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2840 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Romain Pierrat

Transport equation for the time correlationfunction of scattered field in dynamic turbid media

Romain Pierrat1

1Laboratoire Kastler Brossel, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France(romain.pierrat@spectro.jussieu.fr)

Received July 9, 2008; revised September 3, 2008; accepted September 4, 2008;posted September 15, 2008 (Doc. ID 98541); published October 24, 2008

I derive a transport equation for the time correlation function in transmission and reflexion and inside a turbidmedium. This equation goes beyond the diffusion approximation that is at the root of the well-establisheddiffusing-wave spectroscopy technique. It takes into account all the transport regimes from ballistic to diffusiveand the relaxation in direction at each scattering event. The derivation is based on a generalized form of theBethe–Salpeter equation coupled to a generalized form of the scattering operator. The method presented can beeasily adapted to compute the correlation function in systems with several time scales encountered, for ex-ample, in biology and polymer physics. The obtained equation is easily solvable numerically using a MonteCarlo scheme. © 2008 Optical Society of America

OCIS codes: 030.1640, 030.5620, 290.7050, 290.4210, 170.3660.

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. INTRODUCTIONrobing the dynamics of light scattering in a complex me-ium has become a very active field of research [1]. In par-icular, two important techniques have emerged. Theuasi-elastic light scattering method (QELS) [2,3], devel-ped in the 1970s, became a standard tool to characterizecatterers’ internal properties in a very dilute solution.he diffusing-wave spectroscopy method (DWS) [4–6], de-eloped ten years later, is now a major technique to char-cterize optically the dynamics of the scatterers in a com-lex medium under multiple scattering, in particular inoft matter [7–10] or in biological tissues [11]. These ex-mples demonstrate the considerable interest of temporaluctuations of intensity to obtain information on a com-lex medium structure. Nevertheless, these techniquesuffer from strong limitations. In particular, their do-ains of validity are confined to the single scattering re-

ime for QELS and to the diffusive regime for DWS.oreover, it is well-known that the diffusion approxima-

ion overestimates the short paths contribution [12,13]nd does not take into account properly the boundary con-itions [14,15]. In the context of biomedical imaging, it isf primary importance to consider the multiple scatteringut nondiffusive regime [1], in particular in a view of mul-iscale imaging.

To improve the DWS technique, in particular in theultiple scattering regime, investigations have been

riven using the telegrapher equation [16]. Recently, Car-inati et al. [17] proposed an approach based on the ra-

iative transfer equation (RTE) to correct the behavior ofhe time correlation function of the field for small pathsnd for long and rare paths with few scattering eventsi.e., in media where the optical thickness is too small forhe diffusion approximation to be valid). The idea is to re-lace the probability density P�s� of having a path of

1084-7529/08/112840-6/$15.00 © 2

ength s computed in the diffusion approximation with�s�P�n ,s�, which is the probability density of having aath of length s computed by the RTE times the probabil-ty density of having n scattering events for a path ofength s—which is a Poissonian distribution. In particu-ar, this work remains based on the previous work of Pinet al. [5]. This approach has been checked experimentallyery recently in reflection (where the role of shorts pathss predominant) [18]. Experiments have been also done byopescu et al. [19] to show the limitations of the DWS

echnique in the subdiffusive regime.In this paper, I propose a new approach to the problem.

he idea is to derive a transport equation for the time cor-elation function of the field from a generalized Bethe–alpeter equation that takes into account the motion ofhe scatterers. Such a study has been done previously forhe spatial correlation function of the field in a static com-lex medium [20]. The approach presented here is an ex-ension in the case of a dynamic system of techniquessed to derive the RTE in a static system from first prin-iples (Bethe–Salpeter equation, scattering operator,eynman diagrams) [21–27].The main advantage of the derivation from first prin-

iples is that it can be adapted to many situations, suchs stratified systems [28] (by taking into account theoundary conditions properly), or systems in whicheveral time scales are involved [29] (for example, witheveral different types of scatterers).

In Section 2, I describe the Bethe–Salpeter equationnd its spatial and temporal Fourier transform used foronvenience. Then, I derive in Section 3 the expression ofhe mass and vertex operators using a new form of thecattering operator. Finally, in Section 4, I derive theransport equation for the specific intensity (local and di-ectional radiative flux in space and time) and for the

008 Optical Society of America

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Romain Pierrat Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2841

ime autocorrelation function of the field. I conclude byhowing that this equation is solvable very easily with aonte Carlo scheme in Section 5.

. GENERALIZED BETHE–SALPETERQUATION

n the case of fixed scatterers the Bethe–Salpeter equa-ion is usually obtained using a diagrammatic approachrom a microscopic wave equation [22,30,31]. This is anxact equation for the field autocorrelation function. Itontains an operator K depending on four space variablesalled the vertex operator and describing the correlationetween two scattering processes. The idea is to general-ze this equation to the case of moving scatterers. Thiseads to a new vertex operator Km (subscript m means

obile) depending on four space variables and four timeariables. I neglect polarization effects (scalar approxima-ion) and I assume that no source is present (the incidenturrent densities that correspond to the coherent termre missing). Thus the generalized Bethe–Salpeter equa-ion is written

�E�r1,t1�E*�r2,t2�� =� �G�r1,r1�,t1,t1����G*�r2,r2�,t2,t2���

�Km�r1�,�1,r2�,�2,t1�,�1,t2�,�2�

��E��1,�1�E*��2,�2��d3r1�d3r2�d

3

��1d3�2dt1�dt2�d�1d�2, �1�

here E is the electric field and G the Green function inhe scattering medium. The symbol � denotes the conju-ate quantity, and the brackets � � denote the average overn ensemble of realizations of the medium (average overhe positions and over the velocity of the scatterers). Ionsider that the medium is infinite and statistically sta-ionary in space and time, which leads to an invariance ofhe average Green operator �G� and the generalized ver-ex operator Km by translation in space and time. For con-enience, it is easier to work in Fourier space. The con-entions used are the following:

�G�r,t�� =� �G�k,���exp�ik · r − i�t�d3k

8�3

d�

2�, �2�

Mm�r,t� =�Mm�k,��exp�ik · r − i�t�d3k

8�3

d�

2�, �3�

m�r1,�1,r2,�2,t1,�1,t2,�2�

=� Km�k1,�1,k2,�2,�1,�1,�2,�2�

�exp�ik1 · r1 − i�1 · �1 − ik2 · r2 + i�2 · �2�

�exp�− i�1t1 + i�1�1 + i�2t2 − i�2�2�

�d3k1

8�3

d3�1

8�3

d3k2

8�3

d3�2

8�3

d�1

2�

d�2

2�

d�1

2�

d�2

2�, �4�

tm�r1,r2,t1,t2� =� tm�k1,k2,�1,�2�exp�ik1 · r1 − ik2 · r2

− i�1t1 + i�2t2�d3k1

8�3

d3k2

8�3

d�1

2�

d�2

2�. �5�

m and tm are the mass and scattering operators in thease of moving particles. They essentially describe the ex-inction [32,33] and the scattering phenomena, respec-ively. The signs in the exponentials are chosen such thathe vertex operator Km describes correctly the correlationetween the following two scattering processes:

�1,�1 → k1,�1, �2,�2 → k2,�2, �6�

here �1, �2 and k1, k2 are the incident and the emergentave vectors, respectively. �1, �2 and �1, �2 are the inci-ent and the emergent frequencies, respectively. Theranslational invariance in space and time allows me torite the vertex operator in the form

m�k1,�1,k2,�2,�1,�1,�2,�2�

= 8�3��k1 − �1 − k2 + �2�2����1 − �1 − �2 + �2�

�K̃m�k1,�1,k2,�2,�1,�1,�2,�2�. �7�

he Dyson equation [32,33] gives the expression of the av-rage Green function from the mass operator:

�G�k,��� =1

�2/c02 − k2 − Mm�k,��

=1

keff2 ��� − k2

, �8�

here c0 and keff are, respectively, the light velocity in theost medium and the wave vector in the effective homo-eneous medium. An exact form of the generalized Bethe–alpeter equation for moving particles in an infinite me-ium is given by inserting Eqs. (2)–(4), (7), and (8) intoq. (1):

−2��

c02 + 2k · q + Mmk +

q

2,� +

�

2 − Mm* k −

q

2,� −

�

2�f�q,k,�,��

= ��Gk +q

2,� +

�

2 −�G*k −q

2,� −

�

2 � � K̃mk +q

2,� +

q

2,k −

q

2,� −

q

2,� +

�

2,�� +

�

2,� −

�

2,�� −

�

2�f�q,�,�,���

d3�

8�3

d��

2�, �9�

w

it

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wta

Bt�

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K

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K

w

2842 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Romain Pierrat

here the function

f�q,k,�,�� =�Ek +q

2,� +

�

2E*k −q

2,� −

�

2 �10�

s the Fourier form of the spatiotemporal correlation func-ion of the field.

. SCATTERING, MASS, AND VERTEXPERATORS

n this section, I derive the expression of the scattering,ass, and vertex operators in the case of moving par-

icles. The system is assumed to be diluted enough for therst-order diagrammatic expansion of the operators to bealid. I denote by t �k ,k� ,�� the Fourier transform of thecattering operator for a given particle. k and k� are thencident and scattered wave vectors, respectively. For theake of simplicity, I assume that all the scattering par-icles are identical in the system. Let us consider a mov-ng particle of speed v constant during the whole scatter-ng process. I denote by tm

v �k ,k� ,� ,��� the Fourierransform of the scattering operator for the moving par-icle. Thus, the incident and scattered fields are relatedy

Esca�k,�� =� G0�k,��tmv �k,k�,�,���Einc�k�,���

d3k�

8�3

d��

2�,

here G0 is the Green function in the host medium (sys-em without the scattering particles). Considering nowhis expression in the particle frame, we have

Esca�k,�� =� G0�k,� + k · v�tmv �k,k�,� + k · v,��

+ k� · v�Einc�k�,���d3k�

8�3

d��

2�, �11�

here E is the field in the particle frame. In this frame,he particle is fixed and the relation between the incidentnd the scattered fields can be written as

Esca�k,�� =� G0�k,��t�k,k�,��Einc�k�,��d3k�

8�3 . �12�

y identifying Eqs. (11) and (12) and by assuming thathe velocity of the particle is weak enough to have k ·v

� in the Green operator, we have

tmv �k,k�,�,��� = 2�t�k,k�,� − k · v����� − � − �k� − k� · v�.

�13�

f � is not a resonant frequency of the particle, I can alsossume that k ·v�� in the scattering operator t becausehe variations of this function are sufficiently small. Theass operator is now approximated by the first term of its

iagrammatic expansion (Foldy–Twersky approximation)rovided that the medium is sufficiently diluted. In thatase, the mass operator is the sum over all the N scatter-rs of the averaged scattering operator for all accessibleositions, times, and velocities:

Mm�r − r�,t − t�� = �i=1

N � tmv �r − ri,r� − ri,t − ti,t� − ti�

�P�ri,ti�g�v�d3ridtidv, �14�

here P�r , t� is the probability density of having a scat-erer at the position r and at the time t, and g�v� is theistribution function of the particles velocity. This is aormalized probability density,

� g�v�dv = 1. �15�

ote that g�r� depends only on v2. If we assume that thearticles are uniformly distributed in space and time,�r , t�=1/ �VT�, where V is the volume of the system andthe time window. Actually, the system is infinite, and

ime passes to infinity. In these conditions, the mass op-rator in the Fourier domain reads

Mm�k,�� = limT→�

T � tmv �k,k,�,��g�v�dv, �16�

here is the density of scatterers. This leads to the finalxpression of the mass operator if we note thatimT→� 2����−�� /T=1,

Mm�k,�� = t�k,k,�� = M�k,��, �17�

here M is the mass operator in the case of fixed par-icles. M and Mm are identical, which means that Mm isot affected by the scatterers’ motion. This result de-cribes the fact that the behavior of the average electriceld is the same whatever the speed of the particles. Forhe vertex operator, I keep only the first term of its dia-rammatic expansion (ladder approximation) just as forhe mass operator; the approximation leads to the sumver all the scatterers of the average for all accessible po-itions, times, and velocities of the scattering operatororrelation function. It reads in the Fourier space

˜m�k1,�1,k2,�2,�1,�1,�2,�2�

= limT→�

T � tmv �k1,�1,�1,�1�tm

v*�k2,�2,�2,�2�g�v�dv.

eplacing the expression of the scattering operator in thexpression of the vertex operator leads to

˜mk +

q

2,� +

q

2,k −

q

2,� −

q

2,

� +�

2,�� +

�

2,� −

�

2,�� −

�

2= tk +

q

2,� +

q

2,� +

�

2t*k −q

2,� −

q

2,� −

�

2�

1

�k − ��g̃�� − �

�k − �� ,

here g̃ is given by

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Tpo

wuptl(s

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�

Tst

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Romain Pierrat Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2843

g̃�w� =� ��w −w · v

w �g�v�dv, �18�

ith w= �w�. Finally, the vertex operator reads

K̃m =K̃

�k − ��g̃�� − �

�k − �� , �19�

here K̃ is the vertex operator in the case of fixed par-icles. In the following, I will use Eqs. (17) and (19) toake explicit the operators in the Bethe–Salpeter equa-

ion (9).

. TOWARD A TRANSPORT EQUATIONo derive a transport equation first of all for the specificntensity and second for the time correlation function ofhe field, I have to consider long time scales compared tohe frequency of the wave (i.e., ���) and large spacecales compared to the wavelength (i.e., q�k ,) in Eq.9). Physically, this means that the temporal variations ofhe specific intensity I�r ,u , t ,�� (local and directional ra-iative flux) are slow compared to the temporal variationsf the wave (i.e., of the order of the period of the wave).imilarly the spatial variations of the specific intensityre small compared to the spatial variations of the wavei.e., of the order of the wavelength). These conditions areulfilled if the frequency � is not near a resonance of thecattering particles and if the medium is diluted enough.sing these approximations coupled to expressions (17)nd (19), the Bethe–Salpeter equation (9) reduces to

−2��

c02 + 2k · q + M�k,�� − M*�k,���f�q,k,�,��

= ��G�k,��� − �G*�k,���� � K̃�k,�,k,�,��

�k − ��

�g̃�� − �

�k − ��f�q,�,�,���d3�

8�3d��. �20�

he last approximation concerns the averaged Greenropagator. If I assume I�keff�2�R�keff�2=kr

2 in Eq. (8) Ibtain

�G�k,��� = PV� 1

kr2 − k2� − i���kr

2 − k2�, �21�

here PV is the principal value operator and k=ku withbeing the unit vector along the direction k. This ap-

roximation is valid for a dilute medium and means thathe extinction length is large compared to the wave-ength. The Dirac-delta function allows me to write Eq.20) for a fixed k=kr. I define the Fourier transform of thepecific intensity by

f�q,k,�,�� = ��kr − k�I�q,u,�,��. �22�

he specific intensity is then the Fourier transform of thepatiotemporal field autocorrelation function, which is theo-called Wigner transform of the field. This expressioneads to the well-known Walther formula [34]. Equation20) becomes

− i��

c02kr

+ iu · q −I�M�kru,���

kr�I�q,u,�,��

=1

16�2 � K̃�kru,kru�,kru,kru�,��

kr�u − u��g̃ �� − �

kr�u − u���I�q,u,�,���du�d��. �23�

his expression allows me to define the extinction �e andcattering �s coefficients and the phase function p, respec-ively, by

�e��� = −I�M�kru,���

kr, �24�

�s���p�u,u�,�� =K̃�kru,kru�,kru,kru�,��

4�. �25�

et us recall that the extinction coefficient represents thettenuation of a beam by scattering and absorption. Thus,he absorption coefficient is given by �a���=�e����s��� and the associated absorption, scattering, and ex-

inction mean free path are la���=1/�a���, ls���1/�s���, and le���=1/�e���, respectively. The phase

unction describes the part of the incident beam in the di-ection u� scattered in the direction u. Usually, the fre-uency shift at each scattering event is small enough toonsider that the radiative properties of the medium re-ain constant. This would not be the case of course for a

requency near a resonance of the particles. With thesessumptions and approximating the speed of light in theystem by c=c0

2kr /�, the RTE for the specific intensity inirect space reads

1

c

�

�t+ u · �r + �e�I�r,u,t,��

=�s

4�� p�u,u��

kr�u − u��g̃ �� − �

kr�u − u��I�r,u�,t,���du�d��.

�26�

quation (26) has the same structure as the well-knownTE as derived by Chandrasekhar [35] except that it ex-ibits a frequency coupling. This coupling is due to theotion of the scatterers. Due to this motion, a Doppler

hift occurs at each scattering event, which induces ahange of frequency for the scattered photon. In the casef a static system, the function g̃ is a Dirac-delta functionnd the frequency coupling disappears, resulting recoveryf the well-known RTE.

I will now derive the same equation for the time auto-orrelation function of the field g1. The specific intensityn the Fourier space is given by Eqs. (10) and (22). Te-ious but straightforward spatiotemporal Fourier trans-orms and integration gives the field temporal autocorre-ation function in term of the specific intensity

wta=f

�

wmdB

wd

Iptp

Eldvp

s

(c

Bftcb

mt[

5TEsthtwfpMa

6IpsTesC

ATPLf(a

R

1

2844 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Romain Pierrat

g1�r,t,�� =�Er,t +�

2E*r,t −�

2 = kr

2�4�

I�r,u�,t,��du,

here r is the position, t the time, and � the correlationime. This expression allows me to define a directionalnd temporal autocorrelation function by g1�r ,u , t ,��kr

2I�r ,u , t ,��. The RTE for this function is given by therequencial Fourier transform of Eq. (26), which yields

1

c

�

�t+ u · �r + �e�g1�r,u,t,��

=�s

4�� p�u,u��g5 �kr�u − u����g1�r,u,t,��du�, �27�

here g̃̃ is the Fourier transform of g̃. Equation (27) is theain result of this paper. It is valid whatever the velocity

istribution. For the particular case of a Maxwell–oltzmann distribution, we have

g�v� =1

���2��3exp�−

v2

2�2� , �28�

hich is a Gaussian distribution of null average and stan-ard deviation �. This gives

g5 �kr�u − u���� = exp�− �2kr2�u − u��2�2�. �29�

n the case of Brownian motion, the mean-square dis-lacement is � r2�=6DB� where DB is the Brownian mo-ion diffusion coefficient. This leads to the following ex-ression for g̃̃:

g5 �kr�u − u���� = exp�− 2DBkr2�u − u��2��. �30�

quations (27) and (30) simply express that the decorre-ation for each scattering event is given by an exponentialecay. Actually, this equation is more powerful than pre-iously developed theories [5,16,17] because we can com-ute easily the correlation function

(1) for all positions r;(2) for all directions u;(3) for all times t;(4) inside or outside (by transmission or reflection) the

cattering medium;(5) from the single-scattering to the diffusive regime.

Moreover, I did not make the assumption

1. of small correlation time;2. of the average angle relaxation �u−u��2�2�1−g�

where g is the anisotropy factor, i.e., the average of theosine of the scattering angle);

3. of n=s / ls scattering events for a path of length s.

Of course, under these assumptions and in the case ofrownian motion, I recover the standard DWS theory [5]

rom the more general equations derived here. Note thathe derivation given in this paper is the simplest one. Itan be adapted to the case of stratified systems (humanrain, for example), systems with more than one type of

oving particles (more than one time scale) [29], and sys-ems illuminated near a resonance (radiation trapping)36].

. MONTE CARLO SCHEME TO SOLVE THERANSPORT EQUATIONquation (27) can be solved easily using a Monte Carlocheme. The idea is to use a classical scheme that is usedo solve the RTE for fixed particles [37,38]. Then, we justave to multiply the result by the exponential correlationime decay for each scattering events. This is the reasonhy the CPU time needed to compute the correlation

unction g1 is quite the same as the one needed to com-ute the specific intensity in the case of fixed particles. Aonte Carlo approach is then natural for such a problem

nd will give reliable results for all optical thicknesses.

. CONCLUSIONn a word, I have derived in this work a new way of com-uting the time correlation function of the electric field in-ide or outside a scattering system with moving particles.his derivation is based on a generalized Bethe–Salpeterquation. The method can be adapted to many differentystems and is easily solvable numerically by a Montearlo scheme.

CKNOWLEDGMENTShis work is supported by the European Integratedroject Molecular Imaging, under contract numberSHG-CT-2003-503 259 and by a postdoctoral fellowship

rom the Centre National de la Recherche ScientifiqueCNRS). I thank Dominique Delande, Benoît Grémaud,nd Rémi Carminati for helpful discussions.

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Romain Pierrat Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2845

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