Abstract

A comprehensive physically realizable space, namely, the overall purity index-components of purity (PI4DCP) space is proposed for the characterization of the depolarization caused by random (or deterministic) media. The overall purity index (PI4D)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 (P4D) (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 P4D denotes dimensions of the associated Hermitian covariance matrix of a Mueller matrix, which describes the statistical nature of the Mueller matrix [8]. Therefore, P4D 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 (P1, P2 and P3). 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 P4D via components of purity (CP) in terms of the degree of polarizance (PP) and degree of spherical purity (PS) 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 (PI4D) 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 (1S4D,P4D,PI4D) based on the eigenvalues of the covariance matrix is presented [17]. It should be noted that PP and PS (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 (S4D and PI4D) 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 P4D.

In this paper, a physically realizable purity index - components of purity (PI4DCP) space is proposed which comprises of P4D as a combination of the two components of purity (CP) representing the basis of a coordinate plane whose normal is the overall purity index (PI4D). 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 PI4DCP space can be generated by writing PS and PP in polar coordinates with the azimuth ranging from 0 to π/2, and satisfying a physical constraint on the values of PP 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 PI4DCP space. Furthermore, Monte Carlo simulations based on the sphere-cylinder scattering model (SCSM) are implemented to differentiate media via the PI4DCP space.

2. The PI4DCP 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=M00M^=M00[1m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33].
Here, M^ is the normalized Mueller matrix with mij=Mij/M00 (i,j=0,1,2,3) and m00=1. It is customary to write M^ in the following block form [14,15],
M=[1DTPm].
T denoting the transpose. The diattenuation vector D of M^ is given by [2,14,15]
D=[m01m02m03],
and the polarizance vector P of M^ is expressed as [2,14,15]
P=[m10m20m30].
The magnitudes of these vectors of the M^ give the amount of diattenuation and polarizance by the interacting media, respectively. From the reciprocity constraint of the 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=[m11m12m13m21m22m23m31m32m33].
The Frobenius norm of the 3x3 submatrix can be written as,
m2=i=i3j=13|mij|2=tr(mTm),
with tr standing trace and superscript T being the conjugate transpose. It is related to the degree of spherical purity PS as [2,13–15],
PS=m23,
with PS ranging from 0 to 1. The magnitude of the polarizance and diattenuation vectors are [2,14,15],
P=|P|(i=13mi02),
and,
D=|D|(j=13m0j2).
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 PP [2,14,15],

PP=|P|2+|D|22.

The physically acceptable maximum value of PP for a linear passive medium is 1 that represents a fully pure polarizer, while the minimum value of PP 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 P4D 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 M^ as [12],

P4D=i,j=03mij213.
Therefore, P4Dcan be expressed as a function of PS and PP given as [2,14],

P4D=2PP23+PS2,
PP2(3/2P4D)2+PS2(P4D)2=1.

The above expression is an equation of an ellipse. Since the variables PP and PS, and parameter P4D 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 P4D [2,13]. The range of the degree of polarizance PP from Eq. (13) is 0PP3/2, which shows the non-physical values for PP1. Therefore, an extra constraint to restrict 0PP1, representing a hyperbolic curve has been proposed [14] which is given by,

PP21+3PS22.
Together with the above expression (Eq. (14)) being the extra constraint and Eq. (13) with P4D 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 PP=P4D=1 and PS=1/3 represents a Mueller matrix of an ideal polarizer [14].

On the other hand, a Hermitian (covariance) positive semi-definite (PSD) matrix H(M^), which can be extracted from the elements of the 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 Eij given in the following [2,7],

H(M^)=14i,j=03mijEij=14i,j=03mij(σiσj).
Here σi are the Pauli matrices and the 2 × 2 identity matrix, and stands for Kronecker product. The eigenvalue spectrum of the H(M^) can be written in the following form [13],
1λ0λ1λ2λ30
This arrangement can be exploited to obtain three peculiar quantities that provide complete information of the polarimetric purity of the M^. These quantities are called the indices of polarimetric purity (IPP) [13] which are invariant under the rotational transformation of the M^ and are invariant under more general retarder transformations [2]. The indices of polarimetric purity (IPP) are obtained from the eigenvalues of H(M^) such that [13],
P1=λ0λ1tr(H(M^)),
P2=λ0+λ12λ2tr(H(M^)),
P3=λ0+λ1+λ23λ3tr(H(M^)),
with tr(H(M^)=1. By the convexity property of Mueller matrix, any depolarizing M^ can be decomposed into a maximum of four components whose relative weights are determined by the magnitude of the eigenvalues of the H(M^). The IPP values are restricted by the following inequality [13],
0P1P2P31.
Thus, the overall purity index PI4D is expressed as [16],
PI4D=P12+P22+P323.
And the depolarization index P4D can be expressed as a function of the IPP [13],
P4D=13(2P12+23P22+13P32).
This is the equation (Eq. (22)) for the first octant of a triaxial ellipsoid with PI4D as a parameter, but the IPP follows the inequality Eq. (20), which forms a tetrahedron in the three mutually axes as P1, P2, and P3 called the purity space [13]. Gil [15] further proposed a common purity figure in which a set of five parameters (D,P,P1,P2,P3) are used that classify polarized light scattering media into four main types indicating the rank information of the H(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 M^ and H(M^) in the characteristic decompositions, respectively. The characteristic decomposition contains the M^J1 (pure polarizer), 2D (M^2) and 3D (M^3) unpolarized components, and the M^4 (ideal depolarizer) such that [2],

M^=(P1)M^J1+(P2P1)M^2+(P3P2)M^3+(1P3)M^4.

The PI4DP4D 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 P4D. On the other hand, a graphical representation of P4D via CP with different sources of purity (i.e., PP and PS) [14] are insensitive to the eigenvalues of the covariance matrix. Since PS and PP can be represented in polar coordinates given as PS=P4Dcosφ and PP=3/2P4Dsinφ, satisfying Eq. (13), such that the physical constraint inequality Eq. (14) becomes 3(P4D)2cos2φ1, where the range of the azimuth φ is 0φπ/2. Thus, a physically realizable purity index - components of purity (PI4DCP) space (Fig. 1) may be constructed by continuously varying φ from 0 to π/2. Note that at φ=0, the PI4DPS plane with PS=P4D and PP=0 is obtained; while at φ=π/2, the plane becomes PI4DPP plane with PS=0. However, at φ=π/2, the constraint inequality of Eq. (14) restricts 0PP1/2 and 0PI4D2/3. For all continuous values of the azimuth, the PI4DCP space is obtained through sweeping the first octant counter-clockwise with the PI4DP4D plane along elliptical trajectories whose continuum projection on the coordinate plane PI4D=0 yields the common purity figure or (PP – PS) plane [14].

 figure: Fig. 1

Fig. 1 Graphical representation of polarimetric purity via PI4DCP space

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To each value of P4D in the two-dimensional PI4DP4D plane, there are a set of values for PI4D. Nevertheless, in the PI4DCP space, the values of P4D are shown by elliptical curves. Each elliptical curve has a constant value for P4D=P4D(Ps,PP) 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 PI4DCP space

The sets of points are grouped according to a constant value of P4D such as the elliptical curves bounded between points A3 and A4 to points A1 and A2; are represented by an elliptical surface A1A2A4A3A1 parallel to the PI4D-axis (Fig. 1). Any point representing a Mueller matrix on this surface has P4D = 1/3. Similarly, any point on the surface B1B2B4B3B1 parallel to the PI4D-axis has the value P4D=1/3.

Point O

The Muller matrix of the ideal depolarizer is designated by point O (Fig. 1). Mueller matrices at this point have P1=P2=P3=PP=Ps=0. Hence, P4D=0, then the Mueller matrix is given by,

M^=[10T003×3].

The set A(A1,A2,A3,A4)

The set A has the same value for the depolarization index, i.e., P4D=1/3. Mueller matrices at the points A1(0,1/3, 1/3), A3(0,1/3,1/3), and on the curve A1A3; and A2(1/3,0, 1/3), A4(1/3,0, 1/3), and on the curve A2A4 are written respectively as [2,15],

M^=[1DTP03×3],
M^=[10T0m].
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) PP=0 and PS ranging from 0 to 1. However, points on the elliptical curve A1A2 are achieved by the IPP as P1­=P2=0 and P3=1, therefore, the curve at PI4D=1/3 holds the following equation [14,15],

2PP2+3PS2=1/3.

The Mueller matrices of the points on this curve are expressed as M^=M^3 by the characteristic decomposition. The points on the elliptical curve A3A4 have the IPP as P1=P2=P3=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(B1,B2,B3,B4,B5 )

The general Mueller matrices for the B1 and B3; and the B2 and B4 belong to the same as Eq. (25) and Eq. (26), respectively. However, at the points B1 and B3, the Mueller matrices are equivalent to M^=M^2 and at the B3 and B4, M^=(M^J1+2M^4)/3 by the characteristic decomposition. A typical example of the Mueller matrix located on the point B1 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],

[1/21/2001/21/20000000000][1000000000000000]=12[1000100000000000].
In this case, the IPP with P1=0  and P2=P3=1, showing that the final matrix M^ is equivalent toM^=M^2, with PP=1/2 and Ps=0. Then the values for PI4D and P4D are 2/3 and 1/3, respectively. The elliptical curve B1B2 at PI4D=2/3 follows the equation,

2PP2+3PS2=1.

The set C(C1,C2)

The point C1(0,1,1) is not physically realizable while the Mueller matrix at C1(1,0,1) with P4D=PI4D=1  can be expressed as a pure retarder [2],

M^=[10T0mR].
This represents a Mueller matrix of non-depolarizing medium (a pure retarder), which is a pure component (M^J1) in the characteristic decomposition.

The point D

This point belongs to an ideal polarizer with coordinates D(1,1/3,1) whose Mueller matrix with the IPP as P1=P2=P3=1 and P4D=PI4D=1  is given by [2],

M^=[1DTPPDT].
Any point on the curve C2D (excluding the pointD) following the equation 2PP2+3Ps2=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 PI4DCP space

There are six faces of the PI4DCP 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 OA1A2O

When assuming the IPP as, P1=P2=0and 0P31, the points lie on the surface OA1A2O with 0PI4D1/3 and 0P4D1/3. The characteristic decomposition entails the Mueller matrix of any point on this surface,

M^=P3M^3+(1P3)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(M^))=4. In the mould of the components of purity, the points on this surface have different elliptical curves restricted to 02PP2+3Ps21/3 with 0Ps1/3 and 0PP1/6.

The face A1A2B3B1A1

The points on the face A1A2B3B1A1 can be realized by considering the IPP values asP1=0, 0P21, and P3=1, with 1/3PI4D2/3 and 1/3P4D1/3. The elliptical curves are restricted by inequality as 1/32PP2+3Ps21. The ranges of Ps and Pp are 0Ps1/3 and 0PP1/2. 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,

M^=P2M^2+(1P2)M^3.

The face B1B2C2DB1

The IPP values with, 0P11 and P2=P3=1, populate the surface B1B2C2DB1. The ranges of P4D and PI4D are 1/3P4D1 and 2/3PI4D1. The elliptical curves follow the inequality given as,

12PP2+3PS23,
with the extra constraint such that for Ps=0, the PP values with 1/2PP<1 are nonphysical Mueller matrices. This restriction is the consequence of the Eq. (14) [14,15]. For the range of PI4D values 2/3PI4D 1, the hyperbolic curves form a hyperbolic surface which is represented by B1DB3B1. From the statistical point of view with the characteristic decomposition, the points on the face B1B2C2DB1 can be decomposed as a linear combination of the spectral components with the one and two equiprobable eigenvalues such that,

M^=P1M^J1+(1P1)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 OB3DC2O

When all the IPP values are equal such that 0P1=P2=P31 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 M^=P1(M^J1)+(1P3)M^4. The equivalent representation of this face corresponds to the maximum entropy curve of the entropy-depolarization diagram [19]. The ranges of PI4D and P4D=2PP2+3Ps2/3 are between 0 and 1.

The side faces OA1B1B3O and OA2B2C2O

The side faces OA1B1B3O and OA2B2C2O of the space have different sets of the IPP values with former 0P4D=PP1/2, Ps=0 and the latter has0P4D=Ps1, PP=0,. The region B1C1B3B1 with Ps=0 is physically not achievable. The face OA2B2C2O can be decomposed into four Mueller matrices which may completely be described by the two-dimensional purity-depolarization plane with P4D=PS, whereas the extra constraint by Eq. (14) on the face OA1B1C1O with PS=0 excludes the non-physical region B1C1B3B1. It is worth mentioning that the type-I depolarizers [21] fall under a category that lies on the side face OA2B2C2O with P4D=Ps 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 PP=1/2, 0Ps1/6 and 0.5951PI4D0.8292.

2.3 Volume elements in the PI4DCP space

The regions that classify scattering media by the rank information of the H(M^) in the two-dimensional common purity figure (Ref. [14]) are represented by volume elements in the PI4DCP space such that the points of the volume elements, B1B2B4C2DB3B1 A1A2A4C2DB3A3A1, and OC2DB3O represent the two-dimensional, three-dimensional, and four-dimensional media, respectively, while the curve DC2 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 OB2B1O, A2C2DB5A1A2, OB2C2DB1O, and OA2C2DB5B3O 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.

Tables Icon

Table 1. The regions of the PI4DCP space characterized by the characteristic decomposition

It worth remarking that the set of Mueller matrices generated for Rayleigh spheres in Ref [16]. lie on the surface B1B2C2DB1 of Fig. 1, which belongs to two-dimension media, whereas the points show a monotonic decrease for multiple scattering Rayleigh spheres with PP=0 on the plane OA2B2C2O.

3. Analysis of depolarization caused by media via the PI4DCP space

To interpret the information obtained from the PI4DCP space, two examples of Mueller matrices from the literature are demonstrated in the following.

  • 1) A computed Mueller matrix M^a(θs=60o,φs=0o,θd=40o,φd=300o) of a medium that represents dipole scattering by a needle spheroid [22,23] whose dielectric constant ε is considered to be 4+i0.8. The polar angle θs and azimuth angle φs represent the angles of the scattered light. The incident light is impinging from the normal direction (along with the z-axis). The angles θd, φd belongs to the orientation of the particle in Ref [22,23]. The calculated matrix is given as [24],
    M^a=[10.88170.24110.10210.79240.78710.40270.01250.45180.55520.19920.10140.11830.06100.10140.3924].

    The point corresponding to this Mueller matrix lies on the curve C2D of the PI4DCP space with values (PS=0.6616,PP=0.9198,PI4D=1.0000). The value of PI4D=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 PI4DP4D or S4DP4D diagram [16,19]. Nevertheless, the information on the sources of purity can graphically be drawn by the first two coordinates of the PI4DCP space. The values of PP=0.9198 and Ps=0.6603 show that the 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 M^b of a diffracting holographic volume grating [22,23] is shown below,
    M^b=[10.25780.07160.11710.25770.99500.05400.02350.01480.03450.67900.49080.08490.08830.47520.6620].

    The measured M^b matrix, however, does not hold the following inequality for any Mueller matrix to be physically realizable [25],

    m00m11m22+m330.

    There is a negative value in the eigenvalue spectrum of H(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 M^b is given by,

    M^b=[10.24230.05760.10510.25630.95140.04450.03340.00020.03570.63880.46510.07530.07130.46770.6596].

    The point corresponding to M^b lies at (0.8550, 0.2687, 0.9393) on the PI4DCP 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 M^b shows a low depolarization and is given by,

    M^b=0.8244[10.29330.03050.09710.29940.99350.04260.07790.05150.04970.76950.55230.05840.02770.55170.7763]+0.1592[10.00770.19210.4630.06620.81020.05260.18040.24660.03060.01240.04560.16120.28580.08820.1438]+0.0165[10.10580.11280.10930.06800.20280.06280.13030.18070.02590.15050.15460.9070.18110.06600.1915].

    On the other hand, the contributions of PP and PS 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., P4D=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 PP andPS) by plotting on the PI4DCP 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 PI4DCP space. It can be seen from Fig. 2 that these scatterers have less significant values for PP 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 PS and PP contributing to P4D along with PI4D for the system at the said three angles.

 figure: Fig. 2

Fig. 2 The points corresponding to M^a and 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.

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Tables Icon

Table 2. The values of the components of purity (PPand PS) along with the overall purity index (PI4D) at the said three angles of the fibrous scatterers.

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, 2ψ+k2ψ=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 PP values of the Rayleigh and Mie scatterers. However, a slight increase from 0.5839 to 0.6057 in PS 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 B1B2B4C2DB3B1 close to the upper surface. This region belongs to the two-dimensional media. The IPP values of the polarized light scattered by cylinders are0P1<P2P31, 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.

 figure: Fig. 3

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

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For the Rayleigh spheres with multiple scatterings, the points lie on the face OA2B2C2O moving towards the point O by increasing the scattering coefficient, while the coordinates of the PI4DCP 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 OB1B2O of the PI4DCP space, which is populated by the Mie backscattering spheres. The scattering media with mixed spheres and cylinders also reside deep inside the volume element OB1B2O and exhibit the monotonic decrease in the values of Ps, PP, and PI4D 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 PI4DCP space, which also indicates the structural properties of the specimen. Therefore, the PI4DCP 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 throughPs, PP.

6. Summary

The purity-depolarization PI4DCP 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 PI4DCP space is proposed, which is bounded in the three mutually orthogonal axes as Ps, PP and PI4DCP.

The PI4DCP space is a physically realizable space in R3 such that R={xR|0x1}, where, x are Ps, PP, and PI4D. The PI4DCP space contains positive-valued portions (first octant) of the elliptical trajectories parallel to the PsPP plane, whereas the PI4D-axis is taken normal to the PsPP 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 PI4DCP 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 PI4DCP 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 PI4DCP 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 PI3DCP space, where PI3D and CP [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]  

References

  • View by:

  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]

2019 (5)

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]

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

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]

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]

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

2017 (1)

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]

2016 (2)

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

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

2014 (1)

2013 (2)

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]

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]

2011 (2)

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]

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

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2005 (2)

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]

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]

2004 (1)

1996 (1)

1994 (1)

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]

1993 (1)

1986 (1)

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

Aiello, A.

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]

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]

Barakat, R.

Bernabeu, E.

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

Bicout, D.

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]

Brosseau, C.

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]

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]

Campos, J.

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]

Chang, Y.

Chen, D.

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]

Chen, F.

Chipman, R. A.

De Martino, A.

Dong, Y.

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]

Du, E.

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]

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]

Durfort, M.

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]

Elson, D.

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]

Escalera, J. C.

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]

Gao, W.

Garcia-Caurel, E.

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]

Garnatje, T.

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]

Gil, J. J.

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]

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

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

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]

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

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

Guo, Y.

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]

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]

Guyot, S.

He, H.

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]

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]

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]

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]

He, Y.

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]

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]

Huang, Y. S.

Jiang, X.

Kleinschmit, M. W.

Li, D.

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]

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]

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]

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]

Li, P.

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]

Li, W.

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]

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]

Liao, R.

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]

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]

Liu, S.

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]

Liu, T.

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]

Lizana, A.

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]

Lu, S. Y.

Lv, D.

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]

Ma, H.

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]

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]

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]

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]

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]

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]

Nee, S. M. F.

Nee, T. W.

Ossikovski, R.

Puentes, G.

Qi, J.

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]

Qiu, Z.

San José, I.

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]

Shahriar, M. S.

Sheng, W.

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]

Tariq, A.

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]

Van Eeckhout, A.

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]

Vidal, J.

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]

Vizet, J.

Voigt, D.

Woerdman, J. P.

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]

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]

Wu, J.

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]

Yang, D. M.

Yun, T.

Zeng, N.

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]

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]

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]

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]

J. Biomed. Opt. (1)

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]

J. Opt. Soc. Am. A (9)

Opt. Acta (Lond.) (1)

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

Opt. Commun. (2)

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]

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

Opt. Express (3)

Opt. Lett. (2)

Photonics (1)

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]

Photonics Lasers Med. (1)

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]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

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[Crossref] [PubMed]

Phys. Rev. Lett. (2)

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]

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]

PLoS One (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]

Other (7)

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

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

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

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

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

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

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).

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Figures (3)

Fig. 1
Fig. 1 Graphical representation of polarimetric purity via P I 4D CP space
Fig. 2
Fig. 2 The points corresponding to M ^ a and 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.
Fig. 3
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

Tables (2)

Tables Icon

Table 1 The regions of the P I 4D CP space characterized by the characteristic decomposition

Tables Icon

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

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

M= M 00 M ^ = M 00 [ 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 ].
M =[ 1 D T P m ].
D=[ m 01 m 02 m 03 ],
P=[ m 10 m 20 m 30 ].
m=[ m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ].
m 2 = i=i 3 j=1 3 | m ij | 2 = tr( m T m) ,
P S = m 2 3 ,
P=| P |( i=1 3 m i0 2 ),
D=| D |( j=1 3 m 0j 2 ).
P P = | P | 2 + | D | 2 2 .
P 4D = i,j=0 3 m ij 2 1 3 .
P 4D = 2 P P 2 3 + P S 2 ,
P P 2 ( 3/2 P 4D ) 2 + P S 2 ( P 4D ) 2 =1.
P P 2 1+3 P S 2 2 .
H( M ^ )= 1 4 i,j=0 3 m ij E ij = 1 4 i,j=0 3 m ij ( σ i σ j ) .
1 λ 0 λ 1 λ 2 λ 3 0
P 1 = λ 0 λ 1 tr(H( M ^ )) ,
P 2 = λ 0 + λ 1 2 λ 2 tr(H( M ^ )) ,
P 3 = λ 0 + λ 1 + λ 2 3 λ 3 tr(H( M ^ )) ,
0 P 1 P 2 P 3 1.
P I 4D = P 1 2 + P 2 2 + P 3 2 3 .
P 4D = 1 3 ( 2 P 1 2 + 2 3 P 2 2 + 1 3 P 3 2 ) .
M ^ =( P 1 ) M ^ J1 +( P 2 P 1 ) M ^ 2 +( P 3 P 2 ) M ^ 3 +( 1 P 3 ) M ^ 4 .
M ^ =[ 1 0 T 0 0 3×3 ].
M ^ =[ 1 D T P 0 3×3 ],
M ^ =[ 1 0 T 0 m ].
2 P P 2 +3 P S 2 =1/3 .
[ 1/2 1/2 0 0 1/2 1/2 0 0 0 0 0 0 0 0 0 0 ][ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]= 1 2 [ 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ].
2 P P 2 +3 P S 2 =1.
M ^ =[ 1 0 T 0 m R ].
M ^ =[ 1 D T P P D T ].
M ^ = P 3 M ^ 3 +( 1 P 3 ) M ^ 4 .
M ^ = P 2 M ^ 2 +( 1 P 2 ) M ^ 3 .
12 P P 2 +3 P S 2 3,
M ^ = P 1 M ^ J1 +( 1 P 1 ) M ^ 2 .
M ^ a =[ 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 ].
M ^ b =[ 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 ].
m 00 m 11 m 22 + m 33 0.
M ^ b =[ 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 ].
M ^ b =0.8244[ 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 ]+0.1592[ 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 ]+0.0165[ 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 ].

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