Digital in-line holography with a rectangular complex coherence factor

Digital in-line holography with a rectangularcomplex coherence factor

Clément Remacha,1,2,3 Sébastien Coëtmellec,2,4 Marc Brunel,2 and Denis Lebrun2

1Électricité de France Recherche et Développement Mécanique des Fluides, Énergies,Environnement, 6 quai Wattier, Chatou 78600, France

2Département Optique-Lasers, UMR-6614 Complexe de Recherche Interprofessionnel en Aérothermochimie,Avenue de l’Université, Saint-Etienne du Rouvray 76801, France

3e-mail: [email protected]: [email protected]

Received 17 July 2012; accepted 9 August 2012;posted 24 September 2012 (Doc. ID 172795); published 13 November 2012

We propose in this paper the study of a particular spatially partially coherent source applied to digitalin-line holography of dense particle flow. A source with a rectangular complex coherence factor isimplemented. The effects of such a source on the intensity distribution of the diffraction pattern aredescribed. In particular, we show that this type of source allows us to eliminate the diffraction patternalong one axis while all the information about the dimension of the particle is kept along the otherperpendicular axis. So particle images can be well reconstructed along one direction and the specklecan be largely limited. © 2012 Optical Society of AmericaOCIS codes: 030.0030, 090.0090, 100.0100.

1. Introduction

Digital in-line holography (DIH) allows carrying outdiagnostics in three-dimensional (3D) space. Thistechnique has been used in many configurations:in free space [1], in complex flows [2,3], or, more re-cently, to visualize the flow in a micropipe [4]. Theseexamples show the possibilities of this tool. Theimages of particles that seed a fluid flow can be re-constructed by means of DIH in 3D space becausethe position along the optical axis, denoted here ζ,is coded in the recorded hologram. The main diffi-culty of this technique is that the particle densitymust be low, since otherwise the holograms are cor-rupted and the estimation of the ζ coordinate is in-accurate. In this case, speckle noise appears andthe measurement of the position and the diameterof the particles may be inaccurate [5].

To increase the density of the particles, severalapproaches have been proposed. For example, aMach–Zehnder configuration allows reducing theobscuration problems [1]. From the source point ofview, a subpicosecond pulsed laser beam can be usedto restrict the extent of the diffraction pattern [6]. Asubtraction of the image of the objects detected in thereconstructed hologram is another solution forlimiting a part of noise [7]. Here, knowing that thecontrast of the interference fringes is directly depen-dent on the coherence degree of the source, we showthat control of the coherence is a promising way toincrease the particle density. Spatially partially co-herent beams have already been used for holographyin dense particle fields [8–10]. A limitation of thespeckle noise is observed, but the quality of the re-constructed particle images decreases [11]. The firststep to resolve this difficulty has been done in a pre-vious paper [12]. It deals with the quantitative effectof the spatial coherence on the hologram of a particle.

Propagation of spatial coherence has been largelydeveloped by Goodman [13] and by Mandel and

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1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS A147

Wolf [14]. It has been expressed with the fractionalFourier transform [15] or recently by the Wignerfunction [16]. The influence of turbulent media on co-herence also plays an important role in coherencestudies [17,18].

The theoretical development by the authors of [12]allows understanding the influence of spatial coher-ence on the hologram of one opaque disk. Now, theaim of this paper is to find an optimal complex coher-ence factor to reduce the speckle while keepinga good resolution on the diameter of the object. Todo this, in Section 1, a beam with a quasi-rectangular complex coherence factor is studied.First, we recall the way to obtain a rectangular com-plex coherence factor and the needed assumptions tocalculate it. The geometry of rectangular sourcesleads to keeping all high frequencies along one axisto find the position with a maximum of accuracy andto filter the diffraction pattern in the other perpen-dicular direction. Section 2 is devoted to describingthe intensity recorded by a quadratic sensor when anopaque and elliptical disk is illuminated by this kindof source. Amathematical development with an ellip-tic filter and an elliptic object have been proposed.This mathematical development allows us to speedup the numerical simulations. The numerical simu-lation and experimental results are compared inSection 3. Then, in Section 4, the reconstruction of aparticle image by means of the correlation with achirplet function is carried out.

2. Mathematical Method

The optical setup is illustrated in Fig. 1. It is com-posed of a spatially incoherent source. The beampasses through an intensity filter denoted If . The fil-ter enables controlling the mutual intensity, denotedJi, of the incident beam on the object localized at thedistance ζs from the filter. In our case, the object is anopaque and elliptical disk object. It is shown that itcan also model a droplet [19]. A charge-coupled de-vice (CCD) sensor, located at the distance ζe fromthe object, records the intensity distribution, denotedIs, of the hologram.

The equation of thepropagation ofmutual intensity(Eq. (5.6-1) of [13]) has been established under the as-sumption of a quasi-monochromatic and ergodic beam[13,20]. Two other assumptions are introduced: fromEqs. (5.6-4) and (5.6-5) of [13], the width of the sourceand the observation region are relatively small com-pared to the distance along the optical axis, and the

propagation is under the paraxial approximation:only small angles are considered. Now, the expressionof the intensity Is on the CCD sensor can be obtainedin two steps.

In the first step, the mutual intensity is the resultof the propagation of the mutual intensity from thefilter to the object. From Eq. (5.6-8) of [13], theVan Cittert–Zernike theorem allows us to expressthe mutual intensity Ji at the distance ζs from thefilter as

Ji�ξ; ν;Δξ;Δν� �exp

h−

i2π�ξΔξ�νΔν�λζs

iπζ2s

ZR2

If �α; β�

× exp�i2πλζs

�αΔξ� βΔν��dβdα; (1)

where �α; β� are the coordinates in the plane of thefilter and �ξ; ν� are the coordinates in the objectplane. The coordinates Δξ � ξ2 − ξ1, Δν � ν2 − ν1 arethe distances between two points in the object planeand ξ � �ξ1 � ξ2�=2, ν � �ν1 � ν2�=2 are the mean va-lues of the coordinates.

The second step consists in calculating the mutualintensity transmitted by the object, denoted in Fig. 1by Jt, to the CDD sensor plane �x; y�. The equation ofpropagation of the mutual intensity is used withIs�x; y� � Js�x; x; y; y�. From Eq. (5.7-9) of [13], the in-tensity recorded by the camera can be written as

Is�x; y� �1

�λζe�2ZR4

Jt�ξ; ν;Δξ;Δν�

× exp�−i

2πλζe

�ξΔξ� νΔν − xΔξ − yΔν��

× dξdνdΔξdΔν: (2)

From Eq. (5.7-4) of [13], the mutual intensity Jt is afunction of Ji as follows:

Jt�ξ;ν;Δξ;Δν� �P�ξ−

Δξ

2;ν−

Δν

2

�P��ξ�Δξ

2;ν�Δν

2

�×Ji�ξ;ν;Δξ;Δν�; (3)

with P being the transmittance of an opaque andelliptical disk centered at the origin, and P� beingits conjugate. It is defined as a chirped Gaussiansum [21]:

s e

y

x CCD

If ( , ) Ji

Is(x,y)

Spatially incoherent source

Filter Object

Jt

Fig. 1. (Color online) Theoretical setup.

A148 APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

P�ξ; ν� � 1 −

XNn�1

An exp�−Bn

R2 �ξ2 �R2ellν

2��; (4)

with An and Bn dimensionless constants given in[21], Rell the ellipticity ratio [3], and R the particleradius along the ξ-axis in the case where Rell � 1.In this paper, the case of N � 10 is chosen. Thetwo steps of the propagation of the beam are de-scribed by Eqs. (1) and (2) to any filters. In the nextsection, the particular case of a Gaussian filter willbe considered.

A. Spatially Incoherent Source Filtered bya Gaussian Filter

Van Cittert–Zernike’s theorem assumes that thesource is spatially incoherent with constant power,denoted here I0, in the plane of the filter. In ourparticular case, where the filter is Gaussian, fromEq. (29) of [22], the transmittance of the filter is de-fined by

If �α; β� � I0 exp�−

�α2

ω2α� β2

ω2β

��; (5)

where ωβ and ωα are the 1=e2 width of the filter. Bysubstituting Eq. (5) in Eq. (1), Ji can be found. FromEq. (7.4.32) of [23], we get

Ji�ξ;ν;Δξ;Δν� �C1 exp�−i

2π�ξΔξ� νΔν�λζs

�

× exp�−π2ω2

α

�λζs�2Δξ2

�exp

�−

π2ω2β

�λζs�2Δν2

�;

(6)

with C1 � I0ωβωα=ζ2s . The effective coherence area inthe plane of the object is described by Eq. (6). FromEq. (5.2-23) of [13], the modulus of the complexcoherence factor is

jμ�ξ; ν;Δξ;Δν�j � exp�−π2ω2

α

�λζs�2Δξ2

�exp

�−

π2ω2β

�λζs�2Δν2

�;

(7)

where j · j denotes the modulus. In Fig. 2, the modu-lus of the complex coherence factor is illustrated fordifferent filters. The size of the diffraction patterndepends directly on the area of the complex coher-ence factor. So it is an important feature of the beam.Note that this complex coherence factor could bemeasured [24].

The smaller the aperture of the filter, the higher thedistributionof the complex coherence factor.Anexam-pleisgiveninFig.2(a).Thisresult is ingoodagreementwith classical observations. When the filter is a point(i.e., ω very small), the beam becomes coherent,whereas if the source is not filtered (i.e.,ω very large),the beam is spatially incoherent, as one can see inFig. 2(b). Finally, for a filter with a rectangular shape(i.e., high ellipticity, ωα ≪ ωβ), the complex coherentfactor has a rectangular shape too, as one can see inFig. 2(c). This is precisely the case that interests usbecause a good image reconstruction is possible,whereas theareaof thediffractionpattern isminimal.The next sections will show this advantage.

B. Intensity Distribution Recorded by the CCD Camera

By means of the expression for Ji in Eq. (6) and thetransmittance function P of the particle in Eq. (4), therelation for Is, in Eq. (2), can be rewritten as

2 mm

2 mm

(a) (b) (c)

If

| |

Fig. 2. Illustration of the influence of the spatial filtering of an incoherent source on the coherence factor. At the top is the filter function,and at the bottom is the the modulus of the complex coherence factor (jμ�ξ; ν;Δξ;Δν�j) from the center point in the particle plane(ζs � 175 mm) (white � 1, black � 0). (a) ωβ � ωα � 10 μm, (b) ωβ � ωα � 200 μm, and (c) ωα � 10 μm, ωβ � 200 μm.


Is�x;y��C2

ZR4P�ξ−

Δξ

2;ν−

Δν

2

�P��ξ�Δξ

2;ν�Δν

2

�

×exp�−π2ω2

α

�λζs�2Δξ2�

�−i

2πλζe

�ξ

�1�ζe

ζs

�−x��

Δξ

�

×exp�−

π2ω2β

�λζs�2Δν2�

�−i

2πλζe

�ν

�1�ζe

ζs

�−y��

Δν

�×dξdνdΔξdΔν; (8)

withC2 � I0ωβωα=�λζsζe�2. From the definition of PP�

in Eq. (A1) found in Appendix A, the intensity distri-bution recorded by the camera can be expressed asthe sum of four terms:

Is�x; y� � C2fI1�x; y� − �I2�x; y� � I3�x; y�� I4�x; y�g:(9)

The first term can be expressed as follows:

I1�x; y� ��λζe�2�1� ζe

ζs

�2 : (10)

The other terms of Eq. (9) take into account the para-meters of the particle. From Appendix A, the secondterm, I2�x; y�, is equal to

I2�x; y� �XNn�1

Anπ2R2λζeRellBn

�Kαn�−1=2�Kβn�−1=2

× exp�−π2

λζe

x2

Kαn

�exp

�−π2

λζe

y2

Kβn

�; (11)

with

Kαn � π2ζeω2α

λζ2s

2641�

R2B�nζ

2s

�1� ζe

ζs

�2

ω2αBnB�

nζ2e

� iλζ2s�1� ζe

ζs

�πω2

αζe

375;

Kβn � π2ζeω2β

λζ2s

2641�

R2B�nζ

2s

�1� ζe

ζs

�2

R2ellω

2βBnB�

nζ2e


ζs

�πω2

βζe

375:

Note that I3�x; y� is the conjugate of I2�x; y� in thesame manner as Eq. (20) in [12]. Consequently, withthe assumptions Eqs. (B1), (B5), and (B19) (seeAppendix B), Eq. (11) can be rewritten as

I2�x;y�� I3�x;y� �2λζe�1� ζe

ζs

� sin

"π�x2 � y2�λζe�1� ζe

ζs

�#

× exp

"−

π2�ω2αx2 �ω2

βy2�

�λζs�2�1� ζe

ζs

�2

#

×XNn�1

AnπR2

RellBn

× exp�−

R2π2

�λζe�2Bn

�x2 � y2

R2ell

��: (12)

These assumptions basically limit the validity of thisresult to two cases. When the coherence area is verysmall, this result is not valid. The second limit isnear the far-field approximation. Equation (B1) isclose to the Fresnel number when the object is de-scribed by a sum of Gaussians.

To understand the effect of a partially spatiallycoherent source on the reconstruction process, letus note that Eq. (12) is equivalent to (see Appendix C)

I2�x;y�� I3�x;y��2λζe�1�ζe

ζs

�sin"π�x2�y2�λζe�1�ζe

ζs

�#

×exp

"−

π2�ω2αx2�ω2

βy2�

�λζs�2�1�ζe

ζs

�2

#

×F−1

��λζe�2circ

��x2a�R2elly

2a�1=2

R1

��x;y�;

(13)

where the function circ is the circular function de-fined by circ�r� � 1 if 0 ≤ r ≤ 1 and 0 otherwise,F−1 is the inverse Fourier transformation operator,and R1 � R=�λζe�. The understanding of the fourthterm, I4�x; y�, is not straightforward [see Eq. (A9)in Appendix A], but it does not influence the recon-struction in the far-field approximation [12]. It can becompared to the last term of the case of totalcoherence; cf. [25]. So the characteristics of the holo-gram depend on the 3D location, the diameter ofthe particle, and the complex coherence factor μin Eq. (7).

The vergence of the beam in the object plane isdescribed by the phase front variation [26,27]. Theradius of curvature of a spatially partially coherentbeam is obtained from the complex part of the mu-tual intensity. From [18], Section 3, part C, when theradius of curvature is infinity, the complex part of themutual intensity is equal to unity. So, in our case, ifwe assume that the beam is collimated, then thecomplex part of the mutual intensity, i.e.,exp�−i�2π�ξΔξ� νΔν�=λζs�� in Eq. (6), becomes uni-tary. So, Eq. (9) can be written as


Is�x; y� ≈ C2 · �λζe�2 − 2C2

· λζe sin�π�x2 � y2�

λζe

�exp

"−

π2�ω2αx2 � ω2

βy2�

�λζs�2#

× F−1

��λζe�2circ

��x2a �R2elly

2a�1=2

R1

��x; y�:

(14)

Using a collimated beam allows us to easily compareexperimental studies and mathematical results.Equation (14) is quite similar to the description ofthe intensity distribution for an elliptical particle il-luminated by a coherent source [28]. The difference(except the multiplicative term linked to the energy)is the Gaussian function depending on the spatial co-herence of the source, which limits the width of thediffraction pattern. It is worth noting that the case ofa spatially coherent illuminating beam is recoveredin the particular case ωα � ωβ ≈ 0.

3. Numerical Experiment and Experimental Results

Figure 3 shows the experimental setup. A greenlight-emitting diode (Luxeon LXML-PM01-0070) il-luminates the lens (L1) of focal length 25 mm, whichfocuses the beam on the filter. The field in the planeof the filter is not strictly incoherent, as assumed,since it represents a secondary source. However, ifthe output numerical aperture of L1 is large enough,the effective coherence area is much smaller than thearea of the filter. In this case the assumption ofincoherent illumination is valid in practice [12].The filter is composed of a variable slit with micro-metric accuracy in one direction and a fixed slit inthe perpendicular direction. The aperture of thefilter controls the complex coherence factor. Thebeam is collimated by a second lens (L2) with a focallength of 50 mm. The intensity distribution is re-corded by a CCD sensor (BM-500 GE of JAI 2456 ×2056 square pixels of 3.45 μm). The experimental re-sults are compared with the theoretical model inFigs. 4 and 5.

Here, one opaque disk of diameter 70 μm only isused and two slits are tested. The first slit has a sizeof 20 × 200 μm and the other of 50 × 200 μm. In thefirst case, the effective coherence width (here definedas the Gaussian factor) is 1.8 mm along the ξ-axisand 176 μm along the ν-axis, and for the secondcase, 353 μm along the ξ-axis and 176 μm along theν-axis. The transverse intensity profiles obtainedfrom Figs. 4(a) and 4(b) are presented in Fig. 4(c).In the same way, the transverse intensity profiles

LE

D

se

y

x CCD

If ( , ) Ji

Is(x,y)FilterLens Lens

Disk

Jt

L1 L2

Fig. 3. (Color online) Experimental setup. For examplespresentedFigs. 4 and 5 the parameters are filter-L2 �f � 50 mm� �50 mm, L2-Disk � 125 mm, Disk �R � 35 μm�-CDD � 115 mm.

Fig. 4. Comparison between the analytic (a) and dotted curves of (c) and the experimental (b) and solid curves of (c), withωβ � 200 μm, ωα � 20 μm, ζe � 115 mm, ζs � 175 mm, R � 35 μm, and λ � 532 nm.


from Figs. 5(a) and 5(b) are presented in Fig. 5(c). Aswe can see in these figures, the theoretical develop-ments are in good agreement with the experimentalresults and, therefore, our theoretical model isvalidated.

To show the feature of a source with a rectangularcomplex coherence factor, a comparison is made inFig. 6. A target of oil (Di-Ethyl-Hexyl-Sebacate) dro-plets (diameter ≈300 μm) is illuminated by three dif-ferent collimated sources: a common laser source[Fig. 6(a)], a spatially partially coherent source witha circular complex coherence factor [Fig. 6(b)], and aspatially partially coherent sourcewith a rectangularcomplex coherence factor [Fig. 6(c)]. This situationcan be seen in diphasic flows, for example. TheCCD is placed 440 mm from the droplets. From theintensity profile shown in Fig. 6(a), it is difficult toidentify the signal from the background noise whena laser source is used. The intensity field recordedby the CCD camera in the case of the circular complexcoherence factor [Fig. 6(b)] shows that the signal-to-noise ratio is increased even if the high frequencieshave been largely filtered. By using a rectangularcomplex coherence factor, the noise is reduced andthe information along the x-axis is kept, as we cansee in Fig. 6(c). Thus, the signal-to-noise ratio ofthe diffraction pattern is increased with this kind ofsource.

Now, to reconstruct particle images, the correlationof the hologram intensity with a chirplet function isobtained. The next section is devoted to this task.

4. Reconstruction by Correlation with a ChirpletFunction

Digital reconstruction of particle image can be rea-lized in many ways. Generally, the Fresnel integralis used. However, other operators are also consid-ered, such as the fractional Fourier transformation[28] or the wavelet transformation [25]. Since theparticles (or the disk) have circular symmetry, corre-lation and convolution are defined in the same way.We can therefore say that these transformations areconvolution or correlation operators. In this paper,the reconstruction of a particle image is carried outby correlation with real Gaussian chirplets. Let usdenote by Ca the correlation between the intensitydistribution on the hologram plane Is�x; y� and aGaussian chirplet [29]:

Ca�xa;ya��1

a2

ZR2Is�x;y�sin

��xa −x�2��ya −y�2a2

�dxdy:

(15)

In our case, Is�x; y� ≈ C2�I1�x; y� − I2�x; y� − I3�x; y��, sothe correlation becomes

Ca�xa;ya�� πC2 · �ζeλ�2

−C2 ·2λζe

a2

ZR2FO�x;y�sin

�π�x2�y2�

λζe

�

×sin�x2 −2xxa�x2a�y2 −2yya�y2a

a2

�dxdy;

(16)

Fig. 5. Comparison between the analytic (a) and dotted curves of (c) and the experimental (b) and the solid curves of (c), withωα � 50 μm, ωβ � 200 μm, ζe � 115 mm, ζs � 175 mm, R � 35 μm, and λ � 532 nm.


with

FO�x; y� � exp�−

π2�ω2αx2 � ω2

βy2�

�λζs�2�

× F−1

��λζe�2circ

��x2a �R2elly

2a�1=2

R1

��x; y�:

(17)

The reconstruction of the particle image is based onthe fact that the linear chirp of the chirplet function,controlled by the factor a, cancels the linear chirpcontained in the hologram Is. To do this, we know

from [29] that the second or the third term ofEq. (9), containing the linear chirp, allow a digital re-focusing on the image of the particles from recordedholograms. To cancel the quadratic phase (i.e.,the linear chirp), the optimal order a, denoted aopt,is defined by

aopt ��λζeπ

�1=2

: (18)

By recalling that sin�x� � �exp�ix� − exp�−ix��=2i, itis possible to simplify Eq. (16):

Fig. 6. Holograms of oil drops and their profiles (ζs � 200 mm, ζe � 440 mm, and R ≈ 150 μm) illuminated by (a) a coherent source, (b) aspatially partially coherent source with a circular complex coherence factor (ωα � ωβ � 100 μm), and (c) a spatially partially coherentsource with a rectangular complex coherence factor (ωα � 10 μm, ωβ � 200 μm).


Ca�xa;ya��πC2 ·�λζe�2−πC2·F �FO�x;y��

xaπa2

opt;ya

πa2opt

�

·cos�π�x2a�y2a�

λζe

��πC2 ·

ZR2FO�x;y�

×cos�2�x2�y2�−2�xax�yay��y2a�x2a�

λζe

�dxdy:

(19)

The first term is the background in the reconstructedimage. The second term is the reconstruction ofthe image of the object, and the last describes the dif-fraction pattern due to the twin image located ata distance 2ζe from the actual plane �xa; ya� [25].Equation (19) exhibits a bidimensionnal Fouriertransformation with the dilatation factor aopt. Then,with Eq. (18), the reconstructed image of the particleis F �FO�x; y��xa=λζe; ya=λζe�. So, the image diameteris directly linked to the order of the chirplet function.The cosine function is without effect [28]. Using theconvolution theorem of Fourier analysis and usingthe similarity theorem, the second term of Eq. (19)becomes

F �FO�x;y��xaλζe

;yaλζe

��λζe�2circ

��x2a�R2elly

2a�1=2

R

�

��ζ2s

ζ2e

1πωβωα

exp�−x2aζ2sζ2eω

2α

−y2aζ2sζ2eω

2β

�;

(20)

where �� denotes the bidimensional convolution.Equation (20) shows that the reconstructed imagecan be described by the convolution of a Gaussianwith the object function. So the image of the parti-cle is not reconstructed in the broad sense of theterm. The quality of the reconstruction to retrievethe dimension (i.e., the diameter) of the particledepends on the width of the Gaussian comparedto the diameter of the particle. However, in thecase of a coherent source, i.e., ωα → 0 and ωβ → 0,the Gaussian function becomes a Dirac impulse[30], i.e.,

limωα→0ωβ→0

ζ2sζ2e

1πωβωα

exp�−x2aζ2sζ2eω

2α

�� δ�xa; ya�; (21)

where δ�xa; ya� is the Dirac delta impulse. Then,in this case, the object function is recovered. Inthe particular case where the complex coherencefactor is rectangular and very large along one axis,i.e., ωβ ≫ ωα → 0, and using Eq. (21), Eq. (20)becomes

F �FO�x; y��xaλζe

;yaλζe

�� λζe�2circ

��x2a �R2elly

2a�1=2

R

�

� ζsζe

1

ωβπ1=2 exp

�−y2aζ2sζ2eω

2β

�:

(22)

So along the xa-axis, the diameter of the particle isrecovered and along the ya-axis, a gate function isconvolved with a Gaussian.

Fig. 8. Profiles along xa-axis of Fig. 7(a) (dotted curve) andFig. 7(b) (solid curve).

Fig. 7. Reconstruction at ζe � 115 mmof Fig. 4 and its equivalentwith a coherent source. The ellipticity is due to the source, hereRell � 1.

Fig. 9. Reconstruction at ζe � 115 mm of the experimentalholograms of Figs. 4(b) [panel (a)] and 5(b) [panel (b)].


Figure 7(a) is the digital reconstruction of the ho-logram of Fig. 4(a) by recalling that it is the casewhere the complex coherence factor is rectangular,and Fig. 7(b) is the digital reconstruction of thehologram of the particle with the same parameters�λ; ζe� but with a coherent source. As one can see fromFig. 8, the profile along the xa-axis of these two recon-structions gives the same particle diameter.

The image of the object is well reconstructed alongthe x-axis because the thin shape of the rectangularcomplex coherence factor is along the y-axis. Fromthis example, the reconstruction of experimental ho-lograms is illustrated in Fig. 9. Figures 9(a) and 9(b)are obtained from the holograms of Figs. 4(b) and 5(b)with a � 0.14 mm. As one can see again from thesefigures, the shape of the reconstructed images

confirms our theoretical predictions: along the y-axisthe diameter is not recovered, and along the x-axisthe higher the degree of coherence, the better thereconstructed diameter.

5. Conclusion

We have described the digital in-line holograms ofopaque and elliptical particles illuminated by a beamwith a complex coherence factor. For this, a newanalytical solution to the problem of the scalar dif-fraction of a spatially partially coherent beam hasbeen proposed. This kind of source also allowsgreatly limiting the width of the diffraction pattern.This can be an advantage for avoiding the overlap-ping of the hologram patterns that can occur in thecase of high-density particle fields. A larger experi-mental study would be necessary to know the quan-titative impact of this kind of source on the measureof the dense flow. Then, we have demonstratedthat the reconstruction process remains possible asin the case of a coherent beam. In particular, whenthe filtering of the source is rectangular, the 3Dlocation and the diameter of an opaque disk can befound. Finally, digital in-line experiments have beencarried out. A good agreement between the simu-lated intensity distributions and the experimentalresults has been demonstrated. We have shownthat the reconstructions could be performed by

using a correlation with a real Gaussian chirplet.This confirms the theoretical developments andpredictions.

Appendix A: Integral Calculation

The transmittance function in the plane of anopaque and elliptical disk can be described bysums of Gaussians [Eq. (4)], from Eq. (5.7-3) of[13]. To calculate the propagation of the mutualintensity, it is necessary to know the productbetween the transmitted function P along the coordi-nates �ξ1; ν1� and its conjugate P� along the coordi-nates �ξ2; ν2�: so when Δξ � ξ2 − ξ1, ξ � �ξ2 � ξ1�=2,Δν � ν2 − ν1, and ν � �ν2 � ν1�=2, the product PP�

becomes

P�ξ−

Δξ

2;ν−

Δν

2

�P��ξ�Δξ

2;ν�Δν

2

��1−

XNn�1

An exp�−Bn

R2

�ξ2−ξΔξ�Δξ2

4�R2

ellν2−R2

ellνΔν�R2ellΔν2

4

��

−

XNn�1

A�n exp

�−B�

n

R2

�ξ2�ξΔξ�Δξ2

4�R2

ellν2�R2

ellνΔν�R2ellΔν2

4

��

�XN2

n�1

En exp�−Cn

R2 �ξ2�R2ellν

2��−Dn

R2 �ξΔξ�R2ellνΔν�− Cn

4R2 �Δξ2�R2ellΔν2�

�; (A1)

with Cn�10�m−1��q�Pm�N

m�1

Pq�Nq�1 Bm�B�

q, En�10�m−1��q

�Pm�Nm�1

Pq�Nq�1 AmA�

q, and Dn�10�m−1��q�P

m�Nm�1

Pq�Nq�1

B�q−Bm. From Eq. (A1) the intensity distribution

recorded by the camera Eq. (8) can be expressed asthe sum of four terms:

Is�x; y� � C2fI1�x; y� − �I2�x; y� � I3�x; y�� I4�x; y�g:(A2)

In calculating, it is important to note that the realpart of Bn is strictly positive. The equations areGaussian so their integrals are directly found fromEq. (7.4.32) of [23]. The first step is an integrationalong Δξ and Δν, and the second step is an integra-tion along ξ and ν, so

I1�x; y� ��λζs�2πω2

α

�1=2��λζs�2ω2βπ

�1=2

×�

πζ2eω2α

�ζe � ζs�2�1=2� πζ2eω

2β

�ζe � ζs�2�1=2

� �λζe�2�1� ζe

ζs

�2 : (A3)

From Eq. (8), I2�x; y� is


I2�x; y� �ZR4

XNn�1

An exp�ξ2�−Bn

R2

�� ξ

�Δξ

�Bn

R2 − i2πλζe

�1� ζe

ζs

��

× exp�ν2�−R2

ellBn

R2

�� ν

�Δν

�R2

ellBn

R2 − i2πλζe

�1� ζe

ζs

��

× exp�−Δξ2

�π2ω2

α

�λζs�2� Bn

4R2

��exp

�iΔξ

2πλζe

x�

× exp�−Δν2

�π2ω2

β

�λζs�2� R2

ellBn

4R2

��exp

�iΔν

2πλζe

y�dξdνdΔξdΔν: (A4)

The Gaussian integrals in ξ and ν give

I2�x;y� �ZR2

XNn�1

An

�πR2

RellBn

�

× exp�−Δξ2

Kαn

λζe�Δξ

�j2πλζe

x��

× exp�−Δν2

Kβn

λζe�Δν

�j2πλζe

y��

dΔξdΔν; (A5)

with

Kαn � π2ζeω2α

λζ2s

2641�

R2B�nζ

2s

�1� ζe

ζs

�2

ω2αBnB�

nζ2e


ζs

�πω2

αζe

375;

Kβn � π2ζeω2β

λζ2s

2641�

R2B�nζ

2s

�1� ζe

ζs

�2

R2ellω

2βBnB�

nζ2e


ζs

�πω2

βζe

375:

It is now possible to integrate along Δξ and Δν; onlyone condition is necessary. The real part of the factorsof Δξ and Δν square must be negative; this is validbecause R�Bn� > 0, so

I2�x; y� �XNn�1

Anπ2R2λζeRellBn

�Kαn�−1=2�Kβn�−1=2

× exp�−π2

λζe

x2

Kαn

�exp

�−π2

λζe

y2

Kβn

�: (A6)

I3�x; y� can be found using the same method but withthe conjugate factors, and so we have

I3�x; y� � I2�x; y��: (A7)

The last integral to calculate is I4�x; y�; it is yetanother integral of Gaussians in ξ and ν:

I4�x;y��ZR2

XN2

n�1

EnR2π

RellCnexp�KvixnΔξ2�exp

�iΔξ

2πxλζe

�

×exp�KviynΔν2�exp�iΔν

2πyλζe

�dΔξdΔν; (A8)

with the factors

Kvixn � C�n

4CnC�nR2

×

24D2

n � i4πDnR2

�1� ζe

ζs

�λζe

�−4π2R4

�1� ζe

ζs

�2

�λζe�2

35

−π2ω2

α

�λζs�2−

Cn

4R2

and

Kviyn � C�n

4R2R4ellCnC�

n

×

24R2

ellD2n � i

4R2ellR

2Dnπ�1� ζe

ζs

�λζe

�−4π2R4

�1� ζe

ζs

�2

�λζe�2

35 −

π2ω2β

�λζs�2−R2

ellCn

4R2 :

So with the integration along Δξ and Δν, Eq. (A8)becomes

I4�x; y� �XN2

n�1

EnπR2

RellCn

�−π

Kvixn

�1=2�

−π

Kviyn

�1=2

× exp�

π2

�λζe�2x2

Kvixn

�exp

�π2

�λζe�2y2

Kviyn

�: (A9)

Appendix B: Simplification

A. Simplification of jK αnj2 and jK βnj2In Eq. (A6), it is not straightforward to understandthe involvement of the different parameters. To


understand this function, several simplifications arepossible. In the first, in Eq. (A6), we can simplify themodulus of Kαn and Kβn.

1. Imaginary Part

The first assumption is thatI24R2π2B�

n

�1� ζe

ζs

�2

R2ellBnB�

nζ2e

35 ≪

π�1� ζe

ζs

�λζe

: (B1)

When the sum of 10 factors is used to express theparticle transmitted function, the extreme valueof the dimensionless constant given in [21] impliesthat

ℑ�

B�n

BnB�n

� < 0.15; (B2)

where I denotes the imaginary part. If the partneglected is less than 10% of the final value, the as-sumption becomes

0.15R2π2B�

n

�1� ζe

ζs

�2

R2ellBnB�

n�ζe�2< 0.1

π�1� ζe

ζs

�λζe

: (B3)

Another condition on the radius of the object is

RRell

<

"λζe

1.5π�1� ζe

ζs

�#; (B4)

for example, with ζs � ζe � 100 mm, λ � 635 nm andR2

ell � 1, R will be much less than 80 μm.

2. Real Part

The modulus depends only on the imaginary part if itis much more than the real part, so by the same logicas in the previous assumption, it is assumed that

RRell

≪

"λζe

0.25π�1� ζe

ζs

�#1=2

; (B5)

because

R�

B�n

BnB�n

� < 0.25; (B6)

and

ω2βπ

2

�λζs�2≪

π�1� ζe

ζs

�λζe

; (B7)

ωβ <

240.1 λζ2s

�1� ζe

ζs

�πζe

351=2

: (B8)

With our previous numerical values (A1),ωβ < 63 μm. These assumptions allow rewritingthe modulus of Kαn and Kβn as

jK inj � π

�1� ζe

ζs

�; (B9)

with i � α, β. In partially coherent imaging, Eq. (B8)is often valid, but in the situation of low coherence itcannot be used. When the sources are less coherent,another assumption without any dependence on Bncan be made.

B. Square Root Approximation

With the same assumptions, the square roots can berewritten. Remember that a complex square root isequal to

�a� ib�1=2 � ��a2 � b2�1=2 � a

2

�1=2

� isign�b��a2 � b2�1=2 − a

2

�1=2�:

With this assumption, the factor of Eq. (A6) becomes

πλζe�Kαn�−1=2�Kβn�−1=2 � iλζe�

1� ζeζs

� : (B10)

The sign of the roots can not be known, but simula-tions, compared to the coherence case, show that it isvalid when the product of the two roots is positive.

C. Change of Bn

Now, Eq. (A6) can be rewritten by noting

An

Bn� �a� ib�; (B11)

exp�−y2

�R2πiℑ�Bn�

R2ellB

�nBn�λζe�2

�− x2

�R2πiℑ�Bn�B�

nBn�λζe�2��

� �c� id�; (B12)

iλζe�

1� ζeζs

� � ip; (B13)

exp

8<:y2

24 iπ

λζe�1� ζe

ζs

�35� x2

24 iπ

λζe�1� ζe

ζs

�359=; � �e� if �;

(B14)

and a real factor,


C3 � π2R2ζeRell

exp

"−y2

R2πR�Bn�RellB�

nBn�λζe�2− x2

R2πR�Bn�B�

nBn�λζe�2#

× exp

24− π2�ω2

αx2 � ω2βy

2��λζs�

�1� ζe

ζs

�35; (B15)

so Eq. (A6) becomes

I2 �XNn�0

C3ip�a� ib��c� id��e� if �: (B16)

If c=d ≫ a=b and c=d ≫ b=a,

I2 �XNn�0

C3ip�e� if ��a� ib��c� id�

�XNn�0

C3�e� if ��−p�bc� da� � pi�ca − bd��

≈

XNn�0

C3�e� if ��−p�bc − da� � pi�ca� bd��; (B17)

so c� id ≈ c − id; the assumptions are valid if(max�a=b� ≈ max�b=a� < 50):

cd> 500 �> tan

�y2�

R2π2ℑ�Bn�R2

ellB�nBn�λζe�2

�

� x2�R2π2ℑ�Bn�B�

nBn�λζe�2��

<1

500; (B18)

y2�

R2π2ℑ�Bn�R2

ellB�nBn�λζe�2

�� x2

�R2π2ℑ�Bn�B�

nBn�λζe�2�≪ 1; (B19)

R2

R2ell

<0.07�λζe�2

x2max: (B20)

For an image of a particle of 0.5mm (xmax � 0.25 mm)with the previous parameters, R < 67 μm.

D. Simplification of I2 �x ; y� � I3�x ; y�Noting Tn � An exp�−Bn=R2�x2 � R2

elly2�� and PN

n�0

Tn �PNn�0�an � ibn�, then from Eqs. (A7) and

(B17), Eq. (A2) has the form

I2�x; y� � I3�x; y� � −ip�e� if �F−1

XNn�0

Tn

!

� ip�e − if �F−1

XNn�0

T�n

!: (B21)

The object is real, which implies that

Fig. 10. Comparison between the calculations without (a) and the solid curve of (c) and with (a) and the dotted curve of(c) simplification. ω � 80 μm, ζs � 100 mm; ζe � 100 mm; R � 40 μm.


XNn�0

ibn � 0 �XNn�0

−ibn �>XNn�0

Tn �XNn�0

T�n

�> F−1

�XNn�0

Tn

�� F−1

�XNn�0

T�n

�: (B22)

Consequently, from Eq. (B21), the sum of I2 and I3becomes I2�x; y� � I3�x; y� � C3F−1�PN

n�0 Tn��2pf �:

I2�x; y� � I3�x; y� �2λζe�1� ζe

ζs

� sin

"π�x2 � y2�λζe�1� ζe

ζs

�#

× exp

"−

π2�ω2αx2 � ω2

βy2�

�λζs�2�1� ζe

ζs

�2

#

×XNn�1

AnπR2

RellBn

× exp�−

R2π2

�λζe�2Bn

�x2 � y2

R2ell

��:

(B23)

Figure10compares theresultwithout theassumptionand the result with the assumption, for ω � 80 μm,ζs � 100 mm, ζe � 100 mm, and R � 40 μm.

Appendix C: Fourier Transform ofPNn�1 AnπR2=RellBn exp�−R2π2=�λζe �2Bn�x2 � y2=R2

ell��If it is taken into account that

A�XNn�1

AnπR2

RellBnexp

�−

R2π2

Bn�λζe�2�x2� y2

R2ell

��; (C1)

the Fourier transformation of this function is

F �A��xa; ya� �ZR2

XNn�1

AnπR2

RellBn

× exp�−

R2π2

Bn�λζe�2�x2 � y2

R2ell

��

× exp�−2πixax� exp�−2πiyay�dxdy: (C2)

With two Gaussian integrals, the Fourier transfor-mation becomes

F �A��xa;ya��XNn�1

AnπR2

RellBn

�πBn�λζe�2

R2π2

�1=2

×�R2

ellπBn�λζe�2R2π2

�1=2exp

�−x2a

��λζe�2R2

�Bn

�

×exp�−y2a

��λζe�2R2

�BnR2

ell

�; (C3)

with R1 � R=λζe. Equation (C3) can be rewritten

TF�A��xa; ya� � �λζe�2XNn�1

An exp�−Bn

R21

�x2a �R2elly

2a��:

(C4)

Here, An and Bn are the dimensionless constantsgiven in [21], so by definition

B � �λζe�2XNn�1

An exp�−Bn

R21

�x2a �R2elly

2a��

� �λζe�2circ��x2a � y2aR2

ell�1=2R1

�; (C5)

and thus A and B are directly linked by

A � F−1�B��x; y�: �C6�

This work was carried out as part of a Ph.D. fundedby the EDF. The authors thank F. David, J. M. Dorey,the EDF R&D, Mécanique des Fluides, Energies,Environnement, Chatou, France, for supporting thisresearch.

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