## Abstract

We describe a simple technique for simultaneously imaging multiple
layers within an object field onto a single camera. The approach
uses a binary diffraction grating in which the lines are distorted such
that a different level of defocus is associated with each diffraction
order. The design of the gratings is discussed, and their ability
to image multiple object planes is validated
experimentally. Extension of the technique for spherical-aberration
correction is described, and it is shown how the gratings can be used
as part of a wave-front–sensing system.

© 1999 Optical Society of America

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### Equations (18)

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(1)
$${\mathrm{\varphi}}_{m}\left(x,y\right)=\frac{2\mathrm{\pi}m{\mathrm{\Delta}}_{x}\left(x,y\right)}{d},$$
(2)
$$\mathrm{\varphi}\left(r\right)=\frac{2\mathrm{\pi}}{\mathrm{\lambda}}\left[f-{\left({f}^{2}-{r}^{2}\right)}^{1/2}\right],$$
(3)
$$\mathrm{\varphi}\left(r\right)=\frac{2\mathrm{\pi}}{\mathrm{\lambda}}\left(\frac{{r}^{2}}{2f}+\frac{{r}^{4}}{8{f}^{3}}+\frac{{r}^{6}}{16{f}^{5}}+\cdots \right).$$
(4)
$${\mathrm{\Delta}}_{x}\left(x,y\right)=\frac{{W}_{20}d}{\mathrm{\lambda}{R}^{2}}\left({x}^{2}+{y}^{2}\right),$$
(5)
$${\mathrm{\varphi}}_{m}\left(x,y\right)=m\frac{2\mathrm{\pi}{W}_{20}}{\mathrm{\lambda}{R}^{2}}\left({x}^{2}+{y}^{2}\right).$$
(6)
$$\frac{x}{{d}_{0}}+\frac{{W}_{20}\left({x}^{2}+{y}^{2}\right)}{\mathrm{\lambda}{R}^{2}}=n,$$
(7)
$${x}_{n}=-\frac{\mathrm{\lambda}{R}^{2}}{2{W}_{20}{d}_{0}},$$
(8)
$${C}_{n}={\left[\frac{n\mathrm{\lambda}{R}^{2}}{{W}_{20}}+{\left(\frac{\mathrm{\lambda}{R}^{2}}{2{d}_{0}{W}_{20}}\right)}^{2}\right]}^{1/2}.$$
(9)
$$d=\frac{{d}_{0}\mathrm{\lambda}{R}^{2}}{\mathrm{\lambda}{R}^{2}-2{d}_{0}{W}_{20}{x}_{0}}.$$
(10)
$${d}_{min}=\frac{{d}_{0}\mathrm{\lambda}R}{\mathrm{\lambda}R+2{d}_{0}{W}_{20}}.$$
(11)
$${f}_{m}=\frac{{R}^{2}}{2{\mathit{mW}}_{20}},$$
(12)
$${f}_{m}=\frac{{\mathit{fR}}^{2}}{{R}^{2}+2{\mathit{fmW}}_{20}}.$$
(13)
$$\mathrm{\delta}{z}_{m}=-\frac{2{\mathit{mz}}^{2}{W}_{20}}{{R}^{2}+2{\mathit{mzW}}_{20}},$$
(14)
$$\mathrm{\delta}{z}_{m}\approx -2m{\left(\frac{z}{R}\right)}^{2}{W}_{20},$$
(15)
$${\mathrm{\varphi}}_{m}\left(x,y\right)=\frac{2m\mathrm{\pi}{W}_{20}}{\mathrm{\lambda}{R}^{2}}\left[{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}\right],$$
(16)
$${\mathrm{\varphi}}_{m}\left(x,y\right)=\frac{2m\mathrm{\pi}{W}_{20}}{\mathrm{\lambda}{R}^{2}}\left[\left({x}^{2}+{y}^{2}\right)-2{x}_{0}x-2{y}_{0}y+\left(x_{0}{}^{2}+y_{0}{}^{2}\right)\right],$$
(17)
$$\frac{x}{{d}_{0}}+\sum _{j=1}^{\infty}\frac{{W}_{2j,0}{\left({x}^{2}+{y}^{2}\right)}^{j}}{\mathrm{\lambda}{R}^{2j}}=n.$$
(18)
$${x}_{0}=\frac{\mathrm{\lambda}{R}^{2}}{2{d}_{0}{W}_{20}},$$