## Abstract

Wereportthe experimental generation of a family of flattened Gaussian beams with bell-shaped,
flattened, and annular intensity profiles in an electro-optically *Q*-switched Nd:YAG laser with a variable reflectivity mirror of a Gaussian reflectivity profile as an output coupler. The laser beams of different profiles were generated by modifying the resonator magnification. The propagation characteristics of the experimentally generated flat Gaussian beams were found to be in agreement with theory. To the best of our knowledge this is the first time such a family of flattened Gaussian beams is experimentally generated intracavity using a single variable reflectivity mirror.

© 2008 Optical Society of America

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### Equations (19)

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(1)
$${I}_{\text{out}}\left(r\right)=\left\{1-{R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left[-2{\left(\frac{r}{{\omega}_{m}}\right)}^{n}\right]\right\}{I}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left[-2{\left(\frac{r}{{\omega}_{i}}\right)}^{n}\right]\text{,}$$
(2)
$$M=g+\sqrt{{g}^{2}-1},$$
(3)
$$g=2{g}_{1}{g}_{2}-1,$$
(4)
$${g}_{1}=\left(1-\frac{L}{{R}_{1}}\right),$$
(5)
$${g}_{2}=\left(1-\frac{L}{{R}_{2}}\right),$$
(6)
$${\omega}_{i}={\omega}_{m}{\left({M}^{n}-1\right)}^{1/n}.$$
(7)
$${I}_{\text{out}}\left(r\right)=\left\{1-{R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left[-2\beta {\left(\frac{r}{{\omega}_{i}}\right)}^{2}\right]\right\}{I}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left[-2{\left(\frac{r}{{\omega}_{i}}\right)}^{2}\right]\text{,}$$
(8)
$${I}_{\text{out}}\left(r\right)=\left\{1-{R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left[-2\left({M}^{2}-1\right){\left(\frac{r}{{\omega}_{i}}\right)}^{2}\right]\right\}{I}_{0}\times \mathrm{exp}\left[-2{\left(\frac{r}{{\omega}_{i}}\right)}^{2}\right].$$
(9)
$${U}_{N}\left(r,z\right)=A\text{\hspace{0.17em}}\frac{{v}_{0}}{{v}_{N}\left(z\right)}\text{\hspace{0.17em}}\mathrm{exp}\left\{i\left[kz-{\phi}_{N}\left(z\right)\right]\right\}\times \mathrm{exp}\left[\left(\frac{ik}{2{R}_{N}\left(z\right)}-\frac{1}{{{v}_{N}}^{2}\left(z\right)}\right){r}^{2}\right]\times {\displaystyle \sum _{n=0}^{N}{C}_{n}{L}_{n}\left(\frac{2{r}^{2}}{{{v}_{N}}^{2}\left(z\right)}\right)\mathrm{exp}\left[-2in{\phi}_{N}\left(z\right)\right],}$$
(10)
$$R\left(Z\right)=z+\left(\frac{{{z}_{R}}^{2}}{z}\right),$$
(11)
$$v\left(Z\right)={v}_{0}\sqrt{1+{\left(\frac{z}{{z}_{R}}\right)}^{2},}$$
(12)
$$\phi \left(Z\right)={\mathrm{tan}}^{-1}\left(\frac{z}{{z}_{R}}\right),$$
(13)
$${C}_{n}={\left(-1\right)}^{n}{\displaystyle \sum _{m=n}^{N}\left(\begin{array}{c}m\\ n\end{array}\right)\frac{1}{{2}^{m}},}$$
(14)
$${Z}_{R}=\frac{\pi {{v}_{0}}^{2}}{\lambda},$$
(15)
$$I\left(r,z\right)={U}_{N}\left(r,z\right)\left[{U}_{N}\left(r,z\right)\right]*.$$
(16)
$${v}_{0}=\frac{{w}_{0}}{\sqrt{N}}.$$
(17)
$$\frac{\Gamma {\left(\mathbf{N}+1,\mathbf{N}+1\right)}^{2}}{\Gamma {\left(\mathbf{N}+1\right)}^{2}}$$
(18)
$$\pi {{w}_{0}}^{2}/\lambda .$$
(19)
$$T=1-\frac{{R}_{0}}{{M}^{2}}.$$