1. INTRODUCTION

In a variety of problems [1–5], including the study of polarization-sensitive photon-counting statistics, finite-time integrals of instantaneous Stokes parameters occur. In addition, an entirely analogous problem arises in considering the statistical properties of a finite spatial average of polarization speckle. Here, all the measured quantities are always a somewhat smoothed or integrated version in any polarization-related experiments, and the statistical properties of the measured Stokes parameters will be somewhat different than the ideal statistics developed for partially polarized thermal light or polarization speckle [5–11]. Noting the fact that the Stokes parameters cannot be measured at an instant in time and/or an ideal point in space, we have studied the first-order statistics of integrated Stokes parameters and derived approximate solutions of the probability density functions [12]. Similar to the previous study on integrated intensity [13–15], the obtained solutions about integrated Stokes parameters preserved the exact values of the means and variances, but nonetheless involved significant approximations, particularly through the use of “boxcar” approximations to the instantaneous Stokes parameters, and the corresponding higher-order moments will in general not be correct. Therefore, the goal of this paper is to derive “exact” solutions for the probability density functions of integrated Stokes parameters with more accuracy than those we have discussed before. Here the quotation mark around the word “exact” is due to the fact that these solutions ultimately involve numerical approximation, as we will see later.

The study on the “exact” solutions for the probability density functions of the integrated Stokes parameters can be regarded as a generalization and extension of the “exact” solution for the integrated intensity by taking the vector nature of light into account. As for the “exact” solution for the density function of the integrated intensity for fully polarized thermal light and/or laser speckle with a uniform state of polarization, Condie [16] first applied the Karhunen–Loève expansion [17] to calculate the cumulants of the exact distribution and compared them with the cumulants from the approximation approach. Dainty [18] and Barakat [19] applied the same method to find general expressions for the density function of integrated intensity. Barakat [19] and Scribot [20] plotted an “exact” density function of integrated intensity for a slit aperture and a Gaussian correlation function and a sinc correlation function of illumination, respectively. Goodman extended the study of integrated intensity for partially polarized speckle pattern [21]. More recently, the “exact” results for the density function of time-integrated intensity have been developed for imaged speckle patterns in biomedical optics [22].

The purpose of this paper is to derive the “exact” solutions of the probability density functions of the integrated Stokes parameters for partially polarized thermal light or polarization speckle. To facilitate reading by comparison with density functions of the appropriate results derived before [12], we will follow the original descriptions and structure of logic of Goodman’s work because most readers may be familiar with the statistics of integrated intensity of thermal light or integrated speckle well developed in these classic books [4,5,21]. After applying the unitary linear transformation and the Karhunen–Loève expansion to the stochastic electric field, we decompose the light of interest into an infinite number of statistically independent modes and express the integrated Stokes parameters as the sums of infinite numbers of random variables known as a polarization-related mode shape. The exact part of the solutions for the probability density functions of the integrated Stokes parameters have been found as the desired first-order statistics of integrated random phenomena in optics. After introducing some numerical approximations to the exact solutions, we calculate the cumulants of the exact density functions for the integrated Stokes parameters and compare them to the corresponding cumulants obtained from the approximation method. All these results will provide deeper insight into the temporal and/or spatial integrations inherently associated with polarization-sensitive detection of light.

2. UNITARY LINEAR TRANSFORMATION AND THE KARHUNEN–LOÈVE EXPANSION OF STOCHASTIC ELECTRIC FIELD

Before starting our calculation of more accurate probability density functions of integrated Stokes parameters, we first apply the Karhunen–Loève expansion in the context of partially polarized thermal light to obtain the great benefit of uncorrelated expansion coefficients for a stochastic electric field of partially polarized thermal light. Similar results can also be obtained for a stationary polarization speckle. Here, the stochastic electric field in question is a Gaussian random process because the number of independent contributions is usually enormous.

Although a mathematical formalism exists that a sample vector of the stochastic electric field can be expanded in a set of orthonormal vectors [23], the expansion of a sample vector for a stochastic electric field in an infinite number of vectorial bases will make our calculations of the integrated Stokes parameters extremely complicated. On the other hand, the two polarization components of a stochastic electric field are, in general, correlated and its mutual coherence matrix has nonzero off-diagonal elements. Note that the mutual coherence matrix is a Hermitian matrix and a unitary linear transformation exists for its matrix diagonalization [24]. Therefore, a stochastic electric field with two correlated polarization components can always be transformed to an equivalent field with uncorrelated orthogonal components. For the sake of mathematical simplicity and without a loss of generality, the first step in our analysis is to find such a unitary linear transformation that diagonalizes the Hermitian correlation matrix ${\boldsymbol \Gamma}({t_2},{t_1})$, creating two new polarization components that are uncorrelated and, by virtue of the underlying Gaussian statistics, statistically independent [4,5].

Let the set of complex-valued functions $\{{\psi _1}(t),{\psi _2}(t), \cdots ,\def\LDeqbreak{}{\psi _n}(t), \cdots \}$ be orthonormal and complete on the interval $({{- T} / 2},{T / 2})$. Then, the newly transformed equivalent field $\textbf{E}(t)$, which is a well-behaved sample vector for a stochastic electric field, can be expanded on this interval as

These two sets of expansion coefficients $\{b_n^x\}$ and $\{b_n^y\}$ are uncorrelated to each other because of the statistical independence for ${E_x}$ and ${E_y}$ themselves. There are many possible orthogonal expansions that might satisfy the conditions above. However, we wish to choose ones that will be uncorrelated within each set of expansion coefficients, provided the functions ${\psi _n}$ are the solutions of the integral equations

Similarly, the Karhunen–Loève expansion can also be applied to the polarization speckle field across the detector aperture $\Sigma$; that is,

To achieve uncorrelated coefficients, the constraint conditions become

From the two operations introduced above (i.e., the unitary linear transformation followed by the Karhunen–Loève expansion), we can obtain all these expansion coefficients, which are uncorrelated Gaussian random variables, for a stochastic electric field from partially polarized thermal light or polarization speckle.

In the previous investigation [12], we expressed the temporally integrated Stokes vector as

When Eqs. (10) and (11) are written, we have made use of an assumption that the light in question is an ergodic (and therefore, temporally and/or spatially stationary) random process so that the statistics of the integrated Stokes parameters do not depend on the particular observation time and/or observation location.

Substituting Eq. (1) into Eq. (10) or Eq. (7) into Eq. (11), and using the orthonormal properties of the function ${\psi _n}$, we obtain

When Eq. (13) is written, we have made use of the condition in Eq. (6). Thus, with the aid of the unitary linear transformation and the Karhunen–Loève expansion, we have succeeded in decomposing the stochastic electric wave for partially polarized thermal light or polarization speckle into an infinite number of statistically independent modes. Each mode has its individual polarization-related mode shape described by a vector $[{I_n},{Q_n},{U_n},{V_n}]$ with four components of real-valued random variables defined in the similar way as the Stokes parameters but in terms of the complex-valued expansion coefficients: $b_n^x$ and $b_n^y$. Accordingly, each integrated Stokes parameter $S_l^{{\rm Int}}$ for $l = 0 {-}3$ has been expressed as the sum of an infinite number of the corresponding components, respectively.

3. EXACT FORMS OF THE PROBABILITY DENSITY FUNCTIONS OF INTEGRATED STOKES PARAMETERS

In the last section, we applied the Karhunen–Loève expansion to the stochastic electric field and expressed each integrated Stokes parameter $S_l^{\rm{Int}}$ as the sum of an infinite number of the corresponding random variables that are statistically independent with identical statistics. All these random variables share the identical characteristic function, assuming the eigenvalues/mode intensities ${\lambda _n} = \lambda _n^x + \lambda _n^y$ are known.

A. Exact Solution of the Density Function of $S_0^{{\rm Int}}$

In the case of the integrated Stokes parameter $S_0^{\rm{Int}}$ for partially polarized thermal light or polarization speckle with the degree of (ensemble-average) polarization ${\cal P}$, the probability density function of ${I_n}$ must be [4,5]

Note from the property that the characteristic function of the sum of independent random variables is the product of the characteristic functions for component random variables [4]. The characteristic function of $S_0^{\rm{Int}}$ must be of the form

If ${\lambda _n}$ are distinct, the inversion of the characteristic function in Eq. (16) yields an exact probability density function for $S_0^{\rm{Int}}$ of the form

When Eq. (17) is obtained, we apply Jordan’s lemma in conjunction with the residue theorem to evaluate the involved integral over the real axis through the sum of the residues [25]. A detailed proof of Eq. (17) can be found in Appendix A. Note that since ${p_{S_0^{\rm{Int}}}}(S_0^{\rm{Int}})$ is a probability density function, its area must be unity, from which it follows that $\sum\nolimits_{n = 0}^\infty {{{(2{\cal P})}^{- 1}}[c_n^{{S_0}}(1 + {\cal P}) - d_n^{{S_0}}(1 - {\cal P})]} = 1$.

Based on the exact density function above, the mean and variance of $S_0^{{\rm Int}}$ are, therefore,

We turn now to finding the exact probability density function of the integrated Stokes parameter $S_0^{{\rm Int}}$ for two limiting cases. When ${\cal P} = 1$ for polarized thermal light or uniform polarization speckle, the exact probability density function of $S_0^{{\rm Int}}$ can be found through a proper limiting argument; that is,

Because $p_{S_0^{\text {Int }}}\!\left(S_0^{\text {Int }} \left| \mathcal{P}\right.=1\right)$ is a probability density function with its area being unity, it follows that $\sum\nolimits_{n = 0}^\infty {\underline c _n^{{S_0}}} = 1$. As it should be, Eqs. (20) and (21) share the same expressions as those derived for the integrated intensity of polarized thermal light or fully polarized speckle [4,5,14,15,21].

As for the opposite case when ${\cal P} = 0$, we cannot find the desired expression through limiting the arguments of Eqs. (17) and (18). Note that the random variable ${I_n}$ must obey the statistics of instantaneous intensity for unpolarized thermal light or isotropic polarization speckle [4,11]. We have

The corresponding characteristic function of ${I_n}$ for ${\cal P} = 0$ is the Fourier transform of ${p_{{I_n}}}({I_n})$; that is,

Because the integrated Stokes parameter $S_0^{{\rm Int}}$ has been expressed as the sum of an infinite number of statistically independent random variables ${I_n}$, the characteristic function of $S_0^{{\rm Int}}$ is accordingly given by

Then, the probability density function of the integrated Stokes parameter $S_0^{{\rm Int}}$ for unpolarized thermal light or isotropic polarization speckle with ${\cal P} = 0$ is

When Eq. (25) is derived, we have also made use of Jordan’s lemma and the residue theorem to evaluate the involved integral in the inverse Fourier transform of Eq. (24). Since $p_{S_0^{\text {Int }}}\!\left(S_0^{\text{Int} } \left| \mathcal{P}\right.=0\right)$ is a probability density function with its area being unity, we have $\sum\nolimits_{n = 0}^\infty {\underline{\underline c} _{\,n}^{{S_0}}} = 1$.

B. Exact Solution of the Density Function of $S_1^{{\rm Int}}$

To find the exact solution for the integrated Stokes parameter $S_1^{{\rm Int}}$, we start with the probability density function of the random variable ${Q_n}$ taking the form [6,12,26]

Therefore, the characteristic function of $S_1^{{\rm Int}}$, which is the sum of an infinite number of independent random variables ${Q_n}$ with their corresponding mode intensities ${I_n} = {\lambda _n}$ and their distributions given in Eq. (28), must be the form

Taking the inverse Fourier transform of the characteristic function in Eq. (29) and making use of Jordan’s lemma and the residue theorem [25], we have the exact solution for the probability density function of the integrated Stokes parameter $S_1^{{\rm Int}}$; that is,

Note that since ${p_{S_1^{\rm{Int}}}}(S_1^{\rm{Int}})$ is also a probability density function, its area must be unity, from which it follows that $\sum\nolimits_{n = 0}^\infty {{2^{- 1}}[c_n^{{S_1}}(1 + {\cal P}) + d_n^{{S_1}}(1 - {\cal P})]} = 1$.

From the density function above, the mean and variance of $S_1^{{\rm Int}}$ can be readily shown to be

For polarized thermal light or uniform polarization speckle with ${\cal P} = 1$, the exact density function of $S_1^{{\rm Int}}$ can be found through a proper limiting argument of Eqs. (30) and (31); that is,

On the other hand, the exact density function of $S_1^{{\rm Int}}$ for unpolarized thermal light or isotropic polarization speckle with ${\cal P} = 0$ reduces to

For the same reason that $p_{S_1^{\text {Int }}}\!\left(S_1^{\text {Int }} \left| \mathcal{P}\right.=0\right)$ is a probability density function with its area being unity, we have $\sum\nolimits_{n = 0}^\infty {\underline{\underline c} _{\,n}^{{S_1}}} = 1$.

C. Exact Solutions of the Density Functions of $S_2^{{\rm Int}}$ and $S_3^{{\rm Int}}$

For the equivalent stochastic electric field obtained by diagonalization of the correlation matrix, the probability density functions of ${S_2}$ and ${S_3}$ have an identical expression [6,12,26]. Then the random variables ${U_n}$ and ${V_n}$ should share the same distribution, written as

Since ${F_n}$ for $F = U,V$ are independent random variables, and the integrated Stokes parameters $S_2^{{\rm Int}}$ or $S_3^{{\rm Int}}$ are the sums of an infinite number of the corresponding independent random variables ${U_n}$ or ${V_n}$ with their corresponding mode intensities ${I_n} = {\lambda _n}$ and their corresponding distributions given in Eq. (38), therefore, the characteristic function of $S_k^{{\rm Int}}$ for $k = 2,3$ is accordingly given by

In a similar way, the inversion Fourier transformation of the characteristic function above yields the exact probability density function for $S_k^{{\rm Int}}$, taking the form

Because ${p_{S_k^{\rm{Int}}}}(S_k^{\rm{Int}})$ is a probability density function with the unity area, we have $\sum\nolimits_{n = 0}^\infty {c_n^{{S_k}}} = 1$. Meanwhile, the mean and variance of $S_k^{{\rm Int}}$ become

Here, the zero means of the integrated Stokes parameters $S_k^{{\rm Int}}$ for $k = 2,3$ can be considered as the direct consequence of the symmetric distributions of ${S_k}$ after the unitary linear transformation has been applied for diagonalizing the correlation matrix.

For polarized thermal light or uniform polarization speckle with ${\cal P} = 1$, the exact solution of the probability density function of the integrated Stokes parameter $S_k^{{\rm Int}}$ approaches the Dirac delta function through a proper limiting argument; that is, $\mathop {\lim}\limits_{{\cal P} \to 1} {p_{S_k^{{\rm Int}}}}(S_k^{{\rm Int}}) = \delta (S_k^{{\rm Int}})$ for $k = 2,3$. On the other hand, when ${\cal P} = 0$ for unpolarized thermal light or isotropic polarization speckle, the expression of Eq. (40) reduces to

As expected for unpolarized thermal light or isotropic polarization speckle, the exact solutions for the probability density functions of the Stokes parameters $S_1^{{\rm Int}},S_2^{{\rm Int}}$, and $S_3^{{\rm Int}}$ share the identical expressions shown in Eqs. (35) and (43) since the joint probability density $p({S_1},{S_2},{S_3})$ has a typical shape of spherical symmetry when ${\cal P} = 0$ [11].

So far, all the exact forms of the solutions for the probability density functions of integrated Stokes parameters have been obtained and Eqs. (17), (30), and (40) can be understood as one of the main results of this paper. Further progress requires solutions of the integral equations for a specific correlation matrix, detection interval, or detector aperture of interest, so that the eigenvalues $\lambda _n^x$ and $\lambda _n^y$ can be specified.

4. APPROXIMATIONS TO THE EXACT SOLUTIONS

As shown in the last section, the derivations of the probability density functions for the integrated Stokes parameters to this point have been exact. While finding the eigenvalues always involves some level of approximation, including numerical calculation of the eigenvalues and finite terms of selection from an infinite number of series. Nonetheless, the density functions derived by this method will usually be more accurate than the approximate density functions derived earlier [12]. As pointed out by Goodman [4,5], the higher-order moments derived from the approximate approaches will in general not be correct, although all these approximate density functions have been chosen to have the true means and variances. Here, we are about to present some results derived from the exact density functions with more accurate high-order moments.

To make further progress in specifying ${p_{S_l^{{\rm Int}}}}(S_l^{{\rm Int}})$ with $l = 0 {-} 3$ for partially polarized thermal light, it is necessary to find the eigenvalues $\lambda _n^x$ and $\lambda _n^y$, which in turn requires that we assume correlation functions ${\Gamma _{\textit{xx}}}$ and ${\Gamma _{\textit{yy}}}$ in the diagonal correlation matrix, or equivalently power spectral densities ${{\cal G}_{\textit{xx}}}$ and ${{\cal G}_{\textit{yy}}}$ in the diagonal matrix of the power spectral density for partially polarized thermal light with statistically independent polarization components. The most widely studied case is that of the rectangular power spectral densities

Accordingly, the orthonormal functions ${\psi _n}$ and the constants $\lambda _n^x$ and $\lambda _n^y$ must be eigenfunctions and eigenvalues of the integral equations

A similar analysis can also be applied to polarization speckle when a one-dimensional (1D) spot shape with a narrow strip of light for illumination on the depolarizing diffuser [27,28] and a 1D slit aperture for a detector have been chosen.

After some changes of variables, the 1D eigen-equation can be rewritten in a unified way as

The solutions of the integral equation above have been widely studied in the literature with the real-valued eigenfunctions known as prolate spheroidal wave functions and the eigenvalues available in both graphical and tabular forms [29–31]. Both $\tilde \lambda _n^q$ for $q = x,y$ and ${\psi _n}$ depend not only on $n$ and ${\cal P}$ but also on the parameter $c$ given by

Figure 1 shows a plot of the values of the first five pairs of eigenvalues $\tilde \lambda _n^x$ and $\tilde \lambda _n^y$ obtained with a $4000 \times 4000$ matrix, where ${\cal P}$ is chosen to be 0.5.

 

Fig. 1. Plots of the first five pairs of eigenvalues obtained from a discretized version of Eq. (46) as a function of $c$ when ${\cal P} = 0.5$.

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5. COMPARISON OF THE “EXACT” AND APPROXIMATE SOLUTIONS

Once the eigenvalues (i.e., ${\lambda _n} = \lambda _n^x + \lambda _n^y$) are in hand, it is then possible to plot the “exact” solutions for the probability density functions of the integrated Stokes parameters derived before and to compare them with the corresponding distributions obtained in the earlier approximate analysis [12] without recourse to these eigenvalues.

Thus, we plot together the density functions of the integrated Stokes parameters defined by

The comparison between the “exact” and approximate probability density functions for the integrated Stokes parameters are shown in Figs. 2–4. The results for ${p_{S_0^{{\rm Int}}}}(S_0^{{\rm Int}})$ at ${\cal P} = 0.95$ in Fig. 2(c) look similar to those for the integrated intensity of fully polarized light with ${\cal P} = 1$ found by Scribot [20]. We can see that when $c$ is small or ${{\cal M}_{{S_0}}} \to 0$, both “exact” and approximate distributions resemble the familiar distributions of the instantaneous Stokes parameters [6,26]; however, when $c$ is very large or ${{\cal M}_{{S_0}}} \to \infty$, both approaches provide their distributions with the common limits of the delta functions. For large but finite values of $c$ or ${{\cal M}_{{S_0}}}$, all these probability density functions derived by both methods approach Gaussian distributions, centered on $\overline {S_l^{{\rm Int}}}$, which is a consequence of the central limit theorem.

 

Fig. 2. Approximate (dashed lines) and “exact” (solid lines) probability density functions for integrated Stokes parameter $S_0^{{\rm Int}}$ for various values of $c$ and different degrees of polarization: (a) ${\cal P} = 0$; (b) ${\cal P} = 0.5$; and (c) ${\cal P} = 0.95$.

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Fig. 3. Approximate (dashed lines) and “exact” (solid lines) probability density functions for integrated Stokes parameter $S_1^{{\rm Int}}$ for various values of $c$ and different degrees of polarization: (a) ${\cal P} = 0$; (b) ${\cal P} = 0.5$; and (c) ${\cal P} = 0.95$.

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Fig. 4. Approximate (dashed lines) and “exact” (solid lines) probability density functions for integrated Stokes parameter $S_k^{{\rm Int}}$ with $k = 2,3$ for various values of $c$ and different degrees of polarization: (a) ${\cal P} = 0$; (b) ${\cal P} = 0.5$; and (c) ${\cal P} = 0.95$.

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Through the comparison between the “exact” and approximate results, several significant differences have been spotted due to the different methods for choosing ${{\cal M}_{{S_0}}}$ and $\overline {S_0^{{\rm Int}}}$. The first obvious distinction occurs when the range of $c$ is near unity. Meanwhile, we can also observe significant differences in the density functions of the integrated Stokes parameter $S_1^{{\rm Int}}$ when ${\cal P} = 0.5$. It should be pointed out that the finite number of eigenvalues to retain also gives rise to the accuracy of the degree of polarization and the subsequent evaluation for the coefficients $c_n^{{S_l}}$ and $d_n^{{S_l}}$ since the exact solution of ${\cal P}$ can be expressed in terms of $c_n^{{S_l}}$ and $d_n^{{S_l}}$ from Eqs. (19), (32), and (42) based on its definition ${\cal P} = {{\sqrt {{{\overline {S_1^{\rm{Int}}}}^2} + {{\overline {S_2^{\rm{Int}}}}^2} + {{\overline {S_3^{\rm{Int}}}}^2}}} / {\overline {S_0^{\rm{Int}}}}}$. During the preparation of Figs. 2–4, ${\cal P}$ has been chosen as a parameter for plotting the approximated and “exact” probability density functions. Noting the fact that the density functions for the integrated Stokes parameters derived using two approaches look similar, as shown in Figs. 2–4, and they have the same means and variances, the use of the approximate density functions can be justified in most problems.

Before closing this section, we can provide some insight into the relationship between the approximate and the “exact” solutions. Similar to the conclusions drawn by Condie [16], Scribot [20], and Barakat [19] for integrated intensity, the approximate density functions of integrated Stokes parameters may be regarded as the results of assuming that for integer ${{\cal M}_{{S_0}}}$, all eigenvalues with index $n$ smaller than ${{\cal M}_{{S_0}}}$ are taken to be unity, and all those with index $n$ equal to or larger than ${{\cal M}_{{S_0}}}$ are taken to be zero. On the other hand, the “exact” solutions of the density functions for integrated Stokes parameters use calculated, distinct eigenvalues ${\lambda _n} = \lambda _n^x + \lambda _n^y$ to obtain the weightings $c_n^{{S_l}}$ and $d_n^{{S_l}}$, arriving at solutions that are sums of large (ideally infinite) number of negative exponential components and/or exponential components. For example, as Fig. 1 illustrates for the case of ${{\cal M}_{{S_0}}} = 3$ or $c = 3.88$ (indicated by the dash dotted line), the approximate solution replaces the true values of the eigenvalues $(\tilde \lambda _0^x + \tilde \lambda _0^y,\tilde \lambda _1^x + \tilde \lambda _1^y,\tilde \lambda _2^x + \tilde \lambda _2^y)$ by unity, and all higher-index eigenvalues by zero, whereas their true values are shown in the figure.

6. CONCLUSION

Because Stokes parameters cannot be measured at an instant in time or at an ideal point, the measured quantities are always a somewhat smoothed or integrated version in any polarization-related experiments. Note from our previous investigation that the approximation forms of the probability density functions of integrated Stokes parameters involve significant approximations and have less accuracy in their higher-order moments. In this paper, we derive “exact” solutions for the probability density functions of integrated Stokes parameters, which are based on the unitary linear transformation and the Karhunen–Loève expansion of the stochastic electric field. After the introduction of a polarization-related mode shape in terms of a set of the Stokes-parameters-like random variables, we derive the exact part of the solutions for the probability density functions of the integrated Stokes parameters and provide their corresponding means and variances. Some numerical approximations to the exact solutions and comparison to the approximation solutions have been performed for these integrated quantities of physical interests. Together with our previous investigation of the approximate forms of the density functions, all these results constitute a complete analysis of the statistical properties of the integrated Stokes parameters in partially polarized thermal light or polarization speckle, and therefore provide a deeper insight into the temporal or spatial integration associated intrinsically with polarization-sensitive light detection. Furthermore, the proposed polarization-related mode shape enables a complete specification of state of polarization for each mode when a stochastic electric field is decomposed and introduces new opportunities to explore other random polarization phenomena in statistical optics.