superresolution micro-

scope using electrical superposition of holograms,” Appl. Opt.

13, 406–408 (1973).

30. M. Ueda, T. Sato, and M. Kondo, “Superresolution by multiple

superposition of images holograms having different carrier

frequencies,” Opt. Acta 20, 403–410 (1973).

31. X. Chen and S. R. Brueck, “Imaging interferometric lithogra-

phy approaching the resolution limit of the optics,” Opt. Lett.

24, 124–126 (1999).

32. F. Le Clerc, M. Gross, and L. Collot, “Synthetic-aperture ex-

20 February 2007 ? Vol. 46, No. 6 ? APPLIED OPTICS999

Page 8

periment in the visible with on-axis digital heterodyne holog-

raphy,” Opt. Lett. 26, 1550–1552 (2001).

33. J. R. Hassig, “Digital off-axis holography with synthetic aper-

ture,” Opt. Lett. 27, 2179–2181 (2002).

34. C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck, “Imaging

interferometric microscopy,” Opt. Lett. 28, 1424–1426 (2003).

35. V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia,

“Single step superresolution by interferometric imaging,”

Opt. Express 12, 2589–2596 (2004).

36. V. Mico, Z. Zalevsky, and J. Garcia, “Superesolution optical

system by common path interferometry,” Opt. Express 14,

5168–5177 (2006).

37. V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, “Su-

perresolved imaging in digital holography by superposition of

tilted wavefronts,” Appl. Opt. 45, 822–828 (2006).

38. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holog-

raphy for quantitative phase contrast imaging,” Opt. Lett. 24,

291–293 (1999).

39. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous

amplitude-contrast and quantitative phase-contrast micros-

copy by numerical reconstruction of Fresnel off-axis holo-

grams,” Appl. Opt. 38, 6994–7001 (1999).

40. G. Indebetouw, A. El Maghnouji, and R. Foster, “Scanning

holographic microscopy with transverse resolution exceeding

the Rayleigh limit and extended depth of focus,” J. Opt. Soc.

Am. A 22, 892–898 (2005).

41. G. Indebetouw and W. Zhong, “Scanning holographic micros-

copy of three-dimensional fluorescent specimens,” J. Opt. Soc.

Am. A 23, 1699–1707 (2006).

42. G. Indebetouw, Y. Tada, and L. Leacock, “Quantitative phase

imaging with scanning holographic microscopy:

assessment,” Biomed. Eng. Online 5, doi:10.1186/1475-925x-

5-63 (2006).

43. G. Indebetouw, W. Zhong, and D. Chamberlin-Long, “Point-

spread function synthesis in scanning holographic microscopy,”

J. Opt. Soc. Am. A 23, 1708–1717 (2006).

44. G. Indebetouw, “A posteriori quasi-sectioning of the three-

dimensional reconstructions of scanning holographic micros-

copy,” J. Opt. Soc. Am. A 23, 2657–2661 (2006).

45. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging

properties of scanning holographic microscopy,” J. Opt. Soc.

Am. A 17, 380–390 (2000).

46. J.Rosen,G.Indebetouw,andG.Brooker,“Homodynescanning

holography,” Opt. Express 14, 4280–4285 (2006).

47. I. Yamaguchi and T. Zhang, “Phase-shifting digital hologra-

phy,” Opt. Lett. 22, 1268–1270 (1997).

48. I. Yamaguchi, J.-I. Kato, S. Ohta, and J. Mizuno, “Image for-

mation in phase-shifting digital holography and application to

microscopy,” Appl. Opt. 40, 6177–6186 (2001).

49. J. Swoger, M. Martinez-Corral, J. Huysken, and E. H. K.

Stelzer, “Optical scanning holography as a technique for

high-resolution three-dimensional biological microscopy,” J.

Opt. Soc. Am. A 19, 1910–1918 (2002).

50. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.

(McGraw-Hill, 1966).

51. C.W. McCutchen,“Generalized

dimensional diffraction images,” J. Opt. Soc. Am. 54, 240–244

(1964).

52. M. Gu, Advance in Optical Imaging Theory (Springer,

2000).

experimental

aperture and three-

1000APPLIED OPTICS ? Vol. 46, No. 6 ? 20 February 2007