## Abstract

A simple geometrical model was developed for calculation of the
contrast of a polychromatic image-plane speckle pattern from a source
of light with high spatial coherence. It is based on counting the
number of independent speckle patterns that contribute to a given point
in the image plane. This results in a simple equation for the
contrast as a function of imaging geometry; relative orientation of the
projection direction, observation direction, and specimen normal;
bandwidth of the light source; and surface roughness. Its validity
was established by comparison with an exact solution: rms errors in
the calculated contrast were only 0.033 over a wide range of parameter
values likely to be encountered in practice.

© 1999 Optical Society of America

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### Equations (13)

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(1)
$$\mathrm{\u220a}=\frac{C\mathrm{\lambda}}{2\mathrm{\pi}sin{u}_{0}sin2\mathrm{\theta}},$$
(2)
$${s}_{0}\left(k\right)=\frac{1}{{\left(2\mathrm{\pi}\right)}^{1/2}W}exp\left[-{\left(k-{k}_{0}\right)}^{2}/2{W}^{2}\right],$$
(3)
$${C}^{2}\left({\mathrm{\theta}}_{1},{\mathrm{\theta}}_{2}\right)={\iint}_{-\infty}^{\infty}\frac{{S}^{2}}{2\mathrm{\pi}{W}^{2}}exp\left[-\frac{{\left({k}_{1}-{k}_{0}\right)}^{2}+{\left({k}_{2}-{k}_{0}\right)}^{2}}{2{W}^{2}}\right]\times exp\left[-{\mathrm{\sigma}}^{2}{\left(cos{\mathrm{\theta}}_{1}+cos{\mathrm{\theta}}_{2}\right)}^{2}\times {\left({k}_{2}-{k}_{1}\right)}^{2}/2\right]\mathrm{d}{k}_{1}\mathrm{d}{k}_{2},$$
(4)
$$S=a/b={\mathrm{\lambda}}_{2}/{\mathrm{\lambda}}_{1},0d\le b-a=b/a={\mathrm{\lambda}}_{1}/{\mathrm{\lambda}}_{2},b-a\le d0=\left\{{a}^{2}{cos}^{-1}\left[\left({a}^{2}+{d}^{2}-{b}^{2}\right)/2a|d|\right]+{b}^{2}{cos}^{-1}\times \left[\left({b}^{2}+{d}^{2}-{a}^{2}\right)/2b|d|\right]-|d|{\left[{a}^{2}-{\left({a}^{2}+{d}^{2}-{b}^{2}\right)}^{2}/{\left(2d\right)}^{2}\right]}^{1/2}\right\}/\left(\mathrm{\pi}\mathit{ab}\right),|b-a||d|b+a=0,b+a\le |d|$$
(5)
$$d=\frac{1}{2\mathrm{\pi}}\left(\frac{sin{\mathrm{\theta}}_{2}}{cos{\mathrm{\theta}}_{1}}-tan{\mathrm{\theta}}_{1}\right)\left({k}_{2}-{k}_{1}\right),$$
(6)
$$\mathrm{\delta}={l}_{c}/2cos\mathrm{\theta}$$
(7)
$$s=1.22\mathrm{\lambda}R/2Q$$
(8)
$$z=rsin\left[\mathrm{\psi}+{tan}^{-1}\left(h/\mathrm{\rho}\right)\right],$$
(9)
$$N=1+\frac{2rsin\left[\mathrm{\psi}+{tan}^{-1}\left(h/\mathrm{\rho}\right)\right]}{\mathrm{\delta}}.$$
(10)
$$N\approx 1+\frac{4\mathrm{\rho}\left[\mathrm{\psi}+\left(h/\mathrm{\rho}\right)\right]}{{l}_{c}}.$$
(11)
$$N\approx 1+\mathrm{\alpha}\prime s\mathrm{\psi}W+\mathrm{\beta}\prime \mathrm{\sigma}W,$$
(13)
$$\mathrm{\psi}=\left({\mathrm{\theta}}_{2}-{\mathrm{\theta}}_{1}\right)/2,\mathrm{\theta}=\left({\mathrm{\theta}}_{2}+{\mathrm{\theta}}_{1}\right)/2.$$