## Abstract

We propose and demonstrate the improvement of conventional Galilean refractive beam shaping system for accurately generating near-diffraction-limited flattop beam with arbitrary beam size. Based on the detailed study of the refractive beam shaping system, we found that the conventional Galilean beam shaper can only work well for the magnifying beam shaping. Taking the transformation of input beam with Gaussian irradiance distribution into target beam with high order Fermi-Dirac flattop profile as an example, the shaper can only work well at the condition that the size of input and target beam meets *R*
_{0}≥1.3*w*
_{0}. For the improvement, the shaper is regarded as the combination of magnifying and demagnifying beam shaping system. The surface and phase distributions of the improved Galilean beam shaping system are derived based on Geometric and Fourier Optics. By using the improved Galilean beam shaper, the accurate transformation of input beam with Gaussian irradiance distribution into target beam with flattop irradiance distribution is realized. The irradiance distribution of the output beam is coincident with that of the target beam and the corresponding phase distribution is maintained. The propagation performance of the output beam is greatly improved. Studies of the influences of beam size and beam order on the improved Galilean beam shaping system show that restriction of beam size has been greatly reduced. This improvement can also be used to redistribute the input beam with complicated irradiance distribution into output beam with complicated irradiance distribution.

© 2011 OSA

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### Equations (13)

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(1)
$$z\left(r\right)={\displaystyle {\int}_{0}^{r}{\left\{\left({n}^{2}-1\right)+{\left[\frac{\left(n-1\right)d}{h\left(x\right)-x}\right]}^{2}\right\}}^{-1/2}dx}$$
(2)
$$Z\left(R\right)={\displaystyle {\int}_{0}^{R}{\left\{\left({n}^{2}-1\right)+{\left[\frac{\left(n-1\right)d}{{h}^{-1}\left(x\right)-x}\right]}^{2}\right\}}^{-1/2}dx},$$
(3)
$${f}_{phase1}(r)=\frac{2\pi \left[{z}_{edge}-z\left(r\right)+nz\left(r\right)\right]}{\lambda}$$
(4)
$${f}_{phase2}\left(R\right)=\frac{2\pi \left[n{Z}_{edge}-nZ\left(R\right)+Z\left(R\right)\right]}{\lambda},$$
(5)
$${P}_{input}\left(r\right)=\mathrm{exp}\left(\frac{-2{r}^{2}}{{w}_{0}^{2}}\right)$$
(6)
$${P}_{output}\left(r\right)={\left\{1+\mathrm{exp}\left[\beta \left(\frac{r}{{R}_{0}}-1\right)\right]\right\}}^{-1},$$
(7)
$$Error=\sqrt{{\displaystyle \iint {\left[{I}_{output}\left(x,y\right)-{I}_{t\mathrm{arg}et}\left(x,y\right)\right]}^{2}dxdy}},$$
(8)
$$\{\begin{array}{c}\text{\hspace{0.17em}}z\left({r}_{1}\right)={\displaystyle {\int}_{0}^{{r}_{1}}{\left\{\left({n}^{2}-1\right)+{\left[\frac{\left(n-1\right)d}{h\left(x\right)-x}\right]}^{2}\right\}}^{-1/2}dx}\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}0<{r}_{1}<{h}_{1}\\ z\left({r}_{2}\right)=z\left({h}_{1}\right)-{{\displaystyle {\int}_{{h}_{1}}^{{r}_{2}}\left\{\left({n}^{2}-1\right)+{\left[\frac{\left(n-1\right)d}{h\left(x\right)-x}\right]}^{2}\right\}}}^{-1/2}dx\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}{h}_{1}<{r}_{2}<{h}_{2}\end{array}$$
(9)
$$\{\begin{array}{c}\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}Z\left({R}_{1}\right)={{\displaystyle {\int}_{0}^{{R}_{1}}\left\{\left({n}^{2}-1\right)+{\left[\frac{\left(n-1\right)d}{{h}^{-1}\left(x\right)-x}\right]}^{2}\right\}}}^{-1/2}dx\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}0<{R}_{1}<{H}_{1}\\ Z\left({R}_{2}\right)=Z\left({H}_{1}\right)-{{\displaystyle {\int}_{{H}_{1}}^{{R}_{2}}\left\{\left({n}^{2}-1\right)+{\left[\frac{\left(n-1\right)d}{{h}^{-1}\left(x\right)-x}\right]}^{2}\right\}}}^{-1/2}dx\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}{H}_{1}<{R}_{1}<{H}_{2}\end{array},$$
(10)
$${f}_{phase1}\left(r\right)=\frac{2\pi \left[{z}_{\mathrm{max}}-z\left(r\right)+nz\left(r\right)\right]}{\lambda}$$
(11)
$${f}_{phase2}\left(R\right)=\frac{2\pi \left[n{Z}_{\mathrm{max}}-nZ\left(R\right)+Z\left(R\right)\right]}{\lambda},$$
(12)
$$SE=\frac{a{\left\{{\displaystyle {\int}_{0}^{a}{\left[{P}_{output}(r)-\frac{2}{{a}^{2}}{\displaystyle {\int}_{0}^{a}{P}_{output}(r)rdr}\right]}^{2}rdr}\right\}}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\sqrt{2}{\displaystyle {\int}_{0}^{a}{P}_{output}(r)rdr}}$$
(13)
$$\eta =\frac{2\pi {\displaystyle {\int}_{0}^{a}{P}_{output}(r)rdr}}{{W}_{input}},$$