Abstract

A theoretical model for reconstruction errors in the Hartmann-Shack wave-front sensor was constructed. The measured pattern of a regular grid of spots can be analyzed by their individual centroiding or by a global Fourier demodulation. We investigate pixelization errors in the camera and Poisson errors in the camera pixels. We show that by the Fourier demodulation technique it is possible to overcome pixelization errors which occur in the traditional centroid technique. By supporting simulations we show that the two methods coincide for an infinite number of pixels per spot.

© 2006 Optical Society of America

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References

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  1. I. Ghozeil, in: Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978) Chap. 10.
  2. R. K. Tyson, Principles of Adaptive Optics, Second ed. (Academic, New York, 1998).
  3. R. K. Tyson, ed., Adaptive Optics Engineering Handbook (Marcel Decker, New York, 2000).
  4. Y. Carmon and E. N. Ribak, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Comm. 215, 285-288 (2003).
    [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. Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4657 (2004).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).
  8. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
  9. W. J. Tango and R. Q. Twiss, "Michelson stellar interferometry," in Progress in Optics XVII, E. Wolf, ed. (North Holland, 1980) 239-277.
  10. E. Ribak and E. Leibowitz, "Shearing stellar interferometer: 1. digital data analysis scheme," Appl. Opt. 24, 3088-3093 (1985).
    [CrossRef] [PubMed]
  11. M. M. Colavita, "Fringe visibility estimators for the Palomar testbed interferometer," Pub. Astron. Soc. Pac. 111, 111-117 (1999).
    [CrossRef]
  12. J. A. Zadnik and J. W. Beletic, "Effect of CCD readout noise in astronomical speckle imaging," Appl. Opt. 37, 361-368 (1998).
    [CrossRef]
  13. S. B. Howell, B. Koehn, E. Bowell, and M. Hoffman, "Detection and measurement of poorly sampled point sources imaged with 2-D array," Astron. J. 112, 1302-3611 (1996).
    [CrossRef]

Appl. Opt. (2)

E. Ribak and E. Leibowitz, "Shearing stellar interferometer: 1. digital data analysis scheme," Appl. Opt. 24, 3088-3093 (1985).
[CrossRef] [PubMed]

J. A. Zadnik and J. W. Beletic, "Effect of CCD readout noise in astronomical speckle imaging," Appl. Opt. 37, 361-368 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Carmon and E. N. Ribak, "Fast Fourier demodulation," Appl. Phys. Lett. 84, 4656-4657 (2004).
[CrossRef]

Astron. J. (1)

S. B. Howell, B. Koehn, E. Bowell, and M. Hoffman, "Detection and measurement of poorly sampled point sources imaged with 2-D array," Astron. J. 112, 1302-3611 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

Opt. Comm. (1)

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

Progress in Optics XVII (1)

W. J. Tango and R. Q. Twiss, "Michelson stellar interferometry," in Progress in Optics XVII, E. Wolf, ed. (North Holland, 1980) 239-277.

Pub. Astron. Soc. Pac. (1)

M. M. Colavita, "Fringe visibility estimators for the Palomar testbed interferometer," Pub. Astron. Soc. Pac. 111, 111-117 (1999).
[CrossRef]

Statistical Optics (1)

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

Other (4)

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

I. Ghozeil, in: Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978) Chap. 10.

R. K. Tyson, Principles of Adaptive Optics, Second ed. (Academic, New York, 1998).

R. K. Tyson, ed., Adaptive Optics Engineering Handbook (Marcel Decker, New York, 2000).

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Figures (2)

Fig. 1.
Fig. 1.

Schematic illustration of the optical system.

Fig. 2.
Fig. 2.

Comparison of the position errors between the centroid and Fourier demodulation methods for various sub-aperture sizes, as a function of the average number of photons per spot. The Fourier method (without read out noise) does not depend on the number of pixels per lenslet.

Equations (40)

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F mn = T R + ( n 1 ) d R + nd R + ( m 1 ) d R + md I ( x , y ) dxdy = T R + ( n 1 ) d R + nd I ( x ) dx R + ( m 1 ) d R + md I ( y ) dy ,
Q ( x ) = 1 2 erfc ( x 2 ) = x 1 2 π exp ( α 2 2 ) .
F nm = T { Q [ ( n 1 ) μ ρ ] Q [ ρ ] } { Q [ ( m 1 ) μ ρ ] Q [ ρ ] } ,
n 2 = n = 1 N m = 1 N n 2 F nm =
T m = 1 N { Q [ ( m 1 ) μ ρ ] Q [ ρ ] } n = 1 N n 2 { Q [ ( n 1 ) μ ρ ] Q [ ρ ] } =
T [ Q ( ρ ) Q ( ρ ) ] { n = 2 N ( 2 n 1 ) Q [ ( n 1 ) μ ρ ] + Q ( ρ ) N 2 Q ( ρ ) } =
[ Q ( ρ ) Q ( ρ ) ] 1 { n = 2 N ( 2 n 1 ) Q [ ( n 1 ) μ ρ ] N 2 Q ( ρ ) } .
σ x ̂ 2 = E x ̂ d ( N 2 , N 2 ) 2 = 2 d 2 ( E [ n 2 ] N 2 4 ) .
σ x ̂ 2 = 2 σ 2 μ 2 n = 1 N ( 2 n 1 ) Q [ ( n 1 ) μ ρ ] N 2 Q ( ρ ) Q ( ρ ) Q ( ρ ) 2 σ 2 ρ 2 .
E [ X i ] 2 = K 2 i = 1 K E [ X ] i 2 = σ X i 2 K .
σ X ̂ 2 | K = E [ X 2 ] + E [ Y 2 ] = 2 σ X ̂ i 2 K .
σ x 2 = K = 1 σ x i 2 K e λ λ K K ! .
β = K = 1 1 K · λ K K ! .
= K = 1 λ K 1 K ! = K = 0 λ K 1 K ! 1 λ = e λ 1 λ
σ x 2 = σ x i 2 exp ( λ ) [ Ei ( λ ) ln ( λ ) ] ,
σ X ̂ 2 = [ Ei ( λ ) ln ( λ ) ] exp ( λ ) 2 σ 2 { μ 2 n = 1 N ( 2 n 1 ) Q [ ρ + ( n 1 ) μ ] N 2 Q ( ρ ) Q ( ρ ) Q ( ρ ) ρ 2 } .
I ( x , y ) = V ( x , y ) [ 2 cos ( kx F ϕ x ) cos ( ky F ϕ y ) ] ,
I nm = W L 2 + ( n 1 ) d L 2 + nd I ( x ) dx L 2 + ( m 1 ) d L 2 + md I ( y ) dy .
I D = W n = 1 M m = 1 M { [ I ( x , y ) Λ ( x d , y d ) ] δ ( x nd + H ) δ ( y md + H ) }
= ( x L , y L ) n = m = { [ I ( x , y ) Λ ( x d , y d ) ] δ ( x nd + H ) δ ( y md + H )) } ,
Λ ( a , b ) = { 1 ( a , b ) ( 1 2 , 1 2 ) × ( 1 2 , 1 2 ) 0 otherwise } .
I D ( x , y ) = W Λ ( x H L , y H L ) [ I x y Λ ( x H d , y H d ) ] × n m δ ( x nd ) δ ( y md ) .
I ˜ D ( u , v ) = A B C .
A = exp [ 2 H ( u + v ) ] L 2 sinc ( Lu ) sinc ( Lv ) ,
B = W I ˜ ( u , v ) exp [ 2 H ( u + v ) ] d 2 sinc ( du ) sinc ( dv ) .
C = d 2 n = m = δ ( x n d ) δ ( y m d ) .
K ̃ ( u , v ) = C ̃ x exp [ i π ( M 1 ) d ( u + v + 1 D ) ] d 2 sinc ( du + 1 N ) sinc ( dv ) .
K ( u , v ) = C x exp [ i π ( M 1 ) N ] δ ( x H ) d 2 Λ ( x d , y d ) exp ( i 2 πx D ) .
K x y = δ ( x H ) d 2 exp N d 2 d 2 d 2 d 2 exp i 2 π ( x x ' ) D C x x x ' y y ' dx ' dy ' =
= δ ( x H ) d 2 exp N y d 2 y + d 2 x d 2 x + d 2 exp i 2 πx ' D C x x ' y ' dx ' dy ' .
K x y = δ ( x H ) d 2 exp N V ( r ) exp ( iF ϕ x ) y d 2 y + d 2 x d 2 x + d 2 exp i 2 πx ' D dx ' dy ' = δ ( x H ) d 2 V ( r ) N d 2 π sin π N exp [ i ( F ϕ x + 2 πx D π N ) ] .
F ϕ x = K ( x + H , y ) ( 2 πx D π N ) .
E [ X i 2 ] = 1 K i 2 j = 1 K i E [ x i , j 2 ] = σ x i , j 2 K i = π 2 6 3 k x 2 K i ,
σ x i , j 2 = x 2 P ( x ) dx = π k x π k x x 2 P ( x ) dx = k 2 π π k x π k x x 2 [ 1 + cos ( k x x ) ] dx = k x 2 π [ 2 3 ( π k x ) 3 4 π k x 3 ] = π 2 6 3 k x 3 .
σ θ i | K i = k x σ x = 2 ( π 2 3 2 ) K i .
σ θ 2 = K = 1 2 ( π 2 6 ) 3 K exp ( λ ) λ K K ! = 2 ( π 2 6 ) exp ( λ ) 3 K = 1 1 K λ K K ! .
σX γ = 1 k x 2 ( π 2 3 2 ) exp ( λ ) [ Ei ( λ ) In ( λ ) ] .
X TOT = X RoN λ RoN + X γ λ γ λ RoN + λ γ
X γwR = X TOT ( λ RoN + λ γ ) X RoN λ RoN λ RoN
σ X γwR 2 = σ X Ron 2 λ Ron + σ X γ 2 λ γ λ γ = N 2 R N λ γ σ X RoN 2 + σ X γ 2

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