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

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References

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  1. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997).
  2. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).
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    [CrossRef]
  4. Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4567 (2004).
    [CrossRef]
  5. A. Talmi and E. N. Ribak, "Direct demodulation of Hartmann-Shack patterns," J. Opt. Soc. Am. A 21, 632-639 (2004).
    [CrossRef]
  6. A. Talmi and E. N. Ribak, "Wavefront reconstruction from its gradients," J. Opt. Soc. Am. A 23, 288-297 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  8. K. R. Freischlad and C. L. Koliopoulos, "Modal estimation of a wavefront from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-1861 (1986).
    [CrossRef]
  9. P. M. Prieto, F. Vargas-Marin, S. Goelz, and P. Artal, "Analysis of the performance of the Hartmann-Shack sensor in the human eye," J. Opt. Soc. Am. A 17, 1388-1398 (2000).
    [CrossRef]

2006 (1)

2004 (2)

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4567 (2004).
[CrossRef]

A. Talmi and E. N. Ribak, "Direct demodulation of Hartmann-Shack patterns," J. Opt. Soc. Am. A 21, 632-639 (2004).
[CrossRef]

2003 (1)

Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
[CrossRef]

2000 (1)

1991 (1)

1986 (1)

Artal, P.

Carmon, Y.

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4567 (2004).
[CrossRef]

Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
[CrossRef]

Freischlad, K. R.

Goelz, S.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

Koliopoulos, C. L.

Prieto, P. M.

Ribak, E. N.

A. Talmi and E. N. Ribak, "Wavefront reconstruction from its gradients," J. Opt. Soc. Am. A 23, 288-297 (2006).
[CrossRef]

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4567 (2004).
[CrossRef]

A. Talmi and E. N. Ribak, "Direct demodulation of Hartmann-Shack patterns," J. Opt. Soc. Am. A 21, 632-639 (2004).
[CrossRef]

Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215, 285-288 (2003).
[CrossRef]

Roddier, C.

Roddier, F.

Talmi, A.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997).

Vargas-Marin, F.

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Figures (5)

Fig. 1
Fig. 1

(a) Simulated image and (b) its Fourier transform. The distance of the first sidelobes from the Fourier origin is equal to the number of spots across the pattern. Fourier modulation consists in isolating each circled sidelobe, centering it, and transforming it back to get the corresponding slope component. The resultant wavefronts are from five processing methods (c) traditional centroiding, (d) convolution, (e) smoothing, (f) Fourier demodulation, and (g) fast Fourier demodulation. All the methods consisted of first obtaining the wavefront slopes. In (c), Zernike polynomials were fit to the slopes, in (d)–(f) integrating the slopes in the Fourier domain, and in (g) all processing was achieved in the Fourier domain. Note the different residual tilts.

Fig. 2
Fig. 2

Defining a centrosymmetric pupil: (a) original image, (b) calculated pupil edge, (c) calculated box containing all the image points.

Fig. 3
Fig. 3

Comparison of wavefronts: (a) reference image, (b) part of the image used in analysis from the eye's measured spot pattern, (c)–(g) results for the five methods corresponding to Figs. 1(c)–1(g).

Fig. 4
Fig. 4

As in Fig. 3 but with the bottom edge cut by the camera. The fast Fourier demodulation picked up the opposite sidelobes and inverted the sign.

Fig. 5
Fig. 5

Comparison of the Zernike values for the five methods as in Figs. 2–4. Shown from top are the absolute values of (a) defocus, (b) spherical aberration, (c) vertical astigmatism, (d) vertical coma, (e) vertical trefoil. The apertures were limited to 2.70, 3.50, 3.40, 3.04, 3.30, 3.06, and 2.98 mm. The largest pupils show the largest errors, as more boundary values were included.

Equations (11)

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I 0 ( x , y ) = m , n a m , n ( cos   2 π m x / P + cos   2 π n y / P ) ,
I 0 ( x , y ) cos k x + cos k y .
I ( x , y ) cos [ k x + F W x ( x , y ) ] + cos [ k y + F W y ( x , y ) ] ,
{ I ( x , y ) } { exp ( i F W x ) } δ ( u k , v ) + { exp ( i F W x ) } × δ ( u + k , v ) + { exp ( i F W y ) } δ ( u , v k ) + { exp ( i F W y ) } δ ( u , v + k ) ,
{ I x ( x , y ) } { exp ( i F W x ) } δ ( u , v )
{ W ( x , y ) } = [ i u { W x ( x , y ) } + i v { W y ( x , y ) } ] / ( u 2 + v 2 ) .
exp ( i F W x ) 1 + i F W x ,
{ I c x ( x , y ) } = { exp ( i F W x ) } ,
δ ( u , v ) { 1 + i F W x } δ ( u , v ) .
{ i F W x ( x , y ) } [ { exp ( i F W x ) } * { exp ( i F W x ) } ] / 2 .
{ I ( r ) } = [ { I R ( r ) } { I R ( r ) } + i ( { I t ( r ) } + { I t ( r ) } ) ] / 2 ,

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