Optical microscopy is limited by the small depths offocus owing to the high numerical apertures of themicroscope lenses and the high magnification ratios.Therefore the extension of the depth of focus is animportant goal in optical microscopy. In this way it hasbeen demonstrated that the annular filtering processsignificantly increases the depth of focus.1 A wavefrontcoding method has also been proposed in which a non-linear phase plate is introduced into the optical sys-tem.2 Another approach is based on the digitalholography method in which the hologram is recordedwith a CCD camera and the reconstruction is per-formed by computer.3 This method provides an effi-cient tool to refocus, slice by slice, the depth images ofa thick sample by implementing the optical beam prop-
agation of complex amplitude with discrete implemen-tations of the Kirchhoff–Fresnel (KF) propagationequation.4,5 Digital holographic microscopy (DHM) hasbeen applied in several applications of interest,including refractometry,6 observation of biologicalsamples,7–12 living-cell culture analysis,12–14 and ve-locimetry.15 Because the complex-amplitude signal isaccurately determined with a digital holography setup,it is a very flexible tool for implementing powerfulprocessing of the holographic information or of the pro-cessed images. Among the most significant examplesare methods to improve the digital holographic recon-struction,14 to control the image size as a functionof distance and wavelength,16 to perform three-dimensional (3D) pattern recognition,17–19 to processthe border artifacts,20,21 to eliminate twin-image noisefor several configurations,22,23 and to perform quanti-tative phase-contrast imaging.7,13,14,24 It must also beemphasized that a very flexible optical scanning ho-lography approach was introduced in applications, forexample, in remote sensing25 and 3D pattern recogni-tion.17 To our knowledge, this last reference describesthe first digital holographic technique for 3D imagerecognition. We developed a DHM with a partial co-herent illumination in a Mach–Zehnder configurationto reduce the coherent noise inherent to laser illumi-nation.8 With regard to the other microscopic methods,a decisive advantage of the DHM is its capacity ofrecording in a short time the volume information
F. Dubois ([email protected]), N. Callens, C. Yourassowsky, andO. Monnom are with Université Libre de Bruxelles, Chimie PhysiqueCP 165-62, Avenue Franklin Roosevelt 50, B1050 Brussels, Bel-gium. N. Callens, M. Hoyos, and P. Kurowski are with the Labo-ratoire de Physique et Mécanique des Milieux Hétérogènes, UnitéMixte de Recherche–Centre National de la Recherche Scientifique7636, Ecole Supérieure de Physique et Chimie Industrielles, 10Rue Vauquelin, 75231 Paris Cedex 05, France.
Received 13 May 2005; revised 18 August 2005; accepted 18August 2005.
needed for further processing, making it possible toinvestigate fast phenomena with reduced time distor-tions. This is why DHM was used to perform 3D ve-locimetry, in which it increases the flexibility ofclassical methods used in particle holography.26 Forthis purpose, the simplest experimental approach con-sists of using in-line holography in which the recordedinterference pattern results from the direct coherentsuperposition of the field diffracted by the particles andthe illuminating beam.27,28 Although in-line hologra-phy is the only realistic approach for many applica-tions, the reconstruction suffers from the twin-imageeffect that motivated the improvement of digital holo-gram processing.29 In-line DHM was also implementedwith a partially coherent source.30 The DHM that wedeveloped8,31 used the phase-stepping method thatrecords several interference images with small incre-mental phase shifts. This method introduces a speedlimitation when dynamical phenomena are studied. Inthe present paper we investigate the use of the DHM,based on the Mach–Zehnder configuration, workingwith a partial spatial coherent source to study flows ofmicroscopic particles. Because we are working withdynamic phenomena, it is crucial to have a short re-cording time of the holographic information. That im-plies disposal of powerful light illumination to achieveshort camera exposure time and to record the completeholographic information in one video frame. The illu-mination level is obtained by using a laser. The reduc-tion of the spatial coherence is achieved by couplingthe laser source with a moving ground glass as dis-cussed in Section 2. The complete holographic infor-mation is recorded for every recorded frame by usingthe carrier or Fourier-transform method.32,33 The im-plementation of this method is also described in Sec-tion 2. For the experimental demonstration, providedin Section 3, we use a hydrodynamic flow ofmicrometer-size particles, passing through a split-flowlateral-transport thin (SPLITT) separation cell.34 Thisclass of separation device makes possible the continu-ous and rapid fractionation of macromolecular andparticulate materials. The visualization of flowing par-ticles by DHM provides a better understanding of theinfluence of the hydrodynamic effects35,36 on the sepa-ration efficiency, as well as the particle–particle andparticle–wall interactions. DHM permits us to obtainthe particle diameters, their 3D positions in the chan-nel, and an estimation of their mean velocities. In thecase of a mixture of bidisperse particles, DHM can beused to distinguish the two species and their behav-iors.
2. Optical System Description and DigitalHolographic Reconstruction
This section describes the partial spatial coherentmicroscope working in digital holography and themethod to record the complete holographic informa-tion on every single video frame, as is required for thestudy of fast 3D particle flows. The optical setup,shown in Fig. 1, is a modified version of an instru-ment already described elsewhere.31 The coherentsource (a monomode laser diode, � � 635 nm) is
transformed in a partial spatial coherent source byfocusing the beam, performed by lens L1, close to theplane of a rotating ground glass (RGG). The spatialpartial coherence is adjusted by changing the positionof the focused spot with respect to the RGG plane.The lens L2 (focal length of f2 � 50 mm) collimatesthe beam that is divided by beam splitter BS1. Thetransmitted part, the object beam, illuminates thesample S by transmission. When the RGG is stopped,the speckle size in the plane of the sample can bemeasured to control the spatial coherence. For thetest we performed, the speckle size was �30 �m. Theobject beam transmitted by microscope lens L3 (focallength of f3 � 10 mm) is reflected by mirror M2 and bybeam splitter BS2 and is transmitted by lens L5 (focallength of f5 � 150 mm). A pair of lenses L3 and L5perform the image of one plane of the sample on theCCD camera sensor. The reference beam, reflected bybeam splitter BS1 and by mirror M3, is transmittedby microscope lens L4 (focal length of f4 � 10 mm), bybeam splitter BS2, and by lens L5. The referencebeam interferes with the object beam on the CCDsensor. An optical flat, not indicated in Fig. 1, isplaced in front of lens L4 to equalize the optical pathsof the reference and object beams. The camera is aJAI CV-M4 camera with a CCD array of 1280 �1024 pixels. For further processing, a 1024 � 1024pixel window is cropped to match the fast Fourier-transform computation. The camera is adjusted witha 100 �s exposure time. The reference beam isslanted with respect to the object beam in such a waythat a gratinglike thin interference pattern is re-corded on the sensor. This is used to implement theFourier method to compute the complex amplitude ofthe object beam for every recorded frame.32,33
We assume that the object is a complex transpar-ency s�x, y� located in the back focal plane of lens L2,
Fig. 1. Optical setup of the digital holographic microscope. L1,focusing lens; RGG, rotating ground glass for spatial coherencereduction; L2, collimating lens; L3, L4, identical microscope lenses(�20); L5, refocusing lens; CCD, charge-coupled device camerawith the sensor placed in the back focal plane of L5; M1–M3,mirrors; BS1, BS2, beam splitters. The optical compensation in thereference arm is not indicated in the drawing.
where �x, y� represents the spatial coordinates. Asdetailed later in this section, the location of s�x, y� inthe back focal plane of L2 is not a necessary conditionbut simplifies the derivations. We denote the instan-taneous object amplitude incident on s by oi�x, y, t�,where t is the time. Instantaneous means thatoi�x, y, t� is the amplitude corresponding to an arbi-trary but fixed position of the RGG. The relationshipbetween oi�x, y, t� and the amplitude in the RGGplane is described below. At this step we consider thatoi�x, y, t� is a speckle field with a speckle size adjustedby the focusing on RGG. The emergent amplitudeoo�x, y, t� out of the sample is expressed by
oo�x, y, t� � s�x, y�oi�x, y, t�. (1)
Let us denote the instantaneous reference ampli-tude beam by r�x, y, t�. The optical system performsthe superposition of the object and reference beams insuch a way that the amplitude on the CCD sensor canbe expressed by
a�x, y, t� � oo�x, y, t� � r�x, y, t�. (2)
In this expression we did not take into account themagnification of the optical system because it pro-vides a simple scaling effect.
The detection is conducted with an exposure timethat performs a time averaging. Therefore we candefine a mutual coherence function associated withthe amplitude a�x, y, t� expressed by
We now have to evaluate the different terms on theright-hand side of Eq. (4). In the ground-glass plane,the instantaneous amplitude distribution Q is de-scribed by
Q�x, y, t� � P�x, y�N�x, y, t�, (5)
where N�x, y, t� is the random-phase amplitude ofmodulus 1 created by the ground glass and P�x, y� isa real function that defines the amplitude modulationof the spot shape on the ground glass. We assumethat
As the ground glass and the transparency are inthe front focal and back focal planes of L2, respec-tively, there is a Fourier-transformation relationshipbetween Q and oi:
oi�x, y, t� �exp�jk2f2�
j�f2�F�NP� x
�f2,
y�f2
, (7)
where � is the wavelength, j � ��1, k � 2��, and�FW���, �� denotes the Fourier transformation of afunction W�x, y� at the point ��, �� defined by
�FW���, �� ��� dxdyexp� 2j�x� � y��� W�x, y�.
(8)
By combining Eqs. (4) and (6), we find that
�oi*�x2, y2, t�oi�x1, y1, t��t � A�p � p�
�x1 � x2
�f2,
y1 � y2
�f2,
(9)
where A is a constant, the symbol � indicates theconvolution operation, and p is the Fourier transformof P. The reference beam originates from the sameamplitude realization in the RGG plane but is ad-justed, owing to the optical components, in such away that it is seen to be shifted by an amount ��x, �y�in the RGG plane with respect to the object beam.Therefore the instantaneous amplitude distributionto be used for the reference beam is
Q��x, y, t� � P�x � �x, y � �y�N�x � �x, y � �y, t�.(10)
We follow the same steps for the object beam toobtain
where sm�x, y� is the modulus of the transparency and��x, y� is the optical phase.
As it is the intensity that is finally detected, weobtain the following by setting x1 � x2 � x andy1 � y2 � y in Eq. (13):
i�x, y� � g {�s�x, y��2 � 1 � 2sm�x, y�
� cos ��x, y� � 2 �xx � �yy�f2
�}, (15)
with g � A�p � p��0, 0�.At the beginning of the derivation, we assumed
that the object was in the back focal plane of lens 2.This hypothesis is useful only to have an exactFourier-transform relationship between the plane ofthe ground glass and the incident object beam on theobject. However, by invoking the Van Cittert–Zerniketheorem, the mutual coherence function after lens L2is invariant under a translation along the opticalaxis. It results that the position of the object can beout of this plane without restricting the generality ofthe discussion. When the sample is in a differentplane, it will also be out of focus with respect to theimaging system. In that case the amplitude in thedetection plane has to be computed with the KF prop-agation equation. The effect of the partial coherentnature of the beam is to introduce a low spatial fil-tering process, as we demonstrated in another pub-lication.31 It is this effect that reduces the coherentnoise introduced by the deeply out-of-focus objectsand the defects of the optical setup, including theexperimental cell.
Equation (15) shows a fringe-formation processwith a maximum contrast, as we could expect with apurely coherent illumination. This means that thespatial coherent nature of the illumination does notaffect the contrast of the fringes. Therefore we canexpect an efficient computation of the complex am-plitude. For the completeness of the text, we brieflydiscuss the amplitude computation with the Fourier-transform method.32,33 Without restricting the gen-erality of the discussion, we assume that �y � 0 inEq. (15), and we define K � 2�x��f2.
The two-dimensional (2D) Fourier transformation
of the corresponding intensity distribution i�x, y� isgiven by
I�u, v� � g B�u, v� � �u �K2� S*�u, v�
� �u �K2� S�u, v��, (16)
where �u, v� are the spatial frequencies and B�u, v�and S�u, v� are the Fourier transformations of|io�x, y�|2 � 1 and s�x, y�, respectively. Equation (16)shows that the Fourier transformation of i�x, y� iscomposed of three terms: B�u, v�, which is centeredaround �0, 0�, one term that is centerd around�K�2, 0�, and another that is centered around(�K�2, 0). We assumed that the width of each of thethree terms is small in comparison with the shiftdistance K�2 in such a way that there is no overlapamong the three terms. Therefore the third term ofEq. (16) can be isolated by a window function andshifted at the origin �0, 0�. An inverse Fourier trans-formation gives an amplitude signal that is propor-tional to s�x, y�, which is what we are looking for inthe digital holographic reconstruction. For digital re-construction, s�x, y� is sampled with a sampling dis-tance � in both x and y directions:
sd�m, n� � s�m�, n��, (17)
where m and n are intergers varying from 0 toN�1, where N is the total number of sampling pointsby hologram side.
Knowing the amplitude distribution in plane P1allows the digital holographic reconstruction to com-pute the amplitude distribution in plane P2, which isparallel to P1 and separated from it by a distance daccording to
sd��m�, n�� � exp�jkd�[Fm�, n��1 exp �jk�2d
2N 2�2
� �U 2 � V 2�� �FU, V�1sd�m, n��], (18)
where N is the number of pixels in both directionsand m, n, m�, n�, U, and V are integers varying from0 to N � 1. F1 denotes the direct and inverse discreteFourier transformations defined by
�Fm, n1 g�m, n�� �
1N �
k, l�0
N�1
exp 2jN
��mk � nl�� g�k, l�, (19)
where k, l, m, and n are integers so that k, l, m, n� 0, . . . , N � 1.
A ruler with 10 �m spaced divisions was inserted inthe field of view of the DHM to obtain the pixel��mrelation for the digital holograms and to verify thatthe reconstructions are performed without parallaxeffect. With these parameters, the analysis of a holo-gram leads to the relation of 1 pixel width � 0.3 �m.The fringe spacing was adjusted to be approximately6 pixels. To perform the digital holographic recon-struction, we windowed the original 1280 � 1024holograms to 1024 � 1024 pixels. The amplitude ex-traction was performed by software to display theFourier transform intensity to accurately locate thesidelobe to be selected by the Fourier-transformmethod. The sidelobe was selected by multiplying theFourier-transform hologram with a Gaussian func-tion centerd on the lobe to be selected and with ahalf-width equal to the half-distance between thecenters of the zero diffraction order and of the side-lobe. The values of the Gaussian function are set tozero for distances from its center that are larger thanor equal to its width to reduce potential aliasing ef-fects. For display purposes, the reconstructed imagesare resampled to 512 � 512 pixels. Thus, for recon-structed images, the relation becomes 1 pixel width� 0.6 �m. By using a micrometric translation stage,we manually defocus the ruler by accurately knownsteps. On each reconstructed image, we measure thenumber of pixels corresponding to 100 �m. We deter-mined that there is no measurable magnificationchange over the digital holographic reconstruction of234 �m, which corresponds to the thickness of theSPLITT channel used for the particle-flow experi-ment. As the sampling distance is invariant with theuse of Eq. (19), this result was expected.
B. Reconstruction of Size-Calibrated Particles
A SPLITT cell used to separate particles by size isinserted into the sample holder. The cell is a thinribbonlike channel of 234 �m thickness, having at itsextremities two inlets and two outlets that are sepa-rated by splitters. The inlet splitter allows us to sep-arately inject the sample through one inlet and thecarrier liquid through the other inlet. The outlet split-ter divides the flow into two substreams in the out-lets.
To equalize the optical paths of the object and thereference beams, we placed a Plexiglass plate of ad-equate thickness in the reference beam.
Experiments were performed to estimate the accu-racy of the measured size of micrometer-size particlesflowing through the SPLITT channel. To performthese tests, we used different latex bead samples:monodisperse polymer microspheres of 4 and 7 �mdiameter (Duke Scientific, Palo Alto, California) andnominal latex beads of 10 �m (Beckman–CoulterInc., Miami, Florida). Double-distilled de-ionizedwater was used as a carrier fluid for all experiments.An example of recorded hologram is provided inFig. 2(a). The microscope is focused on the front chan-
nel wall. The particles are unfocused. By digital ho-lographic reconstruction, we determine the focusdistance and the size of the suspended particles [Fig.2(b)]. The measured particle sizes are in agree-ment with the specifications: 4.2 0.6 �m, 7.2 0.6 �m, 10.2 0.6 �m. A maximum estimatederror of 0.8 �m is obtained, which is acceptable be-cause the resolution in the reconstructed images is0.6 �m. The DHM allows us to measure the particlefocus distance with an accuracy of 2.6 �m. Thismeans that the particle position over the channelthickness is determined with an accuracy of �1%.
Fig. 2. (a) Digital hologram of size-calibrated 4 �m latex beads flow-ing at a mean velocity of v � 5.34 cm�s in a SPLITT channel. Theparticles are unfocused. The field of view is 300 �m � 300 �m.(b) Numerical refocusing of size-calibrated 4 �m latex beads flow-ing at a mean velocity of v � 5.34 cm�s in the SPLITT channel. Itis the reconstruction of the digital hologram in (a). With respect tothe hologram in (a), there is windowing to keep a field of view of300 �m � 300 �m. The particles have a mean size of 4.2 0.6 �m and a refocus distance of 23.4 2.6 �m. The recon-structed image is processed by a border-processing algorithm.21
The DHM permits us to observe in the SPLITT chan-nel particles of different sizes that are moving towarddifferent transversal positions owing to hydrody-namic effects. When particle size increases, particletransversal migration is also expected to increase.When a mixture of particles of different sizes is in-jected, the two species can then be differentiated andtherefore the respective transverse positions can bedetermined [Figs. 3(a) and 3(b)].
These results illustrate the selectivity inside thechannel and underscore the fact that the DHM allowsus to study the separation processes of micrometer-size objects in SPLITT channels.
The DHM also makes possible a direct assessmentof particle velocities. Under our specific experimentalconditions, when the mean flow velocity v is greaterthan 7 cm�s, some particles have velocities that aretoo high with respect to the camera-exposure timeand the instantaneous particle location becomes im-possible to determine. In this case we obtain particleholograms that are blurred by the motion. We testedthis operating mode on a suspension of 0.1% Lichro-spher Si60 of 5 �m particles (Merck, Darmstadt, Ger-
many), suspended in double-distilled de-ionizedwater [Fig. 4(a)]. Particle traces of different lengthsare observed according to the particle transversal po-sition. As the velocity profile is parabolic, a particleflowing near the channel center moves faster thanwhen near a wall and therefore leaves a longer trace.Using digital holographic reconstruction allowstraces to be refocused and particle velocities to bededuced by measuring their lengths [Fig. 4(b)]. Byassuming that the particle velocity is identical to theflow velocity at that position over the channel thick-ness, one can compare the experimental values withthe theoretical ones. The quantitative agreement isoutlined in Table 1. Although this operation mode isuseful to assess the speed of particles, the refocusingof particle traces should be improved to achieve abetter accuracy. This method seems promising be-cause effective particle velocities will give us the res-
Fig. 3. (a) Latex bead �4 �m�, flowing at a mean velocity ofv � 3.56 cm�s in the SPLITT channel, has a focus distance of25 2.6 �m. (b) Latex bead �7 �m�, flowing at a mean velocityof v � 3.56 cm�s in the SPLITT channel, has a focus distance of50 2.6 �m.
Fig. 4. (a) Blurred digital hologram of 5 �m Lichrospher particlesflowing at a mean velocity of v � 12.46 cm�s in the SPLITT chan-nel. (b) Reconstructed image from the digital hologram of (a) witha refocus distance of 40 �m. We observe a sharp trace of some 5 �mLichrospher particles. The reconstructed image is processed by aborder-processing algorithm.21
idence time of particles inside the channel, which isan important parameter in separation.
4. Concluding Remarks
We have demonstrated the use of DHM coupled witha spatial partial coherent source for the 3D analysisof micrometer-size particles moving in a flow. Thepartial coherent nature of the source makes it possi-ble to eliminate the coherent noise originated fromthe optical defects of the experimental cell and of theoptical system. To record complete holographic infor-mation for every recorded frame, we implemented theFourier-transform method. It is shown that the par-tial coherent nature of the source does not reduce thefringe visibility. Tests on a ruler allowed us to esti-mate that the parallax effect is negligible. Particle-size determinations on calibrated particles were alsoperformed to evaluate the method’s accuracy. Thesystem was tested with a SPLITT cell to measure thesize and the 3D position of particles in motion. Whenthe particles are too fast to provide frozen imageswith respect to the exposure time, a velocity measure-ment based on the length of the holographic blurredimages is proposed.
It has been demonstrated that DHM is capable ofmonitoring the effects that control the transport ofmicrometer-size particles in SPLITT fractionationchannels. It provides a way to visualize and measurethe transversal migration of species into a smallthickness channel. The 3D visualization in time al-lows a better understanding of the separation pro-cesses of micrometer-size objects in SPLITT cells.
This research was supported by the SSTC�ESA–PRODEX (Services Scientifiques Techniques etCulturels�European Space Agency–Programme deDeveloppement d’Experiénces) contract 90171.
References1. T.-C. Poon and M. Motamedi, “Optical�digital incoherent im-
age processing for extended depth of field,” Appl. Opt. 26,4612–4615 (1987).
2. E. R. Dowski, Jr., and W. T. Cathey, “Extended depth of fieldthrough wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
3. U. Schnars and W. Jüptner, “Direct recording of holograms bya CCD target and numerical reconstruction,” Appl. Opt. 33,179–181 (1994).
4. I. Yamaguchi and T. Zhang, “Phase-shifting digital hologra-phy,” Opt. Lett. 22, 1268–1270 (1997).
5. T. Zhang and I. Yamaguchi, “Three-dimensional microscopywith phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
6. M. Sebesta and M. Gustafsson, “Object characterization withrefractometric digital Fourier holography,” Opt. Lett. 30, 471–473 (2005).
7. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holog-raphy for quantitative phase contrast imaging,” Opt. Lett. 24,291–293 (1999).
8. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with digital holography microscope usinga partial spatial coherent source,” Appl. Opt. 38, 7085–7094(1999).
9. G. Indebetouw and P. Klysubun, “Spatiotemporal digital mi-croholography,” J. Opt. Soc. Am. A 18, 319–325 (2001).
10. I. Yamaguchi, J.-I. Kato, S. Otha, and J. Mizuno, “Image for-mation in phase-shifting digital holography and applicationsto microscopy,” Appl. Opt. 40, 6177–6186 (2001).
11. D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M.Deiwick, H. H. Scheld, and G. von Bally, “Lensless Fourierholography for digital holographic interferometry on biologicalsamples,” Opt. Lasers Eng. 36, 241–249 (2001).
12. F. Dubois, C. Yourassowsky, and O. Monnom, “Microscopic enholographie digitale avec une source partiellement cohérente,”in Imagerie et Photonique pour les Sciences du Vivant et laMédecine, M. Faupel, P. Smigielski, and R. Grzymala, eds.(Fontis Media, 2004), pp. 287–302.
13. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery,T. Colomb, and C. Depeursinge, “Digital holographic micros-copy: a noninvasive contrast imaging technique allowing quan-titative visualization of living cells with subwavelength axialaccuracy,” Opt. Lett. 30, 468–470 (2005).
14. D. Carl, B. Kemper, G. Wernicke, and G. von Bally,“Parameter-optimized digital holographic microscope for high-resolution living-cell analysis,” Appl. Opt. 43, 6536–6544(2004).
15. B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D–PIV and temperature measurement using a new CCD basedholographic interferometer,” Flow Meas. Instrum. 7, 1–6(1996).
16. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, andG. Pierattini, “Controlling image size as a function of distanceand wavelength in Fresnel-transform reconstruction of digitalholograms,” Opt. Lett. 29, 854–856 (2004).
17. T.-C. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370–381 (1999).
18. B. Javidi and E. Tajahuerce, “Three-dimensional object recog-nition by use of digital holography,” Opt. Lett. 25, 610–612(2000).
19. F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, and J.-C.Legros, “Pattern recognition with digital holographic micro-scope working in partially coherent illumination,” Appl. Opt.41, 4108–4119 (2002).
20. E. Cuche, P. Marquet, and C. Despeuringe, “Aperture apodiza-tion using cubic spline interpolation: application in digital ho-lography microscopy,” Opt. Commun. 182, 59–69 (2000).
21. F. Dubois, O. Monnom, C. Yourassowsky, and J.-C. Legros,“Border processing in digital holography by extension of thedigital hologram and reduction of the higher spatial frequen-cies,” Appl. Opt. 41, 2621–2626 (2002).
22. S.-G. Kim, B. Lee, and E.-S. Kim, “Removal of bias and the
Table 1. Comparison between the Experimental and TheoreticalValues of the 5 �m Lichrospher Particle Velocities inside the
conjugate image in incoherent on-axis triangular holographyand real-time reconstruction of the complex hologram,” Appl.Opt. 36, 4784–4791 (1997).
23. T.-C. Poon, T. Kim, G. Indebetouw, M. H. Wu, K. Shinoda, andY. Suzuki, “Twin-image elimination experiments for three-dimensional images in optical scanning holography,” Opt.Lett. 25, 215–217 (2000).
24. P. Klysubun and G. Indebetouw, “A posteriori processing ofspatiotemporal digital microholograms,” J. Opt. Soc. Am. A 18,326–331 (2001).
25. T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection andimage recovery in remote sensing by optical scanning holog-raphy,” Opt. Eng. 41, 1331–1338 (2002).
26. C. S. Vikram, Particle Field Holography (Cambridge U. Press,1992).
27. W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer,“Digital in-line holography of microspheres,” Appl. Opt. 41,5367–5375 (2002).
28. W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen,“Tracking particles in four dimensions with in-line holographicmicroscopy,” Opt. Lett. 28, 164–166 (2003).
29. S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of
diffraction patterns directly from in-line holograms with thefractional Fourier transform,” Appl. Opt. 41, 312–319 (2002).
30. L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holo-graphic microscope with light-emitting diode illumination,”Opt. Lett. 29, 1132–1134 (2004).
31. F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, andE. Istasse, “Partial spatial coherence effects in digital holo-graphic microscopy with a laser source,” Appl. Opt. 43, 1131–1139 (2004).
32. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transformmethod of fringe-pattern analysis for computer-based topogra-phy and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
33. T. Kreis, “Digital holographic interference-phase measure-ment using the Fourier-transform method,” J. Opt. Soc. Am. A3, 847–855 (1986).
34. J. C. Giddings, “A system based on split-flow lateral-transportthin (SPLITT) for rapid and continuous particle fractionation,”Sep. Sci. Technol. 20, 749–768 (1985).
35. P. S. Williams, “Particle trajectories in field-flow fractionationand SPLITT fractionation channels,” Sep. Sci. Technol. 29,11–45 (1994).
36. D. Leighton and A. Acrivos, “Viscous resuspension,” Chem.Eng. Sci. 41, 1377–1384 (1986).
