A comprehensive physically realizable space, namely, the overall purity index-components of purity space is proposed for the characterization of the depolarization caused by random (or deterministic) media. The overall purity index (is obtained via indices of polarimetric purity which are incurred by the eigenvalues of the covariance matrix, whereas the components of purity 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.
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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 ) (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 denotes dimensions of the associated Hermitian covariance matrix of a Mueller matrix, which describes the statistical nature of the Mueller matrix . Therefore, can also be obtained from the three invariant indices of polarimetric purity (IPP) that are incurred by the eigenvalues of the covariance matrix . The IPP provides a detailed structure of the polarimetric purity and graphically forms a tetrahedron called the purity space , which is composed of the three mutually orthogonal axes that are represented by the IPP (, and ). 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 via components of purity (CP) in terms of the degree of polarizance and degree of spherical purity is given in a two-dimensional graph  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 , 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 is given by Tariq et al.  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 based on the eigenvalues of the covariance matrix is presented . It should be noted that and (components of purity ) are directly obtained from the elements of a Mueller matrix representing diattenuation-polarizance and depolarization (spherical depolarization), respectively, whereas the eigenvalue-based depolarization metrics and 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 and IPP to the depolarization index .
In this paper, a physically realizable purity index - components of purity space is proposed which comprises of as a combination of the two components of purity representing the basis of a coordinate plane whose normal is the overall purity index . 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 space can be generated by writing and in polar coordinates with the azimuth ranging from 0 to , and satisfying a physical constraint on the values of given in Ref . 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 space. Furthermore, Monte Carlo simulations based on the sphere-cylinder scattering model (SCSM) are implemented to differentiate media via the space.
2. The 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 describes the nature of the scattering medium, which is given as,14,15],2,14,15]2,14,15]2,15]. Submatrix m is expressed as,2,13–15],2,14,15],2,14,15],
The physically acceptable maximum value of for a linear passive medium is 1 that represents a fully pure polarizer, while the minimum value of 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 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 as ,2,14],
The above expression is an equation of an ellipse. Since the variables and , and parameter 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 [2,13]. The range of the degree of polarizance from Eq. (13) is , which shows the non-physical values for . Therefore, an extra constraint to restrict , representing a hyperbolic curve has been proposed  which is given by,Eq. (14)) being the extra constraint and Eq. (13) with as a parameter, the feasible regions of the components of purity of polarized light scattering media have been drawn by Gil  whose origin point corresponds to an ideal depolarizer while the only point of intersection of the hyperbolic curve to the outer elliptical branch with and represents a Mueller matrix of an ideal polarizer .
On the other hand, a Hermitian (covariance) positive semi-definite (PSD) matrix , which can be extracted from the elements of the , 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 given in the following [2,7],13],13] which are invariant under the rotational transformation of the and are invariant under more general retarder transformations . The indices of polarimetric purity (IPP) are obtained from the eigenvalues of such that ,13],16],13],Eq. (22)) for the first octant of a triaxial ellipsoid with as a parameter, but the IPP follows the inequality Eq. (20), which forms a tetrahedron in the three mutually axes as , , and called the purity space . Gil  further proposed a common purity figure in which a set of five parameters are used that classify polarized light scattering media into four main types indicating the rank information of the . The common purity figure is appropriate for understanding the purity structure of Mueller matrices and useful in the characterization and classification of material media . 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  and entropy-depolarization  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 and in the characteristic decompositions, respectively. The characteristic decomposition contains the (pure polarizer), 2D () and 3D () unpolarized components, and the (ideal depolarizer) such that ,
The plane  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 contributing to . On the other hand, a graphical representation of via with different sources of purity (i.e., and )  are insensitive to the eigenvalues of the covariance matrix. Since and can be represented in polar coordinates given as and , satisfying Eq. (13), such that the physical constraint inequality Eq. (14) becomes , where the range of the azimuth is . Thus, a physically realizable purity index - components of purity space (Fig. 1) may be constructed by continuously varying from 0 to π/2. Note that at , the plane with and is obtained; while at , the plane becomes plane with . However, at , the constraint inequality of Eq. (14) restricts and . For all continuous values of the azimuth, the space is obtained through sweeping the first octant counter-clockwise with the plane along elliptical trajectories whose continuum projection on the coordinate plane yields the common purity figure or (PP – PS) plane .
To each value of in the two-dimensional plane, there are a set of values for . Nevertheless, in the space, the values of are shown by elliptical curves. Each elliptical curve has a constant value for that is constructed by the quadratic average of the two CP and has a set of PI4D 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 space
The sets of points are grouped according to a constant value of such as the elliptical curves bounded between points and to points and ; are represented by an elliptical surface parallel to the -axis (Fig. 1). Any point representing a Mueller matrix on this surface has P4D = 1/3. Similarly, any point on the surface parallel to the -axis has the value .
• Point O
The Muller matrix of the ideal depolarizer is designated by point O (Fig. 1). Mueller matrices at this point have . Hence, , then the Mueller matrix is given by,
• The set
The set A has the same value for the depolarization index, i.e., P4D=1/3. Mueller matrices at the points , , and on the curve ; and , , and on the curve are written respectively as [2,15],Eq. (25) is achieved by the contributions of the quadratic average of the diattenuation D and polarizance P vectors , whereas in Eq. (26) and ranging from 0 to 1. However, points on the elliptical curve are achieved by the IPP as P1=P2=0 and , therefore, the curve at holds the following equation [14,15],
The Mueller matrices of the points on this curve are expressed as by the characteristic decomposition. The points on the elliptical curve have the IPP as . 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
The general Mueller matrices for the B1 and B3; and the and belong to the same as Eq. (25) and Eq. (26), respectively. However, at the points and , the Mueller matrices are equivalent to and at the and , by the characteristic decomposition. A typical example of the Mueller matrix located on the point 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 ,
• The set
The point is not physically realizable while the Mueller matrix at with can be expressed as a pure retarder ,
• The point D
This point belongs to an ideal polarizer with coordinates whose Mueller matrix with the IPP as and is given by ,Eqs. (30) and (31).
2.2 Outer faces of the space
There are six faces of the 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
When assuming the IPP as, and , the points lie on the surface with and . The characteristic decomposition entails the Mueller matrix of any point on this surface,
• The face
The points on the face can be realized by considering the IPP values as, , and , with and . The elliptical curves are restricted by inequality as . The ranges of and are and . 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,
• The face
The IPP values with, and , populate the surface . The ranges of and are and . The elliptical curves follow the inequality given as,Eq. (14) [14,15]. For the range of values 1, the hyperbolic curves form a hyperbolic surface which is represented by . From the statistical point of view with the characteristic decomposition, the points on the face 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 2D unpolarized component of the characteristic decomposition, which shows significantly less depolarization.
• The lower face
When all the IPP values are equal such that 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 . The equivalent representation of this face corresponds to the maximum entropy curve of the entropy-depolarization diagram . The ranges of and are between 0 and 1.
• The side faces and
The side faces and of the space have different sets of the IPP values with former , and the latter has ,. The region with is physically not achievable. The face can be decomposed into four Mueller matrices which may completely be described by the two-dimensional purity-depolarization plane with , whereas the extra constraint by Eq. (14) on the face with excludes the non-physical region . It is worth mentioning that the type-I depolarizers  fall under a category that lies on the side face with i.e., the two-dimensional purity-depolarization plane for which an example of the Rayleigh and Mie spherical scatterers has been demonstrated in Ref , whereas the type-II depolarizers  are characterized by a surface in the space with , and .
2.3 Volume elements in the space
The regions that classify scattering media by the rank information of the in the two-dimensional common purity figure (Ref. ) are represented by volume elements in the space such that the points of the volume elements, , and represent the two-dimensional, three-dimensional, and four-dimensional media, respectively, while the curve 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 , , , and 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 . lie on the surface of Fig. 1, which belongs to two-dimension media, whereas the points show a monotonic decrease for multiple scattering Rayleigh spheres with on the plane .
3. Analysis of depolarization caused by media via the space
To interpret the information obtained from the space, two examples of Mueller matrices from the literature are demonstrated in the following.
- 1) A computed Mueller matrix of a medium that represents dipole scattering by a needle spheroid [22,23] whose dielectric constant is considered to be . The polar angle and azimuth angle represent the angles of the scattered light. The incident light is impinging from the normal direction (along with the z-axis). The angles , belongs to the orientation of the particle in Ref [22,23]. The calculated matrix is given as ,
The point corresponding to this Mueller matrix lies on the curve of the space with values . The value of 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 or diagram [16,19]. Nevertheless, the information on the sources of purity can graphically be drawn by the first two coordinates of the space. The values of and show that the 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 of a diffracting holographic volume grating [22,23] is shown below,
The measured matrix, however, does not hold the following inequality for any Mueller matrix to be physically realizable ,
There is a negative value in the eigenvalue spectrum of . 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 . Then, the filtered is given by,
The point corresponding to lies at (0.8550, 0.2687, 0.9393) on the 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 shows a low depolarization and is given by,
On the other hand, the contributions of and 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., .
4. Example from some experimental Mueller matrices
H. He et al.  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 and) by plotting on the 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 space. It can be seen from Fig. 2 that these scatterers have less significant values for 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 , whose formulation was given by solving the Bethe-Salpeter diffusion equation . 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 . 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 may be characterized by a parallel mixture of the four pure retarders. Table 2 shows the values of and contributing to along with 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.  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, , assuming the continuity conditions on the boundaries of the infinitely long cylinders in the cylindrical polar coordinates . 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 values of the Rayleigh and Mie scatterers. However, a slight increase from 0.5839 to 0.6057 in 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 close to the upper surface. This region belongs to the two-dimensional media. The IPP values of the polarized light scattered by cylinders are, 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.
For the Rayleigh spheres with multiple scatterings, the points lie on the face moving towards the point O by increasing the scattering coefficient, while the coordinates of the 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  whose geometrical representation of the depolarization behavior has been given in our previous work . Note that the so-called ‘inaccessible region’ in the work of Puentes et al.  is represented by the volume element of the space, which is populated by the Mie backscattering spheres. The scattering media with mixed spheres and cylinders also reside deep inside the volume element and exhibit the monotonic decrease in the values of , , and 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 space, which also indicates the structural properties of the specimen. Therefore, the 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, .
The purity-depolarization 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 . 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 space is proposed, which is bounded in the three mutually orthogonal axes as , and .
The space is a physically realizable space in such that , where, x are , , and . The space contains positive-valued portions (first octant) of the elliptical trajectories parallel to the plane, whereas the -axis is taken normal to the 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 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 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 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 space, where and  are associated with a 3D polarization state.
National Natural Science Foundation of China (NSFC) (61527826), and Shenzhen Bureau of Science and Innovation (JCYJ20170412170814624 and JCYJ20160818143050110).
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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