## Abstract

A Michelson-type digital speckle photographic system has been proposed in which one light beam produces a Fourier transform and another beam produces an image at a recording plane, without interfering between themselves. Because the optical Fourier transform is insensitive to translation and the imaging technique is insensitive to tilt, the proposed system is able to simultaneously and independently determine both surface tilt and translation by two separate recordings, one before and another after the surface motion, without the need to obtain solutions for simultaneous equations. Experimental results are presented to verify the theoretical analysis.

© 2010 Optical Society of America

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### Equations (7)

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(1)
$${U}_{1}(\xi )=CF\{u(x)\},$$
(2)
$${u}_{2}(x)=u(-x)\mathrm{.}$$
(3)
$$I=|{U}_{1}(\xi ){|}^{2}+|{u}_{2}(x){|}^{2}=|{U}_{1}(\xi ){|}^{2}+|u(-x){|}^{2}={I}_{1}(\xi )+{I}_{2}(-x),$$
(4)
$${u}^{\prime}(x)=u(x-\delta )\mathrm{exp}[j(2\pi /\lambda )\theta x]\mathrm{.}$$
(5)
$${U}_{1}^{\prime}(\xi )=CF\{u(x-\delta )\mathrm{exp}[j(2\pi /\lambda )\theta x]\}={U}_{1}(\xi -\theta f)\mathrm{exp}[-j2\pi (\xi -\theta f)\delta /\lambda f]\mathrm{.}$$
(6)
$${u}_{2}^{\prime}(x)=u(-x+\delta )\mathrm{exp}[j(2\pi /\lambda )\theta x]\mathrm{.}$$
(7)
$${I}^{\prime}=|{U}_{1}^{\prime}(\xi ){|}^{2}+|{u}_{2}^{\prime}(x){|}^{2}\phantom{\rule{0ex}{0ex}}=|{U}_{1}(\xi -\theta f){|}^{2}+|u(-x+\delta ){|}^{2}\phantom{\rule{0ex}{0ex}}={I}_{1}(\xi -\theta f)+{I}_{2}(-x+\delta ),$$