## Abstract

First-order statistics of the speckle phase are studied on an axis of the imaging system under the assumption of Gaussian statistics for the formation of the speckle field. It is found that the speckle phase is most widely distributed at the image plane owing to the noncircularity of the speckle field. Asymmetry of the probability-density distribution of the speckle phase also appears to be due to the inclination of the joint probability density of the complex speckle amplitude. These statistical characteristics of the speckle phase are enhanced for relatively small values of the optical roughness of the object.

© 1985 Optical Society of America

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### Equations (20)

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(1)
$${P}_{r,i}({A}_{r},{A}_{i})=\frac{1}{2\pi {\sigma}_{r}{\sigma}_{i}{(1-{\rho}^{2})}^{1/2}}\times exp\left[-\frac{1}{2(1-{\rho}^{2})}\left(\frac{\mathrm{\Delta}{{A}_{r}}^{2}}{{{\sigma}_{r}}^{2}}-\frac{2\rho \mathrm{\Delta}{A}_{r}\mathrm{\Delta}{A}_{i}}{{\sigma}_{r}{\sigma}_{i}}+\frac{\mathrm{\Delta}{{A}_{i}}^{2}}{{{\sigma}_{i}}^{2}}\right)\right],$$
(2)
$$\begin{array}{ll}\mathrm{\Delta}{A}_{r}={A}_{r}-\u3008{A}_{r}\u3009,\hfill & \mathrm{\Delta}{A}_{i}={A}_{i}-\u3008{A}_{i}\u3009,\hfill \end{array}$$
(3)
$$\delta =\xbd{tan}^{-1}[2\rho {\sigma}_{r}{\sigma}_{i}/({{\sigma}_{r}}^{2}-{{\sigma}_{i}}^{2})].$$
(4)
$$\u3008{A}_{r}\u3009=exp(-{{\sigma}_{\varphi}}^{2}/2),$$
(5)
$$\u3008{A}_{i}\u3009=0,$$
(6)
$${{\sigma}_{i}}^{2}=\frac{1}{2N}[T+\text{sinc}(u)S]exp(-{{\sigma}_{\varphi}}^{2}),$$
(7)
$${{\sigma}_{i}}^{2}=\frac{1}{2N}[T-\text{sinc}(u)S]exp(-{{\sigma}_{\varphi}}^{2}),$$
(8)
$$\rho =-\frac{\text{sinc}(u/2)sin(u/2)S}{{[{T}^{2}-{\text{sinc}}^{2}(u){S}^{2}]}^{1/2}},$$
(9)
$$\begin{array}{ll}T=\text{\u2211}_{n=1}^{\infty}\frac{{({{\sigma}_{\varphi}}^{2})}^{n}}{n!n},\hfill & S=\text{\u2211}_{n=1}^{\infty}\frac{{(-{{\sigma}_{\varphi}}^{2})}^{n}}{n!n},\hfill \\ N={(2f/kw\xi )}^{2},\hfill & u=z/({f}^{2}/k{w}^{2}),\hfill \end{array}$$
(10)
$$\begin{array}{c}\delta =\xbd{tan}^{-1}[tan(-u/2)]=-\xbc(u-2n\pi ),\\ \begin{array}{cc}|u-2n\pi |>\pi ,& n=0,\pm 1,\pm 2,\dots .\end{array}\end{array}$$
(12)
$$C={\sigma}_{y}/{\sigma}_{x}.$$
(13)
$${P}_{\theta}(\theta )=\frac{\nu}{2\pi \tau}\{1+\sqrt{\pi}\zeta exp({\zeta}^{2})[1+\text{erf}(\zeta )]\}exp(-\gamma /2{{\sigma}_{x}}^{2}),$$
(14)
$$\begin{array}{ll}\mu \hfill & =\theta -\delta ,\hfill \\ \nu \hfill & ={\sigma}_{x}/{\sigma}_{y},\hfill \\ \tau \hfill & ={cos}^{2}\mu +{\nu}^{2}{sin}^{2}\mu ,\hfill \\ \beta \hfill & =\u3008{A}_{x}\u3009cos\mu +{\nu}^{2}\u3008{A}_{y}\u3009sin\mu ,\hfill \\ \gamma \hfill & ={\u3008{A}_{x}\u3009}^{2}+{\nu}^{2}{\u3008{A}_{y}\u3009}^{2},\hfill \\ \zeta \hfill & =\beta /\sqrt{2\tau}{\sigma}_{x},\hfill \end{array}$$
(15)
$${(1-{\rho}^{2})}^{-1}\left(\frac{\mathrm{\Delta}{{A}_{r}}^{2}}{{{\sigma}_{r}}^{2}}-\frac{2\rho \mathrm{\Delta}{A}_{r}\mathrm{\Delta}{A}_{i}}{{\sigma}_{r}{\sigma}_{i}}+\frac{\mathrm{\Delta}{{A}_{i}}^{2}}{{{\sigma}_{i}}^{2}}\right)=1.$$
(16)
$$\text{d}\mathrm{\Delta}{A}_{r}/\text{d}\mathrm{\Delta}{A}_{i}=0.$$
(17)
$$\frac{\text{d}\mathrm{\Delta}{A}_{r}}{\text{d}\mathrm{\Delta}{A}_{i}}=\left(\frac{2\rho \mathrm{\Delta}{A}_{r}}{{\sigma}_{r}{\sigma}_{i}}-\frac{2\mathrm{\Delta}{A}_{i}}{{{\sigma}_{i}}^{2}}\right)/\left(\frac{2\mathrm{\Delta}{A}_{r}}{{{\sigma}_{r}}^{2}}-\frac{2\rho \mathrm{\Delta}{A}_{i}}{{\sigma}_{r}{\sigma}_{i}}\right)=0.$$
(18)
$$\mathrm{\Delta}{A}_{i}=\rho {\sigma}_{i}\mathrm{\Delta}{A}_{r}/{\sigma}_{r}.$$
(19)
$$\mathrm{\Delta}{{A}_{r}}^{2}={{\sigma}_{r}}^{2}.$$
(20)
$$\mathrm{\Delta}{{A}_{i}}^{2}={{\sigma}_{i}}^{2}.$$