1. Introduction

In the process of light scattering, the state evolution in the polarization subspace is described by a non-unitary transformation, which, for linear media, can be represented by a 4×4 matrix. Although we may not be able to describe the scattering phenomenon microscopically, the transformation matrix can be readily obtained from a tomographic procedure. When the polarization state of the field is characterized by means of the Stokes parameters, the transformation matrix is known as Mueller matrix. From the properties of the Mueller matrix one can draw insightful information about the underlying system.

The exploitation of polarimetric properties of light has a wide range of applications in a variety of fields, such as photonics technology, astrophysics, biological optics [1], and quantum communication [2], where scattering is linked to decoherence and loss of entanglement.

Sometimes it is interesting to compare two optical devices or media with respect to their depolarizing power. This could be attempted by defining an observable measure of the amount of depolarization imparted by the device or medium to an arbitrary input state. One can immediately realize, however, that this project is impossible, since the degree of polarization of the output state depends crucially on the input, as it is well known. As an alternative, it is possible to impose a partial ordering on the set of Mueller matrices by defining some measure which can grasp the predominant depolarizing behavior of the sample media over the set of all possible input states. Among them, we focus on two popular metrics, namely, the index of depolarization of the medium (D _M) [3] and the entropy of the medium (E _M) [4]. These two quantities are measures of the average depolarizing power of the sample and the average entropy added to the field, respectively [4, 5, 6]. The representation of the depolarization properties of a medium in the D _M×E _M diagram was introduced by Roy-Brehonnet and Le Jeune [4].

Recently, Aiello and Woerdman [6] showed that all possible scattering media satisfy some universal constraints, in the sense that all physically acceptable Mueller matrices are bounded to a relatively small region in the plane D _M×E _M. Each Mueller matrix M has an associated Hermitian matrix H, whose eigenvalues are sufficient to calculate D _M and E _M. Aiello and Woerdman showed that a uniform distribution of eigenvalues leads to a uniform distribution of points in the physically accessible domain of the D _M×E _M plane. In order to test the predicted universal behavior in depolarized light scattering, Puentes et al. performed a series of experiments, where a broad class of scattering media were characterized [5]. By gathering a relatively large ensemble of Mueller matrices, they were able to cover a significant part of the theoretically allowed domain, but with a rather different occupation density than expected from a uniform distribution of eigenvalues. An intriguing feature, though, was the lack of experimental data inside a specific subregion of the physically accessible domain.

In this paper we present an alternative approach to characterize the universality in depolarized light scattering. By focusing on the statistical properties of the Mueller matrices rather than on the eigenvalues of the associated matrices H, we show that the experimental distribution can indeed be recovered. We introduce a simple physical criterion to justify why the empty subregion of the D _M×E _M plane, though allowed, is almost statistically forbidden for some classes of scattering media. Furthermore, with the help of the Lu-Chipman decomposition of Mueller matrices [7], we attempt to attribute a more operational physical meaning to the few sub-domains defined in [6].

The paper is organized as follows: In Sec. 2 we review the theoretical formulation of the problem, introduce the physically accessible region of the D _M×E _M plane and compare with previously obtained experimental results. In Sec. 3 we analyze the depolarization properties of a class of isotropic media and show how to recover the experimental distribution of points. In Sec. 4 we extend the analysis for more general media, via the Lu-Chipman decomposition. We conclude with Sec. 5

2. Entropy-depolarization relation in random light scattering

The state of polarization of a polarized or partially polarized light field can be represented by a Stokes vector S=(S ₀,S ₁,S ₂,S ₃), where S ₀=I ₀ is the intensity of the field, S ₁=I _H-I _V is the difference of intensities in the horizontal and vertical polarization components, S ₂=I ₊₄₅-I _-45, is the difference of intensities in the +45°and -45°polarization components, and S ₃=I _L-I _R has an analogous definition for left and right circular polarization. Any possible transformation on this state can be completely described by a 4×4 Mueller matrix, which links input states to output states via the relation S ^′=MS.

To any given Mueller matrixM a nonnegative Hermitian matrix H is associated according to

where M _ij are the elements of the Mueller matrix, σ_k (k=0…3) are the standard Pauli matrices and the superscript * denotes complex conjugation. H is normalized so that Tr H=1. The nonnegativeness of H assures that its eigenvalues satisfy 0 ≤λ _i≤1, (i=0, …,3). The depolarization strength of a medium can be measured, for instance, by the index of depolarization (D _M) and the entropy (E _M), which are more easily written as a function of the eigenvalues of H:

The physically accessible region in the D _M×E _M diagram can be obtained by randomly choosing the eigenvalues λ _i constrained to Σ³ _i=0 λ _i=1 from a uniform distribution, i.e., from points distributed on the surface of a 4-dimensional hyper-sphere. By following this simulation strategy we can recover the results predicted in [6] for the physical region in the D _M×E _M plane. The result is shown in Fig. 1(a). The curves connecting the cusp points are plotted according to the equations also presented in [6]. Notice how the simulated points uniformly fill the allowed bounded region in the plane.

The predicted behavior of the quantities D _M and E _M was experimentally verified by Puentes et al. [5], where they conclude that their experimental data was able to cover the allowed domain almost completely, except for the region below the inner curve connecting points A and C.

 

Fig. 1. (a) Numerical simulation of the physically accessible region in the plane D _M×E _M. The points were obtained from an ensemble of uniformly distributed eigenvalues λ. The analytical expressions for the boundaries curves can be be found in [6]. (b) Experimental results obtained by G. Puentes et al. in Opt. Lett. 30, 3216 (2005) for the depolarization properties of a broad class of optically scattering media.

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The experimental results obtained by Puentes et al. [5], reproduced in Fig. 1(b), seem to indicate that even for a wide choice of random depolarizing media, some regions in the D _M×E _M diagram, though allowed, are empty. While their results confirm the predicted “depolarization universality in light scattering”, they also raise some questions about the nature of the “forbidden” region below curve AC.

We will argue that the uniform filling is, in fact, not to be expected. By following a simulation criterion grounded on operational assumptions rather then on eigenvalues distributions, we will show that the unoccupied region below curve AC is, in some cases, almost statistically forbidden.

3. Depolarization properties of an ensemble of randomly oriented particles

The problem of simulating Mueller matrices is considerably more involved than selecting a list of acceptable eigenvalues, which is basically a set of four positive numbers whose sum is normalized to one. The Mueller matrix has sixteen free parameters that are restricted by nontrivial constraints. The relationship between elements of the Mueller matrix have been extensively studied and different, but equivalent, sets of conditions have been obtained [10, 11, 12]. In addition, since every sort of depolarizing and non-depolarizing media have their particular Mueller description, it is not necessarily meaningful (nor practicable) to proceed with a fully general analysis. We will consider, instead, a class of scattering media defined by its microscopic properties and then derive the entropy E _M and the index of depolarization D _M for this class of media.

Our study will be based on an ensemble of randomly oriented microscopic particles, which are not too small in comparison with the wavelength of the incident field. We will assume that the particle concentration is not large so we can neglect dense media effects. Within these considerations, the particles can have any size, shape, orientation and distribution; but we will assume further that the medium is isotropic, an isotropy that may be only in the statistical sense. This class of scattering media have been studied extensively and several applications have been discussed [13, 14, 15, 16, 17, 18].

 

Fig. 2. Scattering geometry. A monochromatic light beam propagating along the z axis is scattered by a medium located in its path. The observed scattering direction and the z axis define the scattering plane. The scattered “horizontal” and “vertical” polarization directions H and V are defined as the directions perpendicular and parallel to the scattering plane, respectively, both perpendicular to the scattering direction.

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Following [13], we will show that by symmetry arguments the structure of the Mueller matrix corresponding to this kind of medium can be considerably simplified. Consider the geometry of Fig. 2. A monochromatic light beam propagating along the z axis is scattered by a medium located in its path and we examine the scattered field in a certain direction, which together with the z axis defines the scattering plane. The scattered “horizontal” and “vertical” polarization directions H and V are defined as the directions perpendicular and parallel to the scattering plane, respectively, both perpendicular to the scattering direction. For the incident beam, H coincides with the x direction and V coincides with the y direction. The input and output states of polarization are defined in their local coordinate systems by the Stokes vectors S and S ^′, respectively. They are related by the Mueller coefficients via

For a symmetrical medium the scattering plane is also a symmetry plane, i.e., if we replace the input beam by the symmetrically polarized beam (S ₀,S ₁,-S ₂,-S ₃), the new scattered beam will be symmetrically polarized with respect to the first scattered beam and it will be described by (S′₀,S′₁,-S′₂,-S′₃). This requires that eight of the Mueller coefficients are identically zero, M ₂₀=M ₂₁=M ₃₀=M ₃₁=M ₀₂=M ₀₃=M ₁₂=M ₁₃=0. Two additional constraints can be obtained by considering the reciprocity theorem [13, 14], namely, M ₀₁=M ₁₀ and M ₂₃=-M ₃₂. The Mueller matrix necessary to describe an isotropic media composed by a low-density collection of microscopic non-spherical particles thus becomes

For the special case of spherical particles we also have M ₂₂=M ₃₃ [13, 19]. For the general case of non-spherical particles, one usually finds that these two coefficients have at least the same sign [4]. In the simulations to be described below, these two parameters will be varied independently, but we will require their signs to be the same. The form of the Mueller matrix with M ₂₂=M ₃₃ is also adequate to describe depolarization by scattering from random rough surfaces [20].

We define now our simulation strategy. First we normalize the Mueller matrix with respect to the element M ₀₀ and define m _ij=M _ij/M ₀₀, so that |m _{i j}|≤1. To give an operational meaning to the strategy, we resort to one of the oldest depolarization criteria: the cross-polarization ratio. In a series of papers published in the 1930’s [14, 21, 22] R. Krishnan employed the two easily measurable quantities

to characterize the depolarizing power, where H _h and V _h are the intensities of the horizontal and vertical scattered field when the input field is horizontally polarized. The factors H _v and V _v are similarly defined. These ratios are known as the cross-polarization ratios, which can be rewritten in terms of the normalized Mueller components as

It is natural to expect that both ratios should be limited by ρ _h≤1 and ρ _v≤1, whereas a higher value for these parameters would mean that the medium not only depolarizes but tends to swap the initial H and V intensities. A value greater than unity for these parameters is termed “anomalous depolarization” [21]. As a matter of fact, many experimental and theoretical results up to date show that for disordered media the cross-polarization ratios are never simultaneously greater than unity [14, 15, 16, 17].

We proceed our simulation by randomly choosing the free parameters m ₀₁, m ₁₁, m ₂₂, m ₂₃ and m ₃₃ in the range -1≤m _ij≤1. Naturally, non-physical matrices will be part of that ensemble, as the nontrivial constraints that delimit the physical Mueller matrices are not taken into consideration. We will post-select our results with the following purposes:

1. To assure that the Mueller matrix is physically admissible. This can be done by checking whether the associated Hermitian matrix H is nonnegative [9].

2. To further select the physically realizable matrices by means of the cross-polarization ratio criterion, i.e., to assure that both ρ _h≤1 and ρ _v≤1.

For each Mueller matrix of the post-selected ensemble we obtain the associated Hermitian H matrix according to Eq. (1). After computing its eigenvalues, the entropy of the medium E _M and the index of depolarization D _M can be calculated using Eq. (2) and Eq. (3) respectively. The thus obtained pairs of points are then plotted in the D _M×E _M diagram and shown in Fig. 3(a). One can immediately notice that this strategy yields a distribution of points more consistent with the experimental results of Puentes et al. In particular, the region below the curve AC (previously defined) is scarcely filled. This region, though allowed, is not physically likely to be obtained. This “incomplete” filling is a consequence of the additional physical constraints that we have imposed. In order to check this statement we repeat the same procedure, but post-selecting only media with “anomalous depolarization”, i.e., the set of media for which both cross-polarization ratios are greater than one (ρ _h>1 and ρ _v>1). The result is shown in Fig. 3(b). It is remarkable to notice how this approach provides a set of points that are upper bounded by a curve that was independently obtained in terms of the nature of the eigenvalues. According to these observations, we may provide a physical interpretation to the AC curve: it distinguishes the region of normal depolarization from the “anomalous depolarization” (according to cross-polarization ratio criterion). It should be stressed however, that this criterion does not define an exact partition of the D _M×E _M diagram, as can be verified by the presence of a few points below the AC curve in Fig. 3(a).

 

Fig. 3. Depolarization properties of an ensemble of randomly oriented non-spherical microscopic particles. D _M is the index of depolarization of the medium and E _M the entropy of the medium. After obtaining a set of physically acceptable Mueller matrices we post select the results according to (a) physically realizable matrices as specified by the cross-polarization ratio criterion; both ρ _h≤1 and ρ _v≤1. (b) Matrices for media with “anomalous depolarization”, that is, ρ _h>1 and ρ _v>1.

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It is important to remark that our conclusions are directly linked to the assumed form of the Mueller matrix. In the next section we will extend this analysis to more general matrices. We will not focus in any specific scattering medium, but we will provide an explanation to the experimental data of Puentes et al. in terms of physical parameters of the Lu-Chipman decomposition of Mueller matrices. We also provide a criterion for “anomalous depolarization” in terms of those parameters.

4. Depolarization properties via polar decomposition of Mueller matrices

We will now make use of the polar decomposition of Mueller matrices, originally proposed by Lu and Chipman [7]. Any (normalized) Mueller matrix can be written in the generic form

where d and p are three-component vectors known as diattenuation and polarizance, and m is a 3×3 submatrix. It is possible to decompose it into a sequence of three factors, M=M_ΔM_RM_D, each one representing an elementary operation on the Stokes vector. In this factorization we can split the contributions of the diattenuation (M_D) and retardance (M_R) from the depolarization (M_Δ), with

The matrix M_R represents a purely birefringent element. M_D, which is completely determined by the diattenuation vector d, represents the action of a dichroic element. The modulus of d defines the amount of diattenuation, and its direction defines the polarization direction (in the Poincaré sphere) for maximum transmission. The elements of the submatrix m_d are defined as ${\mu }_{\mathrm{ij}}=d_i{d}_j+\sqrt{1-{\mid \mathbf{d}\mid }^2}\left({\delta }_{\mathrm{ij}}-d_i{d}_j\right)$, where d _i(i=1…3) are the components of d. According to Eq. (8), M_D is completely defined by the first line of the normalized Mueller matrix. M_Δ represents a purely depolarizing element, with m_Δ being a symmetric submatrix. A detailed derivation of these formulas can be found in [7]. Since M_R represents a birefringent element, the submatrix m_R simply implements a rotation on the Poincaré sphere. In addition, the symmetric submatrix m_Δ can be diagonalized by retarder Mueller matrices. These properties allow us to write

M^″ _R=M^′T _RM_R also represents a rotation and M^diag _Δ is a depolarizing Mueller matrix whose submatrix m_Δ=diag{a,b,c} is diagonal.

Although in a different form, we still have the most general Mueller matrix, describing non-depolarizing and depolarizing media of any kind. We will consider only the cases in which the singular values of m_Δ are all equal (a=b=c). The factor M_Δ will therefore represent an isotropic depolarizing media. With certain freedom, we define “anomalous depolarization” in this context those transformations for which the singular value a is negative. This is intuitively reasonable, as the effect of the purely depolarizing Mueller matrix M_Δ=diag{1,-|a|,-|a|,-|a|}, with |a|<1, is to reduce the norm of the Stokes vector (that is, to depolarize) but also to invert the Stokes vector through the origin, for every possible input state. This is not likely to happen in a random medium. If a, b and c are not equal, the Stokes vector undergoes additional operations besides shrinking and inversion. Depending on the values of a, b and c, “anomalous depolarization ” may or may not occur. The restriction a=b=c was made in order to focus the present analysis on the role of “anomalous depolarization”.

We need to establish now a simulation criterion. A physically reasonable criterion is to consider the vectors d and p, the three-dimensional rotation matrix m^′ _R and the singular value a the essential attributes concerning the depolarization properties of a Mueller matrix. The ensemble of Mueller matrices is constructed by randomly choosing these parameters from an uniform distribution. Naturally, weighted distributions could be employed if we had a specific class of scattering systems in mind. Since this is not the case, we proceed with a very regular distribution: (i) we select the vectors d and p from a three-dimensional unit ball (sphere of unitary radius and its interior), (ii) the singular value a is obtained from the interval [0,1], so as to rule out “anomalous depolarization” and (iii) the submatrix m^′ _R is chosen from an ensemble of uniformly distributed three-dimensional real special orthogonal matrices (rotation matrices) [8]. Once we have this ensemble we can use again Eq. (1), (2) and (3) to compute the index of depolarization and the entropy of the medium. We postselect the results by checking whether the associated Hermitian matrix H is nonnegative [9].

 

Fig. 4. Numerical simulations of the depolarization properties of random Mueller matrices obtained according to the polar decomposition criterion (see text). (a) Singular values of the submatrix mΔ are all equal, a=b=c, and nonnegative. We impose further restrictions: (b) modulus of the polarizance vector |p|≤0.1, (c) diattenuation vector|d|≤0.1 and (d) diattenuation vector |d|≥0.9.

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The results of some numerical simulations performed are displayed in figures 4(a) to 4(d). In Fig 4(a) all parameters are free and chosen as specified above. Next we impose further restrictions: Fig. 4(b) represents Mueller matrices with small polarizance (|p|≤0.1), and figures 4(c) and 4(d) elements where the losses due to dichroism are very small (|d|≤0.1) and very large (|d|≥0.9) respectively. Notice how the restrictions imposed on the Mueller matrices affect the density of points through the theoretically allowed region. For example, large dichroism leads to states with a greater degree of polarization, concentrated on the right-lower sub-domain of the plot. The most interesting conclusion, however, is that the region below the AC curve is almost empty, which is again the same pattern experimentally observed. If we relax the restriction on parameter a, allowing it to assume negative values, the region below curve AC will be filled.

This approach, despite not having a direct connection with a specific class of scattering media, is less restrictive than the one discussed in Sec. 3. The physical interpretation of the domain below curve AC as a region of “anomalous depolarization” remains valid in a more general context and explains the incomplete covering of the D _M×E _M domain obtained by Puentes et al.

5. Conclusion

In this work we analyzed the problem of universality in depolarized light scattering in the entropy versus index of depolarization representation. In particular, we extended the theoretical analysis of Aiello and Woerdman [2] in order to better understand the experimental results of Puentes et al. [5]. We showed how a physically grounded criterion, based on the Mueller matrix representation, leads to a statistical distribution of points in the D _M×E _M diagram more consistent with the experimental observations. We explain the origin of of the “inaccessible” region in the depolarization-entropy plane as a region of “anomalous depolarization”. For an ensemble of randomly oriented microscopic particles this concept can be well understood in terms of one the earliest depolarization measures: the cross-polarization ratio. Following, we extended the concept of “anomalous depolarization” to more general media through the polar decomposition of Mueller matrices. The results are self-consistent. Finally, we provided some physical meaning to the sub-domains in the D _M×E _M diagram, as they were originally defined only mathematically.

In this work we did not intend to give a full description of the statistical properties of random Mueller matrices, but to highlight the role of the simulation criteria on the distribution of some depolarization measures. Our analysis could be extended for different media with different depolarization properties by following basically the same strategy.