## Abstract

An analytical method for phase detection in retardation measurements is proposed. The experimental setup is based on a simple linear polariscope with a λ plate. The intensity modulation at the output of the polariscope is measured when the wavelength is changed and a grid of phase-shifted intensity values is recorded. The phase difference between the components of the light propagating along the principal axes of the birefringent sample is determined with Greivenkamp’s algorithm employed in phase-stepping interferometry. Error analysis for the new method is performed. Simplified algorithms for faster data analysis are proposed. The accuracy attained is comparable with the accuracy of known phase-detection methods used in retardation measurements.

© 1994 Optical Society of America

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### Equations (12)

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(1)
$$I(\lambda )=\frac{16}{{\left({n}_{o}+1\right)}^{4}}\left[1-\frac{2{n}_{o}\left({n}_{o}-1\right)}{{n}_{o}+1}\right]{I}_{p}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}\left(\delta /2\right),$$
(2)
$$\delta =\frac{2\pi}{{\lambda}_{0}}\left({n}_{e}-{n}_{o}\right)t,$$
(3)
$$\alpha \left(\Delta \lambda \right)=\frac{2\pi}{{\lambda}_{0}}\Gamma \frac{\Delta \lambda}{{\lambda}_{0}},$$
(4)
$${I}_{i}={I}_{0i}\left[1-\text{cos}\left(\delta -\delta \frac{{\Delta \lambda}_{i}}{{\lambda}_{0}}\right)\right]$$
(5)
$${I}_{0}(\lambda )=\frac{16}{{\left({n}_{o}+1\right)}^{4}}\left[1-\frac{2{n}_{o}\left({n}_{o}-1\right)}{{n}_{o}+1}\right]{I}_{p}.$$
(6)
$${I}_{i}=0.5{I}_{0i}\left\{1-\text{cos}\left[\frac{2\pi}{{\lambda}_{0}}\Gamma +\frac{2\pi}{{\lambda}_{0}}\left(Q-Q\frac{\Delta {\lambda}_{i}}{{\lambda}_{0}}\right)\right]\right\}+{I}_{fi},$$
(7)
$${I}_{i}={a}_{0}+{a}_{1}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}{\delta}_{i}+{a}_{2}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\delta}_{i},$$
(8)
$$\begin{array}{ll}{a}_{0}=\left({I}_{0i}+{I}_{fi}\right)/{I}_{0i},& {a}_{1}=\text{cos}\left(\frac{2\pi}{{\lambda}_{0}}\Gamma \right).\\ {a}_{2}=\text{sin}\left(\frac{2\pi}{{\lambda}_{0}}\Gamma \right),& {\delta}_{i}=\frac{2\pi}{{\lambda}_{0}}\left(Q-Q\frac{{\Delta \lambda}_{i}}{{\lambda}_{0}}\right).\end{array}$$
(9)
$${n}_{e}-{n}_{0}=\frac{{\lambda}_{0}}{2\pi}\phantom{\rule{0.2em}{0ex}}\frac{1}{t}{\text{tan}}^{-1}\left(\frac{{a}_{2}}{{a}_{1}}\right).$$
(10)
$$\varphi ={\text{tan}}^{-1}\left(\Sigma \phantom{\rule{0.2em}{0ex}}{A}_{n}{I}_{n}/\Sigma \phantom{\rule{0.2em}{0ex}}{B}_{n}{I}_{n}\right)$$
(11)
$$\begin{array}{c}\begin{array}{cc}{\alpha}_{i}=\frac{2\pi \left(i-1\right)}{N}\frac{Q}{{\lambda}_{0}},& i=1,...,N.\end{array}\end{array}$$
(12)
$${C}_{12}\left(n\right)=\frac{1}{N}{\displaystyle {\int}_{0}^{N}{I}_{1}(\phi ){I}_{2}\left(\phi -\delta \phi \right)\mathrm{d}\phi},$$