## Abstract

Analysis of Hartmann–Shack wavefront sensors for the eye is traditionally performed by locating and centroiding the sensor spots. These centroids provide the gradient, which is integrated to yield the ocular aberration. Fourier methods can replace the centroid stage, and Fourier integration can replace the direct integration.
The two—demodulation and integration—can be combined to directly retrieve the wavefront, all in the Fourier domain. Now we applied this full Fourier analysis to circular apertures and real images. We performed a comparison between it and previous methods of convolution, interpolation, and Fourier demodulation. We also compared it with a centroid method, which yields the Zernike coefficients of the wavefront.
The best performance was achieved for ocular pupils with a small boundary slope or far from the boundary and acceptable results for images missing part of the pupil. The other Fourier analysis methods had much higher tolerance to noncentrosymmetric apertures.

© 2007 Optical Society of America

Full Article |

PDF Article
### Equations (11)

Equations on this page are rendered with MathJax. Learn more.

(1)
$${I}_{0}\left(x,y\right)={\displaystyle {\sum}_{m\mathrm{\text{,}}n}{a}_{m\mathrm{\text{,}}n}}\left(\text{cos \hspace{0.17em}}\text{2}\pi mx/P+\text{cos \hspace{0.17em}}\text{2}\pi ny/P\right)\text{,}$$
(2)
$${I}_{0}\left(x,y\right)\approx \mathrm{cos}\text{\hspace{0.17em}}kx+\mathrm{cos}\text{\hspace{0.17em}}ky\text{.}$$
(3)
$$I\left(x,y\right)\approx \mathrm{cos}\left[kx+F{W}_{x}\left(x,y\right)\right]+\mathrm{cos}\left[ky+F{W}_{y}\left(x,y\right)\right]\text{,}$$
(4)
$$\mathcal{F}\left\{I\left(x,y\right)\right\}\approx \mathcal{F}\left\{\text{exp}\left(-iF{W}_{x}\right)\right\}\delta \left(u-k,v\right)+\mathcal{F}\left\{\text{exp}\left(iF{W}_{x}\right)\right\}\times \delta \left(u+k,v\right)+\mathcal{F}\left\{\text{exp}\left(-iF{W}_{y}\right)\right\}\delta \left(u,v-k\right)+\mathcal{F}\left\{\text{exp}\left(iF{W}_{y}\right)\right\}\delta \left(u,v+k\right)\text{,}$$
(5)
$$\mathcal{F}\left\{{I}_{x}^{\prime}\left(x,y\right)\right\}\approx \mathcal{F}\left\{\text{exp}\left(-iF{W}_{x}\right)\right\}\delta \left(u,v\right)$$
(6)
$$\mathcal{F}\left\{W\left(x,y\right)\right\}=-\left[iu\mathcal{F}\left\{{W}_{x}\left(x,y\right)\right\}+iv\mathcal{F}\left\{{W}_{y}\left(x,y\right)\right\}\right]/\left({u}^{2}+{v}^{2}\right)\text{.}$$
(7)
$$\text{exp}\left(iF{W}_{x}\right)\approx 1+iF{W}_{x}\text{,}$$
(8)
$$\mathcal{F}\left\{{I}^{cx}\left(x,y\right)\right\}=\mathcal{F}\left\{\text{exp}\left(iF{W}_{x}\right)\right\}\text{,}$$
(9)
$$\delta \left(u,v\right)\approx \mathcal{F}\left\{1+iF{W}_{x}\right\}\delta \left(u,v\right)\text{.}$$
(10)
$$\mathcal{F}\left\{iF{W}_{x}\left(x,y\right)\right\}\approx \left[\mathcal{F}\left\{\text{exp}\left(iF{W}_{x}\right)\right\}-\mathcal{F}*\left\{\text{exp}\left(iF{W}_{x}\right)\right\}\right]/2\text{.}$$
(11)
$$\mathcal{F}\left\{I\prime \left(r\right)\right\}=\left[\mathcal{F}\left\{{I}_{R}^{\prime}\left(r\right)\right\}-\mathcal{F}\left\{{I}_{R}^{\prime}\left(-r\right)\right\}+i\left(\mathcal{F}\left\{{I}_{t}^{\prime}\left(r\right)\right\}+\mathcal{F}\left\{{I}_{t}^{\prime}\left(-r\right)\right\}\right)\right]/2\text{,}$$