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Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach Rachid Elaloufi, Re´mi Carminati, and Jean-Jacques Greffet Laboratoire d’Energe´tique Mole´culaire et Macroscopique, Combustion; Ecole Centrale Paris, Centre National de la Recherche Scientifique, 92295 Chaˆtenay-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 scattering slabs. Solving numerically the time-dependent radiative transfer equation, we obtain the decay time and the effective diffusion coefficient D eff . We observe a nondiffusive behavior for systems whose thickness L is smaller than 8l tr , where l tr is the transport mean free path. We introduce a simple model that yields the position of the transition between the diffusive and the nondiffusive regimes. The size dependence of D eff in the nondiffusive region is strongly affected by internal reflections. We show that the reduction of ⬃50% of D eff that was observed experimentally [Phys. Rev. Lett. 79, 4369 (1997)] can be reproduced by the radiative transfer approach. We demonstrate that the radiative transfer equation is an appropriate tool for studying dynamic light transport in thin scattering systems when coherent effects play no significant role. © 2004 Optical Society of America OCIS codes: 290.1990, 290.7050, 290.4210, 170.5280, 170.3660.
1. INTRODUCTION The study of wave propagation through random media has been an active field of investigation in the past few decades.1 Recently it has attracted considerable interest through the development of mesoscopic physics2,3 and imaging techniques in strongly scattering biological tissues.4,5 Time-resolved techniques using pulse reflection or transmission on short time scales,6 opticalcoherence tomography,7 and diffusing wave 8,9 spectroscopy give promising results. In several other areas, diffusive waves—such as thermal, acoustic, or elastic waves—form the basis of imaging and measurement techniques.10 With the rapid development of microtechnologies and nanotechnologies, understanding the propagation of such waves at short (time and length) scales has become a key issue. Another example is heat conduction at short scales in solids, which can be handled on the basis of a Boltzman transport equation for phonons that undergo scattering, emission, and absorption.11,12 This problem is very similar to that encountered in opticalwave propagation through random media. The study of dynamic wave transport through scattering and absorbing media is considerably simplified when using the diffusion approximation.1 The approximation is widely used in the context of optical imaging through biological tissues.4,5 Yet limitations appear when the size of the system (or the time scale) becomes of the order of the mean free path (or the collision time). Light transport at short time scales in semi-infinite media and thick slabs (with sizes much larger than the transport mean free path) has been studied recently by use of an approximate solution of the radiative transfer equation (RTE).13 In particular the transition from the short-time, quasiballistic 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 the transition between the different regimes versus the length scale of the system. Several experimental results have demonstrated strong deviations from diffusion theory in dynamic light scattering at short-length scales by measuring pulse transmission through thin slabs14–16 or in diffusing wave spectroscopy experiments.17,18 The results in Refs. 14 and 15 show that the effects of the interplay between space and time scales on the validity of diffusion approximation is a difficult issue. On the one hand, experiments reported in Ref. 14 seem to show that diffusion theory gives an accurate prediction for the longtime decay of transmitted pulses for both thin (L ⬍ 10l tr) and thick systems, whereas it fails for the short-time behavior. On the other hand, experiments reported in Ref. 15 show that diffusion theory fails for thin systems (L ⬍ 8l tr), even for the determination of the long-time behavior. The domain of validity of the diffusion approximation for time-dependent transport has been studied by comparison with the prediction of the RTE,19–21 or by using the telegrapher’s equation.22 In particular, the results in Ref. 21 have confirmed that the diffusion approximation is able to predict the long-time behavior of transmitted pulses through systems of size L ⬎ 8l tr , where l tr is the transport mean free path, in agreement with experimental results.15 More recently, the transition from ballistic to diffusive transport was analyzed by solving the Bethe– Salpeter equation for a slab in the lowest-order ladder approximation with isotropic scattering.23 A region of strong deviation from the diffusion approximation was found for 3l tr ⬍ L ⬍ L c where L c is a critical length that depends on the amount of internal reflection at the slab boundaries. The region of nondiffusive transport agrees © 2004 Optical Society of America
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with experimental results.15 Nevertheless, as far as the transition from the ballistic to the diffusive regime is concerned, an overview of the literature leads to the following remarks: 1. The transport model used in Ref. 23 predicts an increase of the effective diffusion coefficient of the slab when the thickness L decreases in the nondiffusive region (sometimes referred to as the anomalous region). This is the opposite of the behavior observed experimentally,15,24 and no convincing argument has been proposed to explain this contradiction. 2. The samples used in practice are usually absorbing and anisotropically scattering. A transport theory able to account for arbitrary scattering and absorbing properties and for rigorous boundary conditions at the slab boundaries is necessary. The RTE seems to be the adequate tool in this context. 3. Last but not least, a simple model explaining (at least qualitatively) the onset of the nondiffusive behavior at small length scales is still missing. In this paper, we present a theoretical and numerical study of the transition from the ballistic to the diffusive regime in light transport through scattering media. Our approach is based on the RTE.25,26 By a numerical study of pulse transmission through scattering slabs, we study the size dependence of the decay time and the effective diffusion coefficient. The RTE approach allows us to handle arbitrary scattering properties of the particles as well as absorption. Internal reflections at the slab boundaries are treated rigorously by means of reflection and transmission factors for both the ballistic and the diffusive part of the light intensity. The results confirm the sensitivity of the effective diffusion coefficient D eff to the level of internal reflection. They also show under which condition a reduction of D eff at small scale is observed, and the experimental results in Ref. 15 are retrieved quantitatively. Finally, a simple analysis based on the dispersion relation of the RTE26 is used to describe the transition from the diffusive to the nondiffusive regime. This crude model allows us to retrieve the critical size at which the transition occurs. The paper is organized as follows. In Section 2 we briefly describe the numerical method used to solve the RTE in a slab geometry. In Section 3 we present numerical calculations of the decay time and effective diffusion coefficient for systems of various sizes and mean (effective) indices of refraction. In Section 4 we discuss the reduction of the effective diffusion coefficient in connection with experimental results. In Section 5 we present the analytical model and use it to discuss the deviation from diffusion theory for small systems. In Section 6 we summarize the main results and give our conclusions.
2. TIME-DEPENDENT RADIATIVE TRANSFER EQUATION We consider a slab of width L with the z axis normal to the boundaries (the strip 0 ⬍ z ⬍ L is filled with the scattering medium). The slab is illuminated from the
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region z ⬍ 0 at normal incidence by a plane-wave pulse. The specific intensity I(z, , t) inside the scattering medium obeys the RTE25,26: 1 I 共 z, , t 兲
t
v
I 共 z, , t 兲
⫹
z
⫽ ⫺共 s ⫹ a 兲 I 共 z, , t 兲 ⫹
s 2
冕
⫹1
⫺1
p 共 0 兲 共 , ⬘ 兲 I 共 z, ⬘ , t 兲 d ⬘ ,
(1)
where v is the energy velocity and ⫽ cos , with the angle between the propagation direction and the z axis. p ( 0 ) is the phase function integrated over the azimuthal angle p ( 0 ) ( , ⬘ ) ⫽ 1/2 兰 02 p(u – u⬘ )d where u and u⬘ are unit vectors corresponding to directions (, ) and (⬘, ⬘). s and a are the scattering and absorption coefficients, respectively. The associated scattering and absorption mean free paths are l a ⫽ a⫺1 and l s ⫽ ⫺1 s . The transport mean free path l tr ⫽ l s /(1 ⫺ g), where g is the anisotropy factor (average cosine of the scattering angle). The real part of the medium’s effective index, accounting for both the homogeneous background medium and the scattering particles, is denoted by n 2 . The halfspaces z ⬍ 0 and z ⬎ L are filled with homogeneous and transparent materials of refractive indices n 1 and n 3 , respectively. To solve Eq. (1), we use the space-frequency method described in Ref. 21. We briefly recall its principle here. A time-domain Fourier transform of Eq. (1) leads to
I 共 z, , 兲 z
冉
⫽ ⫺ s ⫹ a ⫺ i ⫹
s 2
冕
⫹1
⫺1
v
冊
I 共 z, , 兲
p 共 0 兲 共 , ⬘ 兲 I 共 z, ⬘ , 兲 d ⬘ , (2)
where I(z, , ) is the time-domain Fourier transform of I(z, , t). This equation is similar to the static RTE with an effective extinction coefficient ␣ ( ) ⫽ ( s ⫹ a ⫺ i /v). It can be solved numerically by standard methods developed for time-independent problems.27 In assuming illumination by a plane wave (representing, for example, a collimated laser beam), it is useful to separate the ballistic and the diffuse components of the specific intensity inside the medium. One writes I 共 z, , 兲 ⫽ I b⫹共 z, 兲 ␦ 共 ⫺ 1 兲 ⫹ I b⫺共 z, 兲 ␦ 共 ⫹ 1 兲 ⫹ I d共 , , 兲 ,
(3)
where ␦ (x) is the Dirac distribution. For the sake of clarity, the two components of the ballistic intensity propagating toward z ⬎ 0 and z ⬍ 0 have been separated. Inserting Eq. (3) into Eq. (2) leads to dI b⫾共 z, 兲 dz
⫽ ⫺␣ 共 兲 I b⫾共 z, 兲
for the ballistic component and to
(4)
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I d 共 z, , 兲 z
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⫽ ⫺␣ 共 兲 I d 共 z, , 兲 ⫹
s 2
冕
⫹1
⫺1
p 共 0 兲 共 , ⬘ 兲 I d 共 z, ⬘ , 兲 d ⬘
⫹ S 共 z, , 兲
(5)
for the diffuse component. In this last equation, S(z, , ) is a source term that describes the transfer of energy from the ballistic to the diffuse component by scattering. Its expression will be given below. The RTE deals with the specific intensity, which is a directional quantity. Therefore the boundary conditions at the slab interfaces can be accounted for exactly by use of Fresnel reflection and transmission factors. By taking into account the internal reflections, the expressions of the ballistic components inside the slab are I b⫹共 z, 兲 ⫽ T 12共 ⫽ 1 兲 I 0 共 兲 exp关 ⫺␣ 共 兲 z 兴 ⌫,
(6)
I b⫺共 z, 兲 ⫽ T 12共 ⫽ 1 兲 I 0 共 兲 exp关 ⫺␣ 共 兲 ⫻ 共 2L ⫺ z 兲兴 R 23共 ⫽ 1 兲 ⌫,
(7)
where ⌫ ⫽ 关 1 ⫺ R 12( ⫽ 1)R 23( ⫽ 1)exp关⫺2␣()L)兴⫺1 and R ij ( ) and T ij ( ) are the Fresnel reflection and transmission factors in energy at the interface between two media of refractive indices n i and n j . Their expression is given, for example, in Ref. 21. I 0 ( ) is the timedomain Fourier transform of the incident pulse at the boundary z ⫽ 0. The source term in Eq. (5) is given by S 共 z, , 兲 ⫽
s 2 ⫹
p 共 0 兲 共 , 1兲 I b⫹共 z, 兲
s 2
p 共 0 兲 共 , ⫺1 兲 I b⫺共 z, 兲 .
(8)
For the diffuse components of the specific intensity, the boundary conditions at the slab surfaces are I d 共 z ⫽ 0, , 兲 ⫽ R 21共 兲 I d 共 z ⫽ 0, ⫺ , 兲 , for ⬎ 0
(9)
I d 共 z ⫽ L, , 兲 ⫽ R 23共 兩 兩 兲 I d 共 z ⫽ L, ⫺ , 兲 , for ⬍ 0.
(10)
Solving Eq. (5) with the above source term and boundary conditions permits a computation of the diffuse transmitted intensity. The ballistic components (in transmission and reflection) are directly obtained by Eqs. (6) and (7). The total transmitted intensity (ballistic ⫹ diffuse), either directional or angle-integrated,21 is the relevant quantity for the present study.
3. DECAY TIME AND EFFECTIVE DIFFUSION COEFFICIENT In this section we analyze the decay time of pulses transmitted through scattering slabs on the basis of numerical solutions of the time-dependent RTE. In all cases the incident pulse width is 50 fs, which is negligible compared with all time scales of the problem. The total transmitted intensity T(t) angle-integrated over the half-space z ⬎ L is calculated numerically. The decay time is extracted from the long-time exponential behavior exp(⫺t/) of T(t). In Fig. 1 we show the inverse decay time ⫺1 versus the slab thickness L, for different values of the effective index n 2 of the slab. The exterior medium is assumed to be a vacuum (n 1 ⫽ n 3 ⫽ 1). In Fig. 1(a) (isotropic scattering), a size dependence of ⫺1 versus L is visible, which depends strongly on the slab refractive index n 2 . Moreover, the dependence on n 2 is higher for small L. This can be understood with a simple picture. A wave propagating through the slab will undergo both bulk scattering and wall reflections. The latter is very sensitive to the number of internal reflections, which increases with the refractive index n 2 . For thin slabs, wall reflections may dominate over bulk scattering, so that the influence of internal reflections on the decay time becomes predominant. This qualitatively explains the strong influence of the refractive index observed for small L. We shall come back to that point with more refined arguments below. The same results hold for anisotropic scattering in Fig. 1(b). In particular, the behavior for small systems is weakly affected by scattering anisotropy. This is consistent with the fact that the relevant mechanism in this case is surface reflection more than bulk scattering. To characterize more precisely the size dependence of the decay time, an effective diffusion coefficient D eff can
Fig. 1. Inverse decay time versus slab thickness L for different values of the medium effective refractive index n 2 (n 1 ⫽ n 3 ⫽ 1 for the half-spaces z ⬍ 0 and z ⬎ L). The medium parameters are l s ⫽ 0.95 m, l a ⫽ 46.5 m (albedo 0 ⫽ 0.98). (a) g ⫽ 0, (b) g ⫽ 0.4. Phase function: Henyey–Greenstein.
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Fig. 2. Effective diffusion coefficient D eff versus slab thickness L. The medium parameters are the same as in Fig. 1(a) with g ⫽ 0. (a) D eff /L2 versus L; this quantity becomes independent of L for L Ⰷ l tr . (b) D eff versus L.
be introduced.15,23 To proceed, we identify the long-time exponential behavior of the transmitted pulse exp(⫺t/) with that obtained in the diffusion approximation exp(⫺2Deff t/L2)exp(⫺avt) (see, e.g., Ref. 21). This yields the effective diffusion coefficient: D eff ⫽
冉
L2 1
2
冊
⫺ av .
(11)
Note that we have chosen to define D eff with the real length L of the slab instead of an effective length L eff accounting for the boundary conditions used in diffusion theory, as in Refs. 15 and 23. Several reasons justify that choice: 1. The effective length L eff is defined from approximate boundary conditions valid asymptotically for a semiinfinite medium.28 The relevance of such boundary conditions for a thin slab are questionable. 2. For thick systems (L/l tr → ⬁) one always has L ⯝ L eff so that our definition of D eff coincides with that used in Refs. 15 and 23. 3. We emphasize that the final purpose is to study the size dependence that appears in the decay time as a result of nondiffusive behavior for small systems. Therefore one needs to introduce a quantity that suppresses the direct dependence of ⫺1 on absorption [through the exponential decay exp(⫺avt)] and suppresses the diffusiontype dependence proportional to L ⫺2 that remains for infinitely large systems. This is precisely what one obtains by defining the effective diffusion coefficient as in Eq. (11). 4. Finally, we point out that correcting for the dependence on the absorption coefficient a in Eq. (11) does not suppress the dependence of the effective diffusion coefficient on absorption. Indeed, recent studies have shown that the diffusion coefficient depends on absorption in a subtle way and that this dependence is not negligible, for example, in standard situations in biomedical imaging.29–32 In the present study, we focus on the transition between regimes due to multiple scattering and have considered media with low absorption only (although the numerical method allows us to handle arbitrary levels of absorption). In Fig. 2(a), we plot D eff /L2 versus L. This quantity is similar to the inverse decay time ⫺1 , but the influence of
absorption has been subtracted. It exhibits a behavior similar to the decay time in Fig. 1(a), except for thick systems (L Ⰷ l tr) where all curves tend to an asymptotic value that is independent of the refractive index of the scattering medium (i.e., on the level of internal reflection). For small systems (L ⬍ 8 m, which correspond to 7 – 8l tr), the strong dependence on the refractive index confirms the dominant role of internal reflection on the size dependence of the time decay . The variations of D eff versus L are represented in Fig. 2(b). On the curve corresponding to n 2 ⫽ 1, two regimes can be identified. For L ⬍ 7 – 8l tr , a strong dependence on L is visible, and D eff increases with L. For L ⬎ 7 – 8l tr , D eff tends asymptotically to a constant value, which is expected to be the bulk value of the diffusion coefficient. If the effect of absorption on the diffusion coefficient is neglected,29,32 the bulk value in this case is D ⫽ vl tr/3 ⫽ 95 m2 s⫺1 . For higher values of the refractive index n 2 , the general shape of the curves remains the same, but the transition is smoother and the value of D eff is reduced for all values of L. Also note that the effective diffusion coefficient defined in Eq. (11) always increases with L in the region of strong size dependence (L ⬍ 7 – 8l tr). This is expected because internal reflection yields an increase of the decay time and a decrease of D eff [see Eq. (11)]. The results in this section set forth the ability of the RTE to describe the transition from the diffusive regime to a nondiffusive (anomalous) regime of transport when the size of the systems becomes of the order of a few transport mean free paths. In Section 4 we show that experimental results can be reproduced by this approach.
4. COMPARISON WITH EXPERIMENTAL RESULTS The size dependence of the effective diffusion coefficient of a slab was demonstrated experimentally in Ref. 15. In that study the authors chose to define the diffusion coefficient by comparing the long-time exponential decay of the diffuse transmission of short pulses with the solution of the diffusion equation while accounting for extrapolated boundary conditions. For an absorbing medium, this leads to a diffusion coefficient D given by D⫽
冉
2 L eff 1
2
冊
⫺ av .
(12)
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The effective length of the slab that appears in Eq. (12) ¯ )/(1 ⫺ R ¯ ), is L eff ⫽ L ⫹ 2z, where z ⫽ z 0 (1 ⫹ R ¯ z 0 ⫽ 0.71l tr being the extrapolation distance and R the mean reflection coefficient in the diffusion approximation.33 For the numerical calculation of the pulse transmission, we have chosen a configuration that is very close to the experimental one.15 The only difference is that we do not consider a nonsymmetric system (both half-spaces z ⬍ 0 and z ⬎ L are filled with air or a vacuum so that n 1 ⫽ n 3 ⫽ 1). The scattering medium consists of TiO2 particles of radius r ⫽ 95 nm with g ⫽ 0.27, l s ⫽ 0.65 m, and l a ⫽ 200 m (weak absorption, albedo 0 ⫽ 0.997). The central wavelength of the illuminating pulse is ⫽ 780 nm. The effective index of the scattering medium is n 2 ⫽ 1.39. We show in Fig. 3(a) the inverse decay time ⫺1 versus the (real) system size L. The shape of the curve is similar to that observed in Fig. 1(a), namely, a strong size dependence is observed for small systems. This regime appears for L ⬍ 8l tr . The value of the transition (L ⯝ 8l tr) coincides with that observed experimentally.15 In Fig. 3(b), we represent the diffusion coefficient D normalized by its asymptotic value for large L (taken here at L ⫽ 25 m ⯝ 27l tr). This figure should be compared with Fig. 3 in Ref. 15. We see that the reduction of D at small scale, of about 50%, is reproduced by the calculation, when Eq. (12) is used to define D from the decay time , with an effective length L eff ⫽ L ⫹ 2z0 (i.e., with ¯ ⫽ 0). This is surprising because the result in Ref. 15 R ¯ calculated from diffuwas obtained with the value of R 33 ¯ sion theory, which would be R ⫽ 0.49 with our parameters. Nevertheless, as we have already pointed out, the relevance of both the extrapolation distance z 0 and the ¯ for thin slabs is far diffusion-theory reflection factor R from obvious. Moreover, as shown in the inset in Fig. ¯ [i.e., of L in Eq. (12)] com3(b), changing the value of R eff pletely changes the shape of the curve. Although the existence of two regimes and the value of the critical distance (L ⯝ 8l tr) are not affected, one passes from an increase to a decrease of D in the nondiffusive region by ¯ . This extreme sensitivity to R ¯ prevents the changing R
Elaloufi et al.
diffusion coefficient D defined from the effective length ¯ ) from being a robust parameter. From the theoretL eff(R ical point of view, it relies on boundary conditions for diffusion theory that are not well controlled. From the practical point of view, it depends on the precision with which the effective index n 2 of the medium is known. Evaluating n 2 with precision for a dense scattering medium such as that considered here remains a challenging issue. In summary, the results in this section lead to the following important conclusions: (1) The RTE, which appropriately handles the intensity transport outside the diffusive regime as well as the boundary conditions at the slab surfaces, allows us to describe the experimental result presented in Ref. 15. (2) There is a high sensitivity of the diffusion coefficient D defined in Eq. (12) to the effective length L eff . Obtaining a D that increases or decreases with system size L depends on the value of L eff that is used (and no exact determination of L eff can be made). This is in agreement with the fact that the variations of D with the system size in the nondiffusive region are strongly related to internal reflections. (3) The transition 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 for small systems is not due to coherent effects or to dependent scattering, neither of which is accounted for in the present RTE approach.
5. DISPERSION RELATION FOR THE RADIATIVE TRANSFER EQUATION In this section we introduce a simple model to describe the transition between the diffusive and nondiffusive regime. We borrow analytical solutions of the RTE from neutron transport theory.26 In particular, for isotropic scattering, an analysis in terms of modes allows us to calculate analytically the dispersion relation for the timedependent RTE in an infinite medium. To proceed, we look for solutions of the form I(z, , t) ⫽ g( )exp(ikz)exp(st), with k real and s complex. Because 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 TiO2 particles illuminated at ⫽ 780 nm. g ⫽ 0.27, l s ⫽ 0.65 m, l a ⫽ 200 m (albedo 0 ⫽ 0.997). The effective index of the slab is n ⫽ 1.39. (a) ⫺1 versus L, (b) diffusion coefficient D as defined in Ref. 15 normalized by its asymptotic value D 0 ; solid curve, ¯ ⫽ 0. The inset shows the results obtained for different values of R ¯ . Phase function: Mie scattering. R
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Fig. 4. Dispersion relations 关 Re(s) versus k] in an infinite medium for the solutions of the RTE and of the diffusion equation; s and k are in dimensionless units, the reference length scale being L * ⫽ ( s ⫹ a ) ⫺1 and the reference time scale being t * ⫽ L * /v. The medium parameters are l tr ⫽ 0.95 m, 0 ⫽ 0.995 (albedo). (a) s * versus k * for g ⫽ 0. The numerical solution obtained from the RTE is compared with an analytical result valid for isotropic scattering and k * ⬍ 0 /2 and with the solution obtained from the diffusion approximation. (b) s * versus k * for g ⫽ 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 part has the smallest absolute value. The associated decay time is given by ⫽ 关 Re(s)兴⫺1. The analysis in Ref. 26 shows that s ⫽ ⫺共 a ⫹ s 兲 v ⫹
kv tan共 k/ s 兲
,
for 兩 k 兩 ⬍ s /2, (13)
where 0 ⫽ s /( s ⫹ a ) is the albedo for single scattering. For 兩 k 兩 ⬎ s /2 there is a continuum of solutions of s for each value of k. We need only the solution s having the smallest real part, which can be calculated numerically. The numerical calculation consists of solving an eigenvalue problem that follows from the RTE when one uses the discrete ordinate approach.21,27 In particular, this can be done with any type of phase function, allowing us to obtain the results for anisotropic scattering. The same analysis can be done for the diffusion equation,1
u 共 r, t 兲 t
⫺ Dⵜ 2 u 共 r, t 兲 ⫹ a vu 共 r, t 兲 ⫽ 0,
(14)
where u(r, t) is the energy density. By looking for solutions of the form u(r, t) ⫽ u 0 exp(ikz)exp(st), one obtains the dispersion relation for the diffusion equation: s ⫽ ⫺ a v ⫺ k 2 D.
(15)
The dispersion relations of both the RTE [Eq. (13)] and the diffusion equation [Eq. (15)] are shown in Fig. 4. We have represented Re(s) versus k in dimensionless units (see the figure caption for details). For isotropic scattering [Fig. 4(a)] we have represented the dispersion relation of the RTE calculated numerically and by using the analytical formula Eq. (13), as well as the dispersion relation in the diffusion approximation. We see that both curves for the RTE are superimposed for k ⬍ s /2. For larger values of k, the analytical result is no longer valid. The solution s with the smallest real part is computed numerically. We see an abrupt transition in the dependence of s on k. The dispersion relations for the RTE and the diffusion equation coincide for small values of k (k ⬍ 0.7 s if we neglect absorption). For larger values of k, the difference
increases and becomes significant. The spatial dependence of the modes being exp(ikz), large values of k correspond to small systems. Qualitatively, this result shows that the diffusion approximation is able to predict the long-time behavior of the intensity only for large systems. More quantitatively, the transition at k ⯝ 0.7 s corresponds to a system size L ⯝ 2 /k ⯝ 8l s , which is also L ⯝ 8l tr for isotropic scattering. This is in agreement with the transition observed experimentally and numerically.15,23 For anisotropic scattering [Fig. 4(b)], the same observations hold (note that the dispersion relation of the RTE can be calculated only numerically in this case). We still observe the abrupt transition between the domain where a single solution s is obtained for each k (for k * ⬍ 0.55 in dimensionless units) and the domain of the continuum of solutions [for which only those corresponding to the smallest value of Re(s) are represented]. After the transition, s decays more slowly with k. If we compare the curves obtained for the RTE and for the diffusion equation, we see that they coincide for k ⬍ 0.4 s . A substantial difference is observed for k ⬎ 0.5 s , a region for which the diffusion approximation fails to describe intensity transport. The transition occurs in this case for system sizes of the order of L ⫽ 2 /k ⯝ 15l s , which, in terms of transport mean free path gives L ⯝ 8l tr . Once again, this result is in remarkable agreement with that observed in Refs. 15 and 23. The arguments based on the dispersion relation of the modes of an infinite system do not account for the conditions at the system boundaries. Nevertheless, they allow us to describe the position of the transition between the diffusive and the nondiffusive regimes. Therefore, it can be concluded from this analysis that the transition should always appear when the size of the system is reduced, whatever the boundary conditions (and the level of internal reflection). In the absence of absorption, the transition takes place for system sizes of the order of 8l tr . The way the decay time or the effective diffusion coefficient depends on system size L in the nondiffusive region strongly depends on the boundary conditions. This has been shown in Section 2 as well as in the study in Ref. 23. Although these effects are not accounted for in the
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analysis based on the dispersion relation, it is interesting to see which dependence on L is predicted in this approach. To proceed, we start from Eq. (13) by recalling that the decay time ⫽ 关 Re(s)兴⫺1. We now assume that the predominant mode for a slab of size L corresponds to k ⫽ /L. Replacing k by this value in Eq. (13) leads to
冋
⫽ 共 a ⫹ s兲v ⫺
kv tan共 k/ s 兲
册
⫺1
.
(16)
From this expression of the decay time, an effective diffusion coefficient can be introduced by using Eq. (11). One obtains
D eff ⫽
L2
2
sv ⫺
Lv
tan共 / s L 兲
.
(17)
Note that for large systems ( s L → ⬁), one has tan(/sL) ⯝ (/sL) ⫹ (/sL)3/3, so that D eff ⯝ v/(3s). One recovers the well-known result for the bulk diffusion coefficient in a nonabsorbing medium.1 We plot in Fig. 5(a) the variations of ⫺1 versus system size L as predicted in Eq. (16). We see that the result is in good qualitative agreement with the curves obtained in Fig. 1(a), especially with that corresponding to n 2 ⫽ 1 (no internal reflections). In Fig. 5(b), we plot the effective diffusion coefficient given in Eq. (17). We see that the transition between the diffusive regime (D eff independent of L) and the nondiffusive regime is very well reproduced, the transition occuring at L ⯝ 8l tr . This confirms that the origin of the transition lies in the deviation from diffusion theory that is well described by the mode analysis and the dispersion relation (and not in the boundary conditions). Concerning the variation of D eff in the nondiffusive region, the increase that is observed when L decreases disagrees with the result of the full numerical calculations presented in Fig. 2(b), which always shows a decrease of D eff in the nondiffusive region. This result confirms the important role of the boundary conditions at the slab surfaces on the variation of the effective diffusion coefficient in the nondiffusive region.
6. CONCLUSION We have presented a numerical and theoretical study of the transition from the diffusive to the nondiffusive (quasi-ballistic) regime in dynamic light transmission through thin scattering slabs. The analysis is based on the time-dependent RTE. A key feature of this approach is the possibility of writing exactly the boundary conditions at the system surfaces. Also, it allows us to handle any scattering and absorbing properties of the medium. By using numerical calculations of transmitted pulses through slabs of varying thickness L, several results have been obtained: (1) We have shown that the decay time and effective diffusion coefficient display a transition between a diffusive and a nondiffusive regime that occurs for L ⯝ 8l tr . In the nondiffusive region (L ⬍ 8l tr) the effective diffusion coefficient exhibits a strong dependence on L. With this approach, we have reproduced quantitatively an experimental result showing a reduction of about 50% of the effective diffusion coefficient in the nondiffusive region.15 This result demonstrates the relevance of the RTE for studying dynamic light transport through thin scattering systems when coherent effects play no significant role. (2) The deviation from diffusion theory for thin systems that we observe also supports the claims of Ref. 15 that the diffusion approximation is able to describe the long-time behavior of transmitted pulses for thick systems only (L ⬎ 8l tr). (3) The dependence of the effective diffusion coefficient on the slab thickness in the nondiffusive regime is strongly affected by the amount of internal reflection inside the slab. This conclusion is in agreement with a recent theoretical study reported in Ref. 23. (4) Finally, we have presented a simple model based on the dispersion relation of the RTE and of the diffusion equation. This crude model gives an estimate of the critical size L ⯝ 8l tr below which the transport is nondiffusive. We think that this study provides a valuable tool for analyzing experimental data in confined systems (with sizes on the order of the transport mean free path) for which size effects and boundary reflections play a significant role. A particularly important issue is the retrieval of the scattering and absorption coefficients from scattered-light measurements in thin systems. Future 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 slab thickness L obtained from the analytical model Eq. (17). The medium parameters are l tr ⫽ 0.95 m, 0 ⫽ 0.995.
Elaloufi et al.
Corresponding author Remi Carminati’s e-mail address is
[email protected].
Vol. 21, No. 8 / August 2004 / J. Opt. Soc. Am. A
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