## Abstract

The control of two deformable mirrors for compensation of time-varying fluctuations in the complex field that results from wave propagation through a turbulent medium is considered. Iterative vector space projection methods are utilized to determine the control commands to be applied to the two deformable mirrors. Convergence of the iterative algorithm is accelerated when the algorithm is initialized, at each measurement period, with the values for the phase commands obtained from the previous measurement period. Furthermore, it is found that, if the sample frequency is sufficiently greater than the Greenwood frequency, then only a single iterative step at each measurement period is required to obtain good compensation of both amplitude and phase fluctuations.

© 2002 Optical Society of America

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### Equations (9)

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(1)
$${T}_{z}\left[\xb7\right]={\mathcal{F}}^{-1}\left[\mathcal{F}\left(\xb7\right)exp\left(i\mathrm{\pi}\mathrm{\lambda}z{\overline{\mathrm{\kappa}}}^{2}\right)\right],$$
(2)
$$S=\frac{|\int \mathrm{d}{\overline{r}}_{1}{U}_{b}\left({\overline{r}}_{1}\right)exp\left[i{\mathrm{\varphi}}_{1}\left({\overline{r}}_{1}\right)\right]{T}_{z}\left\{{U}_{r}*\left({\overline{r}}_{2}\right)exp\left[-i{\mathrm{\varphi}}_{2}\left({\overline{r}}_{2}\right)\right]\right\}{|}^{2}}{|\int \mathrm{d}{\overline{r}}_{1}{U}_{b}\left({\overline{r}}_{1}\right){U}_{b}*\left({\overline{r}}_{1}\right)||\int \mathrm{d}{\overline{r}}_{2}{U}_{r}\left({\overline{r}}_{2}\right){U}_{r}*\left({\overline{r}}_{2}\right)|},$$
(3)
$$S\propto \u3008{U}_{b}exp\left(i{\mathrm{\varphi}}_{1}\right),{T}_{z}\left[{U}_{l}exp\left(i{\mathrm{\varphi}}_{2}\right)\right]\u3009.$$
(4)
$${T}_{z,\mathrm{\alpha}}\left[\xb7\right]={\mathcal{F}}^{-1}\left\{\mathcal{F}\left[\xb7\right]exp\left[i\mathrm{\pi}\mathrm{\lambda}z{\overline{\mathrm{\kappa}}}^{2}-\mathrm{sign}\left(\mathrm{\alpha}\right)\times {\left(\frac{\mathrm{\pi}\mathrm{\alpha}}{2}\right)}^{2}{\overline{\mathrm{\kappa}}}^{2}\right]\right\}.$$
(5)
$${\stackrel{\u02c6}{S}}_{\mathrm{\alpha}}\propto \u3008{U}_{b}exp\left(i{\mathrm{\varphi}}_{1}\right),{T}_{z,\mathrm{\alpha}}\left[{U}_{l}exp\left(i{\mathrm{\varphi}}_{2}\right)\right]\u3009.$$
(6)
$${\mathrm{\varphi}}_{1}=\mathrm{arg}{U}_{b}*{T}_{-z,\mathrm{\alpha}}\left[{U}_{r}exp\left(i{\mathrm{\varphi}}_{2}\right)\right].$$
(7)
$${\mathrm{\varphi}}_{2}=\mathrm{arg}{U}_{r}*{T}_{z,\mathrm{\alpha}}\left[{U}_{b}exp\left(i{\mathrm{\varphi}}_{1}\right)\right].$$
(8)
$${\mathrm{\varphi}}_{2}\left(k\right)=\mathrm{arg}{U}_{r}*\left(k\right){T}_{z,\mathrm{\alpha}}\left\{{U}_{b}\left(k\right)exp\left[i{\mathrm{\varphi}}_{1}\left(k-1\right)\right]\right\},$$
(9)
$${\mathrm{\varphi}}_{1}\left(k\right)=\mathrm{arg}{U}_{b}*\left(k\right){T}_{-z,\mathrm{\alpha}}\left\{{U}_{r}\left(k\right)exp\left[i{\mathrm{\varphi}}_{2}\left(k\right)\right]\right\}.$$