## Abstract

A novel design method is presented for a simple laser beam shaper. Unlike earlier reports and designs based on the 2-element model, we prove it is possible to convert a laser beam from a non-uniform profile to a uniform flat-top distribution with one single aspherical lens.

©2003 Optical Society of America

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### Equations (12)

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(1)
$$R={\left[\left(\frac{2}{{E}_{o}}\right){\int}_{0}^{{r}_{0}}{E}_{i}\left(r\right)r\mathit{dr}\right]}^{\frac{1}{2}}$$
(2)
$${\left(z\text{'}\right)}^{4}\left[{\gamma}_{1}^{2}{\left(R-r\right)}^{2}+\left({\gamma}_{1}^{2}-1\right){\left(Z-z\right)}^{2}\right]-{\left(z\text{'}\right)}^{3}\left[2\left(R-r\right)\left(Z-z\right)\right]$$
(2)
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-{\left(z\text{'}\right)}^{2}\left(1-{\gamma}_{1}^{2}\right)\left[{\left(R-r\right)}^{2}+{\left(Z-z\right)}^{2}\right]$$
(2)
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-z\text{'}\left[2\left(R-r\right)\left(Z-z\right)\right]-{\left(R-r\right)}^{2}=0$$
(3)
$$C\left(\mathit{cons}\phantom{\rule{.2em}{0ex}}\mathrm{tan}\phantom{\rule{.2em}{0ex}}t\right)={n}_{1}{t}_{1}+\left({Z}_{0}-{t}_{1}\right){n}_{0}+{n}_{2}{t}_{2}$$
(3)
$$={n}_{1}z+{n}_{0}{\left[{\left(R-r\right)}^{2}+{\left(Z-z\right)}^{2}\right]}^{\frac{1}{2}}+{n}_{2}\left({Z}_{0}+{t}_{2}-Z\right)$$
(4)
$$Z={\left({n}_{2}^{2}-{n}_{0}^{2}\right)}^{-1}\{\left[\left({n}_{1}{n}_{2}-{n}_{0}^{2}\right)z+{n}_{2}C\right]\pm {n}_{0}[{\left(C+{n}_{1}{\gamma}_{1}{n}_{2}z\right)}^{2}$$
(4)
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\left({n}_{2}^{2}-{n}_{0}^{2}\right){\left(R-r\right)}^{2}{]}^{\frac{1}{2}}\}$$
(5)
$$Z\text{'}=\frac{z\text{'}{\gamma}_{2}\left\{{\gamma}_{1}-{\left[1+{\left(z\text{'}\right)}^{2}\left(1-{\gamma}_{1}^{2}\right)\right]}^{\frac{1}{2}}\right\}}{\{1+{\left(z\text{'}\right)}^{2}-{\gamma}_{1}{\gamma}_{2}{\left(z\text{'}\right)}^{2}}$$
(5)
$$-{\gamma}_{2}{\left[1+{\left(z\text{'}\right)}^{2}\left(1-{\gamma}_{1}^{2}\right)\right]}^{\frac{1}{2}}\}$$
(6)
$$z\text{'}=\left\{\left(R-r\right)\left(Z-z\right)\pm {\gamma}_{1}\left(R-r\right){\left[{\left(Z-z\right)}^{2}+{\left(R-r\right)}^{2}\right]}^{\frac{1}{2}}\right\}$$
(6)
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}/\left[\left({\gamma}_{1}^{2}-1\right){\left(Z-z\right)}^{2}+{\gamma}_{1}^{2}{\left(R-r\right)}^{2}\right]$$