## Abstract

Gradient-index lenses are samples whose special characteristics must be taken into account to design the optical polariscopes that can be applied in the evaluation of their birefringence. We discuss the main infidelity sources that modify the conoscopic patterns when a traditional polariscopic setup is used.

© 2002 Optical Society of America

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### Equations (25)

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(1)
$${\mathbf{M}}_{G}\left(r,\mathrm{\theta}\right)=\left[\begin{array}{cc}cos\frac{{\mathrm{\varphi}}_{r}}{2}+icos2\mathrm{\theta}sin\frac{{\mathrm{\varphi}}_{r}}{2}& isin2\mathrm{\theta}sin\frac{{\mathrm{\varphi}}_{r}}{2}\\ isin2\mathrm{\theta}sin\frac{{\mathrm{\varphi}}_{r}}{2}& cos\frac{{\mathrm{\varphi}}_{r}}{2}-icos2\mathrm{\theta}sin\frac{{\mathrm{\varphi}}_{r}}{2}\end{array}\right],$$
(2)
$${n}^{2}\left(r\right)=N_{0}{}^{2}-{B}^{2}{r}^{2},$$
(3)
$$x={x}_{0}cos\overline{z}+\frac{{p}_{0}}{B}sin\overline{z},y={y}_{0}cos\overline{z}+\frac{{q}_{0}}{B}sin\overline{z},$$
(4)
$$\overline{z}=\left(B/{l}_{0}\right)z.$$
(5)
$${p}_{0}=n\left({r}_{0}\right)cos\mathrm{\alpha},{q}_{0}=n\left({r}_{0}\right)cos\mathrm{\beta},{l}_{0}=n\left({r}_{0}\right)cos\mathrm{\gamma},$$
(6)
$$cos\mathrm{\alpha}=\frac{\mathrm{d}r}{\mathrm{d}x},cos\mathrm{\beta}=\frac{\mathrm{d}r}{\mathrm{d}y},cos\mathrm{\gamma}=\frac{\mathrm{d}r}{\mathrm{d}z}.$$
(7)
$$x={x}_{0}cos\overline{z}+\frac{sin\mathrm{\xi}}{B}sin\overline{z},$$
(8)
$$\overline{z}=\frac{\mathit{Bz}}{{\left(N_{0}{}^{2}-{B}^{2}r_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}^{1/2}}.$$
(9)
$$x=\frac{sin\mathrm{\xi}}{B}sin\overline{z},$$
(10)
$$\overline{z}=\frac{\mathit{Bz}}{{\left(N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}^{1/2}}.$$
(11)
$${L}_{f}=\frac{2\mathrm{\pi}}{B}\sqrt{N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}},$$
(12)
$$\mathrm{\xi}\prime =\mathrm{arcsin}\left\{\frac{-sin\mathrm{\xi}}{{\left(N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}^{1/2}}cos\left[\frac{\mathit{Bz}}{{\left(N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}^{1/2}}\right]\right\}.$$
(13)
$$x={x}_{0}cos\overline{z},$$
(14)
$$\overline{z}=\frac{\mathit{Bz}}{{\left(N_{0}{}^{2}-{B}^{2}x_{0}{}^{2}\right)}^{1/2}}.$$
(15)
$${L}_{c}=\frac{2\mathrm{\pi}}{B}\sqrt{N_{0}{}^{2}-{B}^{2}x_{0}{}^{2}},$$
(16)
$$\mathrm{\xi}\prime =\mathrm{arcsin}\left\{\frac{-{x}_{0}B}{{\left(N_{0}{}^{2}-{B}^{2}x_{0}{}^{2}\right)}^{1/2}}cos\left[\frac{\mathit{Bz}}{{\left(N_{0}{}^{2}-{B}^{2}x_{0}{}^{2}\right)}^{1/2}}\right]\right\}.$$
(17)
$$\frac{1}{4}\left[\begin{array}{cc}1& \mp i\\ \pm 1& -i\end{array}\right]{M}_{G}\left(r,\mathrm{\theta}\right)\left[\begin{array}{c}1\\ -i\end{array}\right]=\left\{\begin{array}{c}\frac{1}{2}sin\frac{{\mathrm{\varphi}}_{r}}{2}\left(sin2\mathrm{\theta}+icos2\mathrm{\theta}\right)\left[\begin{array}{c}1\\ 1\end{array}\right]\\ \frac{1}{2}cos\frac{{\mathrm{\varphi}}_{r}}{2}\left[\begin{array}{c}1\\ -1\end{array}\right]\end{array}\right..$$
(18)
$$I\propto \left\{\begin{array}{c}{sin}^{2}\frac{{\mathrm{\varphi}}_{r}}{2}\\ {cos}^{2}\frac{{\mathrm{\varphi}}_{r}}{2}\end{array}\right..$$
(19)
$${\mathrm{\varphi}}_{r}=\left(2m+1\right)\mathrm{\pi},{\mathrm{\varphi}}_{r}=2m\mathrm{\pi},$$
(20)
$${\mathrm{\varphi}}_{r}=\frac{2\mathrm{\pi}}{\mathrm{\lambda}}\mathrm{OPD}\left(r,z\right),$$
(21)
$$\mathrm{OP}\left(r,z\right)=\frac{1}{{l}_{0}}{\int}_{0}^{z}{n}^{2}\left(r\right)\mathrm{d}z,$$
(22)
$${\mathrm{OP}}_{R,T}\left(r,z\right)=\left[\frac{N_{0}{}^{2}}{{\left(N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}^{1/2}}-\frac{{sin}^{2}\mathrm{\xi}}{2\left(N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}\right]z+\frac{{sin}^{2}\mathrm{\xi}}{4{B}_{R,T}}sin\frac{2{B}_{R,T}z}{{\left(N_{0}{}^{2}-{sin}^{2}\mathrm{\xi}\right)}^{1/2}},$$
(23)
$${\mathrm{OP}}_{R,T}\left(r,z\right)=z\left[\frac{N_{0}{}^{2}-\left(B_{R,T}{}^{2}x_{0}{}^{2}/2\right)}{{\left(N_{0}{}^{2}-B_{R,T}{}^{2}x_{0}{}^{2}\right)}^{1/2}}\right]-\frac{{B}_{R,T}x_{0}{}^{2}}{4}\times sin\frac{2{B}_{R,T}z}{{\left(N_{0}{}^{2}-B_{R,T}{}^{2}x_{0}{}^{2}\right)}^{1/2}}.$$
(24)
$$\mathrm{OPD}\left(r,z\right)={\mathrm{OP}}_{R}\left(r,z\right)-{\mathrm{OP}}_{T}\left(r,z\right).$$
(25)
$$\mathrm{OPD}\left(r,z\right)=\left(m+\frac{1}{2}\right)\mathrm{\lambda},\mathrm{OPD}\left(r,z\right)=m\mathrm{\lambda},$$