## Abstract

Adaptive optics takes its servo feedback error cue from a wavefront sensor.
The common Hartmann–Shack spot grid that represents the wavefront slopes is usually analyzed by finding the spot centroids. In a novel application,
we used the Fourier decomposition of a spot pattern to find deviations from grid regularity. This decomposition was performed either in the Fourier domain or in the image domain, as a demodulation of the grid of spots. We analyzed the system, built a control loop for it, and tested it thoroughly. This allowed us to close the loop on wavefront errors caused by turbulence in the optical system.

© 2007 Optical Society of America

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### Equations (61)

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(6)
{H}^{\#}={\left(H\prime H\right)}^{-1}H\prime
(7)
{H}^{\#}\left(H{r}_{a}-w\right)={r}_{a}-{H}^{\#}w
(12)
{C}_{1}\left(z\right)
(13)
30\text{\hspace{0.17em} Hz}
(14)
{C}_{1}\left(z\right)
(17)
$$E\left(z\right)=S\left(z\right)\left[{R}_{a}\left(z\right)-D\left(z\right)\right]+T\left(z\right){H}^{\#}N\left(z\right)\text{,}$$
(18)
S\left(z\right)={\left[I+{C}_{1}\left(z\right)\right]}^{-1}
(19)
T\left(z\right)={\left[1+{C}_{1}\left(z\right)\right]}^{-1}{C}_{1}\left(z\right)=I-S\left(z\right)
(20)
{C}_{1}\left(z\right)
(22)
{\mathrm{lim}}_{t\to \infty}\text{\hspace{0.17em}}{e}_{z}\left(t\right)=0
(26)
{C}_{1}\left(1\right)
(27)
T\left({\mathrm{e}}^{\mathrm{j}\mathit{\theta}}\right)
(28)
\theta \text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\left[-\pi ,\pi \right]
(29)
\left|T\left(-1\right)\right|<\alpha
(31)
\left|T\left(z\right)\right|
(32)
$${C}_{1}\left(z\right)={z}^{-1}{k}_{p}\left(1+\frac{{k}_{a}z}{z-1}\right)I,$$
(35)
{C}_{1}\left(z\right)
(36)
$$\begin{array}{c}{k}_{p}>-1,\hfill \end{array}\begin{array}{c}{k}_{p}{k}_{a}>0,\hfill \end{array}$$
(37)
$$\begin{array}{c}{k}_{p}\left(2+{k}_{a}\right)<\frac{2\alpha}{1+\alpha},\hfill \end{array}$$
(38)
$$\begin{array}{c}\begin{array}{cc}{k}_{p}\left(2+{k}_{a}\right)>-\frac{2\alpha}{1-\alpha}\text{,}& \text{if \hspace{0.17em}}\alpha <\frac{1}{2}.\end{array}\hfill \end{array}$$
(39)
\left({k}_{p},{k}_{a}\right)
(40)
\alpha \text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\infty
(43)
25\text{\hspace{0.17em}}x\text{-}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{e}
(44)
25\text{\hspace{0.17em}}y\text{-}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{e}
(47)
{e}^{\mathrm{j}{k}_{0}x}
(50)
80\text{\hspace{0.17em} ms}
(51)
\text{30 \hspace{0.17em} Hz}
(52)
\text{600 \hspace{0.17em} ms}
(53)
\text{2500 \hspace{0.17em} ms}
(54)
\left({k}_{p},{k}_{a}\right)
(55)
\left({k}_{p},{k}_{a}\right)
(56)
0.3\text{\hspace{0.17em} \mu m}
(57)
\text{15 \hspace{0.17em} mm}