## Abstract

The gyrator transform (GT) promises to be a useful tool in image processing, holography, beam characterization, mode transformation, and quantum information. We introduce what we believe to be the first flexible optical experimental setup that performs the GT for a wide range of transformation parameters. The feasibility of the proposed scheme is demonstrated on the gyrator transformation of Hermite–Gaussian modes. For certain parameters the output mode corresponds to the Laguerre–Gaussian one.

© 2007 Optical Society of America

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### Equations (9)

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(1)
$${g}_{o}\left({\mathbf{r}}_{o}\right)={R}^{\alpha}\left[{g}_{i}\left({\mathbf{r}}_{i}\right)\right]\left({\mathbf{r}}_{o}\right)=\frac{1}{\mid \mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha \mid}\int \int {g}_{i}({x}_{i},{y}_{i})\mathrm{exp}\left(i2\pi \frac{({x}_{o}{y}_{o}+{x}_{i}{y}_{i})\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\alpha -({x}_{i}{y}_{o}+{x}_{o}{y}_{i})}{\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha}\right)\mathrm{d}{x}_{i}\mathrm{d}{y}_{i},$$
(2)
$${g}_{o}({x}_{o},{y}_{o})=\mathrm{exp}(-i\pi \frac{{x}_{o}^{2}+{y}_{o}^{2}-2{x}_{o}{y}_{o}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2\phi}{\lambda f}){g}_{i}({x}_{o},{y}_{o}),$$
(3)
$${g}_{1}({x}_{1},{y}_{1})=\frac{1}{i\lambda z}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\pi \frac{{x}_{1}^{2}+{y}_{1}^{2}}{\lambda z}\right)\int \int {g}_{i}({x}_{i},{y}_{i})\mathrm{exp}(-i2\pi \frac{{x}_{1}{x}_{i}+{y}_{1}{y}_{i}-{x}_{i}{y}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{1}}{\lambda z})\mathrm{d}{x}_{i}\mathrm{d}{y}_{i}.$$
(4)
$${g}_{2}({x}_{2},{y}_{2})=-\frac{1}{\mid 2\lambda z\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{2}\mid}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\pi \frac{{x}_{2}^{2}+{y}_{2}^{2}}{\lambda z}\right)\int \int {g}_{i}({x}_{i},{y}_{i})\mathrm{exp}\left(i\pi \frac{2{x}_{i}{y}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{1}-({x}_{2}{y}_{i}+{x}_{i}{y}_{2}+{x}_{i}{y}_{i}+{x}_{2}{y}_{2})\u2215\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{2}}{\lambda z}\right)\mathrm{d}{x}_{i}\mathrm{d}{y}_{i}.$$
(5)
$${g}_{o}({x}_{o},{y}_{o})=-\frac{1}{\mid 2\lambda z\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{2}\mid}\int \int {g}_{i}({x}_{i},{y}_{i})\mathrm{exp}\left(i2\pi \frac{({x}_{o}{y}_{o}+{x}_{i}{y}_{i})(2\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{1}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{2}-1)-({x}_{o}{y}_{i}+{x}_{i}{y}_{o})}{2\lambda z\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{2}}\right)\mathrm{d}{x}_{i}\mathrm{d}{y}_{i}.$$
(6)
$$\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{1}=\mathrm{cot}(\alpha \u22152),$$
(7)
$$\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2{\phi}_{2}=\left(\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha \right)\u22152.$$
(8)
$${\mathrm{HG}}_{m,n}(\mathbf{r};w)={2}^{1\u22152}\frac{{H}_{m}\left(\sqrt{2\pi}\frac{x}{w}\right){H}_{n}\left(\sqrt{2\pi}\frac{y}{w}\right)}{\sqrt{{2}^{m}m!w}\sqrt{{2}^{n}n!w}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{\pi}{{w}^{2}}{\mathbf{r}}^{2}),$$
(9)
$${\mathrm{LG}}_{p,l}^{\pm}(\mathbf{r};w)={w}^{-1}\sqrt{\frac{\mathrm{min}(m,n)!}{\mathrm{max}(m,n)!}}{\left(\sqrt{2\pi}(\frac{x}{w}\pm i\frac{y}{w})\right)}^{l}{L}_{p}^{l}\left(\frac{2\pi}{{w}^{2}}{\mathbf{r}}^{2}\right)\mathrm{exp}(-\frac{\pi}{{w}^{2}}{\mathbf{r}}^{2}),$$