A model of multivariate Gaussian statistics has been applied to study the higher-order statistics of the polarization speckle at two spatial or temporal points. Based on the Gaussian assumption for the random electric field, the joint probability density functions of the Stokes parameters and the parameters characterizing the polarization ellipse for the produced random polarization fields at two points are obtained. Subsequently, the corresponding statistics of an isotropic polarization speckle at two points have been investigated to obtain the joint and conditional probability densities of these random variables.
1. INTRODUCTION
Due to its theoretical importance and practical interest, the subject of laser speckle has been explored extensively since continuous lasers became available in the early 1960s [1–3]. Although the majority of studies on laser speckle has treated random optical fields as scalar fields with their focus being placed on the statistical properties and applications of the intensity distribution, there has been an increasing interest in polarization speckle presented in terms of the random Stokes parameters due to the vector nature of the electromagnetic fields [4–19]. In wide areas of practical applications such as in biomedical imaging and metrology, the statistical properties of polarization speckle play critical roles in describing the fluctuation of the state of polarization, and a large amount of information about the scale sizes (coarseness) of polarization speckle and the physical properties of a measured sample, such as dichroism and birefringence, can be deduced.
In previous studies on polarization speckle, most attention has been limited to its statistical properties at a single point in space or a single point in time for a dynamically changing one. Such considerations are sufficient to describe the fluctuations of the state of polarization but are insufficient to describe other fundamental properties of polarization speckle including but not limited to the spatial structure for polarization modulation. Therefore, study of the joint statistical properties of polarization speckle at two points, rather than one, becomes indispensable. Although the mutual coherence matrix and the generalized Stokes parameters are important higher-order statistical properties of a polarization speckle pattern at two points [20], they do not, in themselves, provide a complete statistical description of the random polarization phenomenon. Accordingly, we will derive the joint probability density functions of the Stokes parameters at two points. Here, the two values of polarization speckle can represent the Stokes parameters at two points in space, at two points in time, or at a single point in two different polarization speckle patterns.
The purpose of this paper is to apply the multivariate Gaussian statistics to the two polarization components of a polarization speckle pattern for investigation of the joint probability density functions of the Stokes parameters and the parameters characterizing the polarization ellipse at two points. We first review some basic concepts describing the random polarization states of the stochastic electric fields and a set of mathematical representations to specify the polarized light. Under the application of multivariate Gaussian statistics, we make our attempt to study the multidimensional statistics of polarization speckle. Based on an ideal of decomposition of a polarization speckle into their two statistically independent polarization components, we provide the joint probability density functions for the Stokes parameters and the parameters characterizing the polarization ellipse for the polarization speckle at two points. Following this, we examine the statistics of an isotropic polarization speckle to show how the correlation coefficient affects its multidimensional statistics.
2. BACKGROUND
Before explaining the statistics of the Stokes parameters in polarization speckle, we must first briefly review a set of mathematical representations of the random walks that underlie the random polarization field phenomenon. Usually, a typical random electric field complex vector for $\vec E = ({\tilde E_x},{\tilde E_y})$ could be described mathematically as follows: ${\tilde E_k} = {A_k}\exp (i{\theta _k})$, $(k = x,y)$, where ${\tilde E_k}$ is the complex-valued polarization component of the stochastic electric vector, ${A_k}$ represents the magnitude of the complex resultant, and ${\theta _k}$ represents the phase of the resultant.
The polarization state of the light can be described by the Stokes parameters, which can be given as [3,20,21]
(1a)$${S_0} = {\left| {{{\tilde E}_x}} \right|^2} + {\left| {{{\tilde E}_y}} \right|^2} = A_x^2 + A_y^2 = {I_x} + {I_y},$$
(1b)$${S_1} = {\left| {{{\tilde E}_x}} \right|^2} - {\left| {{{\tilde E}_y}} \right|^2} = A_x^2 - A_y^2 = {I_x} - {I_y},$$
(1c)$${S_2} = \tilde E_x^*{\tilde E_y} + \tilde E_y^*{\tilde E_x} = 2{A_x}{A_y}\cos ({\theta _y} - {\theta _x}),$$
(1d)$${S_3} = i(\tilde E_x^*{\tilde E_y} - \tilde E_y^*{\tilde E_x}) = 2{A_x}{A_y}\sin ({\theta _y} - {\theta _x}),$$
(2a)$${S_1} = {S_0}\cos (2\chi)\cos (2\varphi),$$
(2b)$${S_2} = {S_0}\cos (2\chi)\sin (2\varphi),$$
(2c)$${S_3} = {S_0}\sin (2\chi),$$
3. MULTIVARIATE GAUSSIAN STATISTICS
The underlying statistical model for a polarization speckle is that of a circular complex Gaussian random process, with the real and imaginary parts for both polarization components being real-valued, jointly Gaussian random processes. It is therefore necessary to begin our discussion from multivariate Gaussian distributions. Let ${}_1{\tilde E_x},{}_1{\tilde E_y},{}_2{\tilde E_x},{}_2{\tilde E_y}$ represent the complex-valued polarization components of polarization speckle field with the left subscript “1” or “2” indicating the point ${\vec r_1}$ or ${\vec r_2}$ in space, or alternatively at point ${t_1}$ or ${t_2}$ in time. In addition, the real and imaginary parts of the complex field are represented by its superscript $r$ or $i$, i.e., ${}_n E_k^r = {\mathop{\rm Re}\nolimits} \{{}_n{\tilde E_k}\}$ and ${}_n E_k^i = {\mathop{\rm Im}\nolimits} \{{}_n{\tilde E_k}\}$, for $(n = 1,2)$. In the case of interest here, the column vector $\vec u$ for eight-dimensional real-valued Gaussian random variables is given by [2]
(3)$$\vec u = {\left\{{{}_1E_x^r,{}_1E_x^i,{}_1E_y^r,{}_1E_y^i,{}_2E_x^r,{}_2E_x^i,{}_2E_y^r,{}_2E_y^i} \right\}^{\rm T}},$$
where a superscript “${\rm T}$” indicates a matrix transpose.To find the characteristic function and the probability density function for this eight-dimensional set of Gaussian random variables, we will start from the $8 \times 8$ covariance matrix given by
(4)$$C = \left[{\begin{array}{*{20}{c}}{\overline {{{({}_1E_x^r)}^2}}}&{\overline {{}_1E_x^r{}_1E_x^i}}& \cdots &{\overline {{}_1E_x^r{}_2E_y^i}}\\{\overline {{}_1E_x^i{}_1E_x^r}}&{\overline {{{({}_1E_x^i)}^2}}}& \cdots &{\overline {{}_1E_x^i{}_2E_y^i}}\\ \vdots & \vdots & \vdots & \vdots \\{\overline {{}_2E_y^i{}_1E_x^r}}&{\overline {{}_2E_y^i{}_1E_x^i}}& \cdots &{\overline {{{({}_2E_y^i)}^2}}}\end{array}} \right],$$
with the overbar indicating a statistical expection from an ensemble average [2]. For a zero means Gaussian random process, i.e., $\overline {{}_n E_k^r} = \overline {{}_n E_k^i} = 0$, we have (5)$$\begin{split}&\overline {{{({}_1E_x^r)}^2}} = \overline {{{({}_1E_x^i)}^2}} = \overline {{{({}_2E_x^r)}^2}} = \overline {{{({}_2E_x^i)}^2}} = \sigma _x^2\,,\\&\overline {{{({}_1E_y^r)}^2}} = \overline {{{({}_1E_y^i)}^2}} = \overline {{{({}_2E_y^r)}^2}} = \overline {{{({}_2E_y^i)}^2}} = \sigma _y^2\,,\\&\overline {{}_1E_x^r{}_1E_x^i} = \overline {{}_2E_x^r{}_2E_x^i} = \overline {{}_1E_y^r{}_1E_y^i} = \overline {{}_2E_y^r{}_2E_y^i} = 0\,,\\&\overline {{}_1E_x^r{}_2E_x^r} = \overline {{}_1E_x^i{}_2E_x^i} = \sigma _x^2\rho _{{xx}}^c(\Delta)\,,\\&\overline {{}_1E_x^r{}_2E_x^i} = - \overline {{}_1E_x^i{}_2E_x^r} = \sigma _x^2\rho _{{xx}}^s(\Delta)\,,\\&\overline {{}_1E_y^r{}_2E_y^r} = \overline {{}_1E_y^i{}_2E_y^i} = \sigma _y^2\rho _{{yy}}^c(\Delta)\,,\\&\overline {{}_1E_y^r{}_2E_y^i} = - \overline {{}_1E_y^i{}_2E_y^r} = \sigma _y^2\rho _{{yy}}^s(\Delta)\,,\\&\overline {{}_1E_x^r{}_2E_y^r} = \overline {{}_1E_x^i{}_2E_y^i} = \sigma _x\sigma _y\rho _{{xy}}^c(\Delta)\,,\\&\overline {{}_1E_x^r{}_2E_y^i} = - \overline {{}_1E_x^i{}_2E_y^r} = \sigma _x\sigma _y\rho _{{xy}}^s(\Delta),\\&\overline {{}_1E_y^r{}_2E_x^r} = \overline {{}_1E_y^i{}_2E_x^i} = \sigma _x\sigma _y\rho _{{yx}}^c(\Delta),\\&\overline {{}_1E_y^r{}_2E_x^i} = - \overline {{}_1E_y^i{}_2E_x^r} = \sigma _x\sigma _y\rho _{{yx}}^s(\Delta),\end{split}$$
where $\sigma _k^2$ is the common variance of ${}_nE_k^m$ $(k = x,y;m = r,i;n = 1,2)$, and $\rho _{{\!jk}}^c$ and $\rho _{{\!jk}}^s$ are the real and imaginary parts, respectively, of a complex correlation coefficient of two polarization components $(j,k = x,y)$ defined by $\rho _{{\!jk}}^c + i\rho _{{\!jk}}^s = \overline {{}_1{{\tilde E}_j^*}{}_2\tilde E_k} /\sqrt {\overline {|{}_1{{\tilde E}_j}{|^2}} \,\,\overline {|{}_2{{\tilde E}_k}{|^2}}}$ with $\Delta$ being the spatial or temporal difference of these two points. With substitution of these relations specified in Eq. (5) into the covariance matrix in Eq. (4), the probability density function can now be written as (6)$$p(\vec u) = \frac{1}{{{{(2\pi)}^{{4}}}{{\left| C \right|}^{1/2}}}}\exp \left[{- \frac{1}{2}{{\vec u}^{\rm{T}}}{C^{- 1}}\vec u} \right].$$
Here $| C |$ is the determinant of the covariance matrix, and ${C^{- 1}}$ is the corresponding inverse matrix. A joint density function of the Stokes parameters can be found by substitution of the expressions of ${}_nE_k^m$ with ${}_n{S_l}(l = 0 \sim 3)$ and multiplying the resulting expression by the Jacobian of the transformation. However, the involved multiple integral of the resulting expression seems too hard to yield a closed-form expression for the probability density function for the Stokes parameters at two points.
Turning attention instead to the polarization matrix $J$, which plays a fundamental role in describing the polarization properties of the polarization speckle, we have
(7)$$J = \left[{\begin{array}{*{20}{c}}{\overline {\tilde E_x^*{{\tilde E}_x}}}&{\overline {\tilde E_x^*{{\tilde E}_y}}}\\{\overline {\tilde E_y^*{{\tilde E}_x}}}&{\overline {\tilde E_y^*{{\tilde E}_y}}}\end{array}} \right].$$
Note the fact that the polarization matrix is Hermitian and therefore can always be diagonalized due to the existence of a unitary matrix for diagonalization. After being diagonalized, the polarization matrix has the eigenvalues of $\lambda {_1}$ and $\lambda {_2}$ and may be rewritten in the form [22]
(8)$$\begin{split}{J^D}& = \left[{\begin{array}{*{20}{c}}{{\lambda _1}}&0\\0&{{\lambda _2}}\end{array}} \right] = \left[{\begin{array}{*{20}{c}}{{\lambda _1}}&0\\0&0\end{array}} \right] + \left[{\begin{array}{*{20}{c}}0&0\\0&{{\lambda _2}}\end{array}} \right]\\&{ = }\,\,J_x^D + J_y^D\,.\end{split}$$
Presented in this form, the original polarization speckle with its polarization matrix $J$ may be regarded as made up of an coherent superposition of two statistically independent fully polarized laser speckle fields with their relative average intensities of $\lambda _1$ and $\lambda {_2}$, where the first speckle field with its polarization matrix $J_x^D$ is fully polarized in the $\hat x$ direction, and the second speckle field with its polarization matrix $J_y^D$ is fully polarized in the $\hat y$ direction. The unitary matrix used for the polarization matrix diagonalization can be interpreted as a combination of a coordinate rotation and a relative retardation of the two polarization components. To obtain the desired analytical expressions for the statistics of polarization speckle at two points, we can reduce the difficulity in the Stokes analysis by performing a coordinate transform (or, in experimental terms, making use of an appropriate combination of wave plates). Similar to any coordinate transformation where the functions before and after the transformation usually have different expressions, while the underlying physical properties remain the same, such mathematical treatment can effectively simplify our statistical analysis for removing the unwanted correlation coefficients $\rho _{{xy}}^{\,c},\rho _{{xy}}^{\,s},\rho _{{yx}}^{\,c}$, and $\rho _{{yx}}^{\,s}$ in Eq. (5) without altering the intrinsic statistical properties of the polarization speckle. Similar analysis has been applied successfully to understand the statistics of polarization speckle produced by a constant polarization phasor plus a random polarization phasor sum [18].
4. STATISTICS OF POLARIZATION SPECKLE AT TWO POINTS
In analogy to the derivation of the joint probability density functions of intensity and phase at two points for laser speckle [3,23], we will specify the joint probability density functions of the Stokes parameters at two points for polarization speckle. Without loss of generality, we can start our analysis after a coordinate transformation has been applied to the multidimensional random variables for the stochastic electric fields to decompose the original polarization speckle into two statistically independent speckle fields with their linear polarization along the $\hat x$ and $\hat y$ directions, respectively. Due to the fact that the joint probability density functions for independent random variables are the product of their individual probability density functions, we can express
(9)$$\begin{split}&p{\rm{(}}{}_1E_x^r{\rm{,}}{}_1E_x^i{\rm{,}}{}_1E_y^r{\rm{,}}{}_1E_y^i{\rm{,}}{}_2E_x^r{\rm{,}}{}_2E_x^i{\rm{,}}{}_2E_y^r{\rm{,}}{}_2E_y^i{\rm{)}} \\[-4pt]&\quad= p{\rm{(}}{}_1E_x^r{\rm{,}}{}_1E_x^i{\rm{,}}{}_2E_x^r{\rm{,}}{}_2E_x^i{\rm{)}}p{\rm{(}}{}_1E_y^r{\rm{,}}{}_1E_y^i{\rm{,}}{}_2E_y^r{\rm{,}}{}_2E_y^i{\rm{),}}\end{split}$$
where $p{\rm{(}}{}_1E_k^r{\rm{,}}{}_1E_k^i{\rm{,}}{}_2E_k^r{\rm{,}}{}_2E_k^i{\rm{)}}$ is given by [3] (10)$$\begin{split}p({}_1E_k^r,{}_1E_k^i,{}_2E_k^r,{}_2E_k^i) & = \frac{1}{{4{\pi ^2}\sigma _k^4[1 - {{(\rho _{{kk}}^c)}^2} - {{(\rho _{{kk}}^s)}^2}]}}\\[-4pt]& \times \exp \left\{{- \frac{{{{({}_1E_k^r)}^2} + {{({}_1E_k^i)}^2} + {{({}_2E_k^r)}^2} + {{({}_2E_k^i)}^2}}}{{2\sigma _k^2[1 - {{(\rho _{{kk}}^c)}^2} - {{(\rho _{{kk}}^s)}^2}]}}} \right\}\\[-4pt]& \times \exp \left\{{\frac{{{}{\rho _{{kk}}^c}[({}_1E_k^r)({}_2E_k^r) + ({}_1E_k^i)({}_2E_k^i)]}}{{\sigma _k^2[1 - {{(\rho _{{kk}}^c)}^2} - {{(\rho _{{kk}}^s)}^2}]}}} \right\}\\[-4pt]& \times \exp \left\{{\frac{{{}{\rho _{{kk}}^s}[({}_1E_k^r)({}_2E_k^i) - ({}_1E_k^i)({}_2E_k^r)]}}{{\sigma _k^2[1 - {{(\rho _{{kk}}^c)}^2} - {{(\rho _{{kk}}^s)}^2}]}}} \right\}.\end{split}$$
The amplitude ${A_k}$ defined by ${A_k} = \sqrt {{{(E_k^r)}^2} + {{(E_k^i)}^2}}$ and the phase ${\theta _k}$ defined by ${\theta _k} = \arctan (E_k^i/E_k^r)$ are of chief interest, indicating the length and phase of the $k$-polarization component of a polarization speckle, respectively. In a similar way as Eq. (9), we can also write a joint probability density function of amplitudes and phases of a polarization speckle as
(11)$$\begin{split}&p{\rm{(}}{}_1{A_x}{\rm{,}}{}_1{\theta _x}{\rm{,}}{}_2{A_x}{\rm{,}}{}_2{\theta _x}{\rm{,}}{}_1{A_y}{\rm{,}}{}_1{\theta _y}{\rm{,}}{}_2{A_y}{\rm{,}}{}_2{\theta _y}{\rm{)}}\\[-4pt]&\quad = p{\rm{(}}{}_1{A_x}{\rm{,}}{}_1{\theta _x}{\rm{,}}{}_2{A_x}{\rm{,}}{}_2{\theta _x}{\rm{)}}p{\rm{(}}{}_1{A_y}{\rm{,}}{}_1{\theta _y}{\rm{,}}{}_2{A_y}{\rm{,}}{}_2{\theta _y}{\rm{),}}\end{split}$$
where $p{\rm{(}}{}_1{A_k}{\rm{,}}{}_1{\theta _k}{\rm{,}}{}_2{A_k}{\rm{,}}{}_2{\theta _k}{\rm{)}}$ is given by [3] (12)$$\begin{split}&p({}_1{A_k}{\rm{,}}{}_1{\theta _k}{\rm{,}}{}_2{A_k}{\rm{,}}{}_2{\theta _k}) = \frac{{{}_1{A_k}{}_2{A_k}}}{{4{\pi ^2}\sigma _k^4{\rm{(}}1 - \eta _k^2{\rm{)}}}}\\[-4pt]&\quad \times \exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_k}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_k}{\rm{)}}}^2} - 2{}_1{A_k}{}_2{A_k}{\eta _k}{\rm{\cos(}}{\phi _k}{ + }{}_1{\theta _k} - {}_2{\theta _k}{\rm{)}}}}{{2\sigma _k^2{\rm{(}}1 - \eta _k^2{\rm{)}}}}} \right\}\end{split}$$
with $\rho _{{kk}}^c = {\eta _k}{\cos}{\phi _k}$ and $\rho _{{kk}}^s = {\eta _k}{\sin}{\phi _k}$. Thus, we have (13)$$\begin{split} & p({}_1{A_x}{\rm{,}}{}_1{\theta _x}{\rm{,}}{}_2{A_x}{\rm{,}}{}_2{\theta _x},{}_1{A_y}{\rm{,}}{}_1{\theta _y}{\rm{,}}{}_2{A_y}{\rm{,}}{}_2{\theta _y})\\[-4pt]& = \frac{{{}_1{A_x}{}_2{A_x}{}_1{A_y}{}_2{A_y}}}{{16{\pi ^4}\sigma _x^4\sigma _y^4{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\[-4pt]& \times \exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_x}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_x}{\rm{)}}}^2} - 2{}_1{A_x}{}_2{A_x}{\eta _x}{\rm{\cos(}}{\phi _x}{ + }{}_1{\theta _x} - {}_2{\theta _x}{\rm{)}}}}{{2\sigma _x^2{\rm{(}}1 - \eta _x^2{\rm{)}}}}} \right\}\\[-4pt]& \times \exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_y}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_y}{\rm{)}}}^2} - 2{}_1{A_y}{}_2{A_y}{\eta _y}{\rm{\cos(}}{\phi _y}{ + }{}_1{\theta _y} - {}_2{\theta _y}{\rm{)}}}}{{2\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}} \right\},\end{split}$$
for ${}_1{A_k},{}_2{A_k} \ge 0$ and ${-}\pi \le {}_1{\theta _k},{}_2{\theta _k} \le \pi$, and we have zero otherwise. Note the fact that only the terms ${}_n{\theta _x} - {}_n{\theta _y}$ for $n = 1,2$ appear in the expressions for the Stokes parameters. Therefore, our next transformations are ${}_n{\theta _x} - {}_n{\theta _y} = {}_n\psi$ and ${}_n{\theta _x} + {}_n{\theta _y} = {}_n\xi$ for $(- 2\pi \le {}_n\psi \le 2\pi)$ and $(- 2\pi \le {}_n\xi \le 2\pi)$ with the Jacobian $\| J \| = 0.25$. To find the joint probability density functions of ${}_n{A_x},{}_n{A_y},{}_n\psi$, we must evaluate (14)$$\begin{split} &p({}_1{A_x}{\rm{,}}{}_1{A_y}{\rm{,}}{}_1\psi {\rm{,}}{}_2{A_x}{\rm{,}}{}_2{A_y}{\rm{,}}{}_2\psi) \\[-4pt]&\quad = \int_{- 2\pi}^{2\pi} {\int_{- 2\pi}^{2\pi} {p{\rm{(}}{}_1{A_x}{\rm{,}}{}_1\psi {\rm{,}}{}_2{A_x}{\rm{,}}{}_2\psi {\rm{,}}{}_1{A_y}{\rm{,}}{}_1\xi {\rm{,}}{}_2{A_y}{\rm{,}}{}_2\xi {\rm{)}}}} {\rm{d}}{}_1\xi {\rm{d}}{}_2\xi \\[-4pt]&\quad = \frac{{{}_1{A_x}{}_2{A_x}{}_1{A_y}{}_2{A_y}}}{{16{\pi ^4}\sigma _x^4\sigma _y^4{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\[-4pt]&\qquad \times \exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_x}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_x}{\rm{)}}}^2}}}{{2\sigma _x^2{\rm{(}}1 - \eta _x^2{\rm{)}}}}} \right\}\exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_y}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_y}{\rm{)}}}^2}}}{{2\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}} \right\}\\[-4pt]&\qquad \times \int_{- 2\pi}^{2\pi} {\int_{- 2\pi}^{2\pi} {{e^{\left\{{\frac{{{}_1{A_x}{}_2{A_x}{\eta _x}{\rm{\cos[}}{\phi _x}{ + }0.5{\rm{(}}{}_1\xi - {}_1\psi {\rm{)}} - 0.5{\rm{(}}{}_2\xi - {}_2\psi {\rm{)]}}}}{{\sigma _x^2(1 - \eta _x^2)}}} \right\}}}}} \\[-4pt]&\qquad \times {e^{\left\{{\frac{{{}_1{A_y}{}_2{A_y}{\eta _y}{\rm{\cos[}}{\phi _y}{ + }0.5{\rm{(}}{}_1\xi + {}_1\psi {\rm{)}} - 0.5{\rm{(}}{}_2\xi + {}_2\psi {\rm{)]}}}}{{\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}} \right\}}}{\rm{d}}{}_1\xi {\rm{d}}{}_2\xi .\end{split}$$
To perform the required integration, we make use of the integral identity [24]
(15)$$\int_{- \pi}^\pi {\int_{- \pi}^\pi {\exp \left\{{A\cos {\rm{(}}{\beta _1} - {\beta _2}{ + }\chi {\rm{)}}} \right\}}} {\rm{d}}{\beta _1}{\rm{d}}{\beta _2} = 4{\pi ^2}{I_0}{\rm{(}}A{\rm{)}},$$
where ${I_0}(\cdots)$ is a modified Bessel function of the first kind with zero order, and we rewrite Eq. (14) as (16)$$\begin{split} &p({}_1{A_x}{\rm{,}}{}_1{A_y}{\rm{,}}{}_1\psi {\rm{,}}{}_2{A_x}{\rm{,}}{}_2{A_y}{\rm{,}}{}_2\psi)\\[-4pt]&\quad = \frac{{{}_1{A_x}{}_2{A_x}{}_1{A_y}{}_2{A_y}}}{{4{\pi ^2}\sigma _x^4\sigma _y^4{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\[-4pt]& \qquad\times \exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_x}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_x}{\rm{)}}}^2}}}{{2\sigma _x^2{\rm{(}}1 - \eta _x^2{\rm{)}}}}} \right\}\exp \left\{{- \frac{{{{{\rm{(}}{}_1{A_y}{\rm{)}}}^2} + {{{\rm{(}}{}_2{A_y}{\rm{)}}}^2}}}{{2\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}} \right\}\\[-4pt]& \qquad\times {I_0}\left({\left\{{{{\left[{{\textstyle{{{}_1{A_x}{}_2{A_x}{\eta _x}} \over {\sigma _x^2{\rm{(}}1 - \eta _x^2{\rm{)}}}}}} \right]}^2} + {{\left[{{\textstyle{{{}_1{A_y}{}_2{A_y}{\eta _y}} \over {\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}}} \right]}^2}} \right.} \right.\\[-4pt]&\qquad+\left. {{{\left. { {\textstyle{{2{}_1{A_x}{}_2{A_x}{\eta _x}{}_1{A_y}{}_2{A_y}{\eta _y}\cos {\rm{(}}{\phi _x} - {\phi _y} - {}_1\psi + {}_2\psi {\rm{)}}} \over {\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}}} \right\}}^{1/2}}} \right).\end{split}$$
From the definitions of the Stokes parameters given in Eqs. (1a)–(1d), we are able to perform another transformation, giving the determinant of the corresponding Jacobian matrix of the transformation as
(17)$$\begin{split}\left\| J \right\| &= \left\| {\begin{array}{*{20}{c}}{\partial {}_1{A_x}/\partial {}_1{S_1}}&{\partial {}_1{A_x}/\partial {}_1{S_2}}& \cdots &{\partial {}_1{A_x}/\partial {}_2{S_3}}\\{\partial {}_1{A_y}/\partial {}_1{S_1}}&{\partial {}_1{A_y}/\partial {}_1{S_2}}& \cdots &{\partial {}_1{A_y}/\partial {}_2{S_3}}\\ \vdots & \vdots & \vdots & \vdots \\{\partial {}_2\psi /\partial {}_1{S_1}}&{\partial {}_2\psi /\partial {}_1{S_2}}& \cdots &{\partial {}_2\psi /\partial {}_2{S_3}}\end{array}} \right\|\\[-4pt] &= \left({{{16}}\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} \sqrt {{}_1S_2^2 + {}_1S_3^2}} \right.\\&\quad\sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} {\left. {\sqrt {{}_2S_2^2 + {}_2S_3^2}} \right)^{- 1}}.\end{split}$$
From Eqs. (1), (16), and (17), the desired joint probability density function of the Stokes parameters at two points can be written immediately as
(18)$$\begin{split}& p{\rm{(}}{}_1{S_1}{\rm{,}}{}_1{S_2}{\rm{,}}{}_1{S_3}{\rm{,}}{}_2{S_1}{\rm{,}}{}_2{S_2}{\rm{,}}{}_2{S_3}{\rm{)}}\\[-4pt]& \quad= \left[{{{256}}{\pi ^2}\sigma _x^4\sigma _y^4\left({1 - \eta _x^2} \right)\left({1 - \eta _y^2} \right)\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2}} \right.\\[-4pt]&\qquad\times{\left. { \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2}} \right]^{- 1}}\\[-4pt]&\qquad \times {e^{\left\{{- \frac{{\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} + {}_1{S_1} + \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} + {}_2{S_1}}}{{4\sigma _x^2\left({1 - \eta _x^2} \right)}}} \right\}}}\\[-4pt]&\qquad \times {e^{\left\{{- \frac{{\sqrt {{}_1\!S_1^2 + {}_1\!S_2^2 + {}_1\!S_3^2} - {}_1\!{S_1} + \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} - {}_2{S_1}}}{{4\sigma _y^2\left({1 - \eta _y^2} \right)}}} \right\}}}\\[-4pt]&\qquad \times {I_0}\left({\left[{{\textstyle{{\left({\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} + {}_1{S_1}} \right)\left({\sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} + {}_2{S_1}} \right)\eta _x^2} \over {4\sigma _x^4{{\left({1 - \eta _x^2} \right)}^2}}}}} \right.} \right.\\[-4pt]&\qquad + {\textstyle{{\left({\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} - {}_1{S_1}} \right)\left({\sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} - {}_2{S_1}} \right)\eta _y^2} \over {4\sigma _y^4{{\left({1 - \eta _y^2} \right)}^2}}}}\\[-4pt]&\qquad+\left. {{{\left. { {\textstyle{{[\cos\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)(}}{}_1{S_2}{}_2{S_2} + {}_1{S_3}{}_2{S_3}{\rm{)}} + \sin\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)(}}{}_1{S_2}{}_2{S_3} - {}_1{S_3}{}_2{S_2}{\rm{)]}}{\eta _x}{\eta _y}} \over {2\sigma _x^2\sigma _y^2\left({1 - \eta _x^2} \right)\left({1 - \eta _y^2} \right)}}}} \right]}^{1/2}}} \right).\end{split}$$
To represent the distribution of different states of polarization for polarization speckle in the Stokes space, it will be convenient to find the joint probability density function for intensity ${S_0}$ and two ellipsoidal parameters $\chi$ and $\varphi$ at two points. From Eqs. (2a)–(2c), our next transform with the corresponding $6 \times 6$ Jacobian matrix is given by
(19)$$\begin{split}\left\| J \right\| &= \left\| {\begin{array}{*{20}{c}}{\partial {}_1{S_1}/\partial {}_1{S_0}}&{\partial {}_1{S_2}/\partial {}_1{S_0}}& \cdots &{\partial {}_2{S_3}/\partial {}_1{S_0}}\\{\partial {}_1{S_1}/\partial {}_1\chi}&{\partial {}_1{S_2}/\partial {}_1\chi}& \cdots &{\partial {}_2{S_3}/\partial {}_1\chi}\\ \vdots & \vdots & \vdots & \vdots \\{\partial {}_1{S_1}/\partial {}_2\varphi}&{\partial {}_1{S_2}/\partial {}_2\varphi}& \cdots &{\partial {}_2{S_3}/\partial {}_2\varphi}\end{array}} \right\|\\&= 16{}_1S_0^2{}_2S_0^2\cos {\rm{(}}2{}_1\chi {\rm{)}}\cos {\rm{(}}2{}_2\chi {\rm{)}}{\rm{.}}\end{split}$$
After trigonometric simplication, the joint density function $p {\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}}$ for the polarization speckle at two points is
(20)$$\begin{array}{l}p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}}\\\\\;\; = \frac{{{}_1{S_0}{}_2{S_0} \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}}}}{{{{16}}{\pi ^2}\sigma _x^4\sigma _y^4{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\\\\;\; \times {e^{\left\{{- \,\,\frac{{{}_1{S_0}{\rm{[}}1 + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)]}} + {}_2{S_0}[1 + \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)]}}}}{{4\sigma _x^2{\rm{(}}1 - \eta _x^2{\rm{)}}}}} \right\}}}\\\\\;\; \times {e^{\left\{{- \,\,\frac{{{}_1{S_0}[1 - \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)]}} + {}_2{S_0}{\rm{[}}1 - \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)]}}}}{{4\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}} \right\}}}\\\\\;\; \times {I_0}\left({\,\left[{{\textstyle{{{}_1{S_0}{}_2{S_0}\eta _x^2{\rm{[}}1 + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)][}}1 + \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)]}}} \over {4\sigma _x^4{{{\rm{(}}1 - \eta _x^2{\rm{)}}}^2}}}}} \right.} \right.\\\\\;\; + {\textstyle{{{}_1{S_0}{}_2{S_0}\eta _y^2{\rm{[}}1 - \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)][}}1 - \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)]}}} \over {4\sigma _y^4{{{\rm{(}}1 - \eta _y^2{\rm{)}}}^2}}}}\\\\\;\; + {\textstyle{{{}_1{S_0}{}_2{S_0}{\eta _x}{\eta _y} \cos\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)[}} \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{) +}} \sin\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)]}}} \over {2\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}}\\\\\;\; + \left. {{{\left. {{\textstyle{{{}_1{S_0}{}_2{S_0}{\eta _x}{\eta _y} \sin\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)[}} \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}} - \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}} \sin\!{\rm{(}}2{}_1\chi {\rm{)]}}} \over {2\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}}} \right]}^{1/2}}} \right).\end{array}$$
This equation is valid for ${}_1{S_0},{}_2{S_0} \ge 0$, ${-}\pi /4 \le {}_1\chi ,{}_2\chi \le \pi /4$, and $0 \le {}_1\varphi ,{}_2\varphi \lt \pi$. To investigate the distribution of the two ellipsoidal parameters $\chi$ and $\varphi$ for two points on the Poincaré sphere, we perform the integral with respect to ${}_1{S_0}$ and ${}_2{S_0}$:
(21)$$\begin{split} p{\rm{(}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}}& = \int_0^{+ \infty} {\int_0^{+ \infty}}\\&\quad\times{ {p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)d}}}} {}_1{S_0}{\rm{d}}{}_2{S_0}\\& = 16a{\rm{(4}}{b_1}{b_2} + {c^2}{\rm{)}}/{(4{b_1}{b_2} - {c^2})^3},\end{split}$$
where the parameters $a,{b_1},{b_2}$, and $c$ are given by (22a)$$a = \frac{{\cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}}}}{{{{16}}{\pi ^2}\sigma _x^4\sigma _y^4{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}{\rm{,}}$$
(22b)$$\begin{array}{l}{b_n} = \frac{{1 + \cos\!{\rm{(}}2{}_n\chi {\rm{)}} \cos\!{\rm{(}}2{}_n\varphi {\rm{)}}}}{{4\sigma _x^2{\rm{(}}1 - \eta _x^2{\rm{)}}}} + \frac{{1 - \cos\!{\rm{(}}2{}_n\chi {\rm{)}} \cos\!{\rm{(}}2{}_n\varphi {\rm{)}}}}{{4\sigma _y^2{\rm{(}}1 - \eta _y^2{\rm{)}}}}{\rm{,}}\\\quad \quad {\rm{(}}n = 1,2{\rm{)}}\end{array}$$
(22c)$$\begin{split} c & = \left\{{\frac{{\eta _x^2{\rm{[}}1 + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)][}}1 + \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)]}}}}{{4\sigma _x^4{{{\rm{(}}1 - \eta _x^2{\rm{)}}}^2}}}} \right.\\&\quad + \frac{{\eta _y^2{\rm{[}}1 - \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)][}}1 - \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)]}}}}{{4\sigma _y^4{{{\rm{(}}1 - \eta _y^2{\rm{)}}}^2}}}\\&\quad + \frac{{{\eta _x}{\eta _y} \cos\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)}} \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}}}}{{2\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\&\quad + \frac{{{\eta _x}{\eta _y} \cos\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)}} \sin\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}}}}{{2\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\&\quad + \frac{{{\eta _x}{\eta _y} \sin\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)}} \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}}}}{{2\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}\\&\quad-{\left. { \frac{{{\eta _x}{\eta _y} \sin\!{\rm{(}}{\phi _x} - {\phi _y}{\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}} \sin\!{\rm{(}}2{}_1\chi {\rm{)}}}}{{2\sigma _x^2\sigma _y^2{\rm{(}}1 - \eta _x^2{\rm{)(}}1 - \eta _y^2{\rm{)}}}}} \right\}^{1/2}}.\end{split}$$
When Eq. (21) is derived, we have made use of the integral identity [24]
(23)$$\int_0^{+ \infty} {\int_0^{+ \infty} {xy{e^{- ax - by}}{I_0}(c\sqrt {xy}){\rm{d}}x{\rm{d}}y}} = \frac{{16(4ab + {c^2})}}{{{{(4ab - {c^2})}^3}}}.$$
Equation (21) is the sought after result for the distribution of the two ellipsoidal angles for two points on the surface of a Poincaré sphere, providing a clear image for the statistics of the studied multidimensional phenomenon on polarization speckle.
To find the joint density function for the Stokes parameters ${}_1{S_0}$ and ${}_2{S_0}$ for polarization speckle at two points, we must evaluate
(24)$$\begin{split}p{\rm{(}}{}_1{S_0},{}_2{S_0}{\rm{)}}& = \int_{- \pi /4}^{\pi /4} \int_0^\pi \int_{- \pi /4}^{\pi /4} \int_0^\pi \\&\quad\times{p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi ,{}_2\varphi {\rm{)}}\,} {\rm{d}}{}_1\chi {\rm{d}}{}_1\varphi {\rm{d}}{}_2\chi {\rm{d}}{}_2\varphi .\end{split}$$
The integration is a difficult one because of the quadruple integral involving the modified Bessel function. Note the fact that ${S_0}$ represents the intensity of the stochastic electronic fields, whose two polarization components are statistically independent as explained at the beginning of this section. Therefore, we try to find the joint density function for ${}_1{S_0}$ and ${}_2{S_0}$ from the joint density function of the intensities ${}_n{I_x}$ and ${}_n{I_y}$ of two polarization components, i.e., ${}_n{S_0} = {}_n{I_x} + {}_n{I_y}$ for $(n = 1,2)$. Due to the fact of statistical independence for two polarization components along the $\hat x$ and $\hat y$ directions, we have
(25)$$p{\rm{(}}{}_1{I_x}{\rm{,}}{}_2{I_x}{\rm{,}}{}_1{I_y}{\rm{,}}{}_2{I_y}{\rm{)}} = p{\rm{(}}{}_1{I_x}{\rm{,}}{}_2{I_x}{\rm{)}}p{\rm{(}}{}_1{I_y}{\rm{,}}{}_2{I_y}{\rm{),}}$$
where $p{\rm{(}}{}_1{I_k}{\rm{,}}{}_2{I_k}{\rm{)}}\,{\rm{,}}\,{\rm{(}}k = x,y{\rm{)}}$ is given by [3] (26)$$\begin{split}p{\rm{(}}{}_1{I_k}{\rm{,}}{}_2{I_k}{\rm{)}} &= \frac{1}{{4\sigma _k^4{\rm{(}}1 - \eta _k^2{\rm{)}}}}\\&\quad\times\exp\left\{{- \frac{{{}_1{I_k}{ + }{}_2{I_k}}}{{2\sigma _k^2{\rm{(}}1 - \eta _k^2{\rm{)}}}}} \right\}{I_0}\left\{{\frac{{{\eta _k}\sqrt {{}_1{I_k}{}_2{I_k}}}}{{\sigma _k^2{\rm{(}}1 - \eta _k^2{\rm{)}}}}} \right\},\end{split}$$
for ${}_1{I_k},{}_2{I_k} \ge 0$, and it is zero otherwise.By using the transform between ${}_n{I_x},{}_n{I_y}$ and ${}_n{S_0}$ in Eq. (1a) with the corresponding Jacobian $\| J \| = 0.25$, we can express the joint probability density function $p({}_1{S_0},{}_2{S_0})$ for polarization speckle at two points as the expected convolution. That is,
(27)$$\begin{split} p{\rm{(}}{}_1{S_0}{\rm{,}}{}_2{S_0}{\rm{)}} & = \int_{- {}_2{S_0}}^{{}_2{S_0}} {\int_{- {}_1{S_0}}^{{}_1{S_0}} {p{\rm{(}}{}_1{S_0}{\rm{,}}{}_2{S_0}{\rm{,}}{}_1{S_1}{\rm{,}}{}_2{S_1}{\rm{)d}}{}_1{S_1}{\rm{d}}{}_2{S_1}}} \\& = {\textstyle{1 \over {64\sigma _x^4\sigma _y^4\left({1 - \eta _x^2} \right)\left({1 - \eta _y^2} \right)}}} \\&\quad\times\int_{- {}_2{S_0}}^{{}_2{S_0}} {\int_{- {}_1{S_0}}^{{}_1{S_0}} {{e^{- \frac{{{}_1{S_0} + {}_1{S_1}{ + }{}_2{S_0} + {}_2{S_1}}}{{{{4}}\sigma _x^2\left({1 - \eta _x^2} \right)}}}}{{\mathop{ e}\nolimits} ^{- \frac{{{}_1{S_0} - {}_1{S_1}{ + }{}_2{S_0} - {}_2{S_1}}}{{{{4}}\sigma _y^2(1 - \eta _y^2)}}}}}} \\&\quad \times {I_0}\left\{{{\textstyle{{{\eta _x}\sqrt {{\rm{(}}{}_1{S_0} + {}_1{S_1}{\rm{)(}}{}_2{S_0} + {}_2{S_1}{\rm{)}}}} \over {2\sigma _x^2\left({1 - \eta _x^2} \right)}}}} \right\} \\&\quad\times{I_0}\left\{{{\textstyle{{{\eta _y}\sqrt {{\rm{(}}{}_1{S_0} - {}_1{S_1}{\rm{)(}}{}_2{S_0} - {}_2{S_1}{\rm{)}}}} \over {2\sigma _y^2(1 - \eta _y^2)}}}} \right\}{\rm{d}}{}_1{S_1}{\rm{d}}{}_2{S_1},\end{split}$$
for ${}_1{S_0},{}_2{S_0} \ge 0$, and it is zero otherwise. While this integral does not seem to be expressible in terms of tabulated functions for general cases, it can be evaluated for the isotropic polarization speckle with the results to be shown in the next section.5. JOINT PROBABILITY DENSITY FUNCTIONS FOR ISOTROPIC POLARIZATION SPECKLE AT TWO POINTS
In Section 4 the joint probability density functions of the Stokes parameters and that of the Poincaré spherical coordinates for a polarization speckle at two points have been obtained. In our derivation for these general formats, the variances $\sigma _k^2$ and the correlation coefficients ${\eta _k}$ of the two components of the electic fields are different for an anisotropic polarization speckle. For the special case of isotropic polarization speckle where $\sigma _x^2 = \sigma _y^2 = {\sigma ^2}$ and ${\eta _x} = {\eta _y} = \eta$, we can obtain a simpler expression of these joint probability density functions for the Stokes parameters in Eq. (18), which becomes
(28)$$\begin{split} &p{\rm{(}}{}_1{S_1}{\rm{,}}{}_1{S_2}{\rm{,}}{}_1{S_3}{\rm{,}}{}_2{S_1}{\rm{,}}{}_2{S_2}{\rm{,}}{}_2{S_3}{\rm{)}}\\& = {\left[{{{256}}{\pi ^2}{\sigma ^8}{{\left({1 - {\eta ^2}} \right)}^2}\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2}} \right]^{- 1}}\\& \times \exp\left\{{- {\textstyle{{\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} + \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2}} \over {2{\sigma ^2}\left({1 - {\eta ^2}} \right)}}}} \right\}\\& \times {I_0}\left({{\textstyle{{\eta \sqrt {{\rm{(}}\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} + {}_1{S_1}{}_2{S_1} + {}_1{S_2}{}_2{S_2} + {}_1{S_3}{}_2{S_3}{\rm{)}}}} \over {\sqrt {{2}} {\sigma ^2}\left({1 - {\eta ^2}} \right)}}}} \right).\end{split}$$
When the equation above is derived, we have made use of ${\phi _x} = {\phi _y}$ due to the same correlation coefficient for isotropic polarization speckle. Meanwhile, $p({}_1{S_0},{}_1\chi ,{}_1\varphi ,{}_2{S_0},{}_2\chi ,{}_2\varphi)$ in Eq. (20) can be reduced to
(29)$$\begin{split} &p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}}\\& = \frac{{{}_1{S_0}{}_2{S_0} \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}}}}{{{{16}}{\pi ^2}{\sigma ^8}{{{\rm{(}}1 - {\eta ^2}{\rm{)}}}^2}}}\\ &\quad\times\exp\left\{{- \frac{{{}_1{S_0} + {}_2{S_0}}}{{{{2}}{\sigma ^2}{\rm{(}}1 - {\eta ^2}{\rm{)}}}}} \right\}{I_0}\left(\vphantom{\left[\begin{array}{l}1 + \sin\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}}\\ + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)}}\\{ + } \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}}\end{array} \right]}{\textstyle{\eta \over {\sqrt {{2}} {\sigma ^2}{\rm{(}}1 - {\eta ^2}{\rm{)}}}}}\left\{\vphantom{\left[\begin{array}{l}1 + \sin\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}}\\ + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)}}\\{ + } \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}}\end{array} \right]}{{}_1{S_0}{}_2{S_0} }\right.\right.\\&\quad\times\left.\left.\left[\begin{array}{l}1 + \sin\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}}\\ + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{)}}\\{ + } \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}}\end{array} \right] \right\}^{1/2} \right),\end{split}$$
which is valid for ${}_1{S_0},{}_2{S_0} \ge 0$, ${-}\pi /4 \le {}_1\chi ,{}_2\chi \le \pi /4$, and $0 \le {}_1\varphi ,{}_2\varphi \lt \pi$.Similarly, the distribution of the two ellipsodial parameters for an isotropic polarization speckle at two points can be further simplified from Eq. (21). That is,
(30)$$p{\rm{(}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}} = \frac{{4 \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)(}}1 - {\eta ^2}{{\rm{)}}^2}{\rm{(}}2 + {\eta ^2}\beta {\rm{)}}}}{{{\pi ^2}{{{\rm{(}}2 - {\eta ^2}\beta {\rm{)}}}^3}}}{\rm{,}}$$
for ${-}\pi /4 \le {}_1\chi ,{}_2\chi \le \pi /4$ and $0 \le {}_1\varphi ,{}_2\varphi \lt \pi$, where $\beta$ is given by (31)$$\begin{split}\beta &= 1 + \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\varphi {\rm{)}}\\&\quad+ \cos\!{\rm{(}}2{}_1\chi {\rm{)}} \cos\!{\rm{(}}2{}_1\varphi {\rm{)}} \cos\!{\rm{(}}2{}_2\chi {\rm{)}} \cos\!{\rm{(}}2{}_2\varphi {\rm{) +}} \sin\!{\rm{(}}2{}_1\chi {\rm{)}} \sin\!{\rm{(}}2{}_2\chi {\rm{)}}{\rm{.}}\end{split}$$
For an isotropic polarization speckle, we are seeking the joint density function of the Stokes parameter ${S_0}$ at two points. Again, an analytic solution is elusive, but numerical integration yields results shown in Figs. 1 and 2. Figure 1 illustrates the shape of the normalized joint probability density function $p ({}_1{S_0},{}_2{S_0})$ for various values of $\eta$, and Fig. 2 gives the corresponding contour plots. It can be seen that the shape of the joint density function $p({}_1{S_0},{}_2{S_0})$ in the $({}_1{S_0},{}_2{S_0})$ plane depends markedly on the value of the correlation coefficient $\eta$. As $\eta$ increases, the joint density function for ${S_0}$ at two points approaches a shaped delta function sheet along the line ${}_1{S_0} = {}_2{S_0}$.
In some applications, we are also interested in the difference of the two ellipsoidal parameters $\Delta \chi$ and $\Delta \varphi$ indicating angle spreads. To get the probability density function, let us make the following change of variables: $\Delta \chi = {}_1\chi - {}_2\chi$, $\chi = {}_2\chi$, $\Delta \varphi = {}_1\varphi - {}_2\varphi$, and $\varphi = {}_2\varphi$ with the corresponding Jacobian determinant of the transform being unity. Through a transformation of variables, we find from Eq. (30) that
(32)$$\begin{split} & p{\rm{(}}\Delta \chi {\rm{,}}\Delta \varphi {\rm{,}}\chi {\rm{,}}\varphi {\rm{)}}\\[-4pt]& = \frac{{4 \cos{\rm{[2(}}\chi + \Delta \chi {\rm{)]}} \cos\!{\rm{(}}2\chi {\rm{)(}}1 - {\eta ^2}{{\rm{)}}^2}{\rm{(}}2 + {\eta ^2}\gamma {\rm{)}}}}{{{\pi ^2}{{{\rm{(}}2 - {\eta ^2}\gamma {\rm{)}}}^3}}},\end{split}$$
with $\gamma$ given by (33)$$\begin{split} \gamma & = \cos{\rm{[2(}}\chi + \Delta \chi {\rm{)]}} \cos{\rm{[2(}}\varphi + \Delta \varphi {\rm{)]}} \cos\!{\rm{(}}2\chi {\rm{)}} \cos\!{\rm{(}}2\varphi {\rm{)}}\\[-4pt]& \quad+ \cos{\rm{[2(}}\chi + \Delta \chi {\rm{)]}} \sin{\rm{[2(}}\varphi + \Delta \varphi {\rm{)]}} \cos\!{\rm{(}}2\chi {\rm{)}} \sin\!{\rm{(}}2\varphi {\rm{)}}\\[-4pt]& \quad+ \sin{\rm{[2(}}\chi + \Delta \chi {\rm{)]}} \sin\!{\rm{(}}2\chi {\rm{) + 1}}{\rm{.}}\end{split}$$
Two observations about this result are important. First, for fixed orientation ${\rm{(}}\chi ,\varphi {\rm{)}}$ in the Stokes space, the joint density function depends only on the difference of the two ellipsoidal parameters $\Delta \chi$ and $\Delta \varphi$. A second observation is that the distribution of ${\rm{(}}\Delta \chi ,\Delta \varphi {\rm{)}}$ is jointly dependent on the selection of ${\rm{(}}\chi ,\varphi {\rm{)}}$ since $p{\rm{(}}\Delta \chi ,\Delta \varphi ,\chi ,\varphi {\rm{)}} \ne p{\rm{(}}\Delta \chi ,\Delta \varphi {\rm{)}}p{\rm{(}}\chi ,\varphi {\rm{)}}$. Thus, our interest is to find the conditional joint probability density of $\Delta \chi$ and $\Delta \varphi$ given that the values of $\chi$ and $\varphi$ are known. This density function represented by $p{\rm{(}}\Delta \chi ,\Delta \varphi | {\chi ,\varphi} {\rm{)}}$ can be found using Bayes’ rule [25]:
(34)$$p{\rm{(}}\Delta \chi {\rm{,}}\Delta \varphi \left| {\chi {\rm{,}}\varphi} \right.{\rm{)}} = p{\rm{(}}\Delta \chi {\rm{,}}\Delta \varphi {\rm{,}}\chi {\rm{,}}\varphi {{)/p(}}\chi {\rm{,}}\varphi {\rm{)}}.$$
Note from [18] that $p{\rm{(}}\chi {\rm{,}}\varphi {\rm{)}} = \cos\!{\rm{(}}2\chi {\rm{)}}/\pi$, and the conditional density function in Eq. (37) can be further reduced to
(35)$$p{\rm{(}}\Delta \chi {\rm{,}}\Delta \varphi \left| {\chi {\rm{,}}\varphi} \right.{\rm{)}} = \frac{{4 \cos{\rm{[2(}}\chi + \Delta \chi {\rm{)](}}1 - {\eta ^2}{{\rm{)}}^2}{\rm{(}}2 + {\eta ^2}\gamma {\rm{)}}}}{{\pi {{{\rm{(}}2 - {\eta ^2}\gamma {\rm{)}}}^3}}}.$$
Figure 3 shows plots of the conditional joint density function $p{\rm{(}}\Delta \chi ,\Delta \varphi | {\chi ,\varphi} {\rm{)}}$ for various values of $\eta$ when the coordinates ${\rm{(}}\chi ,\varphi {\rm{)}}\,{ = }\,{\rm{(0,0)}}$ have been fixed along the ${\hat s_1}$ direction in the Stokes space. Note that as $\eta \to 0$, the conditional probability density takes the form $ \cos\!{\rm{(}}2\chi {\rm{)}}/\pi$, indicating that the differences of two ellipsoidal parameters $(\Delta \chi ,\Delta \varphi)$ have a uniform distribution on the spherical surface of the Poincaré sphere. As $\eta \to 1$, the density function approaches a shaped delta function at ${\rm{(}}\Delta \chi ,\Delta \varphi {\rm{)}}\,{ = }\,{\rm{(0,0)}}$. As the correlation coefficient $\eta$ increases, the probability density becomes more and more concentrated about small values of $\Delta \chi$ and $\Delta \varphi$, indicating that these two sets of the ellipsoidal parameters ${\rm{(}}{}_1\chi ,{}_1\varphi {\rm{)}}$ and ${\rm{(}}{}_2\chi ,{}_2\varphi {\rm{)}}$ are more and more likely to be close together.
Since the value of the correlation coefficient $\eta$ has remarkable effect on the shape of the joint density function, we explore a bit more about the properties of the joint probability density functions of an isotropic polarization speckle under two limiting cases.
When the correlation coefficient $\eta$ goes to zero for both of the composing polarization components (as happens, for example, when the two measurement points are far apart), it is easily seen from Eq. (28) that
(36)$$\begin{split}& p{\rm{(}}{}_1{S_1}{\rm{,}}{}_1{S_2}{\rm{,}}{}_1{S_3}{\rm{,}}{}_2{S_1}{\rm{,}}{}_2{S_2}{\rm{,}}{}_2{S_3}{\rm{)}}\\[-4pt]& = {\left[{{{256}}{\pi ^2}{\sigma ^8}\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2}} \right]^{\, - 1}}\\[-4pt] &\quad\times \exp\left\{{- {\rm{(}}\sqrt {{}_1S_1^2 + {}_1S_2^2 + {}_1S_3^2} + \sqrt {{}_2S_1^2 + {}_2S_2^2 + {}_2S_3^2} {\rm{)}}/({{2}}{\sigma ^2})} \right\}\\[-4pt]& = p{\rm{(}}{}_1{S_1}{\rm{,}}{}_1{S_2}{\rm{,}}{}_1{S_3}{\rm{)}}p{\rm{(}}{}_2{S_1}{\rm{,}}{}_2{S_2}{\rm{,}}{}_2{S_3}{\rm{)}}\end{split},$$
with $p({}_n{S_1},{}_n{S_2},{}_n{S_3}) = {(16\pi {\sigma ^4}\sqrt {\sum\nolimits_{l = 1}^3 {{}_nS_l^2}})^{- 1}}\exp \{{- \sqrt {\sum\nolimits_{l = 1}^3 {{}_nS_l^2}} /(2{\sigma ^2})} \}$ for $(n = 1,2)$. In this case, the joint probability density function for the Stokes parameters at two points can be expressed as a product of those for each individual point given in [18], as it should be. Similarly, we obtain the result of $p({}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi)$ from Eq. (29). That is, (37)$$\begin{split} p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}} & = \frac{{{}_1{S_0} \cos\!{\rm{(}}2{}_1\chi {\rm{)}}}}{{{{4}}\pi {\sigma ^4}}}\exp\left\{{- \frac{{{}_1{S_0}}}{{{{2}}{\sigma ^2}}}} \right\}\\&\quad \times \frac{{{}_2{S_0} \cos\!{\rm{(}}2{}_2\chi {\rm{)}}}}{{{{4}}\pi {\sigma ^4}}}\exp\left\{{- \frac{{{}_2{S_0}}}{{{{2}}{\sigma ^2}}}} \right\}\\ & = p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{)}}p{\rm{(}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{),}}\end{split}$$
where $p({}_n{S_0},{}_n\chi ,{}_n\varphi) = {(4\pi {\sigma ^4})^{- 1}}{}_n{S_0}\cos (2{}_n\chi)\exp \{{- {}_n{S_0}/}{(2{\sigma ^2})} \}$ for $(n = 1,2)$ indicates a ball-shaped probability cloud with a uniform distribution of the states of polarization on the Poincaré sphere [18]. Since the isotropic polarization speckle fields at two far apart points are uncorrelated with $\eta = 0$, both intensities and states of polarization at these two points fluctuate independently from each other.On the other hand, if the two measurement points approach each other arbitrarily closely, the joint density functions can be found by proper limiting arguments with $\eta \to 1$. However, calculating the limit of the function in Eq. (27) or Eq. (29) is not easily solvable analytically. Just like the shapes of the joint density functions in Figs. 1(d), 2(d), and 3(d), numerical evaluation with $\eta$ sufficiently close to unity shows that
(38)$$\begin{split} & \mathop {{\lim}}\limits_{\eta \to 1} p{\rm{(}}{}_1{S_1}{\rm{,}}{}_1{S_2}{\rm{,}}{}_1{S_3}{\rm{,}}{}_2{S_1}{\rm{,}}{}_2{S_2}{\rm{,}}{}_2{S_3}{\rm{)}}\\& = {\left[{{{16}}\pi {\sigma ^4}\sqrt {\sum\nolimits_{l = 1}^3 {{}_1S_l^2}}} \right]^{\, - 1}}\exp\left\{{- \sqrt {\sum\nolimits_{l = 1}^3 {{}_1S_l^2}} /({{2}}{\sigma ^2})} \right\}\\&\quad \times {\delta ^3}{\rm{(}}{}_2{S_1} - {}_1{S_1}{\rm{,}}{}_2{S_2} - {}_1{S_2}{\rm{,}}{}_2{S_3} - {}_1{S_3}{\rm{)}}\\& = p{\rm{(}}{}_1{S_1}{\rm{,}}{}_1{S_2}{\rm{,}}{}_1{S_3}{\rm{)}}{\delta ^3}{\rm{(}}{}_2{S_1} - {}_1{S_1}{\rm{,}}{}_2{S_2} - {}_1{S_2}{\rm{,}}{}_2{S_3} - {}_1{S_3}{\rm{)}}\end{split}$$
and (39)$$\begin{split} &\mathop {{\lim}}\limits_{\eta \to 1} p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{,}}{}_2{S_0}{\rm{,}}{}_2\chi {\rm{,}}{}_2\varphi {\rm{)}} \\& = \frac{{{}_1{S_0} \cos\!{\rm{(}}2{}_1\chi {\rm{)}}}}{{{{4}}\pi {\sigma ^4}}}\exp\left\{{- \frac{{{}_1{S_0}}}{{{{2}}{\sigma ^2}}}} \right\}{\delta ^3}{\rm{(}}{}_2{S_0} - {}_1{S_0}{\rm{,}}{}_2\chi - {}_1\chi {\rm{,}}{}_2\varphi - {}_1\varphi {\rm{)}}\\& = p{\rm{(}}{}_1{S_0}{\rm{,}}{}_1\chi {\rm{,}}{}_1\varphi {\rm{)}}{\delta ^3}{\rm{(}}{}_2{S_0} - {}_1{S_0}{\rm{,}}{}_2\chi - {}_1\chi {\rm{,}}{}_2\varphi - {}_1\varphi {\rm{),}}\end{split}$$
where ${\delta ^3}(\cdots)$ denotes the three-dimensional Dirac delta function. When the polarization speckle fields at two arbitrarily close points are always in phase with full coherence, i.e., $\eta = 1$, although their electric fields are randomly changing with space or time at each individual point, their intensities and states of polarization represented by the random variables $({}_1{S_0},{}_1\chi ,{}_1\varphi)$ and $({}_2{S_0},{}_2\chi ,{}_2\varphi)$ fluctuate in a synchronized manner over the entire observation.Before closing the discussion about the joint statistics of the Stokes parameters, the point should be made that we have already applied the concepts of ensemble-average polarization and coherence in the study of polarization speckle [2,26]. Similar to the fluctuating intensity in conventional laser speckle with a uniform state of polarization, the polarization speckle has its unique feature of the fluctuations for the Stokes parameters observed when coherent light is passed through a birefringent polarization scrambler [16,17]. For such a stochastic electric field, it is not difficult to show that the light behind the depolarizing diffuser is entirely coherent and completely polarized by our usual definitions based on time averages [20]. Therefore, rather than averaging with respect to time, we should modify our definitions of the polarization matrix, the mutual coherence matrix, and the generalized Stokes parameters for such light by averaging over an ensemble of statistically similar birefringent polarization scramblers and allowing the concepts of polarization and coherence to be defined for polarization speckle.
6. CONCLUSION
In a certain class of applications, the statistical properties of polarization speckle at two points in space or time are of great interest. In this paper, we have provided the joint probability density functions of the Stokes parameters and the parameters characterizing the stochastic polarization ellipses for the produced polarization speckle under the Gaussian assumpiton of the stochastic electric fields. Whenever possible, the closed-form expressions for the joint density functions of the Stokes parameters at two points in the polarization speckle are provided. With the aid of the numerical evaluations and the figures, the effects of the correlation coefficient on the joint density functions are investigated. These results can be regarded as a development and extension of previous works focusing on multidimensional statistics of conventional laser speckle, which is stochastic scalar field, and will provide a deeper insight into the spatial or temporal changes of polarization states for the random electric fields.
Funding
Scottish Universities Physics Alliance (SSG040).
Acknowledgment
The author thanks Mr. Yongqi Zhang for his help in preparing some of the figures in this work.
Disclosures
The author declares no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
REFERENCES
1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer, 1975).
2. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley-Blackwell, 2015).
3. J. W. Goodman, Speckle Phenomena in Optics, 2nd ed. (SPIE, 2020).
4. F. Fercher and P. F. Steeger, “First-order statistics of stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981). [CrossRef]
5. P. F. Steeger, “Probability density function of the intensity in partially polarized speckle fields,” Opt. Lett. 8, 528–530 (1983). [CrossRef]
6. P. F. Steeger, T. Asakura, K. Zocha, and A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984). [CrossRef]
7. R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985). [CrossRef]
8. R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987). [CrossRef]
9. I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B 42, 2613–2616 (1990). [CrossRef]
10. S. M. Cohen, D. Eliyahu, I. Freund, and M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991). [CrossRef]
11. D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993). [CrossRef]
12. D. Eliyahu, “Statistics of Stokes variables for correlated Gaussian fields,” Phys. Rev. E 50, 2381–2384 (1994). [CrossRef]
13. J. Dupont, X. Zerrad, G. Soriano, S. Liukaityte, and C. Amra, “Polarization analysis of speckle field below its transverse correlation width: application to surface and bulk scattering,” Opt. Express 22, 24133–24141 (2014). [CrossRef]
14. M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015). [CrossRef]
15. J. Dupont and X. Orlik, “Simulation of polarized optical speckle fields: effects of the observation scale on polarimetry,” Opt. Express 24, 11151–11163 (2016). [CrossRef]
16. N. Ma, S. G. Hanson, M. Takeda, and W. Wang, “Coherence and polarization of polarization speckle generated by a rough-surfaced retardation plate depolarizer,” J. Opt. Soc. Am. A 32, 2346–2352 (2015). [CrossRef]
17. J. Ritter, N. Ma, W. Osten, M. Takeda, and W. Wang, “Depolarizing surface scattering by a birefringent material with rough surface,” Opt. Commun. 430, 456–460 (2019). [CrossRef]
18. W. Wang, S. Zhang, and J. Grimble, “Statistics of polarization speckle produced by a constant polarization phasor plus a random polarization phasor sum,” J. Opt. Soc. Am. A 37, 1888–1894 (2020). [CrossRef]
19. Y. Wang, D. C. Louie, J. Cai, L. Tchvialeva, H. Lui, Z. J. Wang, and T. K. Lee, “Deep learning enhances polarization speckle for in vivo skin cancer detection,” Opt. Laser Technol. 140, 107006 (2021). [CrossRef]
20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
21. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
22. E. L. O’Neill, Introduction to Statistical Optics (Dover, 2004).
23. D. Middleton, An Introduction to Statistical Communication Theory (Wiley-IEEE, 1996).
24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
25. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).
26. J. W. Goodman, “Role of coherence concepts in the study of speckle,” Proc. SPIE 194, 86–94 (1979). [CrossRef]
Open Access
Add to BibSonomy
