## Abstract

We present an innovative approach that allows superresolved images to be obtained by axial moving of two gratings and time integrating in the detector plane. The two gratings do not have to be in contact with either the object or the detector, and both are positioned between the object and the image planes. One of the main applications for the proposed approach in contrast to previously discussed time multiplexing superresolving methods is that it may fit well to superresolved imaging of remote objects, since both gratings are not in contact with either the object or the detector planes.

© 2007 Optical Society of America

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### Equations (13)

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(1)
$${U}_{{z}_{1}}\left(x\right)=\int G\left(\nu \right)\mathrm{exp}\left(j\pi \lambda {z}_{1}{\nu}^{2}\right)\mathrm{exp}\left(j2\pi x\nu \right)\mathrm{d}\nu ,$$
(2)
$$\sum _{n}{A}_{n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-j2\pi {\nu}_{0}nx).$$
(3)
$${\widehat{U}}_{0}\left(x\right)=\sum _{n}{A}_{n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-j\pi \lambda {z}_{1}{n}^{2}{\nu}_{0}^{2})\mathrm{exp}(-j2\pi n{\nu}_{0}x)g(x+\lambda {z}_{1}n{\nu}_{0}).$$
(4)
$${\widehat{U}}_{4F}\left({x}^{\prime}\right)=\sum _{n}{A}_{n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(\pi i\lambda {z}_{1}{n}^{2}{\nu}_{0}^{2}\right)\bullet \int \mathrm{rect}\left(\frac{\lambda F\nu}{\Delta}\right)G(-\nu +n{\nu}_{0})\mathrm{exp}[-j2\pi n\lambda {z}_{1}{\nu}_{0}\nu ]\mathrm{exp}\left(j2\pi \nu {x}^{\prime}\right)\mathrm{d}\nu ,$$
(5)
$$\sum _{m}{B}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-j2\pi {\nu}_{0}mx).$$
(6)
$${U}_{4F}\left({x}^{\prime}\right)=\sum _{n,m}{A}_{n}{B}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(j\pi \lambda {\nu}_{0}^{2}({z}_{1}{n}^{2}-{z}_{2}{m}^{2}-2{z}_{1}nm)\right)\bullet \int \mathrm{rect}\left(\frac{\nu +m{\nu}_{0}}{\Delta \u2215\lambda F}\right)G(-\nu +(n-m){\nu}_{0})\mathrm{exp}(-j2\pi \lambda \nu {\nu}_{0}({z}_{1}n+{z}_{2}m))\mathrm{exp}\left(j2\pi \nu {x}^{\prime}\right)\mathrm{d}\nu .$$
(7)
$${z}_{1}=-{z}_{2}=z\left(t\right).$$
(8)
$${U}_{4F}({x}^{\prime},t)=\sum _{n,m}{A}_{n}{B}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left[j\pi \lambda {\nu}_{0}^{2}z\left(t\right){(n-m)}^{2}\right]\bullet \int \mathrm{rect}\left(\frac{\nu +m{\nu}_{0}}{\Delta \u2215\lambda F}\right)G(-\nu +(n-m){\nu}_{0})\mathrm{exp}[-j2\pi \lambda \nu {\nu}_{0}z\left(t\right)(n-m)]\mathrm{exp}\left(j2\pi \nu {x}^{\prime}\right)\mathrm{d}\nu .$$
(9)
$$\frac{1}{T}{\int}_{0}^{T}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left\{j\pi \lambda {\nu}_{0}z\left(t\right)[{\nu}_{0}{(n-m)}^{2}-2\nu (n-m)]\right\}\mathrm{d}t={\delta}_{n,m}.$$
(10)
$$\u27e8{U}_{4F}({x}^{\prime},t)\u27e9=\int \left[\sum _{n}{A}_{n}{B}_{n}\phantom{\rule{0.2em}{0ex}}\mathrm{rect}\left(\frac{\nu +n{\nu}_{0}}{\Delta \u2215\lambda F}\right)\right]G(-\nu )\mathrm{exp}\left(j2\pi \nu {x}^{\prime}\right)\mathrm{d}\nu .$$
(11)
$$T>\frac{1}{\lambda V{\nu}_{0}^{2}},$$
(12)
$$\mathrm{OTF}\left(\nu \right)=\int \sum _{n,{n}^{\prime}}{A}_{n}{B}_{n}{A}_{{n}^{\prime}}^{*}{B}_{{n}^{\prime}}^{*}\mathrm{rect}\left(\frac{\mu +\nu \u22152+n{\nu}_{0}}{\Delta \u2215\lambda F}\right)\mathrm{rect}\left(\frac{\mu -\nu \u22152+{n}^{\prime}{\nu}_{0}}{\Delta \u2215\lambda F}\right)\mathrm{d}\mu .$$
(13)
$${Z}_{c}=\frac{\delta {x}^{2}N}{\lambda}=\frac{{L}_{x}\delta x}{\lambda},$$