## Abstract

Fundamental properties of the phase-modulation ability of a nematic liquid-crystal cell are studied. Based on these phase-modulation properties of the liquid-crystal cell, a new type of speckle-shearing interferometer is proposed and studied experimentally. A liquid-crystal cell is employed as a phase shifter to implement the phase-shifting method for the conventional speckle-shearing interferometer. From the experiments used to measure the deformation of an object, the usefulness of the method is confirmed. Finally, a compensation method for phase-shift error is proposed on the basis of the statistical properties of the fully developed speckle field. In this method the speckle phase is regarded, in a statistical sense, as a standard phase object used to calibrate the measuring system. Experiments to confirm the error-compensation method are performed, and it is shown that the phase-shift error can be determined with an accuracy of as much as λ/100.

© 1991 Optical Society of America

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### Equations (16)

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(1)
$$\begin{array}{c}{I}_{i}(x,y)\hfill & ={|f(x,y)|}^{2}+{|f(x+\mathrm{\Delta}x,y)|}^{2}+2|f(x,y)|\times |f(x+\mathrm{\Delta}x,y)|cos[\varphi (x,y)+{\psi}_{i}],\\ \hfill i& =1,2,3,\hspace{0.17em}{\psi}_{1}=-\psi ,\hspace{0.17em}{\psi}_{2}=0,\hspace{0.17em}{\psi}_{3}=\psi ,\hfill \end{array}$$
(2)
$$\begin{array}{cc}\hfill {{I}_{i}}^{\prime}(x,y)& ={|f(x,y)|}^{2}+{|f(x+\mathrm{\Delta}x,y)|}^{2}+2|f(x,y)|\times |f(x+\mathrm{\Delta}x,y)|cos[\varphi (x,y)+\mathrm{\Delta}\theta (x,y)+{\psi}_{i}],\hfill \\ \hfill i& =1,2,3,\hfill \end{array}$$
(3)
$$\begin{array}{cc}\hfill \mathrm{\Delta}\theta (x,y)& =(4\pi /\mathrm{\lambda})\hspace{0.17em}[w(x,y)-w(x+\mathrm{\Delta}x,y)]\hfill \\ & \cong (4\pi /\mathrm{\lambda})\hspace{0.17em}[\partial w(x,y)/\partial x]\mathrm{\Delta}x.\hfill \end{array}$$
(4)
$$\varphi (x,y)={tan}^{-1}\left\{\frac{[{I}_{1}(x,y)-{I}_{3}(x,y)]\hspace{0.17em}(cos\psi -1)}{[{I}_{1}(x,y)+{I}_{3}(x,y)-2{I}_{2}(x,y)]sin\psi}\right\}.$$
(5)
$$\gamma =\frac{2|f(x,y)|\hspace{0.17em}|f(x+\mathrm{\Delta}x,y)|}{{|f(x,y)|}^{2}+{|f(x+\mathrm{\Delta}x,y)|}^{2}}.$$
(6)
$$\begin{array}{cc}{p}_{\varphi}(\varphi )=1/2\pi ,& -\pi <\varphi \le \pi .\end{array}$$
(7)
$$\begin{array}{cc}\hfill {\psi}_{1}& =-{\psi}^{\prime}=-\psi -\mathrm{\Delta}{\psi}_{s},\hfill \\ \hfill {\psi}_{2}& =0,\hfill \\ \hfill {\psi}_{3}& ={\psi}^{\prime}=\psi +\mathrm{\Delta}{\psi}_{s}.\hfill \end{array}$$
(8)
$$\begin{array}{ll}{I}_{i}(x,y)\hfill & ={|f(x,y)|}^{2}+{|f(x+\mathrm{\Delta}x,y)|}^{2}+2|f(x,y)|\times |f(x+\mathrm{\Delta}x,y)|cos[\varphi (x,y)+{\psi}_{i}],\hfill \\ \hfill i& =1,2,3,\hspace{0.17em}{\psi}_{1}=-{\psi}^{\prime},\hspace{0.17em}{\psi}_{2}=0,\hspace{0.17em}{\psi}_{3}={\psi}^{\prime}.\hfill \end{array}$$
(9)
$$tan{\varphi}^{\prime}=\frac{{I}_{1}-{I}_{3}}{{I}_{1}+{I}_{3}-2{I}_{2}}\frac{cos\psi -1}{sin\psi}.$$
(10)
$$tan\varphi =\frac{{I}_{1}-{I}_{3}}{{I}_{1}+{I}_{3}-2{I}_{2}}\frac{cos{\psi}^{\prime}-1}{sin{\psi}^{\prime}}.$$
(11)
$$\begin{array}{ll}{{p}_{\varphi}}^{\prime}({\varphi}^{\prime})\hfill & ={p}_{\varphi}(\varphi )|\text{d}\varphi /\text{d}{\varphi}^{\prime}|\hfill \\ \hfill & =\frac{1}{\pi}\left|\frac{\alpha}{({\alpha}^{2}-1)cos2\varphi +({\alpha}^{2}+1)}\right|,\hfill \end{array}$$
(12)
$$\alpha =\frac{sin{\psi}^{\prime}}{cos{\psi}^{\prime}-1}\frac{cos\psi -1}{sin\psi}.$$
(13)
$${p}_{{\varphi}^{\prime}}({\varphi}^{\prime})=\alpha /\pi [({\alpha}^{2}-1)cos2{\varphi}^{\prime}+({\alpha}^{2}+1)].$$
(14)
$$T=\frac{1}{2\pi}{\mathit{\int}}_{-\pi}^{\pi}\frac{cos2{\varphi}^{\prime}}{{p}_{{\varphi}^{\prime}}({\varphi}^{\prime})}\text{d}{\varphi}^{\prime}.$$
(15)
$$\alpha =T/\pi +{[{(T/\pi )}^{2}+1]}^{1/2}.$$
(16)
$$\mathrm{\Delta}{\psi}_{s}={sin}^{-1}\frac{1-{\alpha}^{2}}{1+{\alpha}^{2}}.$$