## Abstract

A comprehensive physically realizable space, namely, the overall purity index-components of purity $(P{I}_{4\text{D}}-CP)$ space is proposed for the characterization of the depolarization caused by random (or deterministic) media. The overall purity index ($P{I}_{4\text{D}})$is obtained via indices of polarimetric purity which are incurred by the eigenvalues of the covariance matrix, whereas the components of purity $(CP)$ are the functions of the elements of a Mueller matrix. On the one hand, the proposed space is useful in studying the depolarization caused by material media and on the other hand, it provides information on the diattenuation-polarizance properties of a Mueller matrix. Thus, it gives a remarkable physical insight of the depolarization problem.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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 $({P}_{4\text{D}}$) (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 ${P}_{4\text{D}}$ denotes dimensions of the associated Hermitian covariance matrix of a Mueller matrix, which describes the statistical nature of the Mueller matrix [8]. Therefore, ${P}_{4\text{D}}$ 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 ${P}_{4\text{D}}$ 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 $(P{I}_{4\text{D}})$ 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 $(1-{S}_{4\text{D}},{P}_{4\text{D}},P{I}_{4\text{D}})$ based on the eigenvalues of the covariance matrix is presented [17]. It should be noted that ${P}_{P}$ and ${P}_{S}$ (components of purity $CP$) are directly obtained from the elements of a Mueller matrix representing diattenuation-polarizance and depolarization (spherical depolarization), respectively, whereas the eigenvalue-based depolarization metrics $({S}_{4\text{D}}$ and $P{I}_{4\text{D}})$ 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 $CP$ and IPP to the depolarization index ${P}_{4\text{D}}$.

In this paper, a physically realizable purity index - components of purity $(P{I}_{4\text{D}}-CP)$ space is proposed which comprises of ${P}_{4\text{D}}$ as a combination of the two components of purity $(CP)$ representing the basis of a coordinate plane whose normal is the overall purity index $(P{I}_{4\text{D}})$. 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 $P{I}_{4\text{D}}-CP$ space can be generated by writing ${P}_{S}$ and ${P}_{P}$ in polar coordinates with the azimuth ranging from 0 to $\pi /2$, 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 $P{I}_{4\text{D}}-CP$ space. Furthermore, Monte Carlo simulations based on the sphere-cylinder scattering model (SCSM) are implemented to differentiate media via the $P{I}_{4\text{D}}-CP$ space.

## 2. The $P{I}_{4\text{D}}-CP$ 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,

*T*denoting the transpose. The diattenuation vector

**D**of $\widehat{M}$ is given by [2,14,15]and the polarizance vector

**P**of $\widehat{M}$ is expressed as [2,14,15]The magnitudes of these vectors of the $\widehat{M}$ give the amount of diattenuation and polarizance by the interacting media, respectively. From the reciprocity constraint of the $\widehat{M}$ (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,

*T*being the conjugate transpose. It is related to the degree of spherical purity ${P}_{S}$ as [2,13–15],with ${P}_{S}$ ranging from 0 to 1. The magnitude of the polarizance and diattenuation vectors are [2,14,15],and,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],

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 ${P}_{4\text{D}}$ 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 $\widehat{M}$ as [12],

Therefore, ${P}_{4\text{D}}$can be expressed as a function of ${P}_{S}$ and ${P}_{P}$ given as [2,14],The above expression is an equation of an ellipse. Since the variables ${P}_{P}$ and ${P}_{S}$, and parameter ${P}_{4\text{D}}$ 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 ${P}_{4\text{D}}$ [2,13]. The range of the degree of polarizance ${P}_{P}$ from Eq. (13) is $0\le {P}_{P}\le \sqrt{3/2}$, which shows the non-physical values for ${P}_{P}\ge 1$. Therefore, an extra constraint to restrict $0\le {P}_{P}\le 1$, representing a hyperbolic curve has been proposed [14] which is given by,

Together with the above expression (Eq. (14)) being the extra constraint and Eq. (13) with ${P}_{4\text{D}}$ 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 ${P}_{P}={P}_{4\text{D}}=1$ 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 $H(\widehat{M})$, which can be extracted from the elements of the $\widehat{M}$, 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 ${E}_{ij}$ given in the following [2,7],

*2*D (${\widehat{M}}_{2}$) and

*3*D (${\widehat{M}}_{3}$) unpolarized components, and the ${\widehat{M}}_{4}$ (ideal depolarizer) such that [2],

The $P{I}_{4\text{D}}-{P}_{4\text{D}}$ 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 $(CP)$ contributing to ${P}_{4\text{D}}$. On the other hand, a graphical representation of ${P}_{4\text{D}}$ via $CP$ 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\text{D}}\mathrm{cos}\phi $ and ${P}_{P}=\sqrt{3/2}{P}_{4\text{D}}\mathrm{sin}\phi $, satisfying Eq. (13), such that the physical constraint inequality Eq. (14) becomes $3{({P}_{4\text{D}})}^{2}\mathrm{cos}2\phi \ge -1$, where the range of the azimuth $\phi $ is $0\le \phi \le \pi /2$. Thus, a physically realizable purity index - components of purity $(P{I}_{4\text{D}}-CP)$ space (Fig. 1) may be constructed by continuously varying $\phi $ from 0 to π/2. Note that at $\phi =0$, the $P{I}_{4\text{D}}-{P}_{S}$ plane with ${P}_{S}={P}_{4\text{D}}$ and ${P}_{P}=0$ is obtained; while at $\phi =\pi /2$, the plane becomes $P{I}_{4\text{D}}-{P}_{P}$ plane with ${P}_{S}=0$. However, at $\phi =\pi /2$, the constraint inequality of Eq. (14) restricts $0\le {P}_{P}\le \sqrt{1/2}$ and $0\le P{I}_{4\text{D}}\le \sqrt{2/3}$. For all continuous values of the azimuth, the $P{I}_{4\text{D}}-CP$ space is obtained through sweeping the first octant counter-clockwise with the $P{I}_{4\text{D}}-{P}_{4\text{D}}$ plane along elliptical trajectories whose continuum projection on the coordinate plane $P{I}_{4\text{D}}=0$ yields the common purity figure or (*P _{P} – P_{S}*) plane [14].

To each value of ${P}_{4\text{D}}$ in the two-dimensional $P{I}_{4\text{D}}-{P}_{4\text{D}}$ plane, there are a set of values for $P{I}_{4\text{D}}$. Nevertheless, in the $P{I}_{4\text{D}}-CP$ space, the values of ${P}_{4\text{D}}$ are shown by elliptical curves. Each elliptical curve has a constant value for ${P}_{4\text{D}}={P}_{4\text{D}}({P}_{s},{P}_{P})$ 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 $P{I}_{4D}-CP$ space

The sets of points are grouped according to a constant value of ${P}_{4\text{D}}$ 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 $P{I}_{4\text{D}}$-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 $P{I}_{4\text{D}}$-axis has the value ${P}_{4\text{D}}=\sqrt{1/3}$.

**•** 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, ${P}_{4\text{D}}=0$, then the Mueller matrix is given by,

**•** The set $A$$\left({A}_{1},{A}_{2},{A}_{3},{A}_{4}\right)$

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

**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

*P*

_{1}=

*P*

_{2}=0 and ${P}_{3}=1$, therefore, the curve at $P{I}_{4\text{D}}=\sqrt{1/3}$ holds the following equation [14,15],

The Mueller matrices of the points on this curve are expressed as $\widehat{M}={\widehat{M}}_{3}$ 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$$\left({B}_{1},{B}_{2},{B}_{3},{B}_{4},{B}_{5}\right)$

The general Mueller matrices for the *B*_{1} and *B*_{3}; 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 $\widehat{M}={\widehat{M}}_{2}$ and at the ${B}_{3}$ and ${B}_{4}$, $\widehat{M}=({\widehat{M}}_{J1}+2{\widehat{M}}_{4})/3$ 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],

**•** The set $C$$\left({C}_{1},{C}_{2}\right)$

The point ${C}_{1}\left(0,1,1\right)$ is not physically realizable while the Mueller matrix at ${C}_{1}\left(1,0,1\right)$ with ${P}_{4\text{D}}=P{I}_{4\text{D}}=1$ can be expressed as a pure retarder [2],

This represents a Mueller matrix of non-depolarizing medium (a pure retarder), which is a pure component $({\widehat{M}}_{J1})$ in the characteristic decomposition.**•** The point *D*

*D*

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

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 $P{I}_{4D}-CP$ space

There are six faces of the $P{I}_{4\text{D}}-CP$ 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 $0\le {P}_{3}\le 1$, the points lie on the surface $O{A}_{1}{A}_{2}O$ with $0\le P{I}_{4\text{D}}\le \sqrt{1/3}$ and $0\le {P}_{4\text{D}}\le 1/3$. The characteristic decomposition entails the Mueller matrix of any point on this surface,

Hence, any point on this surface may be characterized by a combination of*3*D and completely depolarized components in the characteristic decomposition with the $\text{rank}\left(H\left(\widehat{M}\right)\right)=4$. In the mould of the components of purity, the points on this surface have different elliptical curves restricted to $0\le 2{P}_{P}{}^{2}+3{P}_{s}{}^{2}\le 1/3$ with $0\le {P}_{s}\le 1/3$ and $0\le {P}_{P}\le 1/\sqrt{6}$.

**•** 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${P}_{1}=0$, $0\le {P}_{2}\le 1$, and ${P}_{3}=1$, with $\sqrt{1/3}\le P{I}_{4\text{D}}\le \sqrt{2/3}$ and $1/3\le {P}_{4\text{D}}\le \sqrt{1/3}$. The elliptical curves are restricted by inequality as $1/3\le 2{P}_{P}{}^{2}+3{P}_{s}{}^{2}\le 1$. The ranges of ${P}_{s}$ and ${P}_{p}$ are $0\le {P}_{s}\le \sqrt{1/3}$ and $0\le {P}_{P}\le 1/\sqrt{2}$. The Mueller matrix of these points on this plane can be decomposed as a linear combination of the *2*D and *3*D components of the characteristic decomposition, written as in the following,

**•** The face ${B}_{1}{B}_{2}{C}_{2}D{B}_{1}$

The IPP values with, $0\le {P}_{1}\le 1$ and ${P}_{2}={P}_{3}=1$, populate the surface ${B}_{1}{B}_{2}{C}_{2}D{B}_{1}$. The ranges of ${P}_{4\text{D}}$ and $P{I}_{4\text{D}}$ are $\sqrt{1/3}\le {P}_{4\text{D}}\le 1$ and $\sqrt{2/3}\le P{I}_{4\text{D}}\le 1$. The elliptical curves follow the inequality given as,

with the extra constraint such that for ${P}_{s}=0$, the ${P}_{P}$ values with $\sqrt{1/2}\le {P}_{P}<1$ are nonphysical Mueller matrices. This restriction is the consequence of the Eq. (14) [14,15]. For the range of $P{I}_{4\text{D}}$ values $\sqrt{2/3}\le P{I}_{4\text{D}}\le $ 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,Thus, a Mueller matrix on this face can be realized as the composition of a completely pure component and a *2*D 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 $0\le {P}_{1}={P}_{2}={P}_{3}\le 1$ 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 $\widehat{M}={P}_{1}\left({\widehat{M}}_{J1}\right)+(1-{P}_{3}){\widehat{M}}_{4}$. The equivalent representation of this face corresponds to the maximum entropy curve of the entropy-depolarization diagram [19]. The ranges of $P{I}_{4D}$ and ${P}_{4D}=\sqrt{2{P}_{P}{}^{2}+3{P}_{s}{}^{2}/3}$ 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 $0\le {P}_{4\text{D}}={P}_{P}\le \sqrt{1/2}$, ${P}_{s}=0$ and the latter has$0\le {P}_{4\text{D}}={P}_{s}\le 1,$ ${P}_{P}=0$,. 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 ${P}_{4\text{D}}={P}_{S}$, 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 ${P}_{4\text{D}}={P}_{s}$ 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 ${P}_{P}=1/2$, $0\le {P}_{s}\le 1/\sqrt{6}$ and $0.5951\le P{I}_{4\text{D}}\le 0.8292$.

#### 2.3 Volume elements in the $P{I}_{4D}-CP$ space

The regions that classify scattering media by the rank information of the $H\left(\widehat{M}\right)$ in the two-dimensional common purity figure (Ref. [14]) are represented by volume elements in the $P{I}_{4\text{D}}-CP$ 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.

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 $P{I}_{4\text{D}}-CP$ space

To interpret the information obtained from the $P{I}_{4\text{D}}-CP$ space, two examples of Mueller matrices from the literature are demonstrated in the following.

- 1) A computed Mueller matrix ${\widehat{M}}_{\text{a}}\left({\theta}_{s}={60}^{\text{o}},{\phi}_{s}={0}^{\text{o}},{\theta}_{d}={40}^{\text{o}},{\phi}_{d}={300}^{\text{o}}\right)$ of a medium that represents dipole scattering by a needle spheroid [22,23] whose dielectric constant $\epsilon $ is considered to be $4+i0.8$. The polar angle ${\theta}_{s}$ and azimuth angle ${\phi}_{s}$ represent the angles of the scattered light. The incident light is impinging from the normal direction (along with the z-axis). The angles ${\theta}_{d}$, ${\phi}_{d}$ belongs to the orientation of the particle in Ref [22,23]. The calculated matrix is given as [24],
The point corresponding to this Mueller matrix lies on the curve ${C}_{2}D$ of the $P{I}_{4\text{D}}-CP$ space with values $({P}_{S}=0.6616,{P}_{P}=0.9198,P{I}_{4\text{D}}=1.0000)$. The value of $P{I}_{4\text{D}}=1$ 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 $P{I}_{4\text{D}}-{P}_{4\text{D}}$ or ${S}_{4\text{D}}-{P}_{4\text{D}}$ diagram [16,19]. Nevertheless, the information on the sources of purity can graphically be drawn by the first two coordinates of the $P{I}_{4\text{D}}-CP$ space. The values of ${P}_{P}=0.9198$ and ${P}_{s}=0.6603$ show that the ${\widehat{M}}_{\text{a}}$ 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 ${\widehat{M}}_{b}$ of a diffracting holographic volume grating [22,23] is shown below,
The measured ${\widehat{M}}_{b}$ matrix, however, does not hold the following inequality for any Mueller matrix to be physically realizable [25],

There is a negative value in the eigenvalue spectrum of $H({\widehat{M}}_{\text{b}})$. 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 ${\widehat{M}}_{b}$ is given by,

$${\widehat{M}}_{b}=\left[\begin{array}{cccc}1& -0.2423& -0.0576& -0.1051\\ -0.2563& 0.9514& 0.0445& 0.0334\\ -0.0002& -0.0357& 0.6388& 0.4651\\ -0.0753& -0.0713& -0.4677& 0.6596\end{array}\right].$$The point corresponding to ${\widehat{M}}_{b}$ lies at (0.8550, 0.2687, 0.9393) on the $P{I}_{4\text{D}}-CP$ 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 ${\widehat{M}}_{b}$ shows a low depolarization and is given by,

$${\widehat{M}}_{b}=0.8244\left[\begin{array}{cccc}1& -0.2933& -0.0305& -0.0971\\ -0.2994& 0.9935& 0.0426& 0.0779\\ -0.0515& -0.0497& 0.7695& 0.5523\\ -0.0584& -0.0277& -0.5517& 0.7763\end{array}\right]+0.1592\left[\begin{array}{cccc}1& 0.0077& -0.1921& -0.463\\ -0.0662& 0.8102& 0.0526& -0.1804\\ 0.2466& 0.0306& 0.0124& 0.0456\\ -0.1612& -0.2858& -0.0882& 0.1438\end{array}\right]+0.0165\left[\begin{array}{cccc}1& -0.1058& -0.1128& -0.1093\\ -0.0680& 0.2028& 0.0628& -0.1303\\ 0.1807& 0.0259& 0.1505& 0.1546\\ -0.907& -0.1811& -0.0660& -0.1915\end{array}\right].$$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., ${P}_{4\text{D}}=0.8827$.

## 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 $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}-CP$ 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 $\text{rank}\left(H\right)=4$ 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 ${P}_{4\text{D}}$ along with $P{I}_{4\text{D}}$ for the system at the said three angles.

## 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, ${\nabla}^{2}\psi +{k}^{2}\psi =0$ , 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 × 10^{7} 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 *2*D and *3*D components of characteristic decomposition.

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 $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}$ 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 $P{I}_{4\text{D}}-CP$ space, which also indicates the structural properties of the specimen. Therefore, the $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}-CP$ space is proposed, which is bounded in the three mutually orthogonal axes as ${P}_{s}$, ${P}_{P}$ and $P{I}_{4\text{D}}-CP$.

The $P{I}_{4\text{D}}-CP$ space is a physically realizable space in ${R}^{3}$ such that $R=\left\{x\in R\text{|}0\le x\le 1\right\}$, where, *x* are ${P}_{s}$, ${P}_{P}$, and $P{I}_{4\text{D}}$. The $P{I}_{4\text{D}}-CP$ space contains positive-valued portions (first octant) of the elliptical trajectories parallel to the ${P}_{s}-{P}_{P}$ plane, whereas the $P{I}_{4\text{D}}$-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 $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}-CP$ 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 $P{I}_{4\text{D}}-CP$ 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 *3*D polarization states of light by $P{I}_{\text{3D}}-CP$ space, where $P{I}_{\text{3D}}$ and $CP$ [31] are associated with a *3*D 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.

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