## Abstract

Holography is reformulated by using the framework of phase-space optics.
The Leith–Upatnieks off-axis reference hologram is compared with precursors,
namely, single-sideband holography. The phase-space representation of complex amplitudes focuses on similarities between different holographic recording schemes and is particularly useful for investigating the degree of freedom and the space–bandwidth product of optical signals and systems. This allows one to include computer-generated holography and recent developments in digital holography in the discussion.

© 2008 Optical Society of America

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### Equations (13)

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(1)
$$W\left(x,\nu \right)={\displaystyle {\int}_{-\infty}^{\infty}u\left(x+\frac{x\prime}{2}\right)u*}\left(x-\frac{x\prime}{2}\right)\text{exp}\left(-i2\pi \nu x\prime \right)\mathrm{d}x\prime .$$
(2)
$${\int}_{-\infty}^{\infty}W\left(x,\nu \right)\mathrm{d}\nu ={\left|u\left(x\right)\right|}^{2},\text{\hspace{1em}}}{\displaystyle {\int}_{-\infty}^{\infty}W}\left(x,\nu \right)\mathrm{d}x={\left|\overline{u}\left(\nu \right)\right|}^{2}\text{.$$
(3)
$${W}_{u}\left(x,\nu \right)={\displaystyle {\int}_{-\infty}^{\infty}{W}_{g}\left(x,\nu \prime \right){W}_{h}\left(x,\nu -\nu \prime \right)\mathrm{d}\nu \prime ={W}_{g}\left(x,\nu \right)\text{\hspace{0.17em}}{\ast}_{\nu}\text{\hspace{0.17em}}{W}_{h}\left(x,\nu \right)}.$$
(4)
$${W}_{{\left|u\right|}^{2}}\left(x,\nu \right)={W}_{u}\left(x,\nu \right)\text{\hspace{0.17em}}{\ast}_{\nu}\text{\hspace{0.17em}}{W}_{u}\left(x\text{,}-\nu \right).$$
(5)
$$W\left(x,\nu ;\text{\hspace{0.17em}}z\right)=W\left(x-\lambda z\nu ,\nu ;\text{\hspace{0.17em}}0\right),$$
(6)
$${W}_{L+}\left(x,\nu \right)={W}_{L-}\left(x,\nu +x/\lambda f\right),$$
(7)
$${W}_{2f}\left(x,\nu \right)={W}_{u}\left(-\lambda f\nu ,x/\lambda f\right).$$
(8)
$$u\left(x\right)=\text{exp}\left[i\text{2}\pi \left(\alpha {x}^{2}+\beta x+\gamma \right)\right].$$
(9)
$${W}_{u}\left(x,\nu \right)=\delta \left(\nu -\alpha x-\beta \right),$$
(10)
$${W}_{O}\left(x,\nu \right)\approx \delta \left[\nu -\frac{1}{2\pi}\text{\hspace{0.17em}}\frac{\partial \varphi \left(x\right)}{\partial x}\right],$$
(11)
$$\Delta {x}_{z}=\Delta x+\lambda z\Delta \nu .$$
(12)
$${S}_{F}=2\left[{S}_{O}+{{N}_{F}}^{-1}\right],$$
(13)
$${N}_{F}={\left(\lambda z\Delta {\nu}^{2}\right)}^{-1}.$$