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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 6 — Mar. 19, 2007
  • pp: 2847–2872
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The influence of the microscopic characteristics of a random medium on incoherent light transport

F. Caton, C. Baravian, and J. Mougel  »View Author Affiliations


Optics Express, Vol. 15, Issue 6, pp. 2847-2872 (2007)
http://dx.doi.org/10.1364/OE.15.002847


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Abstract

In this paper the influence of the microscopic characteristics of a random medium on non polarized, incoherent steady light transport (ISLT) is investigated. After close examination of current diffusion models, the source term in those models is modified, allowing a complete modelling of experimental and simulated radial dependance of backscattered and transmitted intensities for media thicknesses larger than the transport length. The new model only presents an additional source with respect to the elementary point source model. Thanks to more than 200 Monte-Carlo simulations, this parameter is correlated to the backscattering part of the Mie phase function. Incoherent Steady Light Transport measurements on two industrial emulsions at various volume fractions validate experimentally this correlation. This establishes a complete link between the microscopic characteristic of the random medium (size, optical indexes and volume fraction) and its macroscopic description in terms of diffusion and source parameters, openning new potential applications of the ISLT technique to, for example, the evaluation of the particles interaction potential in concentrated suspensions.

© 2007 Optical Society of America

1. Introduction

Incoherent light transport in random scattering media has developed steadily in the last thirty years or so [1

1. L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters,” App. Opt. 15,2059 (1976). [CrossRef]

, 2

2. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]

, 3

3. M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]

, 4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 6

6. J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics,” App. Opt. 37,3586 (1998). [CrossRef]

, 7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

], even though coherent techniques have known a greater success in the last decade. Originally, the development of incoherent light transport techniques was motivated by the determination of the light transport properties of a turbid medium so that embedded objects could be detected through optical tomography (e.g. [8

8. A. Polishchuk, T. Dolne, F. Liu, and R. Alfana, “Averaged and most probable photon paths in random media,” J. Opt. Soc. Am. A 22,430 (1997).

, 9

9. S. Arridge, “Topical review: optical tomography in medical imaging,” Inv. Probl. 15,R41 (1999). [CrossRef]

, 10

10. J. Paasschens, On the transmission of light through random media, Ph.D. thesis, Leiden University, Netherlands (1997).

, 11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

]). Today, there is a continuing emphasis on the development of non-invasive measurement tools to probe a very wide range of materials, such as biological tissues, colloidal suspensions, emulsions, extraplanetarian dust or atmospheric particles. From the direct approach considered originally, a logical alternative to coherent techniques such as Dynamic Wave Spectroscopy is to consider the inverse problem, i.e. to try to obtain some information about the microscopic properties of the turbid medium from the macroscopic measurements of the backscattered or transmitted incoherent light. This approach has been shown to be able to give a non invasive, real time estimation of the average size of a fast evolving random medium under flow ([12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

]), measurement that was previously only feasible with Small Angle X Scattering or Small Angle Neutron Scattering.

At the same time, theoretical papers concerning incoherent light transport focused first on the obtention of easily calculable approximations of the photon fluxes [3

3. M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

] in the framework of radiative transfer theory. The most used approximation consists in an isotropy approximation leading to a diffusion equation (hence “diffusion approximation”) which has recently been shown to be of remarkable application range [4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

]. The use of this powerful approximation has led to several studies concerning the closure of the problem. Indeed, to solve the problem, boundary conditions must be specified for the isotropic intensity (fluence) [2

2. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]

, 4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 13

13. A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]

], and an isotropic light source must be specified [7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

]. The boundary conditions to be used are now well defined and justified (partial current boundaries for the fluence [2

2. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]

, 4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

]). The source definition however is far less clear. Indeed, if there has been some discussion of the best location for a single diffusive point source [14

14. X. Intes, B. L. Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Rand. Media 9,489 (1999). [CrossRef]

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

], there has been little investigation on whether this is justified and how the diffusive source itself should be described. This is quite surprising as any solution of the diffusion equation will depend very strongly on the way the diffuse source is written.

The main scope of this article is thus to investigate the influence of the microscopic characteristics of the random media (mainly particle size, distribution and concentration) on the macroscopic transport of incoherent light, from the experimental, theoretical and numerical simulations point of view.

2. Experimental Setup and Methods

2.1. Experimental Setup

The experimental setup, sketched in Fig. 1, is identical to the one described in [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

]. The first part consists of a laser diode (λ= 635nm) with a collimator, a circular polarizer and a system of polarization conserving mirrors. Two such systems are used, one for backscattering (bottom), the other one for transmission (top). The output intensity is stable within ±10% (manufacturer’s datasheets). The samples are placed on a glass plate 2 cm thick. It is treated, for the laser wavelength, on the bottom to avoid reflections from the air-glass interface and on the top to avoid reflections from the glass-water interface. This ensures index matching boundary conditions.

The second part of the system consists of a digital image acquisition system that is some 15 cm away from the glass plate. An Adimec MX12P digital camera (1024×1024, 12 bit DAC, 65 dB Signal to Noise ratio) is linked to an Imasys Digital Image Acquisition Card controlled by the PC computer. The size of the acquired image of the backscattered or transmitted light is 5 mm, which is very large compared to the impinging laser spot (30 μm). The experiments are performed in a temperature controlled room at 20°C. Thus, the camera noise is held constant within one gray level. The thickness of the medium is controlled by a rheometer [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

] with a typical accuracy of 1 μm.

Usually, 10 images are acquired and averaged in order to decrease the speckle and numerical noise. The grey levels barycenter of the image is then detected and an angular average taken with this barycenter as origin of the polar coordinates. This yields the radial (r) distribution of the angular averaged intensity. This procedure, added to the excellent S/N ratio (65 dB) and linearity of the camera gives a dynamic range in excess of three decades in measured light intensity.

Fig. 1. Sketch of the experimental system.

All these operations are performed using a home build C++ program. The fits were performed using standard least square methods.

2.2. Materials

Three stable oil in water emulsions, prepared by Firmenich S.A, were used for the experiments. The STT043 and STT063 were prepared using standard batches and surfactant emulsifiers, whereas the EC06-01a was prepared using the technique described in [17

17. C. Goubault, K. Pays, D. Olea, P. Gorria, J. Bibette, V. Schmitt, and F. Leal-Calderon, “Shear Rupturing of Complex Fluids: Application to the Preparation of Quasi-Monodisperse Water-in-Oil-in-Water Double Emulsions.” Langmuir 17,5184–5188 (2001). [CrossRef]

] and a different emulsifier. The oil refractive index is 1.45. The size distribution of the emulsions are plotted on Fig. 2 as determined by SALS measurements using a Malvern Mastersizer S. The angular intensity distribution was inverted to give a size distribution using Malvern’s proprietary software. The droplet size distribution for the three emulsions are well described by a log-normal distribution:

12πσdexp((ln(d)ln(d¯))22σ2)
(1)

where σ is the variance of the log-normal distribution and d¯ the average diameter size. This distribution was fitted for each emulsion (lines in Fig. 2), giving an estimate of both the average volumic diameter and the deviation from this size. We also calculated the statistical volume average D[4,3]=Σinidi4Σinidi3 and the uniformity parameter U=1dΣinidi3dd¯Σinidi3 as proposed by [18

18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]

] (ni is the number density).

In table 1 we show the calculated radii and deviations for each emulsion. According to the criteria proposed in [18

18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]

], the emulsions STT046 and STT063 are poly disperse, while the EC06-01a can be considered as “experimentally monodisperse” as U < 0.25.

Fig. 2. Droplets radii distributions for the three emulsions.

Table 1. Size distribution parmeters for the emulsions.

table-icon
View This Table

Finally, the three emulsions can be considered as non absorbing for the wavelength used.

3. Diffusion theory: The double source model

3.1. Radiative transfer basic hypothesis

The propagation of light in a turbid media is modelled, when coherence and polarization effects can be neglected, by the radiative transfer equation for the radiance (also called specific intensity) [7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

]. We will now discuss the relevance of these approximations to our experiments.

3.1.1. Incoherent light transport

As the light source in our setup is both polarized and coherent, some physical justifications of the use of unpolarized incoherent models are needed. The coherence effects that can be observed in a backscattering experiment are of three kinds. First, an essentially coherent field can be observed if the material exhibits strong Anderson localization. Materials that exhibit this phenomena need to be specifically designed as the particles must verify the Ioffe-Regel criterion which states that the scattering length (ls) to wavenumber ratio must be smaller than one : 2πls/λ′ ≤ 1 (λ′ is the wavelength in the suspending medium). Even in the worst case of very high concentrations and particle size of the same order than the wavelength, the Ioffe-Regel criterion is never attained for any of our samples as the minimum scattering length to wavenumber ratio is of 2πlsmin/λ'10 . Therefore, in our case, the scattered intensity field is not entirely coherent, but consists in the superposition of coherent and incoherent light.

The last kind of coherence effects are known as speckles. They arise from the interferences of photons having travelled the same distance deeply in the random medium, and coming back on the detector at the same position with the same wave vector. Those patterns are time dependent for fluid samples, as the positions of the scattering centers fluctuate in time due to Brownian motion. The study of the temporal evolution of those speckles is the core of techniques like Diffusive Wave Spectroscopy. They are visible in the whole backscattered field, but account for a small number of photons relative to incoherent ones. Indeed, the interference condition is more and more difficult to meet as the scattered path length increases. In our setup, the speckles are averaged, and, as the speckles intensity is always positive, they produce small intensity variations representing typically one percent of the incoherent backscattered intensity.

As a conclusion, our setup detects essentially incoherent photons, allowing the use of the scalar radiative transfer theory to model the backscattered intensity.

3.1.2. Non polarized light transport

As long as there is no global orientation of anisotropic objects in the sample or optical anisotropy of the continuous phase (optical activity, birefringence or dichroism), an incoming circularly polarized light (with no analyser on the receiver) will behave rigorously as nonpolarized light [6

6. J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics,” App. Opt. 37,3586 (1998). [CrossRef]

]. As a consequence, the scalar radiative transfer theory should be appropriate to model the intensity backscattered by the random medium as measured by the CCD camera. We will now briefly describe the standard diffusion approximation (see e.g. [1

1. L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters,” App. Opt. 15,2059 (1976). [CrossRef]

, 7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

]), then expose alternative models for the diffusive source.

3.2. Scalar radiative transfer

In this section, the standard radiative transfer and diffusion approximation are presented together with the definitions of the different parameters, based on the description found in Ishi-maru’s book ([7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

]) and Prahl’s thesis [11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

]. The normal to the surface of the medium is defined outwards of the medium, and the vertical direction z is defined as the direction of the incident beam (see Fig. 1). The medium is of depth d in z and is considered as infinite in the other directions (at least one order of magnitude larger than d in the experiments). Since there is no source within the random media, the equation of radiative transfer for the steady state specific intensity I(r,s^) (watts steradian−1m−2) at r propagating in the medium in the direction ŝ is:

ŝI(r,ŝ)=1ltIrŝ+1ls4πp(ŝ,ŝ)I(r,ŝ)dΩ
(2)

where la, ls and lt are respectively the absorption, scattering and total scattering lengths (1lt=1ls+1la) and p(ŝ,ŝ′) is the usual scattering phase function [7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

] from the initial direction ŝ′ to the direction of observation ŝ. r is the spatial position vector, ŝ is the unit direction of propagation vector and dΩ′ the differential solid angle in the direction ŝ′.

This equation can only be solved numerically if no further hypothesis is made. To proceed, a standard isotropy hypothesis (diffusion approximation) is used. This allows the diffuse specific intensity Id to be written, to leading order:

Id(r,ŝ)=14πϕ(r)+34πF(r).ŝ+h.o.t
(3)

where φ(r) is called the “fluence” and F(r) is the “flux”:

ϕ(r)=4πId(r,ŝ)dΩandF(r)=4πId(r,ŝ)ŝdΩ
(4)

Inserting this isotropy approximation in the radiative transfer equation, a diffusion equation is obtained for the isotropic diffuse intensity φ(r) :

Δϕ(r)31laltrϕ(r)=S(r)
(5)

where ltr is the “transport length”:

1ltr=1la+1gls

and g=4πpŝŝŝ.ŝdω4πpŝŝdω is the anisotropy parameter. S(r) is the source of diffuse photons that cannot be described by the “true” ballistic source. So, to close the problem, some assumptions on this diffuse source are necessary, assumptions that will now be discussed in detail.

3.3. The diffuse source problem for a collimated beam

In the following, we suppose that the physical source is an infinitely thin, collimated beam entering vertically in the medium.

3.3.1. Intensity separation hypothesis: the exponential model

In a first approach [4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

, 7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

, 11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

], the specific intensity is separated in two parts. One corresponds to the light diffused by the random medium I d, the rest corresponding to the collimated (ballistic) light I coll: I(r,s^)=Icoll(r,s^)+Id(r,s^) .

The diffuse source S(r) emerges from the addition of this diffuse-ballistic decomposition hypothesis to the diffusion approximation. It consists of a decreasing exponential that starts at the origin r = 0 and extends vertically over a length lt([7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

] p. 181, [4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

, 11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

]):

Srz=S0(g)Exp(zlt)δ(r=0)
(6)

The appeal of this approximation is that the source distribution is directly derived from the above decomposition and feels very rigorous as no further assumptions seems to have been made. Actually, this approximation corresponds to choosing g=0, since the ballistic photons are directly converted into diffusive photons at their first scattering. Thus this approximation should be good for small g and less appropriate for large ones. This model is called “the exponential model” in the following.

3.3.2. Source location hypothesis: the Haskell Model

The alternative approach as proposed by Haskell et al.[5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

] is the opposite assumption. Indeed, it considers that the diffusion equation should be applied only to diffusive photons. As those diffusive photons are created by successive scatterings, and by definition of the diffusion mean free path (=transport length ltr), the source of diffusive photons must be close to z = ltr. This source is simply approximated by a Dirac distribution located in z = ltr and r = 0. A major advantage of this model is that an analytical solution can be found for the reflectance from a semi infinite medium (index matching):

R(ρ)*ltr2=0.0398(1+(rltr)2)32+0.0928(5.444+(rltr)2)32+
+(0.0597(1+(rltr)2)120.0597(5.444+(rltr)2)12)
(7)

3.3.3. The double source model

Our point of view is that both approaches are approximate and that these approximations can be improved. Indeed, the transition from a ballistic photon to a diffuse photon is not instantaneous but requires a certain number of scatterings. So, the ballistic-diffusive decomposition is inaccurate as the diffuse source should extend close to ltr. On the other hand photons that are diffusive (i.e. that have undergone 1/(1-g) scatterings) are not necessarily located in ltr. A proportion of them, sometimes referred to as “short photons” or “snake photons”, may have turned back (backscattered) and could be at distances between 0 and ltr when they become diffusive. As a consequence, we simply model this complex, spatially distributed, axisymmetric source by two sources, one located in z=0, the other one in z = ltr, with a parameter α that represents the respective strength of the two sources.

Further assuming that the source has a small horizontal extent compared to ltr, it can be written as:

Srz=[αδ(z=0)+(1α)δ(z=ltr)]δ(r=0)
(8)

3.4. Boundary conditions

As our boundaries are experimentally index matching, we use the simple Robin boundary conditions [5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

, 4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

, 20

20. H. J. Kopf, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scatters light in thin disordered slabs,” Phys. Rev. Lett. 79,4369 (1997). [CrossRef]

]:

{ϕ(r)ϕ(r)h0z=0inz=0ϕ(r)+ϕ(r)hdz=0inz=d
(9)

h0=23ltr1+R21R1 and hd=23ltr1+R11R2 where R 2 and R 1 are the relevant angle averaged reflection coefficients at the boundary [5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

]. It is important to realize that, when there is a strong index mismatch between the suspended medium and the outside medium, the boundary conditions themselves depend on the details of the scattering [21

21. G. Popescu, C. Mujat, and A. Dogariu, “vidence of scattering anisotropy effects on boundary conditions of the diffusion equation,” Phys. Rev. E 61,04 8264 (2005).

].

3.5. Solution method

The complete set of equations can be non dimensionalized using ltr as the unit length. The governing equation for the non dimensional fluence φ˜(ρ,ξ)=ltr2φ(ρ,ξ) is:

Δϕ˜ρξβ2ϕ˜ρξ=S(ρ)
(10)

with the boundary conditions:

{ϕ˜ρξϕ˜ρξh0ξ=0inξ=0ϕ˜ρξ+ϕ˜ρξhτξ=0inξ=τ
(11)

where ρ= r/ltr, ξ= z/ltr, τ= d/ltr,h˜=hltr and β2 = 3la/ltr.

The scaling obtained above for the fluence will be applied to the experimental data. The exact solution of this steady diffusion problem in both the backscattering and transmission geometries can be found using the Green function described in Ishimaru’s book[7

7. A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

] (pp.184–185) and the numerical method in S. Prahl thesis[11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

]. We provide in Annexe I a Matlab program to calculate the double source model for no absorption and index matching boundary conditions, both in the transmission and backscattering geometry.

3.6. Comparison between the models

The double source model solution for α = 0 corresponds to the hypothesis of a single Dirac source in ltr, whereas the solution for α= 1 corresponds to only a Dirac source at the boundary. A value of a= 0.5 corresponds to two sources with the same strength, experimental values rarely exceeding 0.6. Solutions for both α= 0 and α= 0.5 are plotted on Fig. 3 together with the solution using the exponential source (“expo. model”) for two example values of g (g=0 and g=0.9), and the solution obtained using the extrapolated boundary condition approximation and the Haskell model (i.e. a source in ltr only, see eq. 7, [5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 13

13. A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]

]). The analytical solutions were calculated for a medium depth of 20ltr, except for the Haskell model which considers that the medium is infinitely deep in the positive z direction. The first observation is that, in contradiction with the results of Durian et al. [4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

], the Haskell model compares very well with the α= 0 solution using the Robin boundary conditions. So, the apparent discrepancy between the model using appropriate Robin boundary conditions[4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

] and the extrapolated boundary condition[5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

] does not come from the boundary condition, but from the different modelling of the source!

According to the exponential model, our experimental curves should lie above the exponential model with g = 0 and close to the one for g = 0.9. These behaviours will allow clear discrimination between the models. Finally, the double source and exponential model had to be re-normalized to be compared to the haskell model, because the short photons are taken into account in the first two models.

4. Monte Carlo simulations

Fig. 3. Comparison of the different models. Contiuous line: Haskell model (eq. 7); Squares and triangles: double source model (eq. 8); Dashed and dotted lines: exponential source model (eq. 6)

4.1. Parameters

The “ingredients” of our numerical simulations are now quite standardized [15

15. X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments.” J. Biomedical Opt. 8,608–617 (2003). [CrossRef]

, 16

16. S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39,1580 (2000). [CrossRef]

]. The Mie scattering phase function of a plane electromagnetic wave (EMW) by an individual homogeneous spherical particle depends on two dimensionless parameters, the size parameter x = 2πns a/λ and the refractive index ratio m = np/ns, where ns and np are respectively the optical indexes of the suspending medium and of the particles, a is the radius of the particles and λ is the wavelength of the incoming EMW. For a collection of particles, one more parameter is necessary, the volume fraction. However, in the simulations, we do not take into account the influence of particles pair correlations. So, since ltr1ϕ, the volume fraction is a simple scaling parameter that disappears when lengths are non dimensionnalized by ltr [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

], and the intensity is normalized by ltr2 (see equation 7).

4.2. Monte Carlo Algorithm

Monte Carlo simulations solve the radiative transfer equation (eq. 2) by simulating directly the propagation of incoherent non polarized photon packets. In agreement with the experimental set-up, a photon packet enters the sample vertically at the space origin. The distance between two successive scattering events (iteration i and i+1) is randomly sampled according to −ltlog(rand) with lt = 1/(1/ls + la and rand is a random number with rand ∈]0,1]. Between scattering events, the photon packets propagate balistically. At a scattering event, the photon packet selects a direction defined by a polar angle (θ) and an azimutal angle (φ)according to the chosen phase function p(θ, φ). For non polarized light propagation, the phase function depends only on θ, the azimutal angle φ being uniformly distributed between [0,2π[. Sampling of θ is performed through the following cumulative distribution function: CDF(θ)=2π0θp(θ)sin(θ)dθ=rand , with rand ∈ [0,1]. The discretization step for θ is usually π/200 in our simulations. A decrease of the size of the discretization steps down to π/2000 did not change the simulations result. The outside medium has the same index as the suspending medium (index matching), which mimics the coatings on the glass plate in the experimental system. Photons leaving the random medium almost vertically are detected and stored in two matrices, one for the backscattered photons, the other one for transmitted photons. An important aspect of this simulation is that coherence effects are not accounted for, as we propagate directly the intensity and not the amplitude of an EMW.

About 107 photons per simulation are send in the medium. One simulation lasts typically for a few hours on a pentium 4 PC.

4.3. Phase functions

If the central goal of the simulations is to solve the complete problem with a description of individual scatterings as close to reality as possible, it is important to understand the influence of the modelling of the microscopic properties on macroscopic light transport, i.e. the influecne of the details of the phase function.

When modelling those microscopic scattering events, several phase functions can be chosen. Up to date, simulations of incoherent non polarized light transport used the empirical Henyey-Greenstein (HG) phase function [23

23. L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J 93,70 (1941). [CrossRef]

], or modified versions of it (e.g. the Dirac Eddington approximation [11

11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

]). The main advantage of the Henyey Greenstein phase function is that it is analytical:

pHG(θ)=121g2(12gcos(θ)+g2)32
(12)

Alternatively, the monodisperse Mie phase function seems a more appropriate choice as it is the exact solution of the scattering of an EMW by a spherical particle. However, real suspensions are never perfectly monodisperse, even when called monodisperse (see section 2.2 and [18]). We term those suspensions as “experimentally monodisperse”. To simulate such dispersions, we use the log-normal distribution: 12πσdexp((ln(d)ln(d¯))22σ2 which is widely accepted as representative of experimental size distributions in many dispersions. We also choose σ = 0.2 for the polydispersity parameter as it is smaller than the criteria on the uniformity parameter (U < 0.25, [18

18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]

]). As shown on Fig. 4, the phase function for a theoretically monodisperse suspension is quite different from an experimentally monodisperse one as soon as the size parameter x ≥ 1. The main difference is that the lobes are “smoothed out”.

As can also be seen in Fig. 4, the HG phase function almost never describes properly either the monodisperse or the poly disperse phase functions, whatever the index or the size parameter, especially for the forward and backward scattering part (note that the amplitude of the scattering probability is plotted in log scale, which compresses the differences). Oddly enough, it fails both for very small particles (Fig. 4 top left) and for large ones (Fig. 4 on the right)).

In the following, all three phase functions will be used in the simulations to assess their influence on macroscopic incoherent light transport.

Fig. 4. Example of the differences between the 3 phase functions. The logarithm of the phase function is plotted as a function of the scattering angle. Black curve: monodisperse Mie calculation; red curve: polydisperse Mie calculation with σ= 0.2; blue curve: Henyey-Greenstein approximation.

5. Results

5.1. A Universal Curve?

The diffusion approximation is usually interpreted as a random walk, where the transport length ltr (also called diffusive mean free path: 1ltr=1la+1gls is the size of the random step (see e.g. [3

3. M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]

, 5

5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

, 12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

]). Such an interpretation is often thought to imply that this macroscopic parameter is sufficient to describe completely the macroscopic incoherent light transport. The work in [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

] partly validates this assumption, showing that the radial distribution of backscattered light is indeed universal for r < ltr and can be described using ltr only in this region. However, it was also noted in this work that the intensity distribution for r < ltr is not well described by the Haskell model (eq. 7). In Fig. 3 of [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

] it is plain to see that the low ρ part of the curve depends on the volume fraction, and this discrepancy is usually attributed to a failure of the diffusion approximation.

We performed similar dilution experiments, using a different emulsion (EC06-01a). On Fig. 5 the radial intensity distribution is plotted for several volume fractions (1%, 5%, 10%, 20%, 35% and 60%, top to bottom). The Fig. 6 is obtained as follows (as suggested by the non dimensional diffusion equation). First, ltr and α are measured by fitting the double source model. The radial coordinate is then divided by ltr and the intensity is multiplied by ltr2 , so that, if ltr is the only relevant parameter, all experimental curves should collapse on a single master curve (see eq.7).

Fig. 5. Radial intensity distribution for several dilutions of emulsion EC-06a in dimensional units. The first 4 radial points correspond to the laser beam.

It is clear from Fig. 6 that the curves do collapse on a single master curve for r/ltr > 1 and do not collapse for r/ltr < 1. So, it seems not possible to describe completely the incoherent light transport using only the macroscopic parameter ltr. Also, as suggested already in [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

], the Haskell model cannot describe this low r/ltr part of the backscattered intensity. Before attributing this discrepancy to the failure of diffusive models, one must test whether this variation can be described by the more sophisticated diffusive models, e.g. the double source or the exponential source models. As a reminder, the analytical Haskell model has two independent parameters which are the transport length ltr and the absorption length la, and is valid only for a semi infinite medium in the backscattering geometry.

Fig. 6. Scaled radial intensity distribution for a few dilutions of emulsion EC06-a in non dimensional units. The continuous lines are fits using the double source model (“DSF”). The exponential model (g=0 and g=0.9) and the Haskell model are also displayed.

5.2. Fit of experimental data: first validation of the double source model

5.2.1. Fit of the low r/ltr region

In Fig. 6, the double source model is also plotted for each displayed concentration. The model is able to describe almost perfectly the entire radial evolution of the reflectance, with values of α ranging from 0.125 to 0.6. So the discrepancy noted above was not the consequence of a poor description of the light transport (i.e. the failure of the diffusion approximation), but of an oversimplified description of the diffusive source.

In contrast, the exponential model fails to fit the low ρ= r/ltr curves for volume fractions between 1% and 35%, even using the unrealistic value of g = 0 (lowest possible curve with this model). In this respect, the exponential model does not improve much the elementary Haskell model.

It is also important to note that the low ρ variation is quite complex with a curvature change at low α (see Fig. 7 or 9 for further examples). An arbitrary additional fit parameter cannot describe such a complex evolution, but is very well accounted for by our model, validating our physical approach of the problem.

5.2.2. Accuracy of the model for a finite size media

The double source model (as the exponential model) not only possesses the additional parameter α representing the sources relative strength, but it also includes the possibility of finite medium depth, and thus can also be used to evaluate the amount of transmitted light. The ability of those models to describe backscattering and transmission in finite media has never been tested experimentally, the validation presented in [4

4. D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

] only confronting the exponential model to numerical simulations of the radiative transfer in the backscattering geometry.

Fig. 7. Comparison of the double source and exponential models(g=0) to experiments for several depths for emulsion STT063 at a volume fraction of 63% (ltr = 0.25mm, α= 0.09 and β = 0).

Backscattering.

In Fig. 7 we compare the normalized experimental data in the backscattering geometry for the STT063 emulsion with the double source model and the exponential model for several depths. The data for the largest depth were fitted using the double source model, yielding ltr = 0.25mm, α = 0.09 and β = 0 (no absorption). These values were used for the other depths, which are thus pure predictions. When using the double source model, the model-data agreement is excellent in the whole range, even in the ρ = r/ltr < 1 region. It is indeed close to perfection as the first points correspond to the width of the laser beam (∼ 30μm). The agreement is excellent down to sample thicknesses of about 1.2ltr, the model starting to break down below this thickness (not displayed). This is expected because there is a source in the model at z = ltr which invalidates the model for thicknesses close or smaller than ltr.

We also show in Fig. 7, the best fit from the exponential model for the largest depth (ltr = 0.25mm), which is obtained when choosing an effective g=0. The exponential model is able to fit very well the large ρ part, while it clearly fails at low ρ.

Fig. 8. Comparison of the double source model to experiments in the transmission geometry for several depths.

Transmission.

Using the same sample, we also performed several experiments in the transmission geometry. In Fig. 8, we compare the double source model predictions to experimental transmission data. The parameters used in the model are the ones determined above using the backscattering curves, i.e. ltr = 0.25mm and α = 0.09, so the theoretical intensity distribution presented are predictions (no fitting parameter). The description of the spatial distribution by the model is once more excellent.

5.3. Numerical simulations: an exploration of the microscopic parameter space

5.3.1. Determination of α: fit of the radial intensity distributions

Fig. 9. Intensity curves as a function of the size parameter x. Points corresponds to numerical simulations while lines correspond to the double-source model.

In Fig. 9 a few radial intensity curves for m=1.1 and x ∈ [10 …5] are plotted against ρ = r/ltr. This Fig. shows once again that all curves collapse on a single one from ρ = r/ltr = 1 to ∞. There is the same discrepancy as observed in experiments for ρ < 1.

Every simulation curve was fitted using the double source model, with α as the only free parameter, yielding a value of α for each (x,m) pair, or for each value of g when using the Henyey Greenstein phase function. The corresponding model curves are also plotted on Fig. 9, showing the excellent agreement between the model and the simulations. Once more, this shows that the double source model is able to represent properly the variations in shape of the backscattering curves when changing microscopic parameters. Also, this good representation of the simulations by the model cannot be reduced solely to the addition of an arbitrary fit parameter.

5.3.2. Source relative strength: a measure of the backscattering phase function

We tried to correlate α with several microscopic and macroscopic characteristics of the random medium.

Correlation with the anisotropy parameter g.

The first parameter that seems reasonable to correlate α with, is the anisotropy parameter g. Indeed, as g measures the relative probability of the photons to be scattered either forwards or backwards, it seems reasonable to suppose that it influences the relative source strength α. Figure 10 shows that α is clearly not correlated to the anisotropy parameter g if a real phase function is used. In contrast, the evolution of α when using the Henyey-Greenstein phase function is monotonous. As a consequence, simulating the incoherent light transport using the Henyey-Greenstein phase function is not equivalent to using the Mie phase function whatever the polydispersity of the sample.

Fig. 10. α plotted as a function of the anisotropy parameter g. left) monodisperse phase function. right)polydisperse phase function (σ= 0.2). HG: Henyey Greenstein phase function.

An important consequence of this a priori surprising result is that the detailed microscopic properties of the random medium have a measurable influence on the macroscopic characteristics of the incoherent light transport. In other words, the microscopic properties, which are smoothed out in the universal part of the radial intensity distribution (r/ltr > 1), seem to determine the shape of the low (r/ltr < 1) part of the curve.

Correlation with the size parameter x.

Given the above lack of correlation, a natural idea is to correlate α with the main microscopic parameter: the size parameter x.

Fig. 11. α plotted as a function of the size parameter x. left)monodisperse phase function. right)polydisperse phase function (σ = 0.2).HG corresponds to the Henyey Green-stein phase function.

Correlation with backscattering phase function P(π).

Finally, we tried to correlate α with the amplitude of the phase function at the backscattering angle (π). The correlation displayed in Fig. 12 (left) is already convincing for monodisperse suspensions. It is improved when simulating experimentally monodisperse suspensions with σ= 0.2 (12 (right)).

Fig. 12. α plotted as a function of the backscattering probability P(π). left)monodisperse phase function. right)polydisperse phase function (σ = 0.2).

This correlation supports the heuristic construction of the double-source model presented in the theoretical part of the present article. Indeed, as stated in the model’s description, some photons may have turned back and could be at distances between 0 and ltr when they become diffusive. The proportion of these early backscattered photons is quite logically linked to the scattering probability in the backward direction.

Also, it is quite clear that the diffusive source has a complex spatial distribution and thus cannot be perfectly described by a simple two source model. As a consequence, the correlation is not very good when the spatial distribution of diffusive photons is very different from a dipolar one. Two examples of this are the Henyey-Greenstein phase function for large g, and truly monodisperse suspensions with strong oscillating structure (m = 1.3, x ≈ 4). Interestingly, even the very weak polydispersity used for the “experimentally monodisperse” sample tends to smooth the phase function giving it a simpler, more dipolar structure (see fig. 4).

5.3.3. Conclusions: determination of P(π) and influence of the phase function

We have shown above that the relative source amplitude α of the double source model is well correlated to the backscattering part of the phase function. Better, for experimentally monodis-perse suspensions, the variation of α with p(π) is strictly unique and monotonic. So, we can fit this correlation using an ad hoc function f so that α = f(p(π)). Once a is estimated, this heuristic function allows to determine p(π) for the suspension under consideration.

From a broader point of view, an important and quite surprising result of these simulations is that the details of the phase function have an important impact on the macroscopic transport of light. Indeed, Fig. (11,10,12) show that simulations with the Henyey-Greenstein phase function never give α values smaller than 0.15. In other words simulations with the Henyey-Greenstein phase function will give incorrect results for the low r/lTR part of the backscattered intensity and for media with P(π) < 2.10−2. Thus, using this phase function in Monte Carlo simulations is in a sense a stronger approximation than the diffusion approximation itself.

6. Validation: influence of particle pair correlations

The correlation between α and p(π) established above using numerical simulations must be confirmed experimentally.

Recalling that the phase function itself depends on the particle size parameter x (and its distribution), and on the refractive index ratio m, this may be done by varying either parameters. However, it is not very simple to change the droplet mean size and it is very difficult to keep the same polydispersity. The same holds for the optical index ratio.

However, at large concentrations, the EMW scattering phase function depends in a complex way on the volume fraction (see below). So, an alternative and much simpler experimental way of testing the correlation between α and p(π) is to dilute progressively a stable emulsion, the polydispersity, size and index ratios being kept constant.

6.1. Background: Influence of particle pair correlations on the scattering properties of a random medium

6.1.1. Influence on the phase function

For concentrated suspensions, the scattering of the EMW by a particle can be influenced by the neighbourhing particles, this effect occuring typically when volume fractions are larger than 1%. To take into account these local interference effects, the phase function must be redefined as [24

24. L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]

] (Chapter 8):

pconc(θ)=pMie(θ)S(θ)4πpMie(θ)S(θ)dΩ
(13)

where pconc(θ) corresponds to the phase function in the concentrated case, pMie(θ) corresponds to the dilute Mie calculus, and S(θ) is the structure factor. This structure factor is proportional to the Fourier transform of the particles pair position correlation g 2(r): S(q)=1+0exp(iqr)g2(r)dr where q = 2xsin(θ/2). Note that S(θ) depends on the non dimensional size parameter x. In order to calulate the structure factor, some hypothesis must be made on the interactions between the particles. We choose to use the Percus-Yevick approximation which includes the following hypotheses. The medium is both homogeneous and isotropic and the interaction potential between the particles is approximated by a “hard sphere” potential. This potential is infinite inside the particle and is zero outside. These approximations allow to calculate explicitely the structure factor SPY (θ) [24

24. L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]

] (chapter 8) which depends explicitely on x by definition of q. In Fig. 13, the Percus Yievick structure factor is plotted for several values of x and for ϕ =0.3. The structure factors are almost constant for x = 0.1 and x = 50 which means that, in these cases, it has little influence on the phase function, as the phase function itself is normalized. So, the structure factor influences the phase function essentially when x ≈ 0.5 − 5. This can be seen in Fig. 14 where the phase functions using the Percus Yievick approximation are plotted for x= 0.1,1,5,50, m = 1.1 and ϕ = 0.3.

Finally, as said in the introduction of this paragraph, the pair correlation influence on the phase function increases with the concentration. In Fig. 15 (top left), several phase functions are plotted when increasing concentrations in the case of a size parameter similar to the one of emulsions STT046 and EC06-01a (x = 3) . The structure factor clearly distorts the phase function which acquires a dominant scattering angle for ϕ > 0.3. The evolution of the phase function at the backscattering angle for x = 1 and m = 1.1 is displayed in Fig. 15 (top right), both in the dilute and Percus Yievick approximation. The former curve stays constant while the latter strongly increases versus the concentration.

Fig. 13. Structure factors S(θ) for different size parameters at a volume fraction of ϕ= 0.3.

6.1.2. Influence on the transport parameters

The particle pair correlations have other indirect consequences on scattering. Indeed, a change in the phase function can also change the value of the anisotropy parameter g = ∫ pPY (θ)cos(θ)dΩ in a potentially complex way. In the example shown in Fig. 15, the anisotropy parameter decreases by a factor of 4. For x = 1 (not shown) it can even become negative, this interesting phenomena been studied in some detail in [25

25. L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. J. Saenz, P. Schurtenberger, and F. Scheffold, “Photonic Properties of Strongly Correlated Colloidal Liquids,” Phys. Rev. Letters 93,073 903 (2004). [CrossRef]

].

Another consequence of the influence of the structure factor is that it also changes the scattering length through the scattering cross section which is proportionnal to ∫ pPY(θ)S(θ)dΩ. The evolution of the scattering coefficient divided by the dilute coefficient is plotted in Fig. 15 (bottom right), where it slightly decreases.

So, the transport length ltr=1n(1g)Cscat is affected both through g and through Cscat. Actually both parameters contribute to an increase of the transport length versus the dilute one with increasing concentrations, as we will now see.

6.2. Evolution of the experimental transport length

Several dilutions are performed from the EC06-01a and STT046 raw emulsions (ϕ0 = 0.672). A backscattering experiment is performed for each dilution to obtain the evolution of the transport length ltr and of the source strength parameter α as a function of the volume concentration ϕ. In Fig. 16, the evolution of ltr is plotted versus the volume fraction ϕ, together with the theoretical evolution calculated using the 250nm size for both the dilute theory (dashed line), and the Percus-Yievick calculation (continuous line).

Fig. 14. Dilute (Mie) and P-Y phase functions for 4 different sizes. Top left x=0.1; Bottom left: x=1; Top right, x=5; Bottom right, x=50.

This effect of particle pair correlations is clear to see in Fig. 16, as the evolution with volume fraction of the experimental transport length deviates largely from the dilute curve even at relatively low volume fractions (∼ 5%). For both the EC06-01a and the STT046, the size parameter is about x = 3, which is in the right range to exhibit particle pair correlations at relatively small volume fractions. On the contrary, the STT063 droplet size is too large (x=5.2) to show a pronounced effect (not shown).

The Percus Yievick approximation improves substantially the prediction of the transport length up to volume fraction of about 20%. The evolution of the measured ltr at higher concentration is above both the “Mie” curve and also the “Percus-Yievick” curve, which is probably due to an underestimation of the interaction potential. Indeed, the quite long range electrostatic repulsion is neglected in the hard sphere model used here.

Looking at the curves in more detail, the discrepancy at high volume fractions between experiments and the P-Y prediction is larger for the EC06-01a than for the STT046. This may be the consequence of the different emulsifiers used in the two emulsions or of the different polydispersities.

Fig. 15. Evolution with the volume fraction of: (Top Left) the phase function, (Top Right) p(π), (Bottom Left) the anisotropy parameter g, (Bottom Right) Relative scattering cross section. The size parameter is x=1 for all figures.
Fig. 16. Evolution of the transport length ltr with the volume fraction ϕ for (left) the EC06-01a emulsion and (right) the STT046 emulsion.

6.3. Validation of the α- p(π) correlation

As explained above, given a volume fraction ϕ, a size x and an index ratio m, the Percus-Yievick calculation can give us a prediction of p(π). Using the heuristic function f determined above from the simulations, we get a prediction of the evolution of α versus the concentration for a given size: αxpred(φ) .

In Fig. 17, the experimental values of α obtained by the fit from the steady light transport experiments are plotted together with the predicted curves for sizes of 230nm, 245nm and 255nm.

As long as ϕ < 0.3, the experimental curves lie between the 245nm and the 255nm theoretical curves, actually very close to the 250nm curve. Those size values are in excellent agreement with the average size determined from the SALS measurements (around 250–270nm see section 2.2). Similarly to the transport length measurements, the experimentally determined a deviate from the Percus-Yievick calculation above 30%, the deviation being more pronounced for the monodisperse EC06-01a suspension than for the polydisperse STT046.

These results clearly validate experimentally the correlation between α and the phase function amplitude at the backscattering angle. Furthermore, the correlation seems to be very discriminating as the STT046 and EC06-01a emulsions have almost identical mean sizes. We thus speculate that polydispersity of the sample may have a small but perhaps measurable influence on the source parameter α. To support this speculation, a complete polydisperse calculation of the Percus-Yievick approximation is necessary, calculation which is outside the focus of this paper.

The existence of this correlation opens some very interesting potential developpements, for example for the determination of the structure factor. Indeed, if the wavelength is changed, the size parameter can be scanned, and, as the dilute phase function does not change, would allow a scanning of S(q). Also, an improvement of the size measurement technique described in [12

12. C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

] could be designed.

Fig. 17. Evolution of the transport length ltr with the volume fraction ϕ.

7. Conclusion

In this paper we sought to understand the influence of the microscopic parameters on the macroscopic incoherent steady light transport in a random medium.

After detailing the imperfections of existing diffusion models, we show that an appropriate definition of the diffuse source improves markedly the description of experimentally measured backscattered intensities. Indeed, with only one additionnal parameter with respect to the simple semi-infinite model, the complete backscattering curve extending from 0 to ∞ is successfully modelled while still using the diffusion approximation. The additionnal parameter corresponds to the relative strength between two effective point-like diffusive sources present in the model.

Thanks to a large set of Monte-Carlo simulations performed using either the Henyey green-stein, a monodisperse or a slightly polydisperse Mie phase function, we have shown that this additional parameter does not correlate with the anisotropy or the size parameter, but rather to the amplitude of the scattering phase function at the bacskcattering angle. Using successive dilutions, this correlation was validated experimentally using two concentrated industrial emulsion.

Potentially, the results of this study may be used to extract quantitative informations about the random media under study and should even allow to evaluate the interaction potential between particles in a concentrated suspension.

Appendix A: Calulation of the double source model

In this appendix we present an example MATLAB function to calculate the solution of the double source model for a finite non absorbing medium, with index matching boundaries, together with the Haskell model.

function [rho,signal]=FluxSimple(alpha,depth,flagdirection);

% taup=depth/ltr optical thickness; rho=r/ltr non dimensional radius; xi=z/ltr non dimensional height;

% Reflectance: signal. Nroots: modes number.

% flagdirection=−1 in backscattering and flagdirection=+1 in transmission.

% spatial sampling in log-scale and determination of the number of modes.

drho=0.1; rho=power(10,(-3:drho:3)); h=2/3; Nroots = 10*taup/(pi*drho);

% Calculation of the eigenvalues of the transcendental equation

%(see [11] Appendix A4 for details).

valp=(1:Nroots).*pi/taup;

%Calculus of some parameters. gamman=atan(valp.* h);

Nn2=1./(4.*valp).*(2*valp.*taup+sin(2.*gamman)-sin(2.*(gamman+valp.*taup)));

zn1=alpha*sin(valp.*0+gamman)+(1-alpha)*sin(valp.*(1)+gamman);

if flagdirection==-1;

zn0=sin(valp.*0+gamman); dzn0=valp.*cos(valp.*0+gamman);

I0n=zn0.*zn1./(2*pi*Nn2) ; dxiI0n=dzn0.*zn1./(2*pi*Nn2);

elseif flagdirection==1;

zntau=sin(valp.*taup+gamman); dzntau=valp.*cos(valp.*taup+gamman);

Itaun=zntau.*zn1./(2*pi*Nn2); dxiItaun=dzntau.*zn1./(2*pi*Nn2);

end

%Memory affectation.

I0=zeros(size(rho));

for i=1:Nroots

Bn(i,:)=besselK(0,valp(i).*rho(:)’);

end

%Calculus of the Reflectance

if flagdirection==-1;

%Total fluence in xi=0

dxiI0=mtimes(Bn(:,:)’,dxiI0n(:)); fluxz0=-h/2*dxiI0; signal=-pi*fluxz0;

elseif flagdirection==1;

%Total fluence in xi=tau

dxiItau=mtimes(Bn(:,:)’,dxiItaun(:)); fluxztau=h/2*dxiItau; signal=-pi*fluxztau;

end

signalC(:,1)=rho’; signalC(:,2)=signal;

if flagdirection==-1;

% Plot of the double source and Haskell model in the backscattering geometry.

RH=0.0397887./power(1+power(rho,2),3/2) +0.0928404./power(5.44444+power(rho,2),3/2)+ 0.0596831*(power(1./(1+power(rho,2)),1/2)− 1./power(5.44444+power(rho,2),1/2));

loglog(signalC(:,1),signalC(:,2),signalC(:,1),RH)

axis([0.001 100 0.000001 10])

end

References and links

1.

L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters,” App. Opt. 15,2059 (1976). [CrossRef]

2.

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12,2532 (1995). [CrossRef]

3.

M. Dogariu and T. Asakura, “Reflectance properties of finite-size turbid media.” Waves Rand. Media 4,429–439 (1994). [CrossRef]

4.

D. Durian and J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source.” J. Opt. Soc. Am. A 16,837 (1999). [CrossRef]

5.

R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11,2727 (1994). [CrossRef]

6.

J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics,” App. Opt. 37,3586 (1998). [CrossRef]

7.

A. Ishimaru, Wave Propagation and Scattering in Random Media ( IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).

8.

A. Polishchuk, T. Dolne, F. Liu, and R. Alfana, “Averaged and most probable photon paths in random media,” J. Opt. Soc. Am. A 22,430 (1997).

9.

S. Arridge, “Topical review: optical tomography in medical imaging,” Inv. Probl. 15,R41 (1999). [CrossRef]

10.

J. Paasschens, On the transmission of light through random media, Ph.D. thesis, Leiden University, Netherlands (1997).

11.

S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html(1988).

12.

C. Baravian, F. Caton, and J. Dillet, “Steady light transport under flow: Characterization of evolving dense random media,” Phys. Rev. E 71,066 603 (2005). [CrossRef]

13.

A. Kienle and M. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium.” J. Opt. Soc. Am. A 14,246 (1997). [CrossRef]

14.

X. Intes, B. L. Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Rand. Media 9,489 (1999). [CrossRef]

15.

X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments.” J. Biomedical Opt. 8,608–617 (2003). [CrossRef]

16.

S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39,1580 (2000). [CrossRef]

17.

C. Goubault, K. Pays, D. Olea, P. Gorria, J. Bibette, V. Schmitt, and F. Leal-Calderon, “Shear Rupturing of Complex Fluids: Application to the Preparation of Quasi-Monodisperse Water-in-Oil-in-Water Double Emulsions.” Langmuir 17,5184–5188 (2001). [CrossRef]

18.

F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, “Monodisperse fragmentation in emulsions: Mechanisms and kinetics.” Europhys. Lett. 61,708–714 (2003). [CrossRef]

19.

P. E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. 55,2696–2699 (1985). [CrossRef] [PubMed]

20.

H. J. Kopf, P. de Vries, R. Sprik, and A. Lagendijk, “Observation of anomalous transport of strongly multiple scatters light in thin disordered slabs,” Phys. Rev. Lett. 79,4369 (1997). [CrossRef]

21.

G. Popescu, C. Mujat, and A. Dogariu, “vidence of scattering anisotropy effects on boundary conditions of the diffusion equation,” Phys. Rev. E 61,04 8264 (2005).

22.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1,441–453 (1997). [CrossRef] [PubMed]

23.

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J 93,70 (1941). [CrossRef]

24.

L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]

25.

L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. J. Saenz, P. Schurtenberger, and F. Scheffold, “Photonic Properties of Strongly Correlated Colloidal Liquids,” Phys. Rev. Letters 93,073 903 (2004). [CrossRef]

OCIS Codes
(110.7050) Imaging systems : Turbid media
(290.1350) Scattering : Backscattering
(290.1990) Scattering : Diffusion
(290.4210) Scattering : Multiple scattering
(290.5850) Scattering : Scattering, particles
(290.7050) Scattering : Turbid media

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: August 22, 2006
Revised Manuscript: December 1, 2006
Manuscript Accepted: December 26, 2006
Published: March 19, 2007

Virtual Issues
Vol. 2, Iss. 4 Virtual Journal for Biomedical Optics

Citation
F. Caton, C. Baravian, and J. Mougel, "The influence of the microscopic characteristics of a random medium on incoherent light transport," Opt. Express 15, 2847-2872 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-6-2847


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References

  1. L. Reynolds, C. C. Johnson, and A. Ishimaru, "Diffuse reflectance from a finite blood medium: Applications to modelling of fiber optics catheters," Appl. Opt. 15, 2059 (1976). [CrossRef]
  2. R. Aronson, "Boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2532 (1995). [CrossRef]
  3. M. Dogariu and T. Asakura, "Reflectance properties of finite-size turbid media, "Waves Rand. Media 4, 429-439 (1994). [CrossRef]
  4. D. Durian and J. Rudnick, "Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source," J. Opt. Soc. Am. A 16, 837 (1999). [CrossRef]
  5. R. Haskell, L. Svaasand, T. TSay, T. Feng, and S. McAdams, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727 (1994). [CrossRef]
  6. J. R. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, "Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagostics," Appl. Opt. 37, 3586 (1998). [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, New Jersey and Oxford University Press, 1997).
  8. A. Polishchuk, T. Dolne, F. Liu, and R. Alfana, "Averaged and most probable photon paths in random media," J. Opt. Soc. Am. A 22, 430 (1997).
  9. S. Arridge, "Topical review: optical tomography in medical imaging," Inv. Probl. 15, R41 (1999). [CrossRef]
  10. J. Paasschens, On the transmission of light through random media, Ph.D. thesis, Leiden University, Netherlands (1997).
  11. S. Prahl, Light transport in tissue, Ph.D. thesis, University of Texas, USA, http://www.bme.ogi.edu/ prahl/pubs/abs/prahl88.html (1988).
  12. C. Baravian, F. Caton, and J. Dillet, "Steady light transport under flow: Characterization of evolving dense random media," Phys. Rev. E 71, 066 603 (2005). [CrossRef]
  13. A. Kienle and M. Patterson, "Improved solutions of the steady-state and the time-resolved diffusion equation for reflectance from a semi-infite turbid medium," J. Opt. Soc. Am. A 14, 246 (1997). [CrossRef]
  14. X. Intes, B. L. Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, "Localization of the virtual point source used in the diffusion approximation to model a collimated beam source," Waves Rand. Media 9, 489 (1999). [CrossRef]
  15. X. Wang, L. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: timeresolved Monte Carlo simulations and experiments," J. Biomedical Opt. 8, 608-617 (2003). [CrossRef]
  16. S. Bartel and A. H. Hielscher, "Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media," Appl. Opt. 39, 1580 (2000). [CrossRef]
  17. C. Goubault, K. Pays, D. Olea, P. Gorria, J. Bibette, V. Schmitt, and F. Leal-Calderon, "Shear Rupturing of Complex Fluids: Application to the Preparation of Quasi-Monodisperse Water-in-Oil-in-Water Double Emulsions," Langmuir 17, 5184-5188 (2001). [CrossRef]
  18. F. M. C., F. Leal-Calderon, J. Bibette, and V. Schmitt, "Monodisperse fragmentation in emulsions: Mechanisms and kinetics," Europhys. Lett. 61, 708-714 (2003). [CrossRef]
  19. P. E. Wolf and G. Maret, "Weak Localization and Coherent Backscattering of Photons in Disordered Media," Phys. Rev. Lett. 55, 2696-2699 (1985). [CrossRef] [PubMed]
  20. H. J. Kopf, P. de Vries, R. Sprik, and A. Lagendijk, "Observation of anomalous transport of strongly multiple scatters light in thin disordered slabs," Phys. Rev. Lett. 79, 4369 (1997). [CrossRef]
  21. G. Popescu and C. Mujat and A. Dogariu, "vidence of scattering anisotropy effects on boundary conditions of the diffusion equation," Phys. Rev. E 61, 04 8264 (2005).
  22. A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, "Diffuse backscattering Mueller matrices of highly scattering media," Opt. Express 1, 441-453 (1997). [CrossRef] [PubMed]
  23. L. Henyey and J. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J 93, 70 (1941). [CrossRef]
  24. L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves, Volume II: Numerical Simulations (John Wiley and Sons, 2001). [CrossRef]
  25. L. F. Rojas-Ochoa, J. Mendez-Alcaraz, J. J. Saenz, P. Schurtenberger, and F. Scheffold, "Photonic Properties of Strongly Correlated Colloidal Liquids," Phys. Rev. Lett. 93, 073903 (2004). [CrossRef]

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