## Abstract

The noise characteristics of a compact high-resolution Shack–Hartmann wavefront sensor with an extended pupil to sensor distance have been measured. The standard deviation, *σ*, of the angular position error caused by random noise conforms to theoretical predictions of discrete detector arrays described by ${\sigma}_{\text{tot}}=\lambda /D(\omega /\mathrm{SNR}+\eta /\sqrt{{V}_{s}})$, where *ω* is the position error constant, SNR is the signal-to-noise ratio, *η* is the geometric aperture constant, and ${V}_{s}$ is the sum of the signal’s total counts. The agreement both confirms the theoretical derivation of this useful formula and shows that measurement of ocular aberrations under laboratory conditions at working distances greater than $30\text{\hspace{0.17em}}\mathrm{cm}$ introduces negligible additional error.

© 2010 Optical Society of America

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### Equations (8)

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(1)
$$\sigma =\frac{3\pi}{16}\xb7\frac{\lambda}{D}\xb7\frac{1}{\mathrm{SNR}},$$
(2)
$$({x}_{i},{y}_{i})=(\frac{\sum \sum {x}_{nm}{I}_{nm}}{\sum {I}_{nm}},\frac{\sum \sum {y}_{nm}{I}_{nm}}{\sum {I}_{nm}}),$$
(3)
$$I(\vartheta )={I}_{0}\left(\frac{\mathrm{sin}(\pi {L}_{1}\mathrm{sin}{\theta}_{1}/\lambda )}{\pi {L}_{1}\mathrm{sin}{\theta}_{1}/\lambda}{)}^{2}\right(\frac{\mathrm{sin}(\pi {L}_{2}\mathrm{sin}{\theta}_{2}/\lambda )}{\pi {L}_{2}\mathrm{sin}{\theta}_{2}/\lambda}{)}^{2},$$
(4)
$$I={I}_{0}(-\frac{{r}^{2}}{2{\sigma}^{2}})={I}_{0}(-\frac{(f\mathrm{tan}{\theta}_{1}{)}^{2}}{2(\lambda f/Dn{)}^{2}}),$$
(5)
$${\varphi}_{cs}^{2}=\frac{{G}_{s}^{2}}{{V}_{s}},$$
(6)
$${\varphi}_{cr}^{2}={L}_{1}{L}_{2}\frac{{N}_{r}^{2}({L}_{1}^{2}-1)}{12{V}_{r}^{2}},$$
(7)
$${\sigma}_{\text{tot}}=\frac{\lambda}{D}(\frac{\omega}{\mathrm{SNR}}+\frac{\eta}{{V}_{s}^{(1/2)}}),$$
(8)
$$\omega =(\frac{({L}^{2}-1)}{12{L}^{2}}{)}^{1/2}\frac{l/f}{\lambda /D}\xb7\frac{{L}^{2}{N}_{b}}{({L}^{2}{N}_{b}+{V}_{s}^{1/2})}$$