## Abstract

Here we describe a new method for numerically reconstructing an object with variable viewing angles from its hologram(s) within the Fresnel domain. The proposed algorithm can render the real image of the original object not only with different focal lengths but also with changed viewing angles. Some representative simulation results and demonstrations are presented to verify the effectiveness of the algorithm.

©2002 Optical Society of America

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### Equations (10)

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(1)
$$E({x}_{o},{y}_{o},{z}_{o})=\frac{{\mathit{iE}}_{0}}{\lambda}\int \int \tau \left(x,y\right)\mathrm{exp}\left(\mathit{iky}\phantom{\rule{.2em}{0ex}}\mathrm{sin}\theta \right)\frac{\mathrm{exp}\left[\mathit{ikr}(x,y,{x}_{o},{y}_{o})\right]}{r(x,y,{x}_{o},{y}_{o})}\chi (x,y,{x}_{o},{y}_{o})\mathit{dxdy}$$
(2)
$$r=\sqrt{{\left({z}_{o}-y\mathrm{sin}\theta \right)}^{2}+{\left({x}_{o}-x\right)}^{2}+{\left({y}_{o}-y\mathrm{cos}\theta \right)}^{2}}$$
(3)
$$E({x}_{o},{y}_{o},{z}_{o})=\mathrm{exp}\left({\mathit{ikr}}_{o}\right)\int \int \tau \left(x,y\right)\mathrm{exp}\left[\frac{\mathit{ik}}{2{r}_{o}}\left({x}^{2}+{y}^{2}\right)\right]$$
(3)
$$\phantom{\rule{5.2em}{0ex}}\times \mathrm{exp}\left\{-\frac{\mathit{ik}}{{r}_{o}}\left[{x}_{o}x+{y}_{o}y\mathrm{cos}\theta +\left({z}_{o}-{r}_{o}\right)y\phantom{\rule{.2em}{0ex}}\mathrm{sin}\theta \right]\right\}\mathit{dxdy}$$
(4)
$$\frac{\mathit{ik}}{2{r}_{o}}\left({x}^{2}+{y}^{2}\right)\approx \frac{\mathit{ik}}{2{z}_{o}}\left({x}^{2}+{y}^{2}\right)$$
(5)
$$\xi =\frac{{x}_{o}}{{\mathit{\lambda r}}_{o}}$$
(6)
$$\eta =\frac{1}{\lambda {r}_{o}}\left[{y}_{o}\mathrm{cos}\theta +\left({z}_{o}-{r}_{o}\right)\mathrm{sin}\theta \right]$$
(7)
$$E(\xi ,\eta ,{z}_{o})=\mathrm{exp}\left({\mathit{ikr}}_{o}\right)\int \tau \left(x,y\right)\mathrm{exp}\left[\frac{\mathit{ik}}{2{z}_{o}}\left({x}^{2}+{y}^{2}\right)\right]\mathrm{exp}\left[-i2\pi \mathit{\left(}\mathit{\xi x}+\mathit{\eta y}\right)\right]\mathit{dxdy}$$
(8)
$$\Delta {f}_{{x}_{o}}=\frac{1}{{\mathit{\delta x}}_{o}}\cong \frac{\mathit{Np}}{{\mathit{\lambda z}}_{o}},\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}\Delta {f}_{{y}_{o}}=\frac{1}{{\mathit{\delta y}}_{o}}\cong \frac{\mathit{Np}\phantom{\rule{.2em}{0ex}}\mathrm{cos}\theta}{{\mathit{\lambda z}}_{o}}$$
(9)
$$P=\alpha {z}_{o}\Delta {f}_{{x}_{o}}\xb7\alpha {z}_{o}\Delta {f}_{{y}_{o}}\cong {N}^{2}\phantom{\rule{.2em}{0ex}}\mathrm{cos}\phantom{\rule{.2em}{0ex}}\theta $$