Purity-depolarization relations and the components of purity of a Mueller matrix

Aziz Tariq; Honghui He; Pengcheng Li; Hui Ma

doi:10.1364/OE.27.022645

1. Introduction

The study of polarized light-matter interaction has got a lot of interest because of its potential applications in several fields of science ranging from physical to life sciences. For instance, light depolarization caused by a linear and passive medium can be used to understand the nature of the medium, which may be exploited for the diagnostic of malignancies and cancers [1–5]. The depolarization character of a medium can be studied through the Stokes-Mueller formalism in which a 4 × 1 Stokes vector is transformed by the medium to an emerging 4 × 1 Stokes vector [6,7]. The incident and outgoing Stokes vectors encompass the polarization states of the incident and scattered lights, respectively. The transformation is represented by a 4 × 4 real element matrix called the Mueller matrix which encodes all the polarization altering properties of the medium. One of the advantages of the Stokes-Mueller formalism is that it provides polarimetric information about the medium that depolarizes the incident light, therefore it is favored by experimenters [7,8]. However, to decode these properties, the Mueller matrix of a medium is further transformed or decomposed to physically meaningful parameters or matrices [9–11]. One such parameter is known as the depolarization index $(P_{4 D}$ ) (or the overall degree of polarimetric purity), which can readily be obtained from the squared values of the elements of a given Mueller matrix that characterizes the depolarization behavior of the medium under study [2,12].

The subscript of $P_{4 D}$ denotes dimensions of the associated Hermitian covariance matrix of a Mueller matrix, which describes the statistical nature of the Mueller matrix [8]. Therefore, $P_{4 D}$ can also be obtained from the three invariant indices of polarimetric purity (IPP) that are incurred by the eigenvalues of the covariance matrix [13]. The IPP provides a detailed structure of the polarimetric purity and graphically forms a tetrahedron called the purity space [13], which is composed of the three mutually orthogonal axes that are represented by the IPP ( $P_{1}$ , $P_{2}$ and $P_{3}$ ). Any point on the purity space represents the contributions of the statistical weights of the spectral components of the Mueller matrix of a medium. On the other hand, a physically realizable graphical representation of $P_{4 D}$ via components of purity (CP) in terms of the degree of polarizance $(P_{P})$ and degree of spherical purity $(P_{S})$ is given in a two-dimensional graph [14] called the component of purity figure. Moreover, the combined relations of the set of five invariant quantities: the magnitudes of the diattenuation and polarizance vectors, and the three IPP in a two-dimensional graph called the common purity Figure has been proposed [15], which characterizes the pure, two-dimensional, three-dimensional, and four-dimensional media based on the rank information of the covariance matrix. Recently, a feasible graphical representation of the depolarization relation with the overall purity index $(P I_{4 D})$ is given by Tariq et al. [16] where the said index is defined from the equal quadratic average of the three IPP. More recently, a generalized geometrization of the depolarization through the three depolarization metrics: entropy, depolarization index, and overall purity index $(1 - S_{4 D}, P_{4 D}, P I_{4 D})$ based on the eigenvalues of the covariance matrix is presented [17]. It should be noted that $P_{P}$ and $P_{S}$ (components of purity $C P$ ) are directly obtained from the elements of a Mueller matrix representing diattenuation-polarizance and depolarization (spherical depolarization), respectively, whereas the eigenvalue-based depolarization metrics $(S_{4 D}$ and $P I_{4 D})$ are obtained from the eigenvalues of the covariance matrix. All the above-mentioned graphical illustrations of the depolarization character of material media do not completely circumscribe both contributions of $C P$ and IPP to the depolarization index $P_{4 D}$ .

In this paper, a physically realizable purity index - components of purity $(P I_{4 D} - C P)$ space is proposed which comprises of $P_{4 D}$ as a combination of the two components of purity $(C P)$ representing the basis of a coordinate plane whose normal is the overall purity index $(P I_{4 D})$ . Herein lies an additional advantage of the proposed space over components of the purity figure that the latter is the projection of the former on the coordinate plane. The $P I_{4 D} - C P$ space can be generated by writing $P_{S}$ and $P_{P}$ in polar coordinates with the azimuth ranging from 0 to $π / 2$ , and satisfying a physical constraint on the values of $P_{P}$ given in Ref [14]. The robustness of the proposed model is demonstrated by obtaining the polarimetric information of some computed and experimentally measured Mueller matrices by plotting them on the $P I_{4 D} - C P$ space. Furthermore, Monte Carlo simulations based on the sphere-cylinder scattering model (SCSM) are implemented to differentiate media via the $P I_{4 D} - C P$ space.

2. The $P I_{4 D} - C P$ space

In the Stokes-Mueller formalism, the Stokes vectors incorporate the intensity and the polarization states of the incident and exiting lights and the Mueller Matrix $M$ describes the nature of the scattering medium, which is given as,

M = M_{00} \hat{M} = M_{00} [\begin{matrix} 1 & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{matrix}] .

Here,

\hat{M}

is the normalized Mueller matrix with

m_{i j} = M_{i j} / M_{00}

(i, j = 0, 1, 2, 3)

and

m_{00} = 1

. It is customary to write

\hat{M}

in the following block form [14,15],

\overset{⌢}{M} = [\begin{matrix} 1 & D^{T} \\ P & m \end{matrix}] .

T denoting the transpose. The diattenuation vector D of

\hat{M}

is given by [2,14,15]

D = [\begin{matrix} m_{01} \\ m_{02} \\ m_{03} \end{matrix}],

and the polarizance vector P of

\hat{M}

is expressed as [2,14,15]

P = [\begin{matrix} m_{10} \\ m_{20} \\ m_{30} \end{matrix}] .

The magnitudes of these vectors of the

\hat{M}

give the amount of diattenuation and polarizance by the interacting media, respectively. From the reciprocity constraint of the

\hat{M}

(or

M

), both the vectors give an interchangeable meaning depending upon the direction to which the light interacts with the medium [2,15]. Submatrix m is expressed as,

m = [\begin{matrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{matrix}] .

The Frobenius norm of the 3x3 submatrix can be written as,

{‖ m ‖}_{_{2}} = \sqrt{\sum_{i = i}^{3} \sum_{j = 1}^{3} {| m_{i j} |}^{2}} = \sqrt{t r (m^{T} m)},

with tr standing trace and superscript T being the conjugate transpose. It is related to the degree of spherical purity

P_{S}

as [2,13–15],

P_{S} = \frac{{‖ m ‖}_{_{2}}}{\sqrt{3}},

with

P_{S}

ranging from 0 to 1. The magnitude of the polarizance and diattenuation vectors are [2,14,15],

P = | P | \equiv (\sqrt{\sum_{i = 1}^{3} m_{i 0}^{2}}),

and,

D = | D | \equiv (\sqrt{\sum_{j = 1}^{3} m_{0 j}^{2}}) .

Furthermore, the overall degree of polarizance irrespective of the direction of propagation of the incident Stokes vector is defined by an average measure of the polarizance and diattenuation termed as the overall degree of polarizance

P_{P}

[2,14,15],

P_{P} = \sqrt{\frac{{| P |}^{2} + {| D |}^{2}}{2}} .

The physically acceptable maximum value of $P_{P}$ for a linear passive medium is 1 that represents a fully pure polarizer, while the minimum value of $P_{P}$ being 0 describes the medium with the absence of polarization and diattenuation properties [2,14]. As mentioned in the introduction section that the depolarization index $P_{4 D}$ defined by the average measurement of the intensity of the polarization portion of the incident and exiting beams, so it can be obtained from the elements of the $\hat{M}$ as [12],

P_{4 D} = \sqrt{\frac{\sum_{i, j = 0}^{3} m_{i j}^{2} - 1}{3}} .

Therefore,

P_{4 D}

can be expressed as a function of

P_{S}

and

P_{P}

given as [2,14],

P_{4 D} = \sqrt{\frac{2 P_{P}^{2}}{3} + P_{S}^{2}},

\frac{P_{P}^{2}}{{(\sqrt{3 / 2} P_{4 D})}^{2}} + \frac{P_{S}^{2}}{{(P_{4 D})}^{2}} = 1.

The above expression is an equation of an ellipse. Since the variables $P_{P}$ and $P_{S}$ , and parameter $P_{4 D}$ are positive and real-valued, therefore, Eq. (13) can graphically be represented by the first quadrant of the ellipses with elliptical curves at constant values of $P_{4 D}$ [2,13]. The range of the degree of polarizance $P_{P}$ from Eq. (13) is $0 \leq P_{P} \leq \sqrt{3 / 2}$ , which shows the non-physical values for $P_{P} \geq 1$ . Therefore, an extra constraint to restrict $0 \leq P_{P} \leq 1$ , representing a hyperbolic curve has been proposed [14] which is given by,

P_{P}^{2} \leq \frac{1 + 3 P_{S}^{2}}{2} .

Together with the above expression (Eq. (14)) being the extra constraint and Eq. (13) with

P_{4 D}

as a parameter, the feasible regions of the components of purity of polarized light scattering media have been drawn by Gil [14] whose origin point corresponds to an ideal depolarizer while the only point of intersection of the hyperbolic curve to the outer elliptical branch with

P_{P} = P_{4 D} = 1

and

P_{S} = \sqrt{1 / 3}

represents a Mueller matrix of an ideal polarizer [14].

On the other hand, a Hermitian (covariance) positive semi-definite (PSD) matrix $H (\hat{M})$ , which can be extracted from the elements of the $\hat{M}$ , gives the statistical information of the scattered light such as entropy and depolarization index [6,8,18]. It can be expanded in non-Gell-Mann basis in terms of modified Dirac matrices $E_{i j}$ given in the following [2,7],

H (\hat{M}) = \frac{1}{4} \sum_{i, j = 0}^{3} m_{i j} E_{i j} = \frac{1}{4} \sum_{i, j = 0}^{3} m_{i j} (σ_{i} \otimes σ_{j}) .

Here

σ_{i}

are the Pauli matrices and the 2 × 2 identity matrix, and

\otimes

stands for Kronecker product. The eigenvalue spectrum of the

H (\hat{M})

can be written in the following form [13],

1 \geq λ_{0} \geq λ_{1} \geq λ_{2} \geq λ_{3} \geq 0

This arrangement can be exploited to obtain three peculiar quantities that provide complete information of the polarimetric purity of the

\hat{M}

. These quantities are called the indices of polarimetric purity (IPP) [13] which are invariant under the rotational transformation of the

\hat{M}

and are invariant under more general retarder transformations [2]. The indices of polarimetric purity (IPP) are obtained from the eigenvalues of

H (\hat{M})

such that [13],

P_{1} = \frac{λ_{0} - λ_{1}}{t r (H (\hat{M}))},

P_{2} = \frac{λ_{0} + λ_{1} - 2 λ_{2}}{t r (H (\hat{M}))},

P_{3} = \frac{λ_{0} + λ_{1} + λ_{2} - 3 λ_{3}}{t r (H (\hat{M}))},

with

tr(H (\hat{M}) = 1

. By the convexity property of Mueller matrix, any depolarizing

\hat{M}

can be decomposed into a maximum of four components whose relative weights are determined by the magnitude of the eigenvalues of the

H (\hat{M})

. The IPP values are restricted by the following inequality [13],

0 \leq P_{1} \leq P_{2} \leq P_{3} \leq 1.

Thus, the overall purity index

P I_{4 D}

is expressed as [16],

P I_{4 D} = \sqrt{\frac{P_{1}^{2} + P_{2}^{2} + P_{3}^{2}}{3}} .

And the depolarization index

P_{4 D}

can be expressed as a function of the IPP [13],

P_{4 D} = \sqrt{\frac{1}{3} (2 P_{1}^{2} + \frac{2}{3} P_{2}^{2} + \frac{1}{3} P_{3}^{2})} .

This is the equation (Eq. (22)) for the first octant of a triaxial ellipsoid with

P I_{4 D}

as a parameter, but the IPP follows the inequality Eq. (20), which forms a tetrahedron in the three mutually axes as

P_{1}

,

P_{2}

, and

P_{3}

called the purity space [13]. Gil [15] further proposed a common purity figure in which a set of five parameters

(D, P, P_{1}, P_{2}, P_{3})

are used that classify polarized light scattering media into four main types indicating the rank information of the

H (\hat{M})

. The common purity figure is appropriate for understanding the purity structure of Mueller matrices and useful in the characterization and classification of material media [15]. However, there are more restricting possibilities available for different sets of the IPP that follows the inequality Eq. (20), which may be useful to understand the complete geometrical picture of polarimetric media. In addition, the purity-depolarization [16] and entropy-depolarization [19] diagrams provide the statistical properties of the polarized light scattered from media based on the IPP and eigenvalues which are the coefficients of the components of the

\hat{M}

and

H (\hat{M})

in the characteristic decompositions, respectively. The characteristic decomposition contains the

{\hat{M}}_{J 1}

(pure polarizer), 2D (

{\hat{M}}_{2}

) and 3D (

{\hat{M}}_{3}

) unpolarized components, and the

{\hat{M}}_{4}

(ideal depolarizer) such that [2],

\hat{M} = (P_{1}) {\hat{M}}_{J 1} + (P_{2} - P_{1}) {\hat{M}}_{2} + (P_{3} - P_{2}) {\hat{M}}_{3} + (1 - P_{3}) {\hat{M}}_{4} .

The $P I_{4 D} - P_{4 D}$ plane [16] describes physical constraints on the nature of the depolarizing medium and provides complete information on the structure of polarimetric purity based on IPP of the medium. However, it cannot distinguish different components of purity $(C P)$ contributing to $P_{4 D}$ . On the other hand, a graphical representation of $P_{4 D}$ via $C P$ with different sources of purity (i.e., $P_{P}$ and $P_{S}$ ) [14] are insensitive to the eigenvalues of the covariance matrix. Since $P_{S}$ and $P_{P}$ can be represented in polar coordinates given as $P_{S} = P_{4 D} \cos φ$ and $P_{P} = \sqrt{3 / 2} P_{4 D} \sin φ$ , satisfying Eq. (13), such that the physical constraint inequality Eq. (14) becomes $3 {(P_{4 D})}^{2} \cos 2 φ \geq - 1$ , where the range of the azimuth $φ$ is $0 \leq φ \leq π / 2$ . Thus, a physically realizable purity index - components of purity $(P I_{4 D} - C P)$ space (Fig. 1) may be constructed by continuously varying $φ$ from 0 to π/2. Note that at $φ = 0$ , the $P I_{4 D} - P_{S}$ plane with $P_{S} = P_{4 D}$ and $P_{P} = 0$ is obtained; while at $φ = π / 2$ , the plane becomes $P I_{4 D} - P_{P}$ plane with $P_{S} = 0$ . However, at $φ = π / 2$ , the constraint inequality of Eq. (14) restricts $0 \leq P_{P} \leq \sqrt{1 / 2}$ and $0 \leq P I_{4 D} \leq \sqrt{2 / 3}$ . For all continuous values of the azimuth, the $P I_{4 D} - C P$ space is obtained through sweeping the first octant counter-clockwise with the $P I_{4 D} - P_{4 D}$ plane along elliptical trajectories whose continuum projection on the coordinate plane $P I_{4 D} = 0$ yields the common purity figure or (P_P – P_S) plane [14].

Fig. 1 Graphical representation of polarimetric purity via $P I_{4 D} - C P$ space

Download Full Size | PDF

To each value of $P_{4 D}$ in the two-dimensional $P I_{4 D} - P_{4 D}$ plane, there are a set of values for $P I_{4 D}$ . Nevertheless, in the $P I_{4 D} - C P$ space, the values of $P_{4 D}$ are shown by elliptical curves. Each elliptical curve has a constant value for $P_{4 D} = P_{4 D} (P_{s}, P_{P})$ that is constructed by the quadratic average of the two CP and has a set of PI_4D values normal to them. Thus, each elliptical curve in the space is an iso-depolarization curve. The detailed description of the points, curves, faces, and the volume elements in the space for characterization of the Mueller matrices are given in the following sections.

2.1 Points and elliptical curves on the $P I_{4 D} - C P$ space

The sets of points are grouped according to a constant value of $P_{4 D}$ such as the elliptical curves bounded between points $A_{3}$ and $A_{4}$ to points $A_{1}$ and $A_{2}$ ; are represented by an elliptical surface $A_{1} A_{2} A_{4} A_{3} A_{1}$ parallel to the $P I_{4 D}$ -axis (Fig. 1). Any point representing a Mueller matrix on this surface has P_4D= 1/3. Similarly, any point on the surface $B_{1} B_{2} B_{4} B_{3} B_{1}$ parallel to the $P I_{4 D}$ -axis has the value $P_{4 D} = \sqrt{1 / 3}$ .

• Point O

The Muller matrix of the ideal depolarizer is designated by point O (Fig. 1). Mueller matrices at this point have $P_{1} = P_{2} = P_{3} = P_{P} = P_{s} = 0$ . Hence, $P_{4 D} = 0$ , then the Mueller matrix is given by,

\hat{M} = [\begin{matrix} 1 & 0^{T} \\ 0 & 0_{3 \times 3} \end{matrix}] .

• The set $A$ $(A_{1}, A_{2}, A_{3}, A_{4})$

The set A has the same value for the depolarization index, i.e., P_4D=1/3. Mueller matrices at the points $A_{1} (0, 1 / 3, \sqrt{1 / 3})$ , $A_{3} (0, 1 / 3, 1 / 3)$ , and on the curve $A_{1} A_{3}$ ; and $A_{2} (1 / 3, 0, \sqrt{1 / 3})$ , $A_{4} (1 / 3, 0, 1 / 3)$ , and on the curve $A_{2} A_{4}$ are written respectively as [2,15],

\hat{M} = [\begin{matrix} 1 & D^{T} \\ P & 0_{3 \times 3} \end{matrix}],

\hat{M} = [\begin{matrix} 1 & 0^{T} \\ 0 & m \end{matrix}] .

The matrix Eq. (25) is achieved by the contributions of the quadratic average of the diattenuation D and polarizance P vectors [2], whereas in Eq. (26)

P_{P} = 0

and

P_{S}

ranging from 0 to 1. However, points on the elliptical curve

A_{1} A_{2}

are achieved by the IPP as P₁=P₂=0 and

P_{3} = 1

, therefore, the curve at

P I_{4 D} = \sqrt{1 / 3}

holds the following equation [14,15],

2 P_{P}^{2} + 3 P_{S}^{2} = 1 / 3 .

The Mueller matrices of the points on this curve are expressed as $\hat{M} = {\hat{M}}_{3}$ by the characteristic decomposition. The points on the elliptical curve $A_{3} A_{4}$ have the IPP as $P_{1} = P_{2} = P_{3} = 1 / 3$ . The characteristic decompositions for the points on this curve indicate that their Mueller matrices can be synthesized by the linear combination of pure polarized and totally depolarized components.

• The set $B$ $(B_{1}, B_{2}, B_{3}, B_{4}, B_{5})$

The general Mueller matrices for the B₁ and B₃; and the $B_{2}$ and $B_{4}$ belong to the same as Eq. (25) and Eq. (26), respectively. However, at the points $B_{1}$ and $B_{3}$ , the Mueller matrices are equivalent to $\hat{M} = {\hat{M}}_{2}$ and at the $B_{3}$ and $B_{4}$ , $\hat{M} = ({\hat{M}}_{J 1} + 2 {\hat{M}}_{4}) / 3$ by the characteristic decomposition. A typical example of the Mueller matrix located on the point $B_{1}$ is depolarizing a partially polarized light and the subsequently polarizing it again. It can be done by choosing a depolarizer followed by a linear horizontal polarizer such that [20],

[\begin{matrix} 1 / 2 & 1 / 2 & 0 & 0 \\ 1 / 2 & 1 / 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] = \frac{1}{2} [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] .

In this case, the IPP with

P_{1} = 0

and

P_{2} = P_{3} = 1

, showing that the final matrix

\hat{M}

is equivalent to

\hat{M} = {\hat{M}}_{2}

, with

P_{P} = 1 / 2

and

P_{s} = 0

. Then the values for

P I_{4 D}

and

P_{4 D}

are

\sqrt{2 / 3}

and

\sqrt{1 / 3}

, respectively. The elliptical curve

B_{1} B_{2}

at

P I_{4 D} = \sqrt{2 / 3}

follows the equation,

2 P_{P}^{2} + 3 P_{S}^{2} = 1.

• The set $C$ $(C_{1}, C_{2})$

The point $C_{1} (0, 1, 1)$ is not physically realizable while the Mueller matrix at $C_{1} (1, 0, 1)$ with $P_{4 D} = P I_{4 D} = 1$ can be expressed as a pure retarder [2],

\hat{M} = [\begin{matrix} 1 & 0^{T} \\ 0 & m_{R} \end{matrix}] .

This represents a Mueller matrix of non-depolarizing medium (a pure retarder), which is a pure component

({\hat{M}}_{J 1})

in the characteristic decomposition.

• The point D

This point belongs to an ideal polarizer with coordinates $D (1, \sqrt{1 / 3}, 1)$ whose Mueller matrix with the IPP as $P_{1} = P_{2} = P_{3} = 1$ and $P_{4 D} = P I_{4 D} = 1$ is given by [2],

\hat{M} = [\begin{matrix} 1 & D^{T} \\ P & P \otimes D^{T} \end{matrix}] .

Any point on the curve

C_{2} D

(excluding the point

D

) following the equation

2 P_{P}^{2} + 3 P_{s}^{2} = 3

represents a pure medium. Note that the characteristic decomposition at points C and D does not differentiate media, however, the components of purity express the sources of purity by Eqs. (30) and (31).

2.2 Outer faces of the $P I_{4 D} - C P$ space

There are six faces of the $P I_{4 D} - C P$ space with the three upper faces, a lower face, and the two side faces that can be characterized by some constraints on the IPP values as follows,

• The face $O A_{1} A_{2} O$

When assuming the IPP as, $P_{1} = P_{2} = 0$ and $0 \leq P_{3} \leq 1$ , the points lie on the surface $O A_{1} A_{2} O$ with $0 \leq P I_{4 D} \leq \sqrt{1 / 3}$ and $0 \leq P_{4 D} \leq 1 / 3$ . The characteristic decomposition entails the Mueller matrix of any point on this surface,

\hat{M} = P_{3} {\hat{M}}_{3} + (1 - P_{3}) {\hat{M}}_{4} .

Hence, any point on this surface may be characterized by a combination of 3D and completely depolarized components in the characteristic decomposition with the

rank (H (\hat{M})) = 4

. In the mould of the components of purity, the points on this surface have different elliptical curves restricted to

0 \leq 2 P_{P}^{2} + 3 P_{s}^{2} \leq 1 / 3

with

0 \leq P_{s} \leq 1 / 3

and

0 \leq P_{P} \leq 1 / \sqrt{6}

.

• The face $A_{1} A_{2} B_{3} B_{1} A_{1}$

The points on the face $A_{1} A_{2} B_{3} B_{1} A_{1}$ can be realized by considering the IPP values as $P_{1} = 0$ , $0 \leq P_{2} \leq 1$ , and $P_{3} = 1$ , with $\sqrt{1 / 3} \leq P I_{4 D} \leq \sqrt{2 / 3}$ and $1 / 3 \leq P_{4 D} \leq \sqrt{1 / 3}$ . The elliptical curves are restricted by inequality as $1 / 3 \leq 2 P_{P}^{2} + 3 P_{s}^{2} \leq 1$ . The ranges of $P_{s}$ and $P_{p}$ are $0 \leq P_{s} \leq \sqrt{1 / 3}$ and $0 \leq P_{P} \leq 1 / \sqrt{2}$ . The Mueller matrix of these points on this plane can be decomposed as a linear combination of the 2D and 3D components of the characteristic decomposition, written as in the following,

\hat{M} = P_{2} {\hat{M}}_{2} + (1 - P_{2}) {\hat{M}}_{3} .

• The face $B_{1} B_{2} C_{2} D B_{1}$

The IPP values with, $0 \leq P_{1} \leq 1$ and $P_{2} = P_{3} = 1$ , populate the surface $B_{1} B_{2} C_{2} D B_{1}$ . The ranges of $P_{4 D}$ and $P I_{4 D}$ are $\sqrt{1 / 3} \leq P_{4 D} \leq 1$ and $\sqrt{2 / 3} \leq P I_{4 D} \leq 1$ . The elliptical curves follow the inequality given as,

1 \leq 2 P_{P}^{2} + 3 P_{S}^{2} \leq 3,

with the extra constraint such that for

P_{s} = 0

, the

P_{P}

values with

\sqrt{1 / 2} \leq P_{P} < 1

are nonphysical Mueller matrices. This restriction is the consequence of the Eq. (14) [14,15]. For the range of

P I_{4 D}

values

\sqrt{2 / 3} \leq P I_{4 D} \leq

1, the hyperbolic curves form a hyperbolic surface which is represented by

B_{1} D B_{3} B_{1}

. From the statistical point of view with the characteristic decomposition, the points on the face

B_{1} B_{2} C_{2} D B_{1}

can be decomposed as a linear combination of the spectral components with the one and two equiprobable eigenvalues such that,

\hat{M} = P_{1} {\hat{M}}_{J 1} + (1 - P_{1}) {\hat{M}}_{2} .

Thus, a Mueller matrix on this face can be realized as the composition of a completely pure component and a 2D unpolarized component of the characteristic decomposition, which shows significantly less depolarization.

• The lower face $O B_{3} D C_{2} O$

When all the IPP values are equal such that $0 \leq P_{1} = P_{2} = P_{3} \leq 1$ then the Mueller matrices lie on the lower face of the space that can be characterized by a linear combination of completely pure and the ideally depolarized components such that $\hat{M} = P_{1} ({\hat{M}}_{J 1}) + (1 - P_{3}) {\hat{M}}_{4}$ . The equivalent representation of this face corresponds to the maximum entropy curve of the entropy-depolarization diagram [19]. The ranges of $P I_{4 D}$ and $P_{4 D} = \sqrt{2 P_{P}^{2} + 3 P_{s}^{2} / 3}$ are between 0 and 1.

• The side faces $O A_{1} B_{1} B_{3} O$ and $O A_{2} B_{2} C_{2} O$

The side faces $O A_{1} B_{1} B_{3} O$ and $O A_{2} B_{2} C_{2} O$ of the space have different sets of the IPP values with former $0 \leq P_{4 D} = P_{P} \leq \sqrt{1 / 2}$ , $P_{s} = 0$ and the latter has $0 \leq P_{4 D} = P_{s} \leq 1,$ $P_{P} = 0$ ,. The region $B_{1} C_{1} B_{3} B_{1}$ with $P_{s} = 0$ is physically not achievable. The face $O A_{2} B_{2} C_{2} O$ can be decomposed into four Mueller matrices which may completely be described by the two-dimensional purity-depolarization plane with $P_{4 D} = P_{S}$ , whereas the extra constraint by Eq. (14) on the face $O A_{1} B_{1} C_{1} O$ with $P_{S} = 0$ excludes the non-physical region $B_{1} C_{1} B_{3} B_{1}$ . It is worth mentioning that the type-I depolarizers [21] fall under a category that lies on the side face $O A_{2} B_{2} C_{2} O$ with $P_{4 D} = P_{s}$ i.e., the two-dimensional purity-depolarization plane for which an example of the Rayleigh and Mie spherical scatterers has been demonstrated in Ref [16], whereas the type-II depolarizers [21] are characterized by a surface in the space with $P_{P} = 1 / 2$ , $0 \leq P_{s} \leq 1 / \sqrt{6}$ and $0.5951 \leq P I_{4 D} \leq 0.8292$ .

2.3 Volume elements in the $P I_{4 D} - C P$ space

The regions that classify scattering media by the rank information of the $H (\hat{M})$ in the two-dimensional common purity figure (Ref. [14]) are represented by volume elements in the $P I_{4 D} - C P$ space such that the points of the volume elements, $B_{1} B_{2} B_{4} C_{2} D B_{3} B_{1}$ $A_{1} A_{2} A_{4} C_{2} D B_{3} A_{3} A_{1}$ , and $O C_{2} D B_{3} O$ represent the two-dimensional, three-dimensional, and four-dimensional media, respectively, while the curve $D C_{2}$ belongs to the pure media. Therefore, these volume elements indicate the type of light scattering media based on the IPP and provide information on the sources of purity via the components of purity. Nevertheless, the realization of the volume elements in the space as $O B_{2} B_{1} O$ , $A_{2} C_{2} D B_{5} A_{1} A_{2}$ , $O B_{2} C_{2} D B_{1} O$ , and $O A_{2} C_{2} D B_{5} B_{3} O$ can be used to characterize Mueller matrices representing any of these volumes into three spectral components by the characteristic decomposition, with ranges of the coordinate axes given in Table 1.

Table 1. The regions of the $P I_{4 D} - C P$ space characterized by the characteristic decomposition

View Table | View all tables in this article

It worth remarking that the set of Mueller matrices generated for Rayleigh spheres in Ref [16]. lie on the surface $B_{1} B_{2} C_{2} D B_{1}$ of Fig. 1, which belongs to two-dimension media, whereas the points show a monotonic decrease for multiple scattering Rayleigh spheres with $P_{P} = 0$ on the plane $O A_{2} B_{2} C_{2} O$ .

3. Analysis of depolarization caused by media via the $P I_{4 D} - C P$ space

To interpret the information obtained from the $P I_{4 D} - C P$ space, two examples of Mueller matrices from the literature are demonstrated in the following.

1) A computed Mueller matrix ${\hat{M}}_{a} (θ_{s} = 60^{o}, φ_{s} = 0^{o}, θ_{d} = 40^{o}, φ_{d} = 300^{o})$ of a medium that represents dipole scattering by a needle spheroid [22,23] whose dielectric constant $ε$ is considered to be $4 + i 0.8$ . The polar angle $θ_{s}$ and azimuth angle $φ_{s}$ represent the angles of the scattered light. The incident light is impinging from the normal direction (along with the z-axis). The angles $θ_{d}$ , $φ_{d}$ belongs to the orientation of the particle in Ref [22,23]. The calculated matrix is given as [24], ${\hat{M}}_{a} = [\begin{matrix} 1 & - 0.8817 & - 0.2411 & 0.1021 \\ - 0.7924 & 0.7871 & 0.4027 & - 0.0125 \\ 0.4518 & - 0.5552 & 0.1992 & 0.1014 \\ 0.1183 & - 0.0610 & - 0.1014 & 0.3924 \end{matrix}] .$
The point corresponding to this Mueller matrix lies on the curve $C_{2} D$ of the $P I_{4 D} - C P$ space with values $(P_{S} = 0.6616, P_{P} = 0.9198, P I_{4 D} = 1.0000)$ . The value of $P I_{4 D} = 1$ shows that the dipole scattering by the spheroid at the given angles does not have any effect on the degree of polarization of the incident light. The Mueller matrix has only one pure component in the characteristic decomposition. Note that, it would have been lain on the single point indicating the purity only if we had used the two-dimensional $P I_{4 D} - P_{4 D}$ or $S_{4 D} - P_{4 D}$ diagram [16,19]. Nevertheless, the information on the sources of purity can graphically be drawn by the first two coordinates of the $P I_{4 D} - C P$ space. The values of $P_{P} = 0.9198$ and $P_{s} = 0.6603$ show that the ${\hat{M}}_{a}$ is closer to the point D (a pure polarizer point), thus, the dipole scatterings by a spheroid at the given geometry have prominent polarizance effects than that of retardation.
2) A measured Mueller matrix ${\hat{M}}_{b}$ of a diffracting holographic volume grating [22,23] is shown below, ${\hat{M}}_{b} = [\begin{matrix} 1 & - 0.2578 & - 0.0716 & - 0.1171 \\ - 0.2577 & 0.9950 & 0.0540 & 0.0235 \\ 0.0148 & - 0.0345 & 0.6790 & 0.4908 \\ - 0.0849 & - 0.0883 & - 0.4752 & 0.6620 \end{matrix}] .$
The measured ${\hat{M}}_{b}$ matrix, however, does not hold the following inequality for any Mueller matrix to be physically realizable [25],
$m_{00} - m_{11} - m_{22} + m_{33} \geq 0.$
There is a negative value in the eigenvalue spectrum of $H ({\hat{M}}_{b})$ . This may have arisen because of the experimental errors. The given matrix can be transformed into a physically acceptable matrix by neglecting the contribution of the small negative eigenvalue using the Cloude’s covariance filtering [25]. Then, the filtered ${\hat{M}}_{b}$ is given by,
${\hat{M}}_{b} = [\begin{matrix} 1 & - 0.2423 & - 0.0576 & - 0.1051 \\ - 0.2563 & 0.9514 & 0.0445 & 0.0334 \\ - 0.0002 & - 0.0357 & 0.6388 & 0.4651 \\ - 0.0753 & - 0.0713 & - 0.4677 & 0.6596 \end{matrix}] .$
The point corresponding to ${\hat{M}}_{b}$ lies at (0.8550, 0.2687, 0.9393) on the $P I_{4 D} - C P$ space with IPP (0.8244, 0.9835,1.0000). The coefficients of the components of the characteristic decomposition are (0.8244, 0.1592, 0.0165, 0). Hence, the characteristic decomposition of ${\hat{M}}_{b}$ shows a low depolarization and is given by,
${\hat{M}}_{b} = 0.8244 [\begin{matrix} 1 & - 0.2933 & - 0.0305 & - 0.0971 \\ - 0.2994 & 0.9935 & 0.0426 & 0.0779 \\ - 0.0515 & - 0.0497 & 0.7695 & 0.5523 \\ - 0.0584 & - 0.0277 & - 0.5517 & 0.7763 \end{matrix}] + 0.1592 [\begin{matrix} 1 & 0.0077 & - 0.1921 & - 0.463 \\ - 0.0662 & 0.8102 & 0.0526 & - 0.1804 \\ 0.2466 & 0.0306 & 0.0124 & 0.0456 \\ - 0.1612 & - 0.2858 & - 0.0882 & 0.1438 \end{matrix}] + 0.0165 [\begin{matrix} 1 & - 0.1058 & - 0.1128 & - 0.1093 \\ - 0.0680 & 0.2028 & 0.0628 & - 0.1303 \\ 0.1807 & 0.0259 & 0.1505 & 0.1546 \\ - 0.907 & - 0.1811 & - 0.0660 & - 0.1915 \end{matrix}] .$
On the other hand, the contributions of $P_{P}$ and $P_{S}$ are (0.2687, 0.82550), which indicate that the given system has more spherical depolarization than the degree of polarizance. The quadratic average of these two sources of purity, i.e., $P_{4 D} = 0.8827$ .

4. Example from some experimental Mueller matrices

H. He et al. [26] studied the influence of the orientation of some fibrous scatterers in anisotropic scattering media. They measured backscattering Mueller matrices containing spheres and cylinders (fibrous scatterers) with orientations of silk fibers (cylinders) at 0 ^o, 45°, and 90° with respect to the horizontal (x-axis).

It was observed that the rotation of the fibrous scatterers had caused some periodical variations in the elements of the Mueller matrix of anisotropic media. They implemented simulations based on sphere-cylinder scattering model (SCSM) to correlate the experimental results with the simulations input parameters. Here, these measured experimental Mueller matrices (by taking the average values) are used to understand the influence of the directions of the fibrous scatterers on the structure of polarimetric purity (in terms of IPP) and sources of purity (related to $P_{P}$ and $P_{S}$ ) by plotting on the $P I_{4 D} - C P$ space and using the characteristic decomposition.

Figure 2 shows the three points with different orientation of the fibrous scatterers which are found at slightly different places inside the $P I_{4 D} - C P$ space. It can be seen from Fig. 2 that these scatterers have less significant values for $P_{P}$ which is maximum for the fibers at 45°. Hence, a periodic variation is observed in the degree of polarizance values which can be seen in Table 2. It is worth remarking that these variations cannot be observed in the case of the isotropic medium that may contain spheres because of the absence of the degree of polarizance. Such a case had been studied experimentally [26], whose formulation was given by solving the Bethe-Salpeter diffusion equation [27]. Note that the depolarization behavior of this kind of isotropic spherical scatterers with multiple scatterings using purity-depolarization plane has been reported by us in our previous work [16]. The degree of spherical purity for the anisotropic scattering medium, however, shows a slight increase with the increased orientation angles of the fibers (Table 2). In all these matrices, the degree of spherical purity is comparatively greater than the degree of polarizance. Therefore, the anisotropic media considered here, with the $rank (H) = 4$ may be characterized by a parallel mixture of the four pure retarders. Table 2 shows the values of $P_{S}$ and $P_{P}$ contributing to $P_{4 D}$ along with $P I_{4 D}$ for the system at the said three angles.

Fig. 2 The points corresponding to ${\hat{M}}_{a}$ and ${\hat{M}}_{b}$ are represented by red filled markers o and □, respectively. The average experimental Mueller matrices with the direction of fibrous scatterers at 0°, 45°, and 90° are represented by marker Δ with blue, red, and magenta colors, respectively.

Download Full Size | PDF

Table 2. The values of the components of purity ( $P_{P}$ and $P_{S}$ ) along with the overall purity index $(P I_{4 D})$ at the said three angles of the fibrous scatterers.

View Table | View all tables in this article

5. Monte Carlo simulations based on SCSM

We have conducted Monte Carlo (MC) simulations of polarized photons interacting with some scattering media to simulate the Mueller matrices. Yun et al. [28] have developed a Monte Carlo simulation program for studying the behaviors of the polarization states of light by impinging photons in a scattering medium which consists of spheres and cylinders so that the elements of the Mueller matrix of such medium can be obtained numerically. They have used the analytical solution of the scalar wave equation, $\nabla^{2} ψ + k^{2} ψ = 0$ , assuming the continuity conditions on the boundaries of the infinitely long cylinders in the cylindrical polar coordinates [29]. The program was abbreviated as SCSM: meaning that the sphere-cylinder scattering model.

In the MC simulations of SCSM, three types of scattering media: cylinders, spheres, and spheres mixed with cylinders are considered. The wavelengths of 2 × 10⁷ incident polarized photons are assumed to be 0.63 μm so that the diameters d of 0.1 and 1.1 μm for the scatterers can be associated with the Rayleigh and Mie scatterers in the backscattering detections, respectively. The refractive indices of the sphere and cylinders are chosen to be 1.59 and 1.56, respectively, with 1.33 as the refractive index of the interstitial medium whose thickness is taken as 1 cm. The scattering coefficients of all the scatterers (μs) are taken as the variable of the input parameter of the simulations, which ranges from 2 to 20 cm⁻¹ in steps of 2.

In Fig. 3 the Rayleigh and Mie scatterers are shown by markers as circle (o) and triangles (Δ) with colors red, magenta, and blue representing the cylinders, spheres, and mixed spheres and cylinders, respectively. The increase of the scattering coefficients from 2 to 20 cm⁻¹ in steps of 2 causes the cylinders to a monotonic decrease in $P_{P}$ values of the Rayleigh and Mie scatterers. However, a slight increase from 0.5839 to 0.6057 in $P_{S}$ values by increasing the scattering coefficient from 2 to 20 cm⁻¹ for the Mie scatterers is observed. The cylinders are lying in the region $B_{1} B_{2} B_{4} C_{2} D B_{3} B_{1}$ close to the upper surface. This region belongs to the two-dimensional media. The IPP values of the polarized light scattered by cylinders are $0 \approx P_{1} < P_{2} \approx P_{3} \approx 1$ , which show an absence of the degree of polarization of the Mueller matrices of the cylinders, while an optimum presence of the 2D and 3D components of characteristic decomposition.

Fig. 3 The backscattering Mueller matrices of some spheres (magenta), cylinders (red), and sphere-cylinders (blue) of diameters 0.1 (Rayleigh) and 1.1 (Mie) µm are represented by markers circle and triangle, respectively. The grey shaded region is the non-physical region of the space

Download Full Size | PDF

For the Rayleigh spheres with multiple scatterings, the points lie on the face $O A_{2} B_{2} C_{2} O$ moving towards the point O by increasing the scattering coefficient, while the coordinates of the $P I_{4 D} - C P$ space for the Mie scattering spheres at the scattering coefficient 2 and 20 cm⁻¹ are (0.3203, 0.0011, 0.5521) and (0.3099, 0.0006, 0.4725) exhibiting the monotonic decrease. The Mueller matrix of the Rayleigh spheres scattering depends on the number of scattering events and is obtained by solving Bethe-Salpeter diffusion equation [27] whose geometrical representation of the depolarization behavior has been given in our previous work [16]. Note that the so-called ‘inaccessible region’ in the work of Puentes et al. [30] is represented by the volume element $O B_{1} B_{2} O$ of the $P I_{4 D} - C P$ space, which is populated by the Mie backscattering spheres. The scattering media with mixed spheres and cylinders also reside deep inside the volume element $O B_{1} B_{2} O$ and exhibit the monotonic decrease in the values of $P_{s}$ , $P_{P}$ , and $P I_{4 D}$ for the increasing values of the scattering coefficients of the spheres and cylinders from 2 and 20 cm⁻¹. Note that the three-different media occupy some explicitly defined regions of the $P I_{4 D} - C P$ space, which also indicates the structural properties of the specimen. Therefore, the $P I_{4 D} - C P$ space provides a remarkable geometrical insight of the polarized light scattering from random or deterministic media with complete information of the structure of purity via IPP and the sources of purity through $P_{s}$ , $P_{P}$ .

6. Summary

The purity-depolarization $P I_{4 D} - C P$ plane based on the statistical nature of the associated covariance matrix of a Mueller matrix relates the depolarization index to the overall purity index of the medium [16]. It gives information about the polarimetric purity of a medium with its detailed structure via IPP, but it does not indicate the sources of purity. Since the depolarization index consists of the two components of purity. Geometrically, the information on the sources of purity can be drawn by using the components of purity diagram [14,15] but it does not hold the information of the structure of polarimetric purity in terms of the whole set of constraints in the IPP. Therefore, it is suitable to present a graphical representation which may be useful in studying depolarization properties of scattering media by providing information on both aspects of the depolarization index. In this respect, the $P I_{4 D} - C P$ space is proposed, which is bounded in the three mutually orthogonal axes as $P_{s}$ , $P_{P}$ and $P I_{4 D} - C P$ .

The $P I_{4 D} - C P$ space is a physically realizable space in $R^{3}$ such that $R = {x \in R | 0 \leq x \leq 1}$ , where, x are $P_{s}$ , $P_{P}$ , and $P I_{4 D}$ . The $P I_{4 D} - C P$ space contains positive-valued portions (first octant) of the elliptical trajectories parallel to the $P_{s} - P_{P}$ plane, whereas the $P I_{4 D}$ -axis is taken normal to the $P_{s} - P_{P}$ plane. Thus, the proposed space gives comprehensive information on the depolarization character of the scattering medium under study. To demonstrate the usefulness of the proposed $P I_{4 D} - C P$ space, some computed and experimentally measured Mueller matrices from the literature have been used and plotted. Moreover, the Monte Carlo simulations based on the SCSM have also been implemented to understand the characteristic features of depolarizing media by plotting them on the $P I_{4 D} - C P$ space. The points corresponding to these Mueller matrices have located them in different regions, which are characterized by the components of purity and the characteristic decomposition. Therefore, any point which resides in the space shows complete depolarization character of the corresponding medium. Thence, it is suggested that that the three-dimensional representation as to the $P I_{4 D} - C P$ space is more appropriate for the analysis, interpretation, and classification of the deterministic and random scattering media causing depolarization. It is worth remarking that this scheme of geometrizing could be extended to represent all the polarization properties of the 3D polarization states of light by $P I_{3D} - C P$ space, where $P I_{3D}$ and $C P$ [31] are associated with a 3D polarization state.

Funding

National Natural Science Foundation of China (NSFC) (61527826), and Shenzhen Bureau of Science and Innovation (JCYJ20170412170814624 and JCYJ20160818143050110).

Acknowledgment

A. T. would like to acknowledge the authors of the books (Refs. [2,6]), which he studied throughout his Ph.D. research work.

References

1. A. Van Eeckhout, E. Garcia-Caurel, T. Garnatje, M. Durfort, J. C. Escalera, J. Vidal, J. J. Gil, J. Campos, and A. Lizana, “Depolarizing metrics for plant samples imaging,” PLoS One 14(3), e0213909 (2019). [CrossRef] [PubMed]

2. J. J. Gil and R. Ossikovski, Polarized light and the Mueller matrix approach, (Chemical Rubber Company, 2016).

3. Y. Chang and W. Gao, “Method of interpreting Mueller matrix of anisotropic medium,” Opt. Express 27(3), 3305–3323 (2019). [CrossRef] [PubMed]

4. D. Li, F. Chen, N. Zeng, Z. Qiu, H. He, Y. He, and H. Ma, “Study on polarization scattering applied in aerosol recognition in the air,” Opt. Express 27(12), A581–A595 (2019). [CrossRef] [PubMed]

5. W. Sheng, W. Li, J. Qi, T. Liu, H. He, Y. Dong, S. Liu, J. Wu, D. Elson, and H. Ma, “Quantitative Analysis of 4× 4 Mueller Matrix Transformation Parameters for Biomedical Imaging,” Photonics 6(1), 34 (2019). [CrossRef]

6. C. Brosseau, Fundamentals of polarized light: a statistical optics approach, (Wiley-Interscience, 1998).

7. D. H. Goldstein, Polarized light, (Chemical Rubber Company, 2016).

8. E. L. O’Neill, Introduction to statistical optics, (Courier Corporation, 2004).

9. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

10. H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, and H. Ma, “A possible quantitative Mueller matrix transformation technique for anisotropic scattering media,” Photonics Lasers Med. 2(2), 129–137 (2013). [CrossRef]

11. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007). [CrossRef] [PubMed]

12. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) 33(2), 185–189 (1986). [CrossRef]

13. I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n× n covariance matrices,” Opt. Commun. 284(1), 38–47 (2011). [CrossRef]

14. J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28(8), 1578–1585 (2011). [CrossRef] [PubMed]

15. J. J. Gil, “Structure of polarimetric purity of a Mueller matrix and sources of depolarization,” Opt. Commun. 368, 165–173 (2016). [CrossRef]

16. A. Tariq, P. Li, D. Chen, D. Lv, and H. Ma, “Physically realizable space for the purity-depolarization plane for polarized light scattering media,” Phys. Rev. Lett. 119(3), 033202 (2017). [CrossRef] [PubMed]

17. R. Ossikovski and J. Vizet, “Eigenvalue-based depolarization metric spaces for Mueller matrices,” J. Opt. Soc. Am. A 36(7), 1173–1186 (2019). [CrossRef]

18. R. Barakat and C. Brosseau, “Von Neumann entropy of N interacting pencils of radiation,” J. Opt. Soc. Am. A 10(3), 529–532 (1993). [CrossRef]

19. A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering,” Phys. Rev. Lett. 94(9), 090406 (2005). [CrossRef] [PubMed]

20. R. A. Chipman, Handbook of Optics, (II Ed., M. Bass, Ed.), Chap. 22.

21. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27(1), 123–130 (2010). [CrossRef] [PubMed]

22. S. M. F. Nee, “Decomposition of Jones and Mueller matrices in terms of four basic polarization responses,” J. Opt. Soc. Am. A 31(11), 2518–2528 (2014). [CrossRef] [PubMed]

23. T. W. Nee, S. M. F. Nee, D. M. Yang, and Y. S. Huang, “Scattering polarization by anisotropic biomolecules,” J. Opt. Soc. Am. A 25(5), 1030–1038 (2008). [CrossRef] [PubMed]

24. T. W. Nee, S. M. F. Nee, M. W. Kleinschmit, and M. S. Shahriar, “Polarization of holographic grating diffraction. II. Experiment,” J. Opt. Soc. Am. A 21(4), 532–539 (2004). [CrossRef] [PubMed]

25. S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE 1166, Polarization Considerations for Optical Systems II, (25 January 1990).

26. H. He, N. Zeng, E. Du, Y. Guo, D. Li, R. Liao, Y. He, and H. Ma, “Two-dimensional and surface backscattering Mueller matrices of anisotropic sphere-cylinder scattering media: a quantitative study of influence from fibrous scatterers,” J. Biomed. Opt. 18(4), 046002 (2013). [CrossRef] [PubMed]

27. C. Brosseau and D. Bicout, “Entropy production in multiple scattering of light by a spatially random medium,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(6), 4997–5005 (1994). [CrossRef] [PubMed]

28. T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express 17(19), 16590–16602 (2009). [CrossRef] [PubMed]

29. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, (Wiley. 2008).

30. G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering,” Opt. Lett. 30(23), 3216–3218 (2005). [CrossRef] [PubMed]

31. J. J. Gil, “Components of purity of a three-dimensional polarization state,” J. Opt. Soc. Am. A 33(1), 40–43 (2016). [CrossRef] [PubMed]

Volume Elements	IPP value	P_s	P_P	$P I_{4 D}$	Characteristic decomposition of $\hat{M}$
OB₂B₁O	$0 = P_{1} \leq P_{2} \leq P_{3} \leq 1$	$0 \leq P_{s} \leq \sqrt{1 / 3}$	$0 \leq P_{P} \leq \sqrt{1 / 3}$	$0 \leq P I_{4 D} \leq \sqrt{2 / 3}$	$P_{2} {\hat{M}}_{2} + (P_{3} - P_{2}) {\hat{M}}_{3} + (1 - P_{3}) {\hat{M}}_{4}$
A₂C₂DB₅A₁A₂	$0 \leq P_{1} \leq P_{2} \leq P_{3} = 1$	$1 / 3 \leq P_{s} \leq 1$	$0 \leq P_{P} \leq 1$	$1 / \sqrt{3} \leq P I_{4 D} \leq 1$	$P_{1} {\hat{M}}_{J 1} + (P_{2} - P_{1}) {\hat{M}}_{2} + (1 - P_{2}) {\hat{M}}_{3}$
OA₂C₂DB₅B₃O	$0 \leq P_{1} = P_{2} \leq P_{3} \leq 1$	$0 \leq P_{s} \leq 1$	$0 \leq P_{P} \leq 1$	$0 \leq P I_{4 D} \leq 1$	$P_{1} {\hat{M}}_{J 1} + (P_{3} - P_{1}) {\hat{M}}_{3} + (1 - P_{3}) {\hat{M}}_{4}$
OB₂C₂DB₁O	$0 \leq P_{1} \leq P_{2} = P_{3} \leq 1$	$0 \leq P_{s} \leq 1$	$0 \leq P_{P} \leq 1$	$0 \leq P I_{4 D} \leq 1$	$P_{1} {\hat{M}}_{J 1} + (P_{2} - P_{1}) {\hat{M}}_{2} + (1 - P_{2}) {\hat{M}}_{4}$

Angles	$P_{S}$	$P_{P}$	$P I_{4 D}$
0°	0.1075	0.03506	0.1762
45°	0.1187	0.04683	0.1971
90°	0.1211	0.03713	0.195

Volume Elements	IPP value	P_s	P_P	$P I_{4 D}$	Characteristic decomposition of $\hat{M}$
OB₂B₁O	$0 = P_{1} \leq P_{2} \leq P_{3} \leq 1$	$0 \leq P_{s} \leq \sqrt{1 / 3}$	$0 \leq P_{P} \leq \sqrt{1 / 3}$	$0 \leq P I_{4 D} \leq \sqrt{2 / 3}$	$P_{2} {\hat{M}}_{2} + (P_{3} - P_{2}) {\hat{M}}_{3} + (1 - P_{3}) {\hat{M}}_{4}$
A₂C₂DB₅A₁A₂	$0 \leq P_{1} \leq P_{2} \leq P_{3} = 1$	$1 / 3 \leq P_{s} \leq 1$	$0 \leq P_{P} \leq 1$	$1 / \sqrt{3} \leq P I_{4 D} \leq 1$	$P_{1} {\hat{M}}_{J 1} + (P_{2} - P_{1}) {\hat{M}}_{2} + (1 - P_{2}) {\hat{M}}_{3}$
OA₂C₂DB₅B₃O	$0 \leq P_{1} = P_{2} \leq P_{3} \leq 1$	$0 \leq P_{s} \leq 1$	$0 \leq P_{P} \leq 1$	$0 \leq P I_{4 D} \leq 1$	$P_{1} {\hat{M}}_{J 1} + (P_{3} - P_{1}) {\hat{M}}_{3} + (1 - P_{3}) {\hat{M}}_{4}$
OB₂C₂DB₁O	$0 \leq P_{1} \leq P_{2} = P_{3} \leq 1$	$0 \leq P_{s} \leq 1$	$0 \leq P_{P} \leq 1$	$0 \leq P I_{4 D} \leq 1$	$P_{1} {\hat{M}}_{J 1} + (P_{2} - P_{1}) {\hat{M}}_{2} + (1 - P_{2}) {\hat{M}}_{4}$

Angles	$P_{S}$	$P_{P}$	$P I_{4 D}$
0°	0.1075	0.03506	0.1762
45°	0.1187	0.04683	0.1971
90°	0.1211	0.03713	0.195

Purity-depolarization relations and the components of purity of a Mueller matrix

Abstract

1. Introduction