## Abstract

We present an experimental configuration for phase retrieval from a set of intensity measurements. The key component is a spatial light modulator located in the Fourier domain of an imaging system. It performs a linear filter operation that is associated to the process of propagation in the image plane. In contrast to the state of the art, no mechanical adjustment is required during the recording process, thus reducing the measurement time considerably. The method is experimentally demonstrated by investigating a wave field scattered by a diffuser, and the results are verified by comparing them to those obtained from standard interferometry.

© 2010 Optical Society of America

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### Equations (10)

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(1)
$${u}_{b}(\overrightarrow{x})=({u}_{a}\otimes {h}_{z})(\overrightarrow{x}).$$
(2)
$${\widehat{u}}_{b}(\overrightarrow{\xi})={\widehat{u}}_{a}(\overrightarrow{\xi})\xb7{\widehat{h}}_{z}(\overrightarrow{\xi}),$$
(3)
$${\widehat{h}}_{z}(\overrightarrow{\xi})=\mathrm{exp}\left[i\frac{2\pi z}{\lambda}\sqrt{1-{\lambda}^{2}|\overrightarrow{\xi}{|}^{2}}\right].$$
(4)
$$u(\overrightarrow{v})=\frac{1}{i\lambda f}\xb7\mathcal{F}\{u(\overrightarrow{u})\}\left(\frac{\overrightarrow{v}}{\lambda f}\right).$$
(5)
$${t}_{z}(\overrightarrow{v})=\mathrm{exp}\left[i\frac{2\pi z}{\lambda}\sqrt{1-\frac{1}{{f}^{2}}|\overrightarrow{v}{|}^{2}}\right].$$
(6)
$${\overline{t}}_{z}(\overrightarrow{v})={t}_{z}(\overrightarrow{v})\xb7\sum _{m,n}\delta ({v}_{i}-m\mathrm{\Delta}p,{v}_{j}-n\mathrm{\Delta}p).$$
(7)
$${\overline{t}}_{z}[m,n]=\mathrm{exp}\left[i\frac{2\pi z}{\lambda}\sqrt{1-\frac{\mathrm{\Delta}{p}^{2}}{{f}^{2}}({m}^{2}+{n}^{2})}\right].$$
(8)
$${\tilde{u}}_{m}^{(k)}(\overrightarrow{x})={\mathcal{F}}^{-1}\{{\widehat{u}}_{n}^{(k)}(\overrightarrow{\xi})\xb7{\widehat{h}}_{z}(\overrightarrow{\xi})\}.$$
(9)
$${u}_{m}^{(k)}(\overrightarrow{x})=\sqrt{{I}_{m}(\overrightarrow{x})}\xb7{\tilde{u}}_{m}^{(k)}(\overrightarrow{x})\xb7|{\tilde{u}}_{m}^{(k)}(\overrightarrow{x}){|}^{-1}.$$
(10)
$${A}_{S}(\overrightarrow{x}){A}_{R}(\overrightarrow{x})\mathrm{exp}[-i({\varphi}_{S}(\overrightarrow{x})-{\varphi}_{R}(\overrightarrow{x})+2\pi \overrightarrow{g}\xb7\overrightarrow{x})]={\mathcal{F}}^{-1}\{\mathcal{F}\{{I}_{C}(\overrightarrow{x})\}\xb7{P}_{W}(\overrightarrow{\xi})\}.$$