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892 J. Opt. Soc. Am. A/Vol. 22, No. 5 /May 2005 Indebetouw et al.

Scanning holographic microscopy with transverseresolution exceeding the Rayleigh limit and

extended depth of focus

Guy Indebetouw, Alouahab El Maghnouji, and Richard Foster

Physics Department, Virginia Tech, Blacksburg, Virginia 24061-0435

Received September 17, 2004; revised manuscript received November 10, 2004; accepted November 12, 2004

We demonstrate experimentally that the method of scanning holographic microscopy is capable of producingimages reconstructed numerically from holograms recorded digitally in the time domain by scanning, withtransverse and axial resolutions comparable to those of wide-field or scanning microscopy with the same ob-jective. Furthermore, we show that it is possible to synthesize the point-spread function of scanning holo-graphic microscopy to obtain, with the same objective, holographic reconstructions with a transverse resolutionexceeding the Rayleigh limit of the objective up to a factor of 2 in the limit of low numerical aperture. Theseholographic reconstructions also exhibit an extended depth of focus, the extent of which is adjustable withoutcompromising the transverse resolution. 2005 Optical Society of America

OCIS codes: 090.0090, 100.6640, 110.0180, 110.4850, 180.6900.

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. INTRODUCTIONcanning holographic microscopy is a technique that wasroposed as having the potential of recording the three-imensional (3D) information of thick specimens in aingle two-dimensional (2D) scan, thereby improving theata-acquisition speed compared with methods requiring3D scan.1 Speed is a crucial factor in, for example, the in

ivo study of the dynamics of biological activities. Scan-ing holographic microscopy has other potential advan-ages as well, one of which is the flexibility with which theoint-spread function (PSF) of the imaging mode can bengineered with the method of two-pupil interaction.2

his method allows one to synthesize PSFs that are notonstrained to be real positive, but can be bipolar as well.his broadens considerably the types of imaging mode ac-essible to the instrument (e.g., amplitude contrast, quan-itative phase contrast, fluorescence contrast). Anothernique attribute of scanning holographic microscopy ishat it makes it possible to capture holographic informa-ion of 3D fluorescent structures. Scanning holographicicroscopy is an incoherent holographic process (even if

asers are used as sources for convenience), but becausehe PSF can be bipolar, the method remains quantita-ively sensitive to phase information.3

The holographic approach aims to capture the 3D infor-ation of a specimen in a single shot. The data are then

econvolved a posteriori to reconstruct axial sections.his is opposite to the confocal approach in which sharpxial sections of the specimen are captured one at a time.learly, the single-shot holographic method is not ex-ected to approach the sectioning capability of the confo-al method. It is, however, expected to lead to a significantain in data-acquisition time. The scanning holographicethod is also capable of sectioning, as has been shown

heoretically,4 but the sections must be captured one at aime, as in confocal imaging.

1084-7529/05/050892-7/$15.00 2

The principle of the method has been demonstratedith simulations, and experimentally with macroscopic

bjects, but its capability on a microscopic scale has noteen convincingly demonstrated. The purpose of this pa-er is to provide the experimental demonstration of twossential aspects of scanning holographic microscopy.irst, we want to show that scanning holographic imaging

n its most straightforward implementation, as originallyroposed, leads to images with quality comparable to thatf standard wide-field or scanning microscopy. Next, weant to demonstrate the capability of the method in PSF

ngineering by illustrating an imaging mode leading to aransverse resolution exceeding the Rayleigh limit of thebjective.

The paper is organized as follow. In Section 2 we brieflyeview the principles of scanning holographic microscopy.simplified theory of its originally proposed implementa-

ion is given in Section 3. In Section 4 we describe the the-retical background for scanning holographic microscopyith a transverse resolution exceeding the Rayleigh limit,nd with an extended depth of focus (DOF). In Section 5e give technical details of the experimental setup, andnally the experimental results are discussed in Section.

. PRINCIPLES OF SCANNINGOLOGRAPHIC MICROSCOPY

he basic idea in scanning holographic microscopy is toeverse the order in which conventional images (holo-raphic or not) are being captured in order to take advan-age of imaging parameters that are otherwise not acces-ible. In conventional imaging, an objective is used toroduce a magnified image of a specimen on a pixelated,patially resolving detector (CCD or complementaryetal-oxide semiconductor devices, for example). For ho-

005 Optical Society of America

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Indebetouw et al. Vol. 22, No. 5 /May 2005/J. Opt. Soc. Am. A 893

ographic recording this image must be coherent andade to interfere with a reference beam.5,6 In this way,

ach object voxel is encoded into a spatial pattern thatontains the 3D information of its position in space. Thatoxel can then be reconstructed digitally by a convolutionperation simulating free-space propagation for an appro-riate distance.7 In scanning holographic microscopy, thepatial pattern encoding each object voxel is synthesizedy the interference of two pupil distributions. A constantrequency offset between the illuminations of the pupilsllows one to directly capture the phase of the pattern inhe time domain, using heterodyne methods. The patterns projected through the objective onto the specimen andcanned in a 2D raster. A single nonimaging detectorphotomultiplier, photodiode) is used to capture the time-odulated signal. The 2D raster scan effects a mapping of

he encoding pattern from space domain to time domain,nd the phase of the pattern is extracted by heterodyneethods without any disturbances due to zero-order back-

round or twin-image interference. Figure 1 gives an ideaf the arrangement used to implement this idea. Theardware is discussed in detail in Section 5.

. ORIGINAL IMPLEMENTATIONn the most straightforward implementation of scanningolographic microscopy, as was originally proposed, thewo pupils P1, P2 (Fig. 1) are, respectively, a small aper-ure (pinhole) and a spherical wave. The waves propagat-ng from the pupils are combined by a beam splitter andorm in the plane pattern (see Fig. 1), a magnified versionf the pattern that is to be projected onto the specimen.his pattern is the interference of a plane wave and a co-ropagating spherical wave. The size of the two waves isimited by apertures matching the size of the pupil of thebjective, and the radius of curvature of the sphericalave is chosen to match the numerical aperture of the ob-

ective. In this way, the pattern projected onto the speci-en is a Fresnel pattern with radius a and focal length

0. The numerical aperture of the pattern, sin a=a /z0,atches that of the objective (a is the half cone angle of

he spherical wave). The Fresnel number of the pattern is

ig. 1. Experimental setup. EOPM, electro-optic phase modula-or introducing a frequency difference between the two beams;at, encoding Fresnel pattern projected on the specimen through

he objective; Ps, pupils; BS, beam splitter; BE, beam expander;, mirror; AT, half-wave plate/polarizer attenuator; OBJ, objec-

ive; Ls, lenses; PM, photomultiplier tube detector; AS, aperturetop; C, collecting lens.

=a2 /lz0, where l is the wavelength of the illumination.he Fresnel number is equal to twice the number of in-erference rings observed in the pattern. Of course with arequency offset between the two waves, these rings areemporally modulated and invisible to the eye unless theffset frequency is set to zero.

Without the modulation, the hologram captured after aaster scan of the specimen is equivalent to an in-line Ga-or hologram recorded at a distance z0 from the object.

8

ith modulation and heterodyne detection, one capturesirectly a single-sideband hologram without zero order orwin image. An equivalent way to obtain the single-ideband hologram is to record the entire modulated sig-al with a fast data-acquisition system and to filter outne sideband digitally in the temporal frequency domain.he hologram is then reconstructed numerically.The images reconstructed from such a hologram are ex-

ected to have the same transverse and axial resolutionss those of a conventional wide-field image obtained withhe same objective. That is, the usual Rayleigh limits are

Dx = l/2 sin a,

Dz = l/2s1 cos ad = l/4 sin2 a/2 < l/sin2 a, s1d

ssuming a modest numerical aperture. Theoretical cal-ulations valid for moderate numerical apertures have al-eady been published.3 Extension of these calculations toigh numerical aperture, on the basis of well-documented

iterature,912 is being pursued. Needless to say, scanningolographic microscopy with high-numerical-aperture ob-

ectives involves encoding patterns with high numericalpertures and possibly low Fresnel numbers that must bealculated exactly. If the digital reconstruction of the ho-ogram has to minimize aberrations, the function used forhe reconstruction must include the aberrations of the ob-ective, possibly the aberrations due to the specimen andts environment13,14 and the effects of polarization, andhe vectorial nature of the electromagnetic field.9,10 Ouroal here is to give only a simple, intuitive description ofhe system and to explain in Section 3 how a resolutionxceeding the Rayleigh limit of the objective is attained.o this end, it is sufficient to assume a paraxial system inhich the encoding pattern has a low numerical aperturend a relatively large Fresnel number. Within the limitsf these approximations, the scanning pattern projectednto the specimen can be written as

Asr,zd = uA1sr,zd + A2sr,zdexp siVtdu2, s2d

here sr ,zd are the transverse and axial coordinates inpecimen space, z is measured from the nominal plane ofocus and thus represents a defocus distance, and V is theemporal frequency offset between the two pupils. Themplitude distributions Ajsr ,zd sj=1,2d are given by theourier transforms of the pupils P1snd, P2snd defocused bydistance z. snd is the transverse spatial-frequency coor-

inate in the pupil planes.15

The pupils P1snd, P2snd are arbitrary complex ampli-ude distributions. These distributions can be synthesizedy using masks, refractive or diffractive optical elements,r spatial light modulators, allowing dynamic changes.

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894 J. Opt. Soc. Am. A/Vol. 22, No. 5 /May 2005 Indebetouw et al.

he examples discussed in this paper involve only pupilsith circular symmetry. In this case we have

Ajsr,zd =E0

nmax

J0s2pnrdPjsndexpsiplzn2d2pndn, s3d

here nmax=sin a /l is the cutoff frequency of the pupil ofhe objective and J0 is a zero-order Bessel function of therst kind. After scanning the specimen in a 2D raster, andemodulation of the signal, each specimen voxel of coor-inate sr ,zd is encoded into a complex pattern,

Ssr,zd = A1sr,zdA2*sr,zd, s4d

here * stands for the complex conjugate.Originally11 it was proposed to use two pupils given by

P1snd = dsnd,

P2snd = expsiplz0n2dcircsn/nmaxd, s5d

here dsnd is a delta function approximating the smallinhole aperture, and circsxd=1 for x1 and 0 otherwise.his choice of pupils was dictated by the desire to matchhe performance of scanning holographic microscopy withhat of conventional wide-field microscopy. Within theealm of the stated approximations (low numerical aper-ure and high Fresnel number), it is reasonable to ap-roximate the Fourier transform of a spherical wave ofimited extent by another spherical wave of limitedxtent.16 In these conditions, we have

A1sr,zd < circsr/ad,

A2sr,zd < expf ipr2/lsz0 + zdgcircfr/sin asz0 + zdg.

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ith small numerical apertures, the change of size due tohe defocus distance z can be neglected, and the encodingattern from Eq. (4) is

Ssr,zd > expfipr2/lsz0 + zdgcircfr/sin asz0 + zdg. s7d

The hologram of a specimen is the convolution of theD distribution of that specimen with the encoding pat-ern Ssr ,zd. The reconstruction of the hologram at a focusistance zR is effected digitally by a correlation of the ho-ogram with the encoding pattern at z0+z=zR. The 3DSF of the reconstruction is thus given by

hsr,zd = Ssr,zd ^ Ssr,0d, s8d

here ^ stands for a correlation integral. The transferunction of the system is given by the 2D transverse Fou-ier transform of the PSF. Thus, again within the limit ofalidity of the stated approximations, the transfer func-ion is given by

Hsn;zd < expsipln2zdcircsln/sin ad. s9d

he transfer function represents how a spatial frequencyof the specimen is represented in an image plane with

efocus z. Approximation (9) indicates the following tworoperties. First, the cutoff frequency in the reconstructedmage is nmax=sin a /l and is thus the same as that of thebjective. Consequently, the transverse resolution of the

econstruction is expected to also be the same as that ofhe objective, namely, Dx

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he 3D PSF is again given by Eq. (8), and the resultingransfer function is given by

Hsn,zd < expsipn2z2/2z0dcircsln/2 sin ad. s14d

Comparing this transfer function with that of approxi-ation (9), obtained with an encoding pattern resulting

rom the interference of a planar and a spherical wave,wo remarkable differences can be pointed out. The firstifference is that the cutoff frequency of the transfer func-ion of approximation (14) is twice that of the objective.amely,

ncutoff = 2 sin a/l = 2nmax. s15d

e thus expect a reconstructed image with twice the reso-ution of the objective, namely, Dx

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