## Abstract

A simulation study is presented that evaluates the performance of Hartmann wave-front sensors with measurements obtained with the Fried geometry and the Hutchin geometry. Performance is defined in terms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstruction is used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations is performed to identify the strengths and the weaknesses of Hartmann sensors used in each of the two geometries in the two-dimensional space of the Fried parameter *r*
_{0} and the Rytov parameter. We found that the performance of Hartmann sensors degrades severely when the Rytov number exceeds 0.2 and the ratio *l*/
*r*
_{0} exceeds 1/4 (where *l* is the subaperture side length) because of the presence of branch points in the phase function and the effect of amplitude scintillation on the measurement values produced by the Hartmann sensor.

© 2002 Optical Society of America

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### Equations (10)

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(1)
$$\mathit{\xdb}\left(\overline{r}\prime \right)=\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left[U\left(\overline{r}\right)\right].$$
(2)
$$S\left\{{U}_{1}\left(\overline{r}\right),{U}_{2}\left(\overline{r}\right)\right\}=\frac{{\left|{\displaystyle \sum _{k\in \mathcal{K}\prime}}{U}_{1}\left({\overline{r}}_{k}\right){U}_{2}^{*}\left({\overline{r}}_{k}\right)\right|}^{2}}{\left[{\displaystyle \sum _{k\in \mathcal{K}\prime}}{U}_{1}\left({\overline{r}}_{k}\right){U}_{1}*\left({\overline{r}}_{k}\right)\right]\left[{\displaystyle \sum _{k\in \mathcal{K}\prime}}{U}_{2}\left({\overline{r}}_{k}\right){U}_{2}*\left({\overline{r}}_{k}\right)\right]}.$$
(3)
$${s}_{C}=\frac{\int \mathrm{d}\overline{r}W\left(\overline{r}\right)I\left(\overline{r}\right)\nabla \mathrm{\varphi}\left(\overline{r}\right)}{\int \mathrm{d}\overline{r}W\left(\overline{r}\right)I\left(\overline{r}\right)},$$
(4)
$${s}_{G}=\frac{\int \mathrm{d}\overline{r}W\left(\overline{r}\right)\nabla \mathrm{\varphi}\left(\overline{r}\right)}{\int \mathrm{d}\overline{r}W\left(\overline{r}\right)}.$$
(5)
$$\mathcal{L}\mathcal{S}\left(\mathrm{\varphi}\right)={\left({G}^{T}G\right)}^{-1}{G}^{T}\mathcal{P}\mathcal{V}\left(G\mathrm{\varphi}\right),$$
(6)
$${\mathrm{\varphi}}_{\perp}=\mathcal{P}\mathcal{V}\left\{\mathrm{arg}\left[exp\left(i\mathrm{\varphi}\right)exp\left(-i{\mathrm{\varphi}}_{\Vert}\right)\right]\right\}.$$
(7)
$$\mathcal{S}\left\{exp\left[i{\mathrm{\varphi}}_{\Vert}\left(\overline{r}\right)\right],exp\left[i\mathrm{arg}\left(\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left\{exp\left[i{\mathrm{\varphi}}_{\Vert}\left(\overline{r}\right)\right]\right\}\right)\right]\right\},$$
(8)
$$\mathcal{S}\left\{exp\left[i{\mathrm{\varphi}}_{\perp}\left(\overline{r}\right)\right],exp\left[i\mathrm{arg}\left(\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left\{exp\left[i{\mathrm{\varphi}}_{\perp}\left(\overline{r}\right)\right]\right\}\right)\right]\right\}$$
(9)
$${Q}_{\mathrm{scint},\Vert}=\frac{\mathcal{S}\left\{exp\left[i{\mathrm{\varphi}}_{\Vert}\left(\overline{r}\right)\right],exp\left[i\mathrm{arg}\left(\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left\{A\left(\overline{r}\right)exp\left[i{\mathrm{\varphi}}_{\Vert}\left(\overline{r}\right)\right]\right\}\right)\right]\right\}}{\mathcal{S}\left\{exp\left[i{\mathrm{\varphi}}_{\Vert}\left(\overline{r}\right)\right],exp\left[i\mathrm{arg}\left(\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left\{exp\left[i{\mathrm{\varphi}}_{\Vert}\left(\overline{r}\right)\right]\right\}\right)\right]\right\}},$$
(10)
$${Q}_{\mathrm{scint},\perp}=\frac{\mathcal{S}\left\{exp\left[i{\mathrm{\varphi}}_{\perp}\left(\overline{r}\right)\right],exp\left[i\mathrm{arg}\left(\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left\{A\left(\overline{r}\right)exp\left[i{\mathrm{\varphi}}_{\perp}\left(\overline{r}\right)\right]\right\}\right)\right]\right\}}{\mathcal{S}\left\{exp\left[i{\mathrm{\varphi}}_{\perp}\left(\overline{r}\right)\right],exp\left[i\mathrm{arg}\left(\mathrm{\Omega}_{\mathrm{wfs}}{}^{\mathrm{rec}\prime}\left\{exp\left[i{\mathrm{\varphi}}_{\perp}\left(\overline{r}\right)\right]\right\}\right)\right]\right\}}.$$