## Abstract

Many laser applications require specific irradiance distributions to ensure optimal performance. Geometric optical design methods based on numerical calculation of two plano-aspheric lenses have been thoroughly studied in the past. In this work, we present an alternative new design approach based on functional differential equations that allows direct calculation of the rotational symmetric lens profiles described by two-point Taylor polynomials. The formalism is used to design a Gaussian to flat-top irradiance beam shaping system but also to generate a more complex dark-hollow Gaussian (donut-like) irradiance distribution with zero intensity in the on-axis region. The presented ray tracing results confirm the high accuracy of both calculated solutions and emphasize the potential of this design approach for refractive beam shaping applications.

© 2014 Optical Society of America

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### Equations (25)

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(1)
$${I}_{\mathit{in}}(r)=2/(\pi {w}_{0}^{2})\cdot \text{exp}\left[-2{\left(\frac{r}{{w}_{0}}\right)}^{2}\right]$$
(2)
$${I}_{FL}(R)=\frac{1}{\pi {R}_{FL}^{2}{\left(1+{(R/{R}_{FL})}^{q}\right)}^{1+\frac{q}{2}}}$$
(3)
$$A(r)=\underset{0}{\overset{r}{\int}}{I}_{\mathit{in}}({r}^{\prime})2\pi {r}^{\prime}\hspace{0.17em}d{r}^{\prime}=1-\text{exp}\left[-2{\left(\frac{r}{{w}_{0}}\right)}^{2}\right]$$
(4)
$$B(R)=\underset{0}{\overset{R}{\int}}{I}_{FL}({R}^{\prime})2\pi {R}^{\prime}d{R}^{\prime}={\left[1+{(R/{R}_{FL})}^{-q}\right]}^{(-2/q)}$$
(5)
$$R(r)=\frac{\epsilon {R}_{FL}\sqrt{1-\text{exp}\left[-2{\left(\frac{r}{{w}_{0}}\right)}^{2}\right]}}{\sqrt[q]{1-{\left[1-\text{exp}\left[-2{\left(\frac{r}{{w}_{0}}\right)}^{2}\right]\right]}^{q/2}}}$$
(6)
$${V}_{1}={n}_{2}{\overrightarrow{n}}_{0}\cdot (\overrightarrow{p}-{\overrightarrow{v}}_{0})+\sqrt{{\left((r-R(r)\right)}^{2}+{\left(f(r)-g\left(R(r)\right)\right)}^{2}}$$
(7)
$${D}_{1}=\frac{\partial}{\partial r}{V}_{1}=0$$
(8)
$${D}_{2}={{f}^{\prime}(r)-{\partial}_{r}g(r)|}_{r=R(r)}=0$$
(9)
$$\underset{r\to {r}_{1}}{\text{lim}}\frac{{\partial}^{n}}{\partial {r}^{n}}{D}_{i}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=1,2),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\{n\in {\mathbb{N}}_{0}\}.$$
(10)
$$T(r)=\sum _{i=0}^{k}\left[{a}_{i}(r-{r}_{1})+{b}_{i}(r-{r}_{2})\right]{\left[(r-{r}_{1})(r-{r}_{2})\right]}^{i}$$
(11)
$$f(r)=\sum _{i=0}^{k}\left[{p}_{i}(r-{r}_{1})+{q}_{i}(r-{r}_{2})\right]{\left[(r-{r}_{1})(r-{r}_{2})\right]}^{i}$$
(12)
$$g(r)=\sum _{i=0}^{k}\left[{u}_{i}(r-{r}_{3})+{v}_{i}(r-{r}_{4})\right]{\left[(r-{r}_{3})(r-{r}_{4})\right]}^{i}$$
(13)
$$\begin{array}{cccc}f({r}_{1})={z}_{1}& f({r}_{2})={z}_{2}& {f}^{\prime}({r}_{1})={m}_{1}& {f}^{\prime}({r}_{2})={m}_{2}\\ g({r}_{3})={z}_{3}& g({r}_{4})={z}_{4}& {g}^{\prime}({r}_{3})={m}_{3}& {g}^{\prime}({r}_{4})={m}_{4}\\ & R({r}_{1})={r}_{3}& R({r}_{2})={r}_{4}& \end{array}$$
(14)
$$\underset{r\to {r}_{1}}{\text{lim}}\frac{{\partial}^{n}}{\partial {r}^{n}}{D}_{i}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\underset{r\to {r}_{2}}{\text{lim}}\frac{{\partial}^{n}}{\partial {r}^{n}}{D}_{i}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for}\hspace{0.17em}\hspace{0.17em}(i=1,2),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\{n\in {\mathbb{N}}_{1}\}$$
(15)
$${I}_{\mathit{DHG}}(R)={H}_{0}{\left(\frac{{R}^{2}}{{w}_{1}^{2}}\right)}^{2n}\text{exp}\left[-2{\left(\frac{R}{{w}_{1}}\right)}^{2}\right]$$
(16)
$$B(R)=\underset{0}{\overset{R}{\int}}{I}_{\mathit{DHG}}({R}^{\prime})2\pi {R}^{\prime}d{R}^{\prime}=\frac{\pi {H}_{0}\left({w}_{1}^{4}-\text{exp}\left[-2{\left(\frac{R}{{w}_{1}}\right)}^{2}\right]\hspace{0.17em}\hspace{0.17em}(2{R}^{4}+2{R}^{2}{w}_{1}^{2}+{w}_{1}^{4})\right)}{4{w}_{1}^{2}}$$
(17)
$$r(R)=\sqrt{-\frac{{w}_{0}^{2}}{2}\text{ln}[1-B(R)]}$$
(18)
$$R(r)=\sum _{i=0}^{k}{c}_{i}{r}^{(2i+1)/3}={c}_{0}{r}^{1/3}+{c}_{1}r+{c}_{2}{r}^{5/3}+\dots +{c}_{k}{r}^{(2k+1)/3}$$
(19)
$${V}_{2}={n}_{2}{\overrightarrow{n}}_{0}\cdot (\overrightarrow{p}-{\overrightarrow{v}}_{0})+\sqrt{{(R-r(R))}^{2}+{\left(g(R)-f(r(R))\right)}^{2}}$$
(20)
$${D}_{3}=\frac{\partial}{\partial R}{V}_{2}=0$$
(21)
$${D}_{4}={{g}^{\prime}(R)-{\partial}_{R}f(R)|}_{R=r(R)}=0$$
(22)
$$g(R)=\sum _{i=0}^{k}\left[{u}_{i}(R-{R}_{1})+{v}_{i}(R-{R}_{2})\right]{\left[(R-{R}_{1})(R-{R}_{2})\right]}^{i}$$
(23)
$$\underset{R\to {R}_{1}}{\text{lim}}\frac{{\partial}^{n}}{\partial {R}^{n}}{D}_{i}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\underset{R\to {R}_{2}}{\text{lim}}\frac{{\partial}^{n}}{\partial {R}^{n}}{D}_{i}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for}\hspace{0.17em}\hspace{0.17em}(i=3,4),\hspace{0.17em}\hspace{0.17em}\{n\in {\mathbb{N}}_{1}\}$$
(24)
$$f(R)=\underset{0}{\overset{R}{\int}}{g}^{\prime}\left(R({r}^{\prime})\right)d{r}^{\prime}$$
(25)
$$f(R)={f}_{0}+\sum _{i=1}^{k}{f}_{i}{R}^{(i+2)/3}={f}_{0}+{f}_{1}{R}^{1}+{f}_{2}{R}^{4/3}+{f}_{3}{R}^{5/3}+{f}_{4}{R}^{2}+\dots +{f}_{k}{R}^{(k+2)/3}$$