spatial coherence in strongly scattering media

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Pierrat et al. Vol. 22, No. 11 /November 2005 /J. Opt. Soc. Am. A 2329

Spatial coherence in strongly scattering media

Romain Pierrat, Jean-Jacques Greffet, and Rémi Carminati

Laboratoire d’Énergétique Moléculaire et Macroscopique, Combustion, École Centrale Paris, Centre National de laRecherche Scientifique, 92295 Châtenay-Malabry cedex, France

Rachid Elaloufi

Department of Computer Science, University College London, London WC1E6BT, United Kingdom

Received February 10, 2005; revised manuscript received April 22, 2005; accepted April 28, 2005

We study the spatial coherence of an optical beam in a strongly scattering medium confined in a slab geometry.Using the radiative transfer equation, we study numerically the behavior of the transverse spatial coherencelength in the different transport regimes. Transitions from the ballistic to the diffusive regimes are clearlyidentified. © 2005 Optical Society of America

OCIS codes: 030.1640, 170.5280, 170.7050, 290.4210.

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. INTRODUCTIONight transport in the multiple-scattering regime has be-ome a very active field owing to the rapid development ofiomedical imaging techniques using visible or near-nfrared light.1 In many cases, modeling the intensity pro-le (in space and time) is sufficient in order to analyze the

mages or to improve the experimental setups. Most of theractical approaches rely on the diffusion approximation,hich is valid at large length and time scales (comparedith the transport mean-free path and the collision

ime).2

In some cases, knowledge of the spatial coherence prop-rties of light propagating through a turbid medium is ofrimary importance, e.g., to characterize the beamuality,3 to analyze speckle correlations,1,3 or to model dy-amic experiments based on light fluctuations induced byhe motion of the scatterers, as in diffusing-wavepectroscopy.4 The spatial coherence length can be used toharacterize the quality of a beam transmitted through aurbulent atmosphere.5–7 The beam quality and the sizef the spatially coherent illuminating spot are also keyarameters in optical coherence tomography in mediahere multiple scattering plays a role.8

In this work, we study the spatial coherence of lightropagating through a slab of scattering medium, whoseroperties are similar to that of biological tissues. We in-roduce a practical way to compute the field spatial corre-ation function and the spatial coherence length, both in-ide and outside the medium (in transmission oreflection). We use a numerical solution of the radiativeransfer equation (RTE) to calculate the specific intensity��r ,u� in a slab geometry, from which we derive theross-spectral density and the spatial coherence length.ith this approach, we study the evolution of the spatial

oherence in the scattering medium. In particular, we dis-uss the crossover between the ballistic and the diffusiveegimes.

1084-7529/05/112329-9/$15.00 © 2

. TRANSPORT EQUATION FOR FIELDORRELATION

n this section, we recall the relationship between thepecific intensity and the cross-spectral density. Differentorms of this relationship have been derivedreviously.3,9–14 We introduce the RTE as a transportquation for the specific intensity, defined as the three-imensional Wigner transform of the field.

. Basic Concepts of Spatial Coherence Theorye consider a scattering medium, described statistically

y a random process and illuminated by a monochromaticeld of frequency �. The scattered field u�r , t�=u�r�exp�−2i��t� is itself a random variable. We use the sca-

ar approximation so that we neglect polarization effects.he second-order spatial coherence properties of the fieldre characterized by the cross-spectral density15

W�r1,r2,�� =�−�

+�

�u�r1,t�u*�r2,t + ��exp�2i��d�, �1�

here * denotes the complex conjugate and �¯� denotesn average over an ensemble of realizations of the scat-ering medium. The field is assumed to be statisticallytationary, so that W is independent of time. W is a mea-ure of the field spatial correlation at a given frequency.n a scattering random medium, an exact transport equa-ion for W can be derived from the wave equation.3,16 Thisquation, known as the Bethe–Salpeter equation, is an in-egral equation whose kernel is difficult to compute ex-licitly. We will therefore introduce an approximation tohis transport equation, known as the RTE.

In the following, we introduce the new variables r�r1+r2� /2 and �=r2−r1, and we omit the frequency � for

he sake of brevity. If the random medium is statisticallyomogeneous and isotropic, the cross-spectral density de-ends on �= �r −r � only. One usually defines the spatial
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2330 J. Opt. Soc. Am. A/Vol. 22, No. 11 /November 2005 Pierrat et al.

oherence length lcoh as the FWHM of the degree of spa-ial coherence w��=W�� /W�0�. If the beam is spatiallyoherent, the correlation function is constant, and thuscoh tends to infinity. By contrast, if the beam is incoher-nt, the correlation function is null for ��0, and thus lcohs null. In other situations, the beam is said to be partiallyoherent.

. Link between the Specific Intensity and the Cross-pectral Densityn this subsection, we introduce the specific intensity,hich can be defined as follows2,3,16,17:

W�r,�� =�4�

I��r,u�exp�iku · ��d�, �2�

here k=2� /� is the wavenumber in the medium. Notehat Eq. (2) can be inverted to obtain the specific intensityn the form of a Wigner transform of the field. One obtains

alther’s formula9 as shown in Appendix A. Startingrom the Bethe–Salpeter equation, it has been shown byany authors that I� verifies the RTE.2,3,16–18 The RTE isBoltzmann-type transport equation, which, in the

teady-state regime, can be written as

u · �rI��r,u� = − �a + s�I��r,u�

+s

4��

4�

p�u,u��I��r,u��d��. �3�

n this equation, s is the scattering coefficient, and a ishe absorption coefficient. The associated scattering andbsorption mean-free paths are ls=s

−1 and la=a−1. We

lso define the transport mean-free path l*= ls / �1−g�,here g= �cos � is the average cosine of the scatteringngle.2 p�u ,u�� is the phase function. l* can be inter-reted as the average distance for collimated light to be-ome isotropic because of scattering. The extinction coef-cient is given by e=s+a, and a=s /e is the albedo.This equation was first introduced in astrophysics as a

henomenological equation.19 It was derived from a localadiative energy balance, in which the specific intensitylays the role of a local and directional energy flux. Al-hough the specific intensity, defined in radiometry, is al-ays positive, the expression defined in Eq. (2) can takeegative values. It can be shown that in the limit �→0he specific intensity is always positive and can be inter-reted as a local energy flux.15 Nevertheless, this limit isot necessary for Eqs. (2) and (3) to be valid.3,16 The spe-ific intensity takes negative values when the wave na-ure of the field becomes relevant, as discussed in Ref. 3.ittlejohn and Winston20,21 have shown that the energyux computed using the Wigner transform is always posi-ive. Moreover, the existence of negative values of I� doesot prevent the RTE from being considered a radiometricransport equation.

. PARTICULAR CASE: SLAB GEOMETRY. Geometry of the Systemhe system is described in Fig. 1. It consists of a statisti-ally homogeneous scattering slab of thickness L, which isnfinite along the x and y axes. We will refer to this me-

ium as medium 2. The medium z�0 is denoted by 1, andhe medium z�L is denoted by 3. The real parts of theefractive indices are n1, n2, and n3, respectively, for me-ia 1, 2, and 3. The angles and denote the sphericaloordinate angles. The slab is illuminated from the left byplane wave at normal incidence. In this study we have

onsidered n1=n3=1 and n2�1.

. Cross-Spectral Density in a Slab Geometryhe geometry is translationally invariant along x and ynd is rotationally invariant around the z axis. Assuminghat the phase function depends only on u ·u�, the specificntensity depends on z and =cos only. Moreover, we arenterested in the cross-spectral density in a plane perpen-icular to the direction of propagation (the x–y plane inur case). So the vector � in Eq. (2) belongs to a plane zconstant. In this plane, the cross-spectral density de-ends only on �. Equation (2) becomes

W�z,�� =� I��z,, �exp�ik� · u�dd . �4�

ince � ·u=�u� cos where u� is the projection of u onhe x–y plane, we recognize a Bessel function of order 0efined by

J0�x� =1

2��

0

2�

exp�ix cos �d . �5�

e can cast the cross-spectral density in the form

W�z,�� =�−1

+1

I��z,�J02�

��1 − 2�d, �6�

here u�=1−2. We used the fact that I��z , , � doesot depend on , and we introduced I��z ,��0

2�I��z , , �d =2�I��z , , �. Equation (6) allows us toerive the cross-spectral density in a plane z=constantrom the specific intensity calculated from the RTE. Ow-ng to the linearity of the integral, we can easily split theross-spectral density corresponding to the specific inten-ity propagating in the forward direction (i.e., �0) and

ig. 1. Geometry of the system. Medium 2 is a scattering slab ofhickness L illuminated from the left by a plane wave at normalncidence. The notations used to decompose the specific intensitynto ballistic and diffuse components propagating in the forwardnd backward directions are indicated.

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he one corresponding to the specific intensity in the back-ard direction (i.e., �0).Two particular cases are important to analyze the nu-erical results in Section 4. For a collimated specific in-

ensity [i.e., I��z ,�=I��z��±1�], the spatial correlationunction is constant, and the coherence length tends to in-nity. We retrieve the well-known coherence property ofhe collimated component, which is often called the coher-nt component. For an isotropic specific intensity [i.e.,��z ,�=I��z�], Eq. (6) yields

W�z,�� = 2I��z�sinc2��

�� . �7�

he details of calculation are given in Appendix B. In thisase, the coherence length does not depend on z, and theWHM is given by

lcoh 0.3�. �8�

his expression for the coherence length is a characteris-ic of isotropic radiation. The same result is obtained forlackbody radiation.15 As given in Appendix B, this results also valid when one considers only the propagation ofight in the forward direction (i.e., by applying the opera-or �0

+1 . . .d) or in the backward direction (i.e., by apply-ng the operator �−1

0 . . .d).The purpose of this paper is to study the transition be-

ween these two regimes in a strongly scattering medium.

. Radiative Transfer Equationhe transport equation for I� in a steady-state regime forslab geometry can be written as19

�

��I��,� = − I��,� +

a

2�

−1

+1

p�,��I��,��d�. �9�

n this equation, both the specific intensity I� and thehase function p are integrated over the azimuthal angle. We have introduced the dimensionless distance �=ez.he optical thickness is given by L*=eL.The validity of Eq. (9) in slab geometries has been stud-

ed by comparison with rigorous electromagnetic numeri-

ig. 2. Degree of spatial coherence w�� ,�� inside a semi-infinitenterface � and the distance � between the two observation pointunction with g=0.90 and an albedo a=0.98. (a) Circles correspiffuse component only and is reported on Fig. 3.

al solutions of the multiple-scattering problem.22 It haseen shown that it is valid for both thick and thin slabseven for thickness of the order of one wavelength). Equa-ion (9) can be solved numerically using a discrete-rdinate method.23,24 The boundary conditions at the slabnterfaces are taken into account rigorously using Fresneleflection and transmission factors. In the following, thexpression of the specific intensity in the slab is calcu-ated from Eq. (9).

. NUMERICAL RESULTS. Collimated and Diffuse Beamshe spatial coherence of a collimated beam is very large.y contrast, an isotropic specific intensity yields a coher-nce length of 0.3�. It will be useful to split the specificntensity into two components, collimated and diffuse, inrder to study their contribution to the spatial coherence.

First, we solve the RTE in a slab geometry to obtain�� ,�. Then, using Eq. (6), we compute the cross-pectral density W�� ,��, whose FWHM yields the trans-erse coherence length. We see in Fig. 2(a) that when �ncreases the coherence length decreases. The coherenceength is large close to the medium interface due to theontribution of the ballistic component (propagating inhe forward direction). The contribution of the ballisticomponent decreases exponentially with increasing � dueo scattering and absorption. We also represent in Fig.(b) the degree of spatial coherence (i.e., the normalizedeld correlation function) w�� ,�� inside the scattering me-ium 2, versus �, and the normalized distance from theedium interface �=ez. In this example, the full radia-

ion field is taken into account (with both a forward fieldropagating toward z�0 and a backscattered field). Weee that, at a large depth inside the medium, the correla-ion function exhibits the sinc �2�� /�� structure charac-eristic of isotropic radiation. In this case, the transportegime is diffusive.

We also need to separate the specific intensity propa-ating in the forward direction (i.e., z�0) from the one

ering medium versus the normalized distance from the mediumscattering medium is described by a Henyey–Greenstein phase�c��wc, and the arrow indicates an example of FWHM for the

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ropagating in the backward direction (i.e., z�0). In thisection, we omit the subscript � for the sake of simplicity.e define

I��,� = Ic+�� − 1� + Ic

−�� + 1� + Id��,�, �10�

here

Id+��,� = Id��, � 0�,

Id−��,� = Id��, � 0�. �11�

hese notations are illustrated in Fig. 1. Concerning theross-spectral density, we introduce the collimated �Wc�nd the diffuse �Wd� components:

Wc+�� = Ic

+��,

Wc−�� = Ic

−��,

Wc�� = Wc+�� + Wc

−��, �12�

Wd+��,�� =�

0

+1

Id+��,�J02�

��1 − 2�d,

Wd−��,�� =�

−1

0

Id−��,�J02�

��1 − 2�d,

Wd��,�� = Wd+��,�� + Wd

−��,��. �13�

nserting Eq. (10) into Eq. (6) yields

W��,��

W��,0�=

Wc��

Wc��

Wc��

Wc�� + Wd��,0�+

Wd��,��

Wd��,0�

Wd��,0�

Wc�� + Wd��,0�.

�14�

e can cast Eq. (14) in the form

ig. 3. Spatial coherence length (normalized by the wavelengthropagating in the forward direction inside a semi-infinite scatterhere l* is the transport mean-free path. The scattering mediualues of the anisotropy factor g, an albedo a=0.98, and a refracattered part only and is reported on Fig. 2.

w��,�� = wc�c�� + wd��,��d��, �15�

here wc=1 in order to show explicitly that the cross-pectral degree of coherence is the weighted sum of theontribution of the collimated and diffuse components ofhe field. Considering the forward and backward specificntensity only, we have

W±��,��

W±��,0�=

Wc±��

Wc±��

Wc±��

Wc±�� + Wd

±��,0�

+Wd

±��,��

Wd±��,0�

Wd±��,0�

Wc±�� + Wd

±��,0�. �16�

his can be cast in the form

w±��,�� = wc±�c

±�� + wd±��,��d

±��, �17�

here wc±=1. The coefficients � are weighting factors of

he contribution of ballistic and diffuse components to thepecific intensity. Then we can define the coherenceengths for the ballistic (superscript c) and diffuse (super-cript d) components and for the forward (superscript �)nd backward (superscript �) components of the specificntensity denoted by lcoh

c , lcohd , lcoh

c± , lcohd± , respectively. lcoh

±

orresponds to the beams in the forward or in the back-ard direction. We have obviously lcoh

c =� and lcohc± =�.

hus these quantities will not be used in the following.

. Coherence Length for the Forward Diffuseomponent of the Radiatione show in Fig. 3 the coherence length of the diffuse light

ropagating in the forward direction {ballistic componentemoved [Fig. 3(a)] and included [Fig. 3(b)]} versus theistance z from the medium interface. Different values ofhe anisotropy factor g are considered. We clearly seehree regimes in Fig. 3(a).

• For z� ls, the coherence length of the diffuse compo-ent increases. Note that this behavior is surprising atrst glance because scattering tends to spread the specific

) the diffuse radiation and (b) the diffuse and ballistic radiationdium versus the normalized distance z / l* from the slab interface,escribed by a Henyey–Greenstein phase function with differentindex n2=1. The arrow indicates an example of FWHM for the

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ntensity and thus to decrease the coherence length. Weave derived a simple model, based on the single-cattering approximation of the RTE, which describes thisncrease of the coherence length. This model is describedn Section 5. In simple terms, the origin of this behavior ishat rays propagating in a direction experience an expo-ential decay exp�−� / cos �. It follows that the dampingue to scattering acts as an angular filter that attenuatesarge-angle rays. Note that this is valid only in the single-cattering regime.

• For ls�z� l*, the coherence length decreases, owingo multiple scattering.

• For z� l*, the coherence length reaches a constantalue of the order of � /3 (see Appendix B). This is a fea-ure of the diffusive regime in a weakly absorbing me-ium, in which the radiation is quasi isotropic. The cross-pectral density is then given by Eq. (7).

. Coherence Length for the Forward Componentncluding the Diffuse and Ballistic Radiation

hen only the ballistic component of the specific intensityropagating in the forward direction is considered, the co-erence length lcoh

+ decreases exponentially with z (seeig. 3). Actually, when z tends to 0, we have only a ballis-ic component (the diffuse component tends to 0), so thathe coherence length tends to infinity. In contrast, for z ofhe order of 8l*–10l*, the ballistic component is reduced,nd the diffuse specific intensity is quasi isotropic. Hence

coh+ tends to � /3.

. Coherence Length with Internal Reflectionso far, we have considered the coherence of the field in aandom medium without reflections at the boundaries. Inhis subsection, we consider the coherence length whenhe refractive index n2 is greater than 1 in order to studyhe effect of internal reflections. Figure 4 represents theoherence lengths lcoh

d+ and lcoh+ associated with the forward

omponent of the specific intensity as in Fig. 3 for an in-ex of refraction n2=1.33.In Fig. 4(a) we see that for the diffuse coherence length

cohd+ the variations are the same as in the case without in-ernal reflections. But whatever the value of the aniso-ropy factor g, lcoh

d+ tends to a unique limit, about 0.264�

Fig. 4. Same a

hen z tends to 0. To explain this observation, we plot inig. 5 the variations of the diffuse specific intensity ver-us for z� ls (i.e., at the entrance of the slab). We clearlyee that the contribution of the whole reflection of Id

− (seeection 5) dominates. The fact that the medium is semi-

nfinite produces a quasi-isotropic backscattered specificntensity Id

− for all values of g. Thus the most importantart of Id

+ is a constant (independent of g), and the valuesf the diffuse coherence length are the same.

The full coherence length lcoh+ plotted in Fig. 4(b) does

ot tend to � for z→0 for a weak anisotropy factor. Theimit is finite, and thus the contribution of the collimatedomponent is not dominating.

. SINGLE-SCATTERING MODELo explain quantitatively the increase of the coherenceength of the diffuse radiation at the beginning of the slabi.e., for z� ls), we derived a single-scattering model. We

ake the following assumptions:

1. z� ls: single-scattering assumption.

3 with n =1.33.

ig. 5. Plot of the variations of the specific intensity at the en-rance of the slab. Three zones can be identified: For 2�0, thepecific intensity is quasi isotropic owing to multiple scattering ofight in a semi-infinite slab; for 0�2�l, the specific intensitys also quasi isotropic owing to the whole reflection; and, for 2

l, there are two contributions, which are the partial reflectionf Id

− and the single scattering of the incident specific intensity.

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2. L*�1: we study a semi-infinite medium. Thus thepecific intensity at the entrance of the slab propagatingn the backward direction will be quasi isotropic.

3. a 1: The medium is not (or weakly) absorbing. Thisssumption allows us to write a flux conservation equa-ion at the slab entrance.

4. n2�1: The refractive index can be different from 1.

hese assumptions allow us to derive an analytical formor the diffuse specific intensity propagating in the for-ard direction valid at the entrance of the slab (see Ap-endix C). We report in Fig. 6 the comparison betweenhis analytical model and the full numerical computationor the coherence length. The relative errors are of the or-er of 2% for z�8l*–10l*. The very good agreement showshat the increase in the coherence of the diffuse field is in-eed due to an angular filtering of the specific intensity.

. CONCLUSIONn this paper, we have used the link between the specificntensity and the field spatial correlation function to ana-yze the field spatial coherence in a random medium. Weave calculated numerically the specific intensity in alab of a scattering medium using the RTE.24 We have de-ived the spatial correlation function and the coherenceength from the specific intensity. We have performed nu-

erical experiments on systems with different propertiesnd studied the coherence properties of both the ballisticnd the diffuse components of the beam. In particular, weave studied the evolution of the coherence length versushe transport regime (single scattering, multiple scatter-ng, diffusive). The crossover between the different re-imes has been clearly seen, and the diffusive regime iseached for z�8l*–10l*. The study confirms that the RTEs an appropriate tool to study the transition betweenransport regimes.25,26

Finally, to reproduce the variations of the coherenceength in the presence of reflections at the interface of the

edium (i.e., when n2�1), we have derived an analyticalodel based on the single-scattering approximation to

ig. 6. Comparison between the single-scattering model (solid cuse spatial coherence length in the forward direction normalizedropy factors [(a) g=0.16 and (b) g=0.80], an albedo a=1, and a rehe scale for lcoh

d+ is very small. The relative errors are about 2%

ompute the specific intensity. This model reproduces allhe variations of the coherence length for distances fromhe interface of the order of the transport mean-free path.

PPENDIX A: WALTHER’S FORMULAn this appendix, we invert Eq. (2) to show its equivalenceith Walther’s formula.9 We introduce the decomposition=u�+uzez and �=��+�zez, where ez is a unit vectorlong the Oz axis. We note that

d� =duxduy

�uz�=

d2u�

�uz�. �A1�

hen Eq. (2) becomes

W�r,�� =� exp�ik�zuz�

�uz�I��r,u�exp�ik�� · u��d2u�.

�A2�

he variations of ux and uz are limited because �u�=1.evertheless, we can avoid the problem by replacing the

pecific intensity with 0 for �u��1 and thus perform thentegral over the domain �−� , +��2. Then Eq. (A2) is aourier transform that can be inverted to yield

I��r,u� = k

2��2

�� exp�− ik�zuz�W�r,��

�exp�− ik�� · u��d2��, �A3�

here =cos =uz. And then

I��r,u� = k

2��2

�� W�r,��exp�− ik� · u�d2��,

�A4�

hich is the well-known Walther’s formula.9 EquationA4) shows that the specific intensity is the Fourier trans-orm of the cross-spectral density of the field or the

igner transform of the field.

nd the whole numerical computation (dashed curve) for the dif-e wavelength versus the normalized distance z / ls for two aniso-e index n2=1.33. We used a Henyey–Greenstein phase function.8l*–10l*.

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PPENDIX B: CALCULATION OF THEPATIAL CORRELATION FUNCTION FORN ISOTROPIC SPECIFIC INTENSITYe consider an isotropic specific intensity propagating in

ll directions. After integrating over the azimuthal angle, we obtain a specific intensity independent of . Thuse write

I��,� = I��. �B1�

sing this form in Eq. (6), we obtain

W��,�� = I��−1

+1

J02��

�1 − 2�d. �B2�

irst, we consider in the range �0, +1� to compute theross-spectral density in the forward direction W+�� ,��.e define X=2�� /� and u=1−2; thus d=

udu /1−u2. Equation (B2) now becomes

W+��,X� = I��0

+1

J0�Xu�udu

1 − u2= I��sinc�X�. �B3�

e also have the same result for the cross-spectral den-ity accounting for wave propagation in the backward di-ection,

W−��,X� = I��sinc�X�, �B4�

nd, finally, for all directions, we have

W��,X� = 2I��sinc�X�. �B5�

PPENDIX C: CALCULATION OF THEPECIFIC INTENSITY IN THEINGLE-SCATTERING MODEL

n this appendix, we derive the analytical expression forhe specific intensity that has been used in Section 5. Weenote the cosine of the critical angle by l=1−n1

2 /n22. 1

s the cosine of the angle 1 in the medium number 1, and

2 is the one in the medium 2, so that n11−12

n21−22. We decompose the specific intensity in three

ones:

• For −1�2�0; assumption 2 (i.e., L*�1) allows us torite that Id

−�� ,2�=XIinc with X as an unknown variable.• For 0�2�l; there is total reflection of Id

−. If we ne-lect the part of Id

+ that was present without reflection, weave Id

+�� ,2�l�=XIinc.• For l�2�1; there is partial reflection of Id

−. Weow need to consider the part Id

+� of the specific intensityhat was present without reflection. Thus we haved+�� ,2�l�=X�Iinc+Id

+�� ,2�, where X� is another un-nown.

Figure 5 presents the variations of the specific inten-ity at the entrance of the slab.

. Computation of the Coefficients X and X�ssumption 3 expresses that the flux entering the me-ium is equal to the flux leaving the scattering medium.ssumption 2 implies that this equality can be writtennly for �=0:

�C1�

here R12��1�� is the energy Fresnel factor for the reflec-ion from medium 1 to medium 2 that depends on 1.21d ��1��, which is the transmission factor for the specific

ntensity, is related to the Fresnel energy transmissionactor T21��1�� by24

T21d ��1�� =

n22

n12T21��1��. �C2�

quation (C1) becomes

Id−��,2� =

1 − R12�1�

�0

1

T21d ��1��1d1

Iinc = XIinc. �C3�

o compute X�, we simply write the partial reflection ofhe diffuse specific intensity in the backward direction:

Id−��,2�R21��2�� = X�Iinc, �C4�

hich leads to X�=XR21��2��.

. Computation of Id+�„� ,�2…

o compute the diffuse component of the specific intensityn the forward direction for 2�l and without reflectioni.e., n2=1), we consider the RTE [Eq. (9)] where we splithe specific intensity into its collimated and diffuseomponents.25 Owing to assumption 1, we keep only thexpression of the diffuse specific intensity for −1�2l in the phase function term:

a

2�

−1

1

p�2,2��Id��,2��d2� a

2�

−1

l

p�2,2��XLincd2�

= aIincJ�2�, �C5�

ith

J�2� =X

2�

−1

l

p�2,2��d2� .

or the collimated component of the specific intensity, weeed to take into account the energy Fresnel transmissionactor T12�1�:

Ic+�� = T12�1�exp�− ��. �C6�

hus, Eq. (9) becomes

T

Tc

TeaWtiestalst

ATP2ek

i

R

1

1

1

1

1

1

1

1

1

12

2

2


�Id

+�

��,2� + Id

+��,2� = aIincJ�2� +a

2p�2,1�T12�1�Iinc

�exp�− ��. �C7�

he boundary condition is given by

Id+�0,2� = 0. �C8�

he solution of Equations (C7) and (C8) yields an analyti-al form of the specific intensity:

Id+��,2� =

aIinc

2�1 − 2��2I�2��1 − 2� + p�2,1�T12�1�

�exp�− �� − �2I�2��1 − 2� + p�2,1�T12�1��

�exp−�

2�� . �C9�

he key feature that explains the increase of the coher-nce length is the damping factor exp�−� /2� that acts asn angular filter. Figure 7 shows the origin of this effect.e consider two rays propagating along different direc-

ions. Rays propagate along a length L /2 in the scatter-ng medium and therefore experience an attenuationxp�−� /2�. This is the same effect as the attenuation ofunlight through the atmosphere. Light is strongly at-enuated at sunset �2�1� and almost fully transmittedt noon �2=1�. It follows that this term acts as an angu-ar filter. Finally, Eq. (C9) gives the diffuse specific inten-ity in the forward direction. The coherence length canhus be easily deduced.

Fig. 7. Geometrical filtering effect on the coherence length.

CKNOWLEDGMENTShis work is supported by the European Integratedroject Molecular Imaging, under contract LSHG-CT-003-503 259, and by the Engineering and Physical Sci-nces Research Council grant GR/R86201/01. We ac-nowledge helpful discussions with M. A. Alonso.

Corresponding author Rémi Carminati’s e-mail addresss [email protected].

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spatial coherence in strongly scattering media

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