1.

Introduction

During the 1990s the use of nonpolarized backscattered light has made it possible to optically characterize turbid media with noninvasive measurements. By characterization, we mainly refer to the determination of the mean transport scattering coefficient μ_s ^′ and the absorption coefficient μ_a with spatially resolved techniques.¹ Models² were generally developed from the radiative transfer equation. Then, thanks to polarized light, several studies^{3 4 5} showed that backscattered polarized light contains information that was more localized and dependent on the scattering coefficient μ_s. It was also demonstrated⁶ that linearly polarized light propagates through a smaller region than the circular one. We concentrate here on the second parameter of the backscattered Stokes vector, that is Q, in the case of linearly polarized incident light. This parameter represents the intensity difference between the parallel and perpendicular polarization direction. Parallel (respectively perpendicular) direction means that the analyzer direction is parallel (respectively perpendicular) to the polarizer one. The purpose of this paper is to utilize polarization backscattered patterns from turbid media to determine the scattering coefficient μ_s. The first part presents an experimental test of the method on polystyrene microspheres. Then, in order to determine the limits of the proposed method with respect to the reduced scattering and absorption coefficients, μ_s ^′ and μ_a, respectively, Monte Carlo simulations are realized.

2.

Experimental Method

The experimental setup (see Fig. 1) is composed of a laser diode that has a 670 nm wavelength and a 5 mW power. The beam passes through a polarizer and is directed to the sample. The incidence angle is θ=15°. The sample which is composed of calibrated polystyrene microspheres (Merck) then scatters light to an analyzer positioned in front of a 16 bit CCD camera. The backscattered image is recorded by a CCD matrix (ICX084L HyperHAD, Sony) made of 659×494 pixels, each pixel measuring 7.4 μm.

Figure 1

Schematic diagram of the experimental setup.

Here we only focused on the first and second Stokes parameters I and Q, respectively, using linear polarizers. The Stokes vector is defined as follows:

Figure 2

Experimental Q/I image for a suspension of polystyrene spheres. Parameters: g=0.71, μ_s=59 cm ⁻¹, μ_a=0 cm ⁻¹. Image size=0.57 cm×0.57 cm.

We are interested here in the spatial extent of the vector element Q with respect to I. In other words, we want to know the typical size of the backscattered pattern. We thus integrate the image Q/I over the azimuth angle φ [see Fig. 3(a)] to only have a dependence in r, which is the distance from the entrance point of light. We call this integral q(r):

Figure 3

(a) Azimuth angle φ for a Q/I image and (b) experimental q(r) for two scattering coefficients μ_s=28 and 39 cm⁻¹.

3.

Monte Carlo Simulations

The Monte Carlo code⁸ simulates the propagation of polarized light through a scattering medium having a slab shape, that is composed of homogeneously distributed spheres. This code considers a polarized light source illuminating one point of the turbid medium surface. This Monte Carlo simulation is based on a previous one⁹ that described propagation of unpolarized light thanks to the radiative theory using the absorption and scattering coefficients, μ_a and μ_s, respectively. In addition to the position of the photon and its propagation direction, the photon packet is characterized here by its Stokes vector which is defined in the photon local frame when this photon packet is inside the medium. Thanks to the Mie theory,¹⁰ we know the anisotropy factor g of the spheres and the Mueller matrix for a single scattering. This makes it possible to statistically generate the photon direction for each scattering. Thus, we follow the modification of the Stokes vector and as the photon escapes the medium, either in the backscattering case or the transmission one, this vector is expressed in the detector frame and recorded. Notice that the detector is assumed lying on the medium surface. We end up with a backscattering (and transmission) images for each element of the Stokes vector. An example of backscattering images is given in the Fig. 4 for the I and Q Stokes vector elements. The medium parameters are the following: μ_a=0 cm ⁻¹, μ_s=57 cm ⁻¹, g=0.65, the thickness of the slab is d=0.5 cm, the sphere refraction index is n_s=1.58, these spheres are immersed in a medium with a refraction index of n_m=1.33, and the source wavelength is equal to 670 nm. These images are normalized by the number of the photons used for that simulation, namely 100 millions. In our simulations, we set the value of the reduced scattering coefficient μ_s ^′ equal to 20 cm⁻¹ and we vary the value of the anisotropy factor g from 0.04 (equivalent to μ_s=21 cm ⁻¹) to 0.8 (equivalent to μ_s=84 cm ⁻¹) . The absorption is still null. We follow the evolution of the inverse of the length l_m with respect to the scattering coefficient μ_s. The slab parameters are the following: medium refraction index n_m=1.33, sphere refraction index n_s=1.58, slab thickness d=0.5 cm and the source wavelength is 670 nm. The backscattered Q image is divided by the I image, pixel by pixel, and then integrated to obtain q(r) (see Fig. 5). If we draw the maximum position l_m as a function of μ_s in the case where the anisotropy factor g varies between 0.04 and 0.8, we get the curve of the Fig. 6 where it can be seen a linearity between l_m and μ_s. The corresponding linear regression coefficient r² is equal to 0.985. The evolution of the relationship between l_m and μ_s is shown on Fig. 7 where the reduced scattering coefficient μ_s ^′ has been changed to 10 and 40 cm⁻¹, respectively. We have a linear regression coefficient r² that is equal to 0.995 for μ_s ^′=10 cm ⁻¹ and to 0.977 for μ_s ^′=40 cm ⁻¹. Considering the coefficient r², it is difficult to conclude on the influence of the transport scattering coefficient μ_s ^′ on the linear relationship between 1/l_m and μ_s. But its dependence remains weak compared to the one with the anisotropy factor g.

Figure 4

Simulated backscattered I and Q images for a scattering medium with the following coefficients: μ_a=0 cm ⁻¹, μ_s=57 cm ⁻¹, g=0.65, thickness=0.5 cm, wavelength=670 nm. Image size 0.35 cm×0.35 cm.

Figure 5

Simulated q(r) for different μ_s and with a constant μ_s ^′=20 cm ⁻¹.

Figure 6

1/l_m with respect to μ_s (corresponding anisotropy factor g varies between 0.04 and 0.8). μ_s ^′=20 cm ⁻¹. Linear regression coefficient r²=0.985.

Figure 7

1/l_m with respect to μ_s. For μ_s ^′=10 cm ⁻¹, r²=0.995. For μ_s ^′=40 cm ⁻¹, r²=0.977. The corresponding anisotropy factor g varies between 0.04 and 0.8 for each transport scattering coefficient.

The influence of the absorption coefficient μ_a on the relationship between the maximum position l_m of q(r) is shown on Fig. 8 when μ_s ^′=20 cm ⁻¹, for two examples of absorption: μ_a=1 and 5 cm⁻¹. The linear regression coefficient r² for the first example is 0.966 and 0.659 for the second case. Therefore considering the diminishing linear regression coefficient r², the linearity between 1/l_m and μ_s disappears as the ratio μ_a/μ_s ^′ increases. We notice that as the absorption increases there is a region (corresponding to g<0.7) where the length l_m is smaller than expected.

Figure 8

1/l_m with respect to μ_s for μ_s ^′=20 cm ⁻¹. For μ_a=1 cm ⁻¹, r²=0.966. For μ_a=5 cm ⁻¹, r²=0.659. The corresponding anisotropy factor g varies between 0.04 (μ_s=21 cm ⁻¹) and 0.8 (μ_s=84 cm ⁻¹) for each absorption coefficient.

4.

Discussion

The advantage of this method is its weak dependence with the transport mean free path λ_s ^′ in the case of a negligible absorption coefficient μ_a compared to μ_s ^′. These coefficients can be known, for example, with a reflectance technique using unpolarized light.¹¹ As the anisotropy factor g tends to 1, we see [Fig. 9(left)] that the distance l_m does not vary anymore. Its inverse tends approximately to the value of 80 cm⁻¹ [Fig. 9(right)]. This cannot be explained by the thickness of the slab since the latter has been modified and increased to 10 cm in order to see if it influences the backscattered intensity as the anisotropy factor g increases to 1. The results of the simulation, in the case where g=0.92, give no difference concerning the behavior of the curve q(r), that is the position of the maximum is not altered. Consequently, the distance l_m remains unchanged even if the thickness represents a large number of times (≈2500) the mean free transport λ_s ^′. An explanation can be given by the Mie distribution, that generates the propagation direction of the photons. Indeed if we assume a single scattering, we have the following Mueller matrix:

Figure 9

(Left) q(r) for different μ_s and with a constant μ_s ^′=20 cm ⁻¹, μ_a=0 cm ⁻¹ and (right) 1/l_m with respect to μ_s. μ_s ^′=20 cm ⁻¹ and μ_a=0 cm ⁻¹. Corresponding anisotropy factor g varies between 0.04 (μ_s=21 cm ⁻¹) and 0.93 (μ_s=282 cm ⁻¹).

Figure 10

(Left) Example of Mie intensity drawn in a polar plot and (right) variations of the Mie intensity for θ=180° with respect to g.

We notice that if the integration over the azimuth angle is done on the backscattered image Q normalized by the total backscattered intensity I, the dependence of 1/l_m with respect to μ_s is not seen. We therefore need to combine the spatial evolutions of both Stokes vector element images I and Q by dividing them pixel by pixel. Moreover, since we consider lengths that are of the order of λ_s, an experimental problem that can occur is the entrance diameter of the light beam which will modify the backscattered image near the entry point. Thus the profile of the light source has to be known in order to correctly deconvolute the image and to get rid of the beam finite size influence.

Model describing qualitatively low scattering backscattered polarized light already exists.¹² In order to extend this method for higher anisotropy factors, it is necessary to have a quantitative model that precisely describes the polarized backscattered light.

5.

Conclusion

Thanks to experiments and Monte Carlo simulations, we have seen that the spatial information contained in the backscattered image ratio Q/I allowed to see the influence of the scattering coefficient μ_s and to determine the latter when the anisotropy factor g ranges between 0 and 0.8. A quantitative theoretical model is necessary to understand the multiple scattering regime and to be able the treatment of higher anisotropy factors, in order to be closer to biological tissue conditions.