## Abstract

The characteristics of a fully developed speckle pattern that results from in-plane rotation of a diffuse object observed in an arbitrary observation plane, as described by the spatiotemporal cross-correlation function of the optical intensity in complex *ABCD* optical systems, are derived and discussed. Here we consider off-axis illumination, which is in contrast to previous work in which the illuminating beam was assumed to be parallel to the axis of rotation. The spatiotemporal characteristics of the observed pattern are interpreted in terms of speckle boiling, rotation, and translation. For off-axis illumination it is shown theoretically and experimentally that, for Fourier transform optical systems, in-plane rotation causes the speckles to translate in a direction perpendicular to the direction of surface motion, whereas for an imaging system, the translation is parallel to the direction of surface motion. On this basis, we discuss a novel method, which is independent of both the optical wavelength and the position of the laser spot on the object, for determining either the angular velocity or the corresponding in-plane displacement of the target object.

© 1998 Optical Society of America

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### Equations (26)

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(1)
$${C}_{I}({\mathbf{p}}_{1},{\mathbf{p}}_{2};t,t+\tau )=\frac{\u3008I({\mathbf{p}}_{1},t)I({\mathbf{p}}_{2},t+\tau )\u3009-\u3008I({\mathbf{p}}_{1},t)\u3009\u3008I({\mathbf{p}}_{2},t+\tau )\u3009}{\{[\u3008I({\mathbf{p}}_{1},t{)}^{2}\u3009-\u3008I({\mathbf{p}}_{1},t){\u3009}^{2}][\u3008I({\mathbf{p}}_{2},t+\tau {)}^{2}\u3009-\u3008I({\mathbf{p}}_{2},t+\tau ){\u3009}^{2}]{\}}^{1/2}},$$
(2)
$$\u3008I({\mathbf{p}}_{1},t)I({\mathbf{p}}_{2},t+\tau )\u3009$$
(3)
$$=\u3008U({\mathbf{p}}_{1},t){U}^{*}({\mathbf{p}}_{1},t)U({\mathbf{p}}_{2},t+\tau ){U}^{*}({\mathbf{p}}_{2},t+\tau )\u3009=|\mathrm{\Gamma}({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau ){|}^{2}+\u3008I({\mathbf{p}}_{1},t)\u3009\u3008I({\mathbf{p}}_{2},t+\tau )\u3009,$$
(4)
$$\mathrm{\Gamma}({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau )=\u3008U({\mathbf{p}}_{1},t){U}^{*}({\mathbf{p}}_{2},t+\tau )\u3009.$$
(5)
$${C}_{I}({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau )=\frac{|\mathrm{\Gamma}({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau ){|}^{2}}{\mathrm{\Gamma}({\mathbf{p}}_{1},{\mathbf{p}}_{1};0)\mathrm{\Gamma}({\mathbf{p}}_{2},{\mathbf{p}}_{2};0)}=|\gamma ({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau ){|}^{2},$$
(6)
$$U(\mathbf{p},t)=\int \mathrm{d}\mathbf{r}{U}_{r}(\mathbf{r},t)G(\mathbf{r},\mathbf{p}),$$
(7)
$$G(\mathbf{r},\mathbf{p})=-\frac{\mathit{ik}}{2\pi B}exp\left[-\frac{\mathit{ik}}{2B}(A{\mathbf{r}}^{2}-2\mathbf{r}\xb7\mathbf{p}+D{\mathbf{p}}^{2})\right],$$
(8)
$$\mathrm{\Gamma}({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau )$$
(9)
$$=\iint \mathrm{d}{\mathbf{r}}_{1}\mathrm{d}{\mathbf{r}}_{2}{\mathrm{\Gamma}}_{0}({\mathbf{r}}_{1},{\mathbf{r}}_{2};\tau )G({\mathbf{r}}_{1},{\mathbf{p}}_{1}){G}^{*}({\mathbf{r}}_{2},{\mathbf{p}}_{2}),$$
(10)
$${\mathrm{\Gamma}}_{0}({\mathbf{r}}_{1},{\mathbf{r}}_{2};\tau )=\u3008{U}_{r}({r}_{1},t){U}_{r}^{*}({r}_{2},t+\tau )\u3009={U}_{i}({\mathbf{r}}_{1}){U}_{i}^{*}({\mathbf{r}}_{2})\delta ({\mathbf{r}}_{2}-{\mathbf{r}}_{1}^{\prime}),$$
(11)
$$\mathrm{\Gamma}({\mathbf{p}}_{1},{\mathbf{p}}_{2};\tau )=\int \mathrm{d}\mathbf{r}{U}_{i}(\mathbf{r}){U}_{i}^{*}({\mathbf{r}}^{\prime})\times G(\mathbf{r},{\mathbf{p}}_{1}){G}^{*}({\mathbf{r}}^{\prime},{\mathbf{p}}_{2}).$$
(12)
$${U}_{i}(\mathbf{r})={U}_{0}exp\left[\mathit{ikx}sin\varphi -\left(\frac{{x}^{2}}{r_{s}{}^{2}(sec\varphi {)}^{2}}+\frac{{y}^{2}}{r_{s}{}^{2}}\right)\right],$$
(13)
$$\left(\begin{array}{c}{X}^{\prime}\\ {Y}^{\prime}\end{array}\right)=\left[\begin{array}{cc}cos\omega \tau & -sin\omega \tau \\ sin\omega \tau & cos\omega \tau \end{array}\right]\left(\begin{array}{c}X\\ Y\end{array}\right),$$
(14)
$${\mathbf{r}}^{\prime}=\left(\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}\end{array}\right)=\left(\begin{array}{c}xcos\omega \tau -(y+a)sin\omega \tau \\ xsin\omega \tau +(y+a)cos\omega \tau -a\end{array}\right).$$
(15)
$${C}_{I}(\mathbf{P},\mathbf{p};\tau )=exp\left[-\frac{{p}^{2}}{\rho _{0}{}^{2}}\left(1-\frac{1-cos\omega \tau}{2}\right)\right]\times exp\left[-\frac{2sin\omega \tau}{\rho _{0}{}^{2}}\mathbf{p}\xb7\mathfrak{R}(\mathbf{P}+\mathbf{a}\overline{A})\right]\times exp\left\{-2(1-cos\omega \tau )\times \left[\frac{{a}^{2}}{r_{s}{}^{2}}+\frac{(\mathbf{P}+\mathbf{a}\overline{A}{)}^{2}}{\rho _{0}{}^{2}}\right]\right\},$$
(16)
$${\rho}_{0}={\left[\frac{4|B{|}^{2}}{{k}^{2}r_{s}{}^{2}}+\frac{2}{k}\mathrm{Im}({\mathit{BA}}^{*})\right]}^{1/2},$$
(17)
$$\overline{A}={A}_{r}+\frac{2{B}_{i}}{\mathit{kr}_{s}{}^{2}};$$
(18)
$$\mathfrak{R}=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right].$$
(19)
$${C}_{I}(\mathbf{P},\mathbf{p};\tau )=exp\left[-\left(\frac{\{[\mathbf{p}+\mathfrak{R}(\mathbf{P}+\mathbf{a}\overline{A}){]}_{x}\omega \tau {\}}^{2}}{\rho _{0x}{}^{2}}+\frac{\{[\mathbf{p}+\mathfrak{R}(\mathbf{P}+\mathbf{a}\overline{A}){]}_{y}\omega \tau {\}}^{2}}{\rho _{0}{}^{2}}\right)\right]\times exp\left[-{\left(\frac{a\omega \tau}{{r}_{s}\beta}\right)}^{2}-\frac{(k\alpha {r}_{s}\omega \tau {)}^{2}}{4}+(\omega \tau {)}^{2}\left(\frac{{\beta}^{2}p_{x}{}^{2}}{4\rho _{0x}{}^{2}}+\frac{p_{y}{}^{2}}{4{\beta}^{2}\rho _{0}{}^{2}}\right)\right]\times exp\left[-2{B}_{r}\alpha \left(\frac{(\omega \tau {)}^{2}{P}_{x}}{\rho _{0x}{}^{2}}+\frac{\omega \tau {p}_{y}}{\rho _{0}{}^{2}}\right)\right],$$
(20)
$${\rho}_{0x}={\left[\frac{4|B{|}^{2}}{{k}^{2}{\beta}^{2}r_{s}{}^{2}}+\frac{2}{k}\mathrm{Im}({\mathit{BA}}^{*})\right]}^{1/2}.$$
(21)
$${C}_{I}(\mathbf{P},0;\tau )=exp\left\{-(\omega \tau {)}^{2}\left[\frac{({P}_{x}+\alpha {B}_{r}{)}^{2}}{\rho _{0x}{}^{2}}+\frac{({P}_{y}+a\overline{A}{)}^{2}}{\rho _{0}{}^{2}}+\frac{{a}^{2}}{({r}_{s}\beta {)}^{2}}\right]\right\},$$
(22)
$${C}_{I}(\mathbf{p};\tau )=exp\left[-(\omega \tau {)}^{2}\left(\frac{{a}^{2}}{r_{s}{}^{2}{\beta}^{2}}-\frac{{\beta}^{2}p_{x}{}^{2}}{4\rho _{0x}{}^{2}}-\frac{p_{y}{}^{2}}{4{\beta}^{2}\rho _{0}{}^{2}}\right)\right]\times exp\left\{-\left[\frac{({p}_{x}+\mathit{aA}\overline{\omega}\tau {)}^{2}}{\rho _{0x}{}^{2}}\right]\right\}\times exp\left[-\frac{({p}_{y}+\alpha {B}_{r}\omega \tau {)}^{2}}{\rho _{0}{}^{2}}\right].$$
(24)
$$B={z}_{1}+{z}_{2}(1-{z}_{1}/f),$$
(25)
$${p}_{x0}=-a\overline{A}\theta =-a(1-{z}_{2}/f)\theta ,$$
(26)
$${p}_{y0}=-\alpha {B}_{r}\theta =-sin\varphi ({z}_{1}+{z}_{2}-{z}_{1}{z}_{2}/f)\theta ,$$