## Abstract

The actuator influence functions of a typical deformable mirror are expanded in a Zernike polynomial decomposition. This expansion is then extended to a matrix formalism that describes the modal operation of the mirror. The size of the aperture over which the Zernikes are defined affects the accuracy of the expansion. The optimum size of this aperture is found by minimizing the variance of the wave-front error.

© 1993 Optical Society of America

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### Equations (27)

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(1)
$${\omega}_{m}\phantom{\rule{0.1em}{0ex}}(x,y)=\text{\u2211}_{j=1}^{J}\phantom{\rule{0.3em}{0ex}}{m}_{j}\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.1em}{0ex}}{a}_{j}\phantom{\rule{0.1em}{0ex}},$$
(2)
$${\omega}_{m}=\mathscr{H}\mathbf{\text{a}}\phantom{\rule{0.1em}{0ex}},$$
(3)
$$\u220a=\omega -{\omega}_{m}\phantom{\rule{0.2em}{0ex}}.$$
(4)
$${{\sigma}_{\mathit{\text{wf}}}}^{2}={\u220a}^{t}\phantom{\rule{0em}{0ex}}\u220a,$$
(5)
$$\mathbf{\text{a}}={({\mathscr{H}}^{t}\mathscr{H})}^{-1}{\mathscr{H}}^{t}\phantom{\rule{0em}{0ex}}\omega ,$$
(6)
$${\omega}_{m}=\mathscr{H}\phantom{\rule{0.1em}{0ex}}{({\mathscr{H}}^{t}\mathscr{H})}^{-1}{\mathscr{H}}^{t}\phantom{\rule{0em}{0ex}}\omega .$$
(7)
$${\omega}_{m}={\mathscr{H}}^{\prime}{\mathscr{H}}^{\prime t}\phantom{\rule{0em}{0ex}}\omega ,$$
(8)
$${m}_{j}\phantom{\rule{0.1em}{0ex}}(x,y)=\text{\u2211}_{k\phantom{\rule{0.1em}{0ex}}=0}^{K-1}\phantom{\rule{0.3em}{0ex}}{c}_{\mathit{\text{jk}}}\phantom{\rule{0.1em}{0ex}}{Z}_{k}\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.1em}{0ex}},$$
(9)
$${c}_{\mathit{\text{jk}}}=\frac{1}{\pi}{\mathit{\int}}_{S}{m}_{j}\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.1em}{0ex}}{Z}_{k}\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.1em}{0ex}}\text{d}x\text{d}y\phantom{\rule{0.1em}{0ex}},$$
(10)
$$\mathscr{H}=Z\mathcal{\text{C}}\phantom{\rule{0.1em}{0ex}},$$
(11)
$${\omega}_{m}=Z\mathcal{C}\mathbf{\text{a}}\phantom{\rule{0.2em}{0ex}}.$$
(12)
$$\omega =Z\phantom{\rule{0.1em}{0ex}}\mathbf{\text{I}}\phantom{\rule{0.1em}{0ex}},$$
(13)
$${\omega}_{m}=Z\mathcal{\text{C}}\phantom{\rule{0.1em}{0ex}}{({\mathcal{\text{C}}}^{t}{Z}^{t}Z\mathcal{\text{C}})}^{-1}\phantom{\rule{0.1em}{0ex}}{\mathcal{\text{C}}}^{t}{Z}^{t}Z\phantom{\rule{0.1em}{0ex}}\mathbf{\text{I}}=Z\mathcal{\text{C}}\phantom{\rule{0.1em}{0ex}}{({\mathcal{\text{C}}}^{t}\phantom{\rule{0.1em}{0ex}}\mathcal{\text{C}})}^{-1}\phantom{\rule{0.1em}{0ex}}{\mathcal{\text{C}}}^{t}\mathbf{\text{I}}\phantom{\rule{0.1em}{0ex}},$$
(14)
$$\mathbf{\text{a}}={({\mathcal{\text{C}}}^{t}\phantom{\rule{0.1em}{0ex}}\mathcal{\text{C}})}^{-1}{\mathcal{\text{C}}}^{t}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{I}}\phantom{\rule{0.1em}{0ex}}.$$
(15)
$$\mathbf{\text{I}}=\mathcal{\text{C}}\phantom{\rule{0.1em}{0ex}}{({\mathscr{H}}^{t}\mathscr{H})}^{-1}{\mathscr{H}}^{t}\phantom{\rule{0em}{0ex}}\omega .$$
(16)
$$\mathbf{\text{I}}=\mathcal{\text{C}}\phantom{\rule{0.1em}{0ex}}\mathbf{\text{a}}\phantom{\rule{0.1em}{0ex}}.$$
(17)
$${{\sigma}_{\mathit{\text{wf}}(\text{modal})}}^{2}={\mathbf{\text{I}}}^{t}\phantom{\rule{0.1em}{0ex}}[\mathcal{I}-\mathcal{\text{C}}\phantom{\rule{0.1em}{0ex}}{({\mathcal{\text{C}}}^{t}\mathcal{\text{C}})}^{-1}\phantom{\rule{0.1em}{0ex}}{\mathcal{\text{C}}}^{t}]\phantom{\rule{0.1em}{0ex}}\mathbf{\text{I}}\phantom{\rule{0.1em}{0ex}},$$
(18)
$${{\sigma}_{\mathit{\text{wf}}(\text{zonal})}}^{2}={\omega}^{t}\phantom{\rule{0.1em}{0ex}}[\mathcal{I}-\mathscr{H}\phantom{\rule{0.1em}{0ex}}{({\mathscr{H}}^{t}\mathscr{H})}^{-1}\phantom{\rule{0.1em}{0ex}}{\mathscr{H}}^{t}]\phantom{\rule{0em}{0ex}}\omega .$$
(19)
$$s=\frac{L}{{r}_{\text{aperture}}}\phantom{\rule{0.2em}{0ex}},$$
(20)
$${{\sigma}_{inf}}^{2}=\text{\u2211}_{j=1}^{J}\phantom{\rule{0.1em}{0ex}}{{\sigma}_{j}}^{2}\phantom{\rule{0.1em}{0ex}},$$
(21)
$${\u220a}_{j}={m}_{j}-\text{\u2211}_{k=0}^{K-1}\phantom{\rule{0.2em}{0ex}}{c}_{\mathit{\text{jk}}}{Z}_{k}\phantom{\rule{0.1em}{0ex}},$$
(22)
$${{\sigma}_{j}}^{2}={{\u220a}_{j}}^{t}\phantom{\rule{0em}{0ex}}{\u220a}_{j}\phantom{\rule{0.1em}{0ex}}.$$
(23)
$${c}_{\mathit{\text{jk}}}=\frac{1}{\pi}\mathit{\int}\phantom{\rule{0.2em}{0ex}}\delta \phantom{\rule{0em}{0ex}}(x-{x}_{j},\phantom{\rule{0.1em}{0ex}}y-{y}_{j})\phantom{\rule{0.1em}{0ex}}[m\phantom{\rule{0.1em}{0ex}}(x,y)\u2605{Z}_{k}\phantom{\rule{0.1em}{0ex}}(x,y)]\phantom{\rule{0.1em}{0ex}}\text{d}x\text{d}y\phantom{\rule{0.1em}{0ex}},$$
(24)
$${Z}_{k}\simeq {m}_{j}\u2605{Z}_{K}\phantom{\rule{0.1em}{0ex}},$$
(25)
$${\stackrel{\sim}{c}}_{\mathit{\text{jk}}}={Z}_{k}\phantom{\rule{0.1em}{0ex}}({x}_{j},{y}_{j})\phantom{\rule{0.1em}{0ex}}.$$
(26)
$${Z}_{k}\simeq {\omega}_{m}=\text{\u2211}_{j=1}^{J}\phantom{\rule{0.3em}{0ex}}{\stackrel{\sim}{c}}_{\mathit{\text{jk}}}{m}_{j}\phantom{\rule{0.1em}{0ex}},$$
(27)
$$\mathcal{\text{Z}}\phantom{\rule{0.2em}{0ex}}\simeq \phantom{\rule{0.2em}{0ex}}\mathscr{H}\phantom{\rule{0.1em}{0ex}}{\{{\stackrel{\sim}{c}}_{\mathit{\text{jk}}}\}}^{t}\phantom{\rule{0.1em}{0ex}}.$$