## Abstract

A gradient-index rod whose length is a quarter of the periodic length works as a retroreflective element when its endface is coated with metal. This paper describes the experiment and theory on the reflectivity of such a rod and that of an array comprising seven or nineteen rods when they are illuminated by a Gaussian laser beam. We clarify how the reflectivity depends on the offset of the beam axis and the inclination angle of the axis. The far field pattern of the reflected wave is also investigated.

© 1991 Optical Society of America

Full Article |

PDF Article
### Equations (13)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$\begin{array}{ll}n\left(r\right)={n}_{0}{\left[1-{\left(gr\right)}^{2}\right]}^{1/2},\hfill & 0\le r\le a,\hfill \end{array}$$
(2)
$$R=\left(1/{T}_{H}{R}_{H}{T}_{L}\right)\left({P}_{3}/{P}_{0}\right),$$
(3)
$${L}_{s}=A{\displaystyle {\int}_{C}\phantom{\rule{0.2em}{0ex}}\text{exp}\left[-2{\left(\rho /w\right)}^{2}\right]ds,}$$
(4)
$$\rho ={\left({a}^{2}-2a{x}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\varphi +{x}_{0}^{2}\right)}^{1/2},$$
(5)
$${\varphi}_{1}={\text{tan}}^{-1}\left[{\left({a}^{2}-{x}_{1}^{2}\right)}^{1/2}/{x}_{1}\right],$$
(6)
$${x}_{1}=\left({x}_{0}^{2}+{a}^{2}-2{w}^{2}\right)/2{x}_{0},$$
(7)
$$a-\sqrt{2}w<{x}_{0}<a+\sqrt{2}w.$$
(8)
$$R=\left(1/{R}_{M1}{R}_{M2}{T}_{H}{R}_{H}{T}_{L}\right)\left({P}_{3}/{P}_{0}\right),$$
(9)
$$T=4{n}_{1}\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{i}{\left({n}_{2}^{2}-{n}_{1}^{2}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}{\theta}_{i}\right)}^{1/2}\times {\left[{n}_{1}\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{i}+{\left({n}_{2}^{2}-{n}_{1}^{2}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}{\theta}_{i}\right)}^{1/2}\right]}^{-2}.$$
(10)
$${T}_{1}=4\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{0}{\left({n}_{0}^{2}-{\text{sin}}^{2}{\theta}_{0}\right)}^{1/2}{\left[\text{cos}{\theta}_{0}+{\left({n}_{0}^{2}-{\text{sin}}^{2}{\theta}_{0}\right)}^{1/2}\right]}^{-2}.$$
(11)
$${T}_{2}=4{n}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{1}{\left(1-{n}_{0}^{2}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}{\theta}_{1}\right)}^{1/2}\times {\left[{n}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{1}+{\left(1-{n}_{0}^{2}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}{\theta}_{1}\right)}^{1/2}\right]}^{-2}.$$
(12)
$${n}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}{\theta}_{1}=\text{sin}{\theta}_{0},{n}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{1}={\left({n}_{0}^{2}-{\text{sin}}^{2}{\theta}_{0}\right)}^{1/2}.$$
(13)
$${R}_{d}={R}_{m}{\left(4\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{0}\right)}^{2}\left({n}_{0}^{2}-{\text{sin}}^{2}{\theta}_{0}\right){\left[\text{cos}{\theta}_{0}+{\left({n}_{0}^{2}-{\text{sin}}^{2}{\theta}_{0}\right)}^{1/2}\right]}^{-4}.$$