1430 J. Opt. Soc. Am. A/Vol. 21, No. 8 /August 2004 Elaloufi et al.

Diffusive-to-ballistic transition in dynamic lighttransmission through thin

scattering slabs: a radiative transfer approach

Rachid Elaloufi, Remi Carminati, and Jean-Jacques Greffet

Laboratoire d’Energetique Moleculaire et Macroscopique, Combustion; Ecole Centrale Paris, Centre National de laRecherche Scientifique, 92295 Chatenay-Malabry Cedex, France

Received October 17, 2003; revised manuscript received February 17, 2004; accepted March 23, 2004

We study the deviation from diffusion theory that occurs in the dynamic transport of light through thin scat-tering slabs. Solving numerically the time-dependent radiative transfer equation, we obtain the decay timeand the effective diffusion coefficient Deff . We observe a nondiffusive behavior for systems whose thickness Lis smaller than 8ltr , where ltr is the transport mean free path. We introduce a simple model that yields theposition of the transition between the diffusive and the nondiffusive regimes. The size dependence of Deff inthe nondiffusive region is strongly affected by internal reflections. We show that the reduction of ;50% of Deffthat was observed experimentally [Phys. Rev. Lett. 79, 4369 (1997)] can be reproduced by the radiative trans-fer approach. We demonstrate that the radiative transfer equation is an appropriate tool for studying dy-namic light transport in thin scattering systems when coherent effects play no significant role. © 2004 Op-tical Society of America

OCIS codes: 290.1990, 290.7050, 290.4210, 170.5280, 170.3660.

1. INTRODUCTIONThe study of wave propagation through random mediahas been an active field of investigation in the past fewdecades.1 Recently it has attracted considerable interestthrough the development of mesoscopic physics2,3 and im-aging techniques in strongly scattering biologicaltissues.4,5 Time-resolved techniques using pulse reflec-tion or transmission on short time scales,6 optical-coherence tomography,7 and diffusing wavespectroscopy8,9 give promising results. In several otherareas, diffusive waves—such as thermal, acoustic, or elas-tic waves—form the basis of imaging and measurementtechniques.10 With the rapid development of microtech-nologies and nanotechnologies, understanding the propa-gation of such waves at short (time and length) scales hasbecome a key issue. Another example is heat conductionat short scales in solids, which can be handled on the ba-sis of a Boltzman transport equation for phonons that un-dergo scattering, emission, and absorption.11,12 Thisproblem is very similar to that encountered in optical-wave propagation through random media.

The study of dynamic wave transport through scatter-ing and absorbing media is considerably simplified whenusing the diffusion approximation.1 The approximationis widely used in the context of optical imaging throughbiological tissues.4,5 Yet limitations appear when thesize of the system (or the time scale) becomes of the orderof the mean free path (or the collision time). Light trans-port at short time scales in semi-infinite media and thickslabs (with sizes much larger than the transport meanfree path) has been studied recently by use of an approxi-mate solution of the radiative transfer equation (RTE).13

In particular the transition from the short-time, quasi-ballistic regime to the long-time, diffusive regime has

1084-7529/2004/081430-08$15.00 ©

been described. In the present work, we will focus on thetransition between the different regimes versus thelength scale of the system. Several experimental resultshave demonstrated strong deviations from diffusiontheory in dynamic light scattering at short-length scalesby measuring pulse transmission through thin slabs14–16

or in diffusing wave spectroscopy experiments.17,18 Theresults in Refs. 14 and 15 show that the effects of the in-terplay between space and time scales on the validity ofdiffusion approximation is a difficult issue. On the onehand, experiments reported in Ref. 14 seem to show thatdiffusion theory gives an accurate prediction for the long-time decay of transmitted pulses for both thin(L , 10ltr) and thick systems, whereas it fails for theshort-time behavior. On the other hand, experiments re-ported in Ref. 15 show that diffusion theory fails for thinsystems (L , 8ltr), even for the determination of thelong-time behavior.

The domain of validity of the diffusion approximationfor time-dependent transport has been studied by com-parison with the prediction of the RTE,19–21 or by usingthe telegrapher’s equation.22 In particular, the results inRef. 21 have confirmed that the diffusion approximationis able to predict the long-time behavior of transmittedpulses through systems of size L . 8ltr , where ltr is thetransport mean free path, in agreement with experimen-tal results.15 More recently, the transition from ballisticto diffusive transport was analyzed by solving the Bethe–Salpeter equation for a slab in the lowest-order ladder ap-proximation with isotropic scattering.23 A region ofstrong deviation from the diffusion approximation wasfound for 3ltr , L , Lc where Lc is a critical length thatdepends on the amount of internal reflection at the slabboundaries. The region of nondiffusive transport agrees

2004 Optical Society of America

Elaloufi et al. Vol. 21, No. 8 /August 2004 /J. Opt. Soc. Am. A 1431

with experimental results.15 Nevertheless, as far as thetransition from the ballistic to the diffusive regime is con-cerned, an overview of the literature leads to the followingremarks:

1. The transport model used in Ref. 23 predicts an in-crease of the effective diffusion coefficient of the slabwhen the thickness L decreases in the nondiffusive region(sometimes referred to as the anomalous region). This isthe opposite of the behavior observed experimentally,15,24

and no convincing argument has been proposed to explainthis contradiction.

2. The samples used in practice are usually absorbingand anisotropically scattering. A transport theory ableto account for arbitrary scattering and absorbing proper-ties and for rigorous boundary conditions at the slabboundaries is necessary. The RTE seems to be the ad-equate tool in this context.

3. Last but not least, a simple model explaining (atleast qualitatively) the onset of the nondiffusive behaviorat small length scales is still missing.

In this paper, we present a theoretical and numericalstudy of the transition from the ballistic to the diffusiveregime in light transport through scattering media. Ourapproach is based on the RTE.25,26 By a numerical studyof pulse transmission through scattering slabs, we studythe size dependence of the decay time and the effectivediffusion coefficient. The RTE approach allows us tohandle arbitrary scattering properties of the particles aswell as absorption. Internal reflections at the slabboundaries are treated rigorously by means of reflectionand transmission factors for both the ballistic and the dif-fusive part of the light intensity. The results confirm thesensitivity of the effective diffusion coefficient Deff to thelevel of internal reflection. They also show under whichcondition a reduction of Deff at small scale is observed,and the experimental results in Ref. 15 are retrievedquantitatively. Finally, a simple analysis based on thedispersion relation of the RTE26 is used to describe thetransition from the diffusive to the nondiffusive regime.This crude model allows us to retrieve the critical size atwhich the transition occurs.

The paper is organized as follows. In Section 2 webriefly describe the numerical method used to solve theRTE in a slab geometry. In Section 3 we present numeri-cal calculations of the decay time and effective diffusioncoefficient for systems of various sizes and mean (effec-tive) indices of refraction. In Section 4 we discuss the re-duction of the effective diffusion coefficient in connectionwith experimental results. In Section 5 we present theanalytical model and use it to discuss the deviation fromdiffusion theory for small systems. In Section 6 we sum-marize the main results and give our conclusions.

2. TIME-DEPENDENT RADIATIVETRANSFER EQUATIONWe consider a slab of width L with the z axis normal tothe boundaries (the strip 0 , z , L is filled with thescattering medium). The slab is illuminated from the

region z , 0 at normal incidence by a plane-wave pulse.The specific intensity I(z, m, t) inside the scattering me-dium obeys the RTE25,26:

1

v

]I~z, m, t !

]t1 m

]I~z, m, t !

]z

5 2~ms 1 ma!I~z, m, t !

1ms

2E

21

11

p ~0 !~m, m8!I~z, m8, t !dm8, (1)

where v is the energy velocity and m 5 cos u, with u theangle between the propagation direction and the z axis.p (0) is the phase function integrated over the azimuthalangle p (0)(m, m8) 5 1/2p*0

2pp(u – u8)df where u and u8are unit vectors corresponding to directions (u, f) and(u8, f 8). ms and ma are the scattering and absorption co-efficients, respectively. The associated scattering and ab-sorption mean free paths are la 5 ma

21 and ls 5 ms21.

The transport mean free path ltr 5 ls /(1 2 g), where g isthe anisotropy factor (average cosine of the scatteringangle). The real part of the medium’s effective index, ac-counting for both the homogeneous background mediumand the scattering particles, is denoted by n2 . The half-spaces z , 0 and z . L are filled with homogeneous andtransparent materials of refractive indices n1 and n3 , re-spectively. To solve Eq. (1), we use the space-frequencymethod described in Ref. 21. We briefly recall its prin-ciple here.

A time-domain Fourier transform of Eq. (1) leads to

m]I~z, m, v!

]z5 2S ms 1 ma 2 i

v

v D I~z, m, v!

1ms

2E

21

11

p ~0 !~m, m8!I~z, m8, v!dm8,

(2)

where I(z, m, v) is the time-domain Fourier transform ofI(z, m, t). This equation is similar to the static RTEwith an effective extinction coefficient a(v) 5 (ms 1 ma2 iv/v). It can be solved numerically by standardmethods developed for time-independent problems.27 Inassuming illumination by a plane wave (representing, forexample, a collimated laser beam), it is useful to separatethe ballistic and the diffuse components of the specific in-tensity inside the medium. One writes

I~z, m, v! 5 Ib1~z, v!d ~m 2 1 ! 1 Ib

2~z, v!d ~m 1 1 !

1 Id~t, m, v!, (3)

where d (x) is the Dirac distribution. For the sake ofclarity, the two components of the ballistic intensitypropagating toward z . 0 and z , 0 have been sepa-rated. Inserting Eq. (3) into Eq. (2) leads to

dIb6~z, v!

dz5 2a~v!Ib

6~z, v! (4)

for the ballistic component and to

1432 J. Opt. Soc. Am. A/Vol. 21, No. 8 /August 2004 Elaloufi et al.

m]Id~z, m, v!

]z5 2a~v!Id~z, m, v!

1ms

2E

21

11

p ~0 !~m, m8!Id~z, m8, v!dm8

1 S~z, m, v! (5)

for the diffuse component. In this last equation,S(z, m, v) is a source term that describes the transfer ofenergy from the ballistic to the diffuse component by scat-tering. Its expression will be given below.

The RTE deals with the specific intensity, which is a di-rectional quantity. Therefore the boundary conditions atthe slab interfaces can be accounted for exactly by use ofFresnel reflection and transmission factors. By takinginto account the internal reflections, the expressions ofthe ballistic components inside the slab are

Ib1~z, v! 5 T12~m 5 1 !I0~v!exp@2a~v!z#G, (6)

Ib2~z, v! 5 T12~m 5 1 !I0~v!exp@2a~v!

3 ~2L 2 z !#R23~m 5 1 !G, (7)

where G 5 @1 2 R12(m 5 1)R23(m 5 1)exp@22a(v)L)#21

and Rij(m) and Tij(m) are the Fresnel reflection andtransmission factors in energy at the interface betweentwo media of refractive indices ni and nj . Their expres-sion is given, for example, in Ref. 21. I0(v) is the time-domain Fourier transform of the incident pulse at theboundary z 5 0.

The source term in Eq. (5) is given by

S~z, m, v! 5ms

2p ~0 !~m, 1!Ib

1~z, v!

1ms

2p ~0 !~m, 21 !Ib

2~z, v!. (8)

For the diffuse components of the specific intensity, theboundary conditions at the slab surfaces are

Id~z 5 0, m, v! 5 R21~m!Id~z 5 0, 2m, v!,

for m . 0 (9)

Id~z 5 L, m, v! 5 R23~ umu!Id~z 5 L, 2m, v!,

for m , 0. (10)

Solving Eq. (5) with the above source term and boundaryconditions permits a computation of the diffuse transmit-ted intensity. The ballistic components (in transmissionand reflection) are directly obtained by Eqs. (6) and (7).The total transmitted intensity (ballistic 1 diffuse), ei-ther directional or angle-integrated,21 is the relevantquantity for the present study.

3. DECAY TIME AND EFFECTIVEDIFFUSION COEFFICIENTIn this section we analyze the decay time of pulses trans-mitted through scattering slabs on the basis of numericalsolutions of the time-dependent RTE. In all cases the in-cident pulse width is 50 fs, which is negligible comparedwith all time scales of the problem. The total transmit-ted intensity T(t) angle-integrated over the half-spacez . L is calculated numerically. The decay time t isextracted from the long-time exponential behaviorexp(2t/t) of T(t). In Fig. 1 we show the inverse decaytime t 21 versus the slab thickness L, for different valuesof the effective index n2 of the slab. The exterior mediumis assumed to be a vacuum (n1 5 n3 5 1). In Fig. 1(a)(isotropic scattering), a size dependence of t 21 versus L isvisible, which depends strongly on the slab refractive in-dex n2 . Moreover, the dependence on n2 is higher forsmall L. This can be understood with a simple picture.A wave propagating through the slab will undergo bothbulk scattering and wall reflections. The latter is verysensitive to the number of internal reflections, which in-creases with the refractive index n2 . For thin slabs, wallreflections may dominate over bulk scattering, so that theinfluence of internal reflections on the decay time be-comes predominant. This qualitatively explains thestrong influence of the refractive index observed for smallL. We shall come back to that point with more refinedarguments below. The same results hold for anisotropicscattering in Fig. 1(b). In particular, the behavior forsmall systems is weakly affected by scattering anisotropy.This is consistent with the fact that the relevant mecha-nism in this case is surface reflection more than bulk scat-tering.

To characterize more precisely the size dependence ofthe decay time, an effective diffusion coefficient Deff can

Fig. 1. Inverse decay time versus slab thickness L for different values of the medium effective refractive index n2 (n1 5 n3 5 1 for thehalf-spaces z , 0 and z . L). The medium parameters are ls 5 0.95 mm, la 5 46.5 mm (albedo v0 5 0.98). (a) g 5 0, (b) g 5 0.4.Phase function: Henyey–Greenstein.

Elaloufi et al. Vol. 21, No. 8 /August 2004 /J. Opt. Soc. Am. A 1433

Fig. 2. Effective diffusion coefficient Deff versus slab thickness L. The medium parameters are the same as in Fig. 1(a) with g 5 0. (a)Deff /L2 versus L; this quantity becomes independent of L for L @ l tr . (b) Deff versus L.

be introduced.15,23 To proceed, we identify the long-timeexponential behavior of the transmitted pulse exp(2t/t)with that obtained in the diffusion approximationexp(2p2Deff t/L2)exp(2mavt) (see, e.g., Ref. 21). Thisyields the effective diffusion coefficient:

Deff 5L2

p2 S 1

t2 mav D . (11)

Note that we have chosen to define Deff with the reallength L of the slab instead of an effective length Leff ac-counting for the boundary conditions used in diffusiontheory, as in Refs. 15 and 23. Several reasons justify thatchoice:

1. The effective length Leff is defined from approxi-mate boundary conditions valid asymptotically for a semi-infinite medium.28 The relevance of such boundary con-ditions for a thin slab are questionable.

2. For thick systems (L/ltr → `) one always hasL . Leff so that our definition of Deff coincides with thatused in Refs. 15 and 23.

3. We emphasize that the final purpose is to study thesize dependence that appears in the decay time t as a re-sult of nondiffusive behavior for small systems. There-fore one needs to introduce a quantity that suppresses thedirect dependence of t 21 on absorption [through the ex-ponential decay exp(2mavt)] and suppresses the diffusion-type dependence proportional to L22 that remains for in-finitely large systems. This is precisely what one obtainsby defining the effective diffusion coefficient as in Eq. (11).

4. Finally, we point out that correcting for the depen-dence on the absorption coefficient ma in Eq. (11) does notsuppress the dependence of the effective diffusion coeffi-cient on absorption. Indeed, recent studies have shownthat the diffusion coefficient depends on absorption in asubtle way and that this dependence is not negligible, forexample, in standard situations in biomedicalimaging.29–32

In the present study, we focus on the transition betweenregimes due to multiple scattering and have consideredmedia with low absorption only (although the numericalmethod allows us to handle arbitrary levels of absorp-tion).

In Fig. 2(a), we plot Deff /L2 versus L. This quantity issimilar to the inverse decay time t 21, but the influence of

absorption has been subtracted. It exhibits a behaviorsimilar to the decay time in Fig. 1(a), except for thick sys-tems (L @ ltr) where all curves tend to an asymptoticvalue that is independent of the refractive index of thescattering medium (i.e., on the level of internal reflec-tion). For small systems (L , 8 mm, which correspondto 7 –8ltr), the strong dependence on the refractive indexconfirms the dominant role of internal reflection on thesize dependence of the time decay t. The variations ofDeff versus L are represented in Fig. 2(b). On the curvecorresponding to n2 5 1, two regimes can be identified.For L , 7 –8ltr , a strong dependence on L is visible, andDeff increases with L. For L . 7 –8ltr , Deff tends asymp-totically to a constant value, which is expected to be thebulk value of the diffusion coefficient. If the effect of ab-sorption on the diffusion coefficient is neglected,29,32 thebulk value in this case is D 5 vltr/3 5 95 m2 s21. Forhigher values of the refractive index n2 , the generalshape of the curves remains the same, but the transitionis smoother and the value of Deff is reduced for all valuesof L. Also note that the effective diffusion coefficient de-fined in Eq. (11) always increases with L in the region ofstrong size dependence (L , 7 –8ltr). This is expectedbecause internal reflection yields an increase of the decaytime t and a decrease of Deff [see Eq. (11)].

The results in this section set forth the ability of theRTE to describe the transition from the diffusive regimeto a nondiffusive (anomalous) regime of transport whenthe size of the systems becomes of the order of a fewtransport mean free paths. In Section 4 we show that ex-perimental results can be reproduced by this approach.

4. COMPARISON WITH EXPERIMENTALRESULTSThe size dependence of the effective diffusion coefficient ofa slab was demonstrated experimentally in Ref. 15. Inthat study the authors chose to define the diffusion coef-ficient by comparing the long-time exponential decay ofthe diffuse transmission of short pulses with the solutionof the diffusion equation while accounting for extrapo-lated boundary conditions. For an absorbing medium,this leads to a diffusion coefficient D given by

D 5Leff

2

p2 S 1

t2 mav D . (12)

1434 J. Opt. Soc. Am. A/Vol. 21, No. 8 /August 2004 Elaloufi et al.

The effective length of the slab that appears in Eq. (12)is Leff 5 L 1 2z, where z 5 z0(1 1 R)/(1 2 R),z0 5 0.71ltr being the extrapolation distance and R themean reflection coefficient in the diffusionapproximation.33

For the numerical calculation of the pulse transmis-sion, we have chosen a configuration that is very close tothe experimental one.15 The only difference is that we donot consider a nonsymmetric system (both half-spaces z, 0 and z . L are filled with air or a vacuum so thatn1 5 n3 5 1). The scattering medium consists of TiO2particles of radius r 5 95 nm with g 5 0.27,ls 5 0.65 mm, and la 5 200 mm (weak absorption, albedov0 5 0.997). The central wavelength of the illuminatingpulse is l 5 780 nm. The effective index of the scatter-ing medium is n2 5 1.39.

We show in Fig. 3(a) the inverse decay time t 21 versusthe (real) system size L. The shape of the curve is simi-lar to that observed in Fig. 1(a), namely, a strong size de-pendence is observed for small systems. This regime ap-pears for L , 8ltr . The value of the transition(L . 8ltr) coincides with that observed experimentally.15

In Fig. 3(b), we represent the diffusion coefficient D nor-malized by its asymptotic value for large L (taken here atL 5 25 mm . 27ltr). This figure should be comparedwith Fig. 3 in Ref. 15. We see that the reduction of D atsmall scale, of about 50%, is reproduced by the calcula-tion, when Eq. (12) is used to define D from the decaytime t, with an effective length Leff 5 L 1 2z0 (i.e., withR 5 0). This is surprising because the result in Ref. 15was obtained with the value of R calculated from diffu-sion theory,33 which would be R 5 0.49 with our param-eters. Nevertheless, as we have already pointed out, therelevance of both the extrapolation distance z0 and thediffusion-theory reflection factor R for thin slabs is farfrom obvious. Moreover, as shown in the inset in Fig.3(b), changing the value of R [i.e., of Leff in Eq. (12)] com-pletely changes the shape of the curve. Although the ex-istence of two regimes and the value of the critical dis-tance (L . 8ltr) are not affected, one passes from anincrease to a decrease of D in the nondiffusive region bychanging R. This extreme sensitivity to R prevents the

diffusion coefficient D defined from the effective lengthLeff(R) from being a robust parameter. From the theoret-ical point of view, it relies on boundary conditions for dif-fusion theory that are not well controlled. From thepractical point of view, it depends on the precision withwhich the effective index n2 of the medium is known.Evaluating n2 with precision for a dense scattering me-dium such as that considered here remains a challengingissue.

In summary, the results in this section lead to the fol-lowing important conclusions: (1) The RTE, which ap-propriately handles the intensity transport outside thediffusive regime as well as the boundary conditions at theslab surfaces, allows us to describe the experimental re-sult presented in Ref. 15. (2) There is a high sensitivityof the diffusion coefficient D defined in Eq. (12) to the ef-fective length Leff . Obtaining a D that increases or de-creases with system size L depends on the value of Leffthat is used (and no exact determination of Leff can bemade). This is in agreement with the fact that the varia-tions of D with the system size in the nondiffusive regionare strongly related to internal reflections. (3) The tran-sition between the diffusive regime (D independent of L)and the nondiffusive regime is well described by the RTE.In particular, this means that the size dependence of D forsmall systems is not due to coherent effects or to depen-dent scattering, neither of which is accounted for in thepresent RTE approach.

5. DISPERSION RELATION FOR THERADIATIVE TRANSFER EQUATIONIn this section we introduce a simple model to describethe transition between the diffusive and nondiffusive re-gime. We borrow analytical solutions of the RTE fromneutron transport theory.26 In particular, for isotropicscattering, an analysis in terms of modes allows us to cal-culate analytically the dispersion relation for the time-dependent RTE in an infinite medium. To proceed, welook for solutions of the form I(z, m, t)5 g(m)exp(ikz)exp(st), with k real and s complex. Be-cause our interest is in the long-time decay of the inten-

Fig. 3. Inverse decay time and diffusion coefficient for a slab with parameters similar to those in Ref. 15. The slab contains TiO2particles illuminated at l 5 780 nm. g 5 0.27, ls 5 0.65 mm, la 5 200 mm (albedo v0 5 0.997). The effective index of the slab isn 5 1.39. (a) t 21 versus L, (b) diffusion coefficient D as defined in Ref. 15 normalized by its asymptotic value D0 ; solid curve,R 5 0. The inset shows the results obtained for different values of R. Phase function: Mie scattering.

Elaloufi et al. Vol. 21, No. 8 /August 2004 /J. Opt. Soc. Am. A 1435

Fig. 4. Dispersion relations @Re(s) versus k] in an infinite medium for the solutions of the RTE and of the diffusion equation; s and k arein dimensionless units, the reference length scale being L* 5 (ms 1 ma)21 and the reference time scale being t* 5 L* /v. The mediumparameters are ltr 5 0.95 mm, v0 5 0.995 (albedo). (a) s* versus k* for g 5 0. The numerical solution obtained from the RTE iscompared with an analytical result valid for isotropic scattering and k* , pv0/2 and with the solution obtained from the diffusion ap-proximation. (b) s* versus k* for g 5 0.5; no analytical solution can be found in this case. Phase function: Henyey–Greenstein.

sity, we need to compute the s whose (negative) real parthas the smallest absolute value. The associated decaytime is given by t 5 @Re(s)#21. The analysis in Ref. 26shows that

s 5 2~ma 1 ms!v 1kv

tan~k/ms!, for uku , pms/2,

(13)

where v0 5 ms /(ms 1 ma) is the albedo for single scatter-ing. For uku . pms/2 there is a continuum of solutions ofs for each value of k. We need only the solution s havingthe smallest real part, which can be calculated numeri-cally. The numerical calculation consists of solving an ei-genvalue problem that follows from the RTE when oneuses the discrete ordinate approach.21,27 In particular,this can be done with any type of phase function, allowingus to obtain the results for anisotropic scattering.

The same analysis can be done for the diffusionequation,1

]u~r, t !

]t2 D¹2u~r, t ! 1 mavu~r, t ! 5 0, (14)

where u(r, t) is the energy density. By looking for solu-tions of the form u(r, t) 5 u0 exp(ikz)exp(st), one obtainsthe dispersion relation for the diffusion equation:

s 5 2mav 2 k2D. (15)

The dispersion relations of both the RTE [Eq. (13)] andthe diffusion equation [Eq. (15)] are shown in Fig. 4. Wehave represented Re(s) versus k in dimensionless units(see the figure caption for details). For isotropic scatter-ing [Fig. 4(a)] we have represented the dispersion relationof the RTE calculated numerically and by using the ana-lytical formula Eq. (13), as well as the dispersion relationin the diffusion approximation. We see that both curvesfor the RTE are superimposed for k , pms/2. For largervalues of k, the analytical result is no longer valid. Thesolution s with the smallest real part is computed numeri-cally. We see an abrupt transition in the dependence of son k.

The dispersion relations for the RTE and the diffusionequation coincide for small values of k (k , 0.7ms if weneglect absorption). For larger values of k, the difference

increases and becomes significant. The spatial depen-dence of the modes being exp(ikz), large values of k corre-spond to small systems. Qualitatively, this result showsthat the diffusion approximation is able to predict thelong-time behavior of the intensity only for large systems.More quantitatively, the transition at k . 0.7ms corre-sponds to a system size L . 2p/k . 8ls , which is alsoL . 8ltr for isotropic scattering. This is in agreementwith the transition observed experimentally andnumerically.15,23 For anisotropic scattering [Fig. 4(b)],the same observations hold (note that the dispersion rela-tion of the RTE can be calculated only numerically in thiscase). We still observe the abrupt transition between thedomain where a single solution s is obtained for each k(for k* , 0.55 in dimensionless units) and the domain ofthe continuum of solutions [for which only those corre-sponding to the smallest value of Re(s) are represented].After the transition, s decays more slowly with k. If wecompare the curves obtained for the RTE and for the dif-fusion equation, we see that they coincide fork , 0.4ms . A substantial difference is observed fork . 0.5ms , a region for which the diffusion approxima-tion fails to describe intensity transport. The transitionoccurs in this case for system sizes of the order ofL 5 2p/k . 15ls , which, in terms of transport mean freepath gives L . 8ltr . Once again, this result is in re-markable agreement with that observed in Refs. 15 and23.

The arguments based on the dispersion relation of themodes of an infinite system do not account for the condi-tions at the system boundaries. Nevertheless, they allowus to describe the position of the transition between thediffusive and the nondiffusive regimes. Therefore, it canbe concluded from this analysis that the transition shouldalways appear when the size of the system is reduced,whatever the boundary conditions (and the level of inter-nal reflection). In the absence of absorption, the transi-tion takes place for system sizes of the order of 8ltr .

The way the decay time or the effective diffusion coef-ficient depends on system size L in the nondiffusive re-gion strongly depends on the boundary conditions. Thishas been shown in Section 2 as well as in the study in Ref.23. Although these effects are not accounted for in the

1436 J. Opt. Soc. Am. A/Vol. 21, No. 8 /August 2004 Elaloufi et al.

analysis based on the dispersion relation, it is interestingto see which dependence on L is predicted in this ap-proach. To proceed, we start from Eq. (13) by recallingthat the decay time t 5 @Re(s)#21. We now assume thatthe predominant mode for a slab of size L corresponds tok 5 p/L. Replacing k by this value in Eq. (13) leads to

t 5 F ~ma 1 ms!v 2kv

tan~k/ms!G21

. (16)

From this expression of the decay time, an effective diffu-sion coefficient can be introduced by using Eq. (11). Oneobtains

Deff 5L2

p2msv 2

Lv

p tan~p/msL !. (17)

Note that for large systems (msL → `), one hastan(p/msL) . (p/msL) 1 (p/msL)3/3, so that Deff . v/(3ms).One recovers the well-known result for the bulk diffusioncoefficient in a nonabsorbing medium.1

We plot in Fig. 5(a) the variations of t21 versus systemsize L as predicted in Eq. (16). We see that the result isin good qualitative agreement with the curves obtained inFig. 1(a), especially with that corresponding to n2 5 1 (nointernal reflections). In Fig. 5(b), we plot the effectivediffusion coefficient given in Eq. (17). We see that thetransition between the diffusive regime (Deff independentof L) and the nondiffusive regime is very well reproduced,the transition occuring at L . 8ltr . This confirms thatthe origin of the transition lies in the deviation from dif-fusion theory that is well described by the mode analysisand the dispersion relation (and not in the boundary con-ditions). Concerning the variation of Deff in the nondif-fusive region, the increase that is observed when L de-creases disagrees with the result of the full numericalcalculations presented in Fig. 2(b), which always shows adecrease of Deff in the nondiffusive region. This resultconfirms the important role of the boundary conditions atthe slab surfaces on the variation of the effective diffusioncoefficient in the nondiffusive region.

6. CONCLUSIONWe have presented a numerical and theoretical study ofthe transition from the diffusive to the nondiffusive(quasi-ballistic) regime in dynamic light transmissionthrough thin scattering slabs. The analysis is based onthe time-dependent RTE. A key feature of this approachis the possibility of writing exactly the boundary condi-tions at the system surfaces. Also, it allows us to handleany scattering and absorbing properties of the medium.By using numerical calculations of transmitted pulsesthrough slabs of varying thickness L, several results havebeen obtained: (1) We have shown that the decay timeand effective diffusion coefficient display a transition be-tween a diffusive and a nondiffusive regime that occursfor L . 8ltr . In the nondiffusive region (L , 8ltr) theeffective diffusion coefficient exhibits a strong dependenceon L. With this approach, we have reproduced quantita-tively an experimental result showing a reduction ofabout 50% of the effective diffusion coefficient in the non-diffusive region.15 This result demonstrates the rel-evance of the RTE for studying dynamic light transportthrough thin scattering systems when coherent effectsplay no significant role. (2) The deviation from diffusiontheory for thin systems that we observe also supports theclaims of Ref. 15 that the diffusion approximation is ableto describe the long-time behavior of transmitted pulsesfor thick systems only (L . 8ltr). (3) The dependence ofthe effective diffusion coefficient on the slab thickness inthe nondiffusive regime is strongly affected by theamount of internal reflection inside the slab. This con-clusion is in agreement with a recent theoretical study re-ported in Ref. 23. (4) Finally, we have presented a simplemodel based on the dispersion relation of the RTE and ofthe diffusion equation. This crude model gives an esti-mate of the critical size L . 8ltr below which the trans-port is nondiffusive. We think that this study provides avaluable tool for analyzing experimental data in confinedsystems (with sizes on the order of the transport meanfree path) for which size effects and boundary reflectionsplay a significant role. A particularly important issue isthe retrieval of the scattering and absorption coefficientsfrom scattered-light measurements in thin systems. Fu-ture work will be directed along these lines.

Fig. 5. (a) Decay time versus slab thickness L obtained from the analytical model Eq. (16). (b) Effective diffusion coefficient for slabthickness L obtained from the analytical model Eq. (17). The medium parameters are l tr 5 0.95 mm, v0 5 0.995.

Elaloufi et al. Vol. 21, No. 8 /August 2004 /J. Opt. Soc. Am. A 1437

Corresponding author Remi Carminati’s e-mail addressis remi.carminati@em2c.ecp.fr.

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