## Abstract

Two examples are presented to illustrate the advantages of polarization coded apertures, in which the incoming light will rotate its polarization at a portion of an aperture. In the first example the depth of field of a diffraction limited lens is increased without sacrificing the light throughput; in the second example the axial focal intensity of a pixelated Fresnel zone plate is increased by 100%. Both examples work for linearly polarized or unpolarized illumination.

© 2006 Optical Society of America

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### Equations (8)

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(1)
$$I\left(\rho \right)={\mid {\int}_{0}^{r}r\prime {J}_{0}(2\mathit{\pi r}\prime \frac{\rho}{\mathit{\lambda t}})\mathrm{exp}\left[i\frac{2\mathit{\pi W}}{\lambda}{\left(\frac{r\prime}{R}\right)}^{2}\right]\mathit{dr}\prime \mid}^{2}+{\mid {\int}_{r}^{R}r\prime {J}_{0}(2\mathit{\pi r}\prime \frac{\rho}{\mathit{\lambda t}})\mathrm{exp}\left[i\frac{2\mathit{\pi W}}{\lambda}{\left(\frac{r\prime}{R}\right)}^{2}\right]\mathit{dr}\prime \mid}^{2},$$
(2)
$${E}_{\mathit{mn}}=A{\int}_{\frac{{y}_{n}-a}{2}}^{\frac{{y}_{n}+a}{2}}{\int}_{\frac{{x}_{m}-a}{2}}^{\frac{{x}_{m}-a}{2}}\frac{f}{{f}^{2}+{x}^{2}+{y}^{2}}\mathrm{exp}\left(i\frac{2\pi}{\lambda}\sqrt{{f}^{2}+{x}^{2}+{y}^{2}}\right)\mathit{dxdy},$$
(3)
$${I}_{P}={\mid \sum _{\frac{-\pi}{2}\le {\varphi}_{\mathit{mn}}<\frac{\pi}{2}}{E}_{\mathit{mn}}\mid}^{2}+{\mid \sum _{-\pi \le {\varphi}_{\mathit{mn}}<\frac{-\pi}{2}\mathrm{or}\frac{\pi}{2}\le {\varphi}_{\mathit{mn}}<\pi}{E}_{\mathit{mn}}\mid}^{2}.$$
(4)
$${I}_{P}\simeq 2{\mid \sum _{\frac{-\pi}{2}\le {\varphi}_{\mathit{mn}}<\frac{\pi}{2}}{E}_{\mathit{mn}}\mid}^{2}.$$
(5)
$$t\left(r\right)=\sum _{n=1}^{2N-1}{\left(-1\right)}^{n+1}\mathrm{circ}\left(\frac{r}{{r}_{n}}\right),$$
(6)
$$E\left(z\right)=\sum _{n=1}^{2N-1}{\left(-1\right)}^{n+1}E\left(z,n\right),$$
(7)
$$E\left(z,n\right)={E}_{0}\left[\mathrm{exp}\left(i\frac{2\pi}{\lambda}z\right)-\frac{z}{\sqrt{{z}^{2}+{r}_{n}^{2}}}\mathrm{exp}\left(i\frac{2\pi}{\lambda}\sqrt{{z}^{2}+{r}_{n}^{2}}\right)\right].$$
(8)
$$I\left(z\right)=C{\mid {e}^{i\frac{2\pi}{\lambda}z}+\sum _{n=1}^{2N-1}{\left(-1\right)}^{n}\frac{z}{\sqrt{{z}^{2}+\mathit{n\lambda f}}}\mathrm{exp}\left(i\frac{2\pi}{\lambda}\sqrt{{z}^{2}+\mathit{n\lambda f}}\right)\mid}^{2},$$