## Abstract

The performance of long distance imaging systems is typically degraded by phase errors imparted by atmospheric turbulence. In this paper we apply coherent imaging methods to determine, and remove, these phase errors by digitally processing coherent recordings of the image data. In this manner we are able to remove the effects of atmospheric turbulence without needing a conventional adaptive optical system. Digital holographic detection is used to record the coherent, complex-valued, optical field for a series of atmospheric and object realizations. Correction of atmospheric phase errors is then based on maximizing an image sharpness metric to determine the aberrations present and correct the underlying image. Experimental results that demonstrate image recovery in the presence of turbulence are presented. Results obtained with severe turbulence that gives rise to anisoplanatism are also presented.

© 2009 Optical Society of America

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### Equations (17)

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(1)
$$I={\mid \left(F+G\right)\mid}^{2}$$
(2)
$$={\mid F\mid}^{2}+{\mid G\mid}^{2}+{\mathrm{FG}}^{*}+{F}^{*}G.$$
(3)
$$\mathrm{FT}\left(I\right)=f\otimes {f}^{*}+g\otimes {g}^{*}+f\otimes {g}^{*}+{f}^{*}\otimes g,$$
(4)
$$\mathrm{FT}\left(I\right)=f\otimes {f}^{*}+\delta \left(x\right)+f\left(x-b\right)+{f}^{*}\left(x+b\right).$$
(5)
$$I\left(m,n\right)={\mid \int \int t\left(x,y\right){e}^{i\varphi \left(x,y\right)}h(m,n;x,y)\mathrm{dxdy}\mid}^{2}.$$
(6)
$$\u3008I\left(m,n\right)\u3009=\int \int {\mid t\left(x,y\right)\mid}^{2}{\mid h(m,n;x,y)\mid}^{2}\mathrm{dxdy}.$$
(7)
$$I\left(m,n\right)=V\left(m,n\right){I}_{s}\left(m,n\right).$$
(8)
$$<{I}^{r}\left(m,n\right)>=r!{V}^{r}\left(m,n\right).$$
(9)
$$S=\underset{m,n}{\Sigma}{I}^{2}\left(m,n\right)$$
(10)
$$S=\underset{m,n}{\Sigma}{V}^{2}(m,n){I}_{s}^{2}(m,n),$$
(11)
$$<S>=2\underset{m,n}{\Sigma}{V}^{2}\left(m,n\right).$$
(12)
$${\sigma}^{2}=20\underset{m,n}{\Sigma}{V}^{4}\left(m,n\right)$$
(13)
$$d=\genfrac{}{}{0.1ex}{}{<\mathrm{S\prime}>-<S>}{\sqrt{\mathrm{\sigma \prime}}\sigma},$$
(14)
$$S=\underset{m,n}{\Sigma}{I}^{\beta}\left(m,n\right).$$
(15)
$$S=\genfrac{}{}{0.1ex}{}{\left(\underset{m,n}{\Sigma}{I}^{\beta}\left(m,n\right)\right)}{{\left(\underset{m,n}{\Sigma}I\left(m,n\right)\right)}^{\beta}}.\text{}$$
(16)
$${r}_{0}={\left(0.424{k}^{2}{C}_{n}^{2}L\right)}^{-3/5}.$$
(17)
$${\alpha}_{0}={\left(1.09{k}^{2}{C}_{n}^{2}{L}^{8\u20443}\right)}^{-3\u20445},$$