## Abstract

The spatial depolarization of light emitted by heterogeneous bulks is predicted with exact electromagnetic theories. The sample microstructure and geometry is connected with partial polarization.

© 2015 Optical Society of America

## Introduction

Light polarization and coherence [1–7] have become the focus of numerous works, and were accompanied by the emergence of unified theories [8, 9]. Additional complexity was brought by the introduction of random media, hence mixing spatial and temporal disorders [10–12]. Specific effects were recently emphasized at the speckle size, like the local (temporal) en-polarization of light [13–15], and the multi-scale (spatial) depolarization [11, 12, 16]. Most effects were confirmed by experiment [16–19], which justifies a motivation to go further in these fields.

Far field light scattering has been a powerful tool for decades to analyze complex or random media. For small disturbed media, perturbative theories have shown how the emitted pattern was driven by the Fourier transform of the surface roughness [20–22] or the bulk heterogeneity [23], a result which allowed to directly solve inverse problems and characterize substrates and optical coatings [22, 24]. Arbitrary scattering regimes are much more complex to analyze, due to the presence of integral or coupled equations which connect the fields at different spatial frequencies. In all situations the behavior of the far field pattern is intimately connected with the sample microstructure, which drives the angular, spectral and polarization properties. Recent examples give a detailed analysis of the polarization degree at the speckle size in the far field [11, 12, 16, 25], and emphasize spatial depolarization [11, 16, 26] and temporal repolarization [14, 15] of light, all phenomena which strongly depend on the samples microstructure (roughness, inhomogeneity). Applications were also found for biological tissues and others.

Indeed prediction of the polarization process in disordered media often involve a phenomenological approach, similar to that of Goodman’s phasor model [27] with partially or fully developed speckles. Such description of the scattering process is very practical and allowed to reveal unexpected phenomena such as the en-polarization process [14, 15]; also, it allowed to predict a spectral depolarization [3, 28] responsible for strong modification of the polarization degree (dop) histograms [13], as well as the modification of time coherence [29] in disordered media. However this phenomenological model is valid for highly heterogeneous samples and fails at intermediate electromagnetic regimes, whereas these regimes occur in numerous situations. As an example, electromagnetic theories predict the scattering coefficients to be strongly inter-dependent, except for extreme situations where they can be considered uncorrelated; these last situations are those of highly heterogeneous samples currently described with the phasor model.

Within this framework there is a current need to introduce exact values of the scattering coefficients in the calculation of the polarization process, that is, to provide a more exact link between the sample microstructure and the scattering coefficients. Such data would allow to analyze the gradual transition of polarization degree from a slightly heterogeneous sample to a highly heterogeneous one, hence providing a signature of the complex media together with a key information for optical imaging through these media.

We already performed [11] this kind of study for rough surfaces, that is, for surface scattering. For that we used an exact electromagnetic model based on the frontier integral [30]. Numerical results allowed to emphasize the multi-scale nature of the spatial polarization degree of the speckle emitted by surfaces of increasing slopes or root-mean-squares [11]. In other words, depending on the surface characteristics, it is possible to know from which receiver aperture full polarization starts to fail, due a spatial average process. The theoretical model was one-dimensional since time consuming is prohibitive for high slope 2D surfaces.

However to our knowledge the similar bulk problem has not yet been addressed, even though the analysis of bulk media constitutes a relevant topic for imaging in diffuse environment [31–33]. One question concerns the number of speckle grains whose integration cancels polarization, and the result will depend on the statistics and amplitude of the heterogeneity. To solve this point, an exact electromagnetic bulk model is required to calculate the volume scattering coefficients with accuracy, and to introduce them within the calculation of polarization degree (dop). Compared to the surface problem, this requirement is much more difficult to satisfy, because the basic bulk problem is necessarily at least bi-dimensional, and because the bulk/light interaction involves a larger region (not limited to the surfaces).

Hence this paper is an extension to the bulks of previous works [11] devoted to surface scattering. Here we use an exact electromagnetic model of bulk scattering (based on finite elements) to emphasize the gradual relationship between the sample bulk microstructure and the spatial polarization degree at the speckle size. Diagrams are plotted to emphasize the dispersion of polarization states on the Poincaré sphere, and to calculate the spatial dop versus the bulk heterogeneity. Results allow to calibrate the spatial depolarization process versus sample microstructure (thickness, index heterogeneity), with applications in most polarimetric techniques involved in imaging and diffusive media, spatial and defense, biomedical and biophotonics...

## Principles of spatial depolarization

We start with a brief recall on spatial (global) depolarization, not to be confused with temporal (local) depolarization [3, 11, 19]. Indeed the incident light is assumed to be collimated, fully polarized and coherent (perfectly monochromatic), a situation where temporal depolarization does not occur. The analytical signal of the electromagnetic field is described by two complex polarization modes ${E}_{S}^{0}$ and ${E}_{P}^{0}$. With two non-zero modes (non-linear polarization), the full polarization (temporal and local) of the incident field is equivalent to a unity modulus of the complex correlation coefficient ${\mu}_{0}$ given by:

**ρ**the space coordinates and $\alpha $ the normalization coefficient:

_{0}| = 1 = > dop = 1, with dop the polarization degree.

As for the scattered field denoted$\left({E}_{S},{E}_{P}\right)$, it can be written as the product of the incident field by the scattering matrix$\left({\nu}_{ij}\right)$:

_{ij}the scattering coefficients [19]. This last relationship is valid under the assumption of achromatic scattering coefficients within the laser band-pass [13], in accordance with the assumption of a perfectly monochromatic source.

In a second step the complex correlation coefficient of the scattered field can be classically written as:

Such spatial depolarization is quantitatively defined in a way similar to temporal depolarization, which means that temporal averages are completed by spatial averages. However while temporal average involves a unique scale due to full time integration by the low band-pass optical detectors in the visible range, the spatial average depends on the detection area and hence requires a multi-scale analysis. As a consequence, the complex correlation is extended to a function of the detector aperture $\Delta \Omega $ as follows [11, 16]:

*(MDOP)*defined Eq. (9) as the measurable quantity below:

## The electromagnetic bulk scattering model

In order to avoid a phenomenological model limited to extreme regimes of heterogeneity, we quantify the MDOP with an exact electromagnetic method. To cover most scattering regimes from perturbative inhomogeneities to the resonant domain, a rigorous solution of Maxwell's equations is required. Moreover the complex scattered fields must be perfectly known at all directions in the far field at a speckle size; for this reason the numerical approach has to be quite fast, and we restrict ourselves to the case of two dimensional volumes *n=f(x,z)*, with z the normal direction, which means that the geometry is invariant along the *y* axis.

The problem that we solve at hand is the following. We consider a plane interface at the separation (z = 0) of two non-magnetic materials. The superstrate (incident medium) is air while the substrate is an heterogeneous medium of arbitrary thickness d, with a variation n(x,z) of refractive index for 0 < z < d. For z > d the medium has a constant refractive index of 1, identical to that of the superstrate.

This structure is illuminated by a linearly polarized incident wave (s or p polarization) coming from z = -∝. The incident field is assumed to be an approximate gaussian beam and is represented by a finite sum of plane waves. If ${u}^{t}$designates the tangential component of the total field *i.e.* ${E}^{t}$or ${H}^{t}$ for respectively s or p polarization, it follows the equation:

**,**which we define as:satisfies a radiation condition at infinity.

In order to solve this kind of scattering problem we have first pursued the development of an already existing home made FEM (Finite Elements Method) software based on the resolution of the weak form of the Helmholtz equation and used in different configurations [34]. In this software the unknowns are expanded onto P_{1} Lagrange basis functions associated to a conformal triangular mesh of the scene. The amplitude of the scattered field in a given direction is then computed thanks to the Huygens principle. One of the key point of this method is to properly introduce the sources responsible for the incident field. In the case of an incident field of finite extent, the use of a Bayliss-Turkell boundary conditions on the external boundary yields a good approximation of the incident field [35]. Another method consists in using a scattered field formulation [36] of the problem where a reference field calculated in a closed form way is associated to a geometrical configuration of the setup. In that case the total field can be written as:

In the s polarization case, if one substitutes the whole domain $\Omega $ by a perfect electric conductor the current distribution produced by the incident Gaussian beam on $\Gamma $ is such that it produces a null field inside $\Omega $. If one takes${j}_{\Gamma}$ equal to:

The same kind of result can be obtained for p polarization by using duality principle and assuming in this case that $\Omega $is a perfect magnetic conductor.

#### Numerical considerations

The domain $\Omega $ is a rectangle with size typically equal to $100\lambda \text{\hspace{0.17em}}\times \text{\hspace{0.17em}}50\text{\hspace{0.17em}}\lambda $. A free unstructured mesh generator [38] (gmsh©) is used to dicretize the whole domain. P_{1} Lagrange basis functions are used to discretize the field. The sparse system is solved thanks to a sequential direct sparse solver MUMPS (MUMPS©). The discretization of the whole domain gives around 3.10^{6} degrees of freedom in the linear system. This system is solved in more or less 8 min. per configuration and its resolution uses around 10 GB of RAM.

The computation time for the near to far field transformation is negligible regarding the time spent in the resolution of the linear system. Comparisons of the scattered field thanks to this method with respect to analytical expressions in case of an homogeneous substrate have shown an excellent agreement between both methods.

## Bulk and scattering characteristics

Phase and amplitude of the electromagnetic field scattered by a panel of arbitrary 2D volumes were calculated for the two polarization modes *S* and *P*, where the electric *(S)* or the magnetic *(P)* field is along the invariant y-direction, with z the average sample normal. The bulks are realizations of a stationary stochastic process with Gaussian heterogeneities distribution; also, they have Gaussian autocorrelation functions in both x and z directions. As such, the bulks are statistically characterized by the relative index inhomogeneity ${\delta}_{n}/\left|\overline{n}\right|$ defined Eq. (21), and by their correlation lengths ${L}_{x}$ and ${L}_{z}$ in both ${}_{x}$ and ${}_{z}$ directions. Concerning the bulk geometry, it is 100 λ long while its depth varies in the range 0.1λ to 20 λ . It is illuminated with a Gaussian beam under 15° incidence angle at an arbitrary wavelength λ. The inhomogeneities are assumed to be dielectric (real Δn) and engraved in fused silica whose real refractive index is $\overline{n}$ = 1.49 at this wavelength.

_{x}= 100λ and d

_{z}= d the thickness. As an illustration, three examples of cartography with increasing heterogeneities were generated for the optical indices in Fig. 2.

The complex polarized modes ${E}_{S}$ and ${E}_{P}$ of the scattered field are then calculated for 1800 scattering angles in the *θ* angular range *(0°,180°)* and regularly sampled with a step $\delta \theta ={0.1}^{\xb0}$ .

For each root mean square value of the index heterogeneity, we calculated the corresponding mean value of the Total Integrated Scattering (TIS) versus the layer thickness. Results are given in Fig. 3. The mean free paths L are also given for these samples.

Then in our investigation the scattering angle runs from 18° to 50°, so that the specular region (*θ =15°)* is avoided. Notice that the angular resolution largely resolves the speckle, whose characteristic size calculated as the width at mid-height of the intensity pattern autocorrelation is 2.9° for every couple $\left(\frac{{\delta}_{n}}{\left|\overline{n}\right|},d\right)$.

The noticeable speckle size results from the 100λ illumination spot size, a value that was minimized to avoid prohibitive time calculation. As a consequence, the angular investigation range is limited to less than 5 speckle areas for the bulk analysis. For this reason we will not use speckle histograms, contrary to what was done for the surface study [11]. Moreover, previous studies [11, 26] have shown that the knowledge of polarimetric behavior of the scattered field within less than 5 speckles areas is enough to allow to identify the asymptotic behavior of the multiscale degree of polarization MDOP and more precisely to identify the corresponding macroscopic degree of polarization. At last, for all the bulks considered in this paper, 2 speckles areas correspond to an angular aperture greater than 5°, which is enough to quantify the depolarization issues which could occur for imaging applications.

Notice also that invariance of the bulks in one direction is a severe assumption. Hence in this paper's configuration, no cross-polarization is predicted in the plane of incidence, while it should appear for 3D bulks as soon as the perturbative regime is left. However, the Method of Moments gives the rigorous solution of a 1-D wave scattering problem, with all multiple interactions accurately taken into account, which is the point here. In other words, although enpolarization effects cannot be predicted [13–15] under this assumption, great accuracy can be reached for spatial depolarization [11].

## Dispersion of the polarization states

Now we use numerical calculation to analyze the polarization states emitted from the bulks. Notations are the following:

To discriminate the bulks, two parameters are taken into account, which are the relative index inhomogeneity ${\delta}_{n}/\left|\overline{n}\right|$ and the thickness of the disturbed bulk. Both parameters were studied independently. On the other hand, the two correlation lengths are taken to be L_{x} = L_{z} = 100nm. The thickness varies in the range [1.5λ; 40λ] with a 0.5λ step, while 6 heterogeneities are considered within the interval [0.05; 0.1]. Moreover for each sample, 10 realizations are performed.

As a starting point, the angular variation of the polarization ratio β and the polarimetric phase δ of the resulting fields are plotted Fig. 4 to 7 for 4 representative configurations involving 2 thicknesses (d = 5λ and d = 40λ) and 2 relative inhomogeneities (0.05 and 0.1). Figure 4 and 6 involve 0.05 heterogeneity, while Fig. 6 and 7 are for a 10% heterogeneity. We observe that the root mean square of the polarization ratio increases with thickness, with an average around unity. The same result is obtained for the phase term with an average around zero.

Moreover the phase root mean square clearly increases with the inhomogeneity. With a 5λ thickness, a 10% heterogeneity is required to make the phase uniformly distributed within [0; 2π]; on the other hand, at higher thicknesses, lower heterogeneities allow to approach a uniform phase distribution. Therefore the random nature of the scattered field depends both on the heterogeneity and the thickness, which is a key difference with the surface scattering process [11].

In accordance with the results of Fig. 4-Fig. 7, the polarization states are plotted on the Poincaré sphere in Fig. 8-Fig. 9. Notice at this step that the local (temporal) dop is unity at each point, so that all data points lie at the surface of the sphere. The data cloud on the sphere is directly connected with the sample microstructure and its thickness. Here the angular range includes 3 speckle areas resolved with 100 data points. In Fig. 8 we observe that the dispersion of the states increases with the heterogeneity. Then Fig. 9 shows that dispersion also increases with the thickness.

## Spatial depolarization from the bulks

Figures 8 and 9 have revealed the dispersion of the polarization states of the scattered field in less than 3 speckle areas. However these figures do not visualize the path taken by the polarization state when the scattering angle varies continuously. For that reason a complementary view is given in Fig. 10 and shows the continuous variation of the polarization state (Media 1)

Now from Fig. 11 one can expect polarization to fail when several speckle areas are integrated within the receiver aperture $\Delta \Omega $, as predicted in section 2. To quantify such effect we calculated the multi-scale polarization degree *MDOP*$(\Delta \Omega )$. This function is plotted Fig. 10 for two extreme thicknesses and heterogeneities. In this figure 120 data points (scattering angles) are used; since the speckle area is 29 data points (2.9°), the 120 data points explore 4 speckle areas. The first MDOP value calculated for $\Delta \Omega =0$ is always unity since no spatial integration has occurred, so that the spatial MDOP equals the temporal dop. Then at higher apertures the dop decreases due to the polarization dispersion. At a given heterogeneity (0.05), the effect is more pronounced at high thickness. However when the heterogeneity increases, the effect of thickness vanishes.

## Parametric study

Until now only 2 heterogeneities (0.05 and 0.1) were involved in the polarization analysis from the bulks. To complete the numerical study we studied in this section the variation of normalized root mean squares (polarization ratio, phase) versus the heterogeneity and thickness. Such quantity is calculated for all parameters are follows:

Such root-mean-squares are responsible for polarization reduction due to the spatial average. In Fig. 14 below we have considered the macroscopic dop, that is, the lower MDOP value obtained for the maximum integration (4 speckle areas). As a first result, one can see that this asymptotic value decreases with thickness. Furthermore, the slope of the curve around the origins increases with the heterogeneity.

At last, a linear approximation is proposed Eq. (24) to fit the evolution of the macroscopic degree of polarization as a function of the thickness for each index dispersion ${\delta}_{n}/\left|\overline{n}\right|$. The linear fits are plotted Fig. 15.

## Conclusion

The spatial depolarization of light emitted from inhomogeneous bulks was predicted via exact electromagnetic theories. It was shown how full polarization is progressively lost when the heterogeneity increases, and the transition was predicted with an exact electromagnetic theory taking into account the role of the bulk structural parameters.. In a more general way, the partial degree of polarization was connected to the sample microstructure and geometry. Hence these results devoted to bulk processes can be seen as the continuation of previous results given for surface scattering [11]. We expect the conclusions will be useful in the fields of imaging in complex media, biomedical optics and biophotonics, earth observatory). Moreover, the present work deals with 2D bulks, so that a next step will be to address the 3D geometry which is known to reveal additional phenomena such as temporal repolarization of light [14], a phenomenon which takes its origins in cross-scattering coefficients [15]. The exact calculation of these 3D effects is still in progress but requires huge computational facilities and much more calculation time.

## Acknowledgments

This work was performed in the project TraMEL thanks to the financial support of the French National Research Agency. (project # TRAMEL-10-BLAN-0315)

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