## Abstract

This paper presents analysis of important issues associated with the design of refractive laser beam shaping systems. The concept of “singular radius” is introduced along with solutions to minimize its adverse effect on shaper performance. In addition, the surface boundary constraint is discussed in detail. This study provides useful guidelines to circumvent possible design errors that would degrade the shaper quality or add undesired complication to the system.

© 2008 Optical Society of America

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### Equations (13)

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(1)
$${Z}_{1}\left(r\right)={\int}_{0}^{r}{\left[\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}-{r}_{1}}\right)}^{2}\right]}^{-\frac{1}{2}}d{r}_{1}$$
(2)
$${Z}_{2}\left(r\right)={\int}_{0}^{r}{\left[\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}-{r}_{1}}\right)}^{2}\right]}^{-\frac{1}{2}}d{r}_{2}$$
(3)
$${Z}_{1}\left(r\right)={\int}_{0}^{r}{\left[\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}+{r}_{1}}\right)}^{2}\right]}^{-\frac{1}{2}}d{r}_{1}$$
(4)
$${Z}_{2}\left(r\right)={\int}_{0}^{r}{\left[\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}+{r}_{1}}\right)}^{2}\right]}^{-\frac{1}{2}}d{r}_{2}$$
(5)
$${Z}_{1}\left(r\right)={\int}_{0}^{r}n{[-\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}-{r}_{1}}\right)}^{2}]}^{-\frac{1}{2}}d{r}_{1}$$
(6)
$${Z}_{2}\left(r\right)={\int}_{0}^{r}n{[-\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}-{r}_{1}}\right)}^{2}]}^{-\frac{1}{2}}d{r}_{2}$$
(7)
$${Z}_{1}\left(r\right)={\int}_{0}^{r}n{[-\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}-{r}_{1}}\right)}^{2}]}^{-\frac{1}{2}}d{r}_{1}$$
(8)
$${Z}_{2}\left(r\right)={\int}_{0}^{r}n{[-\left({n}^{2}-1\right)+{\left(\frac{\left(n-1\right)s}{{r}_{2}-{r}_{1}}\right)}^{2}]}^{-\frac{1}{2}}d{r}_{2}$$
(9)
$$g\left(r\right)={g}_{0}\mathrm{exp}(-2{\left(\frac{r}{R}\right)}^{P})$$
(10)
$${g}_{0}=\frac{{2}^{\frac{2}{P}}P}{2\pi {R}^{2}\mathrm{\Gamma}\left(\frac{2}{P}\right)}$$
(11)
$${r}_{1}=h\left({r}_{2}\right)=\sqrt{-\frac{9}{2}\mathrm{ln}\left(1-2\pi {\int}_{0}^{{r}_{2}}g\left(r\right)\mathit{rdr}\right)}$$
(12)
$${r}_{1}-{r}_{2}=\sqrt{\frac{n-1}{n+1}s}$$
(13)
$${r}_{1}+{r}_{2}=\sqrt{\frac{n-1}{n+1}s}$$