## Abstract

Improved quality of phase maps in pulsed digital holographic interferometry is demonstrated by finding the right reconstruction distance. The objective is to improve the optical phase information when the object under study is a phase object and when it is out of focus, leading to low contrast fringes in the phase map. A numerical refocusing is performed by introducing an ideal lens as a multiplication by a phase field in the Fourier domain, and then a region of maximum speckle correlation is found by comparing undisturbed and disturbed subimages in different refocused imaging planes. After finding the right reconstruction distance, a phase map of high visibility is constructed. By this technique a 30% reduction of the phase error for a flow of helium gas and a 50%
reduction of the phase error for a weak thin lens were obtained, which resulted in a significant improvement of the visual appearance of the phase maps.

© 2008 Optical Society of America

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### Equations (7)

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(1)
$$U\left(\xi ,\eta \right)=\mathrm{exp}\left(-ik\sqrt{{f}^{2}+{\xi}^{2}+{\eta}^{2}}\right)$$
(2)
$$\Delta {x}_{i2}=\Delta {x}_{i1}\text{\hspace{0.17em}}\frac{{z}_{2}}{{z}_{1}}\text{,}$$
(3)
$$\gamma =\frac{{\displaystyle \sum _{i}^{p}{\displaystyle \sum _{j}^{p}\left({I}_{1}\left(i,j\right)-{\overline{I}}_{1}\right)\left({I}_{2}\left(i,j\right)-{\overline{I}}_{2}\right)}}}{{\left\{\left[{\displaystyle \sum _{i}^{p}{\displaystyle \sum _{j}^{p}{\left({I}_{1}\left(i,j\right)-{\overline{I}}_{1}\right)}^{2}}}\right]\left[{\displaystyle \sum _{i}^{p}{\displaystyle \sum _{j}^{p}{\left({I}_{2}\left(i,j\right)-{\overline{I}}_{2}\right)}^{2}}}\right]\right\}}^{1/2}}$$
(4)
$$\Delta \varphi =\mathrm{arctan}\left[\frac{\mathrm{Re}\left(s\right)\mathrm{Im}\left(s\prime \right)-\mathrm{Im}\left(s\right)\mathrm{Re}\left(s\prime \right)}{\mathrm{Im}\left(s\right)\mathrm{Im}\left(s\prime \right)+\mathrm{Re}\left(s\right)\mathrm{Re}\left(s\prime \right)}\right]\text{,}$$
(5)
$${\sigma}_{\Delta \varphi}=\sqrt{\frac{{\pi}^{2}}{3}-\pi \text{\hspace{0.17em}}\mathrm{arcsin}\left|\mu \right|+{\mathrm{arcsin}}^{2}\left|\mu \right|-\frac{1}{2}{\displaystyle \text{\hspace{0.17em}}\sum _{n=1}^{\infty}\frac{{\mu}^{2n}}{{n}^{2}}}}\text{,}$$
(6)
$$\gamma \left(q\right)={\left|\frac{{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}P\left(b\right)P*\left(b+{A}_{p}\right)\mathrm{exp}\left[\frac{ik}{L}\text{\hspace{0.17em}}b\left(q-A\right)\right]\mathrm{exp}\left[\frac{ik}{2{L}^{2}}{\left|b\right|}^{2}\left(\alpha -{A}_{Z}\right){\mathrm{d}}^{2}b\right]}}{{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}{\left|P\left(b\right)\right|}^{2}{\mathrm{d}}^{2}b}}\right|}^{2}\text{.}$$
(7)
$$A=-\frac{mL\prime}{k}\text{\hspace{0.17em}}\frac{\partial \varphi}{\partial r}\text{,}$$