Abstract

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15×15-grid Shack–Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than λ130λ150 after 12 iterations and less than λ100 (both for λ=632.8nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.

© 2005 Optical Society of America

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  1. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  2. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  3. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  4. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  5. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  6. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  9. J. Herrmann, “Least-square wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  10. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  11. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  12. K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).
  13. K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave-front difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  14. K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, and G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
  15. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  16. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  17. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  18. F. Roddier, C. Roddier, “Wave-front reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
  19. W. Zou, Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
    [CrossRef]
  20. D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test (European Southern Observatory, Garching, Germany, 1993), pp. 289–292.
  21. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 2nd ed., Vol. 12 of Texts in Applied Mathematics (Springer, New York, 1993), pp. 186, 594, and 635.
  22. D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, New York, 2002), pp. 3, 58, and 548.
  23. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer, New York, 2000).
  24. W. J. Thompson, Computing for Scientists and Engineers (Wiley, New York, 1992), p. 333.
  25. E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice Hall, Englewood Cliffs, N.J., 1988), pp. 134, 164.
  26. R. Kress, Numerical Analysis, Graduate Texts in Mathematics (Springer, New York, 1998), p. 127.

2000 (1)

1991 (1)

1987 (1)

1986 (1)

1982 (1)

1981 (1)

1980 (2)

1979 (2)

1978 (1)

1977 (3)

1975 (1)

1974 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice Hall, Englewood Cliffs, N.J., 1988), pp. 134, 164.

Bulirsch, R.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 2nd ed., Vol. 12 of Texts in Applied Mathematics (Springer, New York, 1993), pp. 186, 594, and 635.

Cubalchini, R.

Fienup, J. R.

Freischlad, K.

K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave-front difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
[CrossRef]

K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, and G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).

Fried, D. L.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Hardy, J. W.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Jiang, S.

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test (European Southern Observatory, Garching, Germany, 1993), pp. 289–292.

Koliopoulos, C. L.

Kress, R.

R. Kress, Numerical Analysis, Graduate Texts in Mathematics (Springer, New York, 1998), p. 127.

Lefebvre, J. E.

Noll, R. J.

Quarteroni, A.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer, New York, 2000).

Rimmer, M. P.

Roddier, C.

Roddier, F.

Sacco, R.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer, New York, 2000).

Saleri, F.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer, New York, 2000).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Shao, L.

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test (European Southern Observatory, Garching, Germany, 1993), pp. 289–292.

Southwell, W. H.

Stoer, J.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 2nd ed., Vol. 12 of Texts in Applied Mathematics (Springer, New York, 1993), pp. 186, 594, and 635.

Su, D.

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test (European Southern Observatory, Garching, Germany, 1993), pp. 289–292.

Thompson, W. J.

W. J. Thompson, Computing for Scientists and Engineers (Wiley, New York, 1992), p. 333.

Watkins, D. S.

D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, New York, 2002), pp. 3, 58, and 548.

Wyant, J. C.

Zhang, Z.

Zou, W.

Appl. Opt. (6)

J. Opt. Soc. Am. (9)

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

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other (9)

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test (European Southern Observatory, Garching, Germany, 1993), pp. 289–292.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 2nd ed., Vol. 12 of Texts in Applied Mathematics (Springer, New York, 1993), pp. 186, 594, and 635.

D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, New York, 2002), pp. 3, 58, and 548.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer, New York, 2000).

W. J. Thompson, Computing for Scientists and Engineers (Wiley, New York, 1992), p. 333.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice Hall, Englewood Cliffs, N.J., 1988), pp. 134, 164.

R. Kress, Numerical Analysis, Graduate Texts in Mathematics (Springer, New York, 1998), p. 127.

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 74–80 (1985).

K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, and G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).

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Figures (12)

Fig. 1
Fig. 1

(a) Ground-truth or original wave front, (b) wave front estimated from measured slope data with the algorithm without iteration, (c) wave-front deviation error computed as the difference between the ground-truth and the estimated wave front.

Fig. 2
Fig. 2

Wave-front estimation schemes. From left to right: the (1) Hudgin, (2) Southwell, and (3) Fried models. In these figures, the small circles symbolize the wave-front values and the small arrows represent the partial derivatives of the wave front.

Fig. 3
Fig. 3

Double sampling grid systems illustrated in the y direction.

Fig. 4
Fig. 4

Domain extension for an irregular-shaped pupil.

Fig. 5
Fig. 5

Flow chart of the Gerchberg-type iterative least-squares wave-front estimation algorithm based on the domain extension technique.

Fig. 6
Fig. 6

(a) 30 - mm -diameter circular pupil without central obstruction shown within the extended domain Ω 1 . (b) The ground-truth wave front within the circular pupil Ω 0 on a vertical scale of ± 1 μ m .

Fig. 7
Fig. 7

Wave-front deviation error (on a vertical scale of ± 1 μ m , λ = 632.8 nm ) for a 30 - mm -diameter circular pupil without obstruction across a sampled 15 × 15 point grid, for the number of iterations (a) i = 0 , rms = λ 16 ; (b) i = 1 , rms = λ 37 ; (c) i = 2 , rms = λ 63 ; (d) i = 3 , rms = λ 86 ; (e) i = 4 , rms = λ 105 ; (f) i = 13 , rms = λ 129 .

Fig. 8
Fig. 8

(a). A 30 - mm diameter circular pupil with a 10% central obstruction. (b). The ground-truth wave front at this pupil on a vertical scale of ± 1 μ m (right).

Fig. 9
Fig. 9

Wave-front deviation error (on a vertical scale of ± 1 μ m , λ = 632.8 nm ) for a 30 - mm -diameter circular pupil sampled with a 15 × 15 grid and with a 10% central obstruction, for the number of iterations (a) i = 0 , rms = λ 14 ; (b) i = 1 , rms = λ 26 ; (c) i = 3 , rms = λ 63 ; (d) i = 5 , rms = λ 107 ; (e) i = 7 , rms = λ 135 ; (f) i = 10 , rms = λ 154 .

Fig. 10
Fig. 10

Plot of RMS deviation errors in units of wavelength as a function of the number of iterations for the two data sets considered.

Fig. 11
Fig. 11

Noise-coefficient limit versus the dimension size of the sampling grid.

Fig. 12
Fig. 12

Normal matrix condition number versus grid dimension size.

Equations (72)

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

s y j = 1 2 ( s y i + s y i + 1 ) ,
s y j = w i + 1 w i a .
w i + 1 w i = a 2 ( s y i + s y i + 1 ) .
w i w i + 1 = a 2 ( s z i + s z i + 1 ) .
[ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 ] [ w 1 w 2 w t w t + 1 w t + 2 w 2 t w m 2 t + 1 w m 2 t + 2 w m t w m t + 1 w m 1 w m ] [ c 2 , 1 c 3 , 2 c t , t 1 c t + 2 , t + 1 c m 1 , m d t + 1 , t d t + 2 , t d 2 t , t d i + t , 1 d m 1 , m t 1 d m , m t ] ,
C W = S ,
c i + 1 , i = a 2 ( s y i + 1 + s y i ) ,
d i + t , i = a 2 ( s z i + 1 + s z i ) .
C T C W = C T S .
s y i + 1 + s y i = e j ,
e j = 2 a ( w i + 1 w i ) , i = 1 , 2 m 1 , but i t , 2 t , 3 t , m .
A 1 S y = E ,
s y i + 2 s y i + 1 = f j + 1 ,
f j + 1 = ( w i + 3 w i + 1 2 a w i + 2 w i 2 a ) = 1 2 a ( w i + 3 w i + 2 w i + 1 + w i ) .
A 2 S y = F .
A S y = U ,
A = [ A 1 A 2 ] ,
S y = [ S y 1 S y 2 , , S y m ] T ,
U = [ E F ] ,
A 1 = diag [ D 1 , D 1 , , D 1 ] ,
A 2 = diag [ D 2 , D 2 , , D 2 ] ,
D 1 = [ 1 1 1 1 1 1 ] ,
D 2 = [ 1 1 1 1 1 1 1 1 ] .
A T A S y = A T U .
s z i + s z i + t = g j , i = 1 , 2 , , m t ,
g j = 2 a ( w i w i + t ) .
B 1 S z = G ,
s z i + t s z 1 + 2 t = h j + t ,
h j + t = 1 2 a ( w i w i + t w i + 2 t + w i + 3 t ) ,
B 2 S z = H .
B S z = V ,
B = [ B 1 B 2 ] ,
S z = [ S z 1 S z 2 S z m ] T ,
V = [ G H ] ,
B 1 = [ I t I t I t I t I t I t I t I t ] ,
B 2 = [ I t I t I t I t I t I t I t I t ] ,
I t = [ 1 1 1 ] t × t .
B T B S z = B T V .
R ω b = 8 ( log 10 ( 1 2 π t 1 ) ) 1 2.94 ( t + 1 ) 3 t .
C T C W = a C T S ,
X 2 = ( X T X ) 1 2
lub 2 ( C ) = max X 0 ( X T C T C X X T X ) 1 2 = [ ρ ( C T C ) ] 1 2 ,
W 2 a [ cond ( C T C ) ] 1 2 lub 2 ( C ) S 2 ,
cond ( C T C ) lub 2 ( C T C ) lub 2 ( C T C ) 1 = ρ ( C T C ) ρ [ ( C T C ) 1 ] .
cond ( C T C ) = λ max λ min .
W 2 a S 2 λ min .
W 2 = m ( 1 m i = 1 m w i 2 ) 1 2 = t σ w ,
S 2 = m ( 1 m i = 1 m s i 2 ) 1 2 = t σ s ,
σ w a σ s λ min 1 2 .
σ w γ σ d ,
γ = λ min 1 2 .
σ w 2 σ d 2 γ 2 = λ min 1 .
σ s 2 σ w 2 1 2 2 = 1 + 2 2 1.71 , t 4 .
σ w 2 σ d 2 γ 2 = { 44 + 28.58 e t 8.925 , t is even 31.875 + 20.61 e t 7.766 , t is odd } .
cond 2 ( C T C ) = { 243.442 + 150.87 e t 7.518 , t is odd 355.157 + 223.750 e t 6.83 , t is even } .
n W y n i
n W y n i + 1 2
w i = w i + 1 2 a 2 W y i + 1 2 + a 2 4 × 2 ! 2 W y 2 i + 1 2 a 3 8 × 3 ! 3 W y 3 i + 1 2 + a 4 16 × 4 ! 4 W y 4 i + 1 2 + O ( a 5 ) ,
w i + 1 = w i + 1 2 + a 2 W y i + 1 2 + a 2 4 × 2 ! 2 W y 2 i + 1 2 + a 3 8 × 3 ! 3 W y 3 i + 1 2 + a 4 16 × 4 ! 4 W y 4 i + 1 2 + O ( a 5 ) .
w i + 1 w i = a W y i + 1 2 + a 3 4 × 3 ! 3 W y 3 i + 1 2 + O ( a 5 ) .
w i + 1 + w i = 2 w i + 1 2 + a 2 4 2 W y 2 i + 1 2 + a 4 8 × 4 ! 4 W y 4 i + 1 2 + O ( a 6 ) .
W y i + 1 + W y i = 2 W y i + 1 2 + a 2 4 3 W y 3 i + 1 2 + O ( a 6 ) .
W y i + 1 2 = 1 2 ( W y i + 1 + W y i ) a 2 8 3 W y 3 i + 1 2 + O ( a 6 ) .
w i + 1 w i = a 2 ( W y i + 1 + W y i ) a 3 12 3 W y 3 i + 1 2 + O ( a 5 ) .
W y i + 1 W y i = a 2 ( 2 W y 2 i + 1 + 2 W y 2 i ) a 3 12 4 W y 4 i + 1 2 + O ( a 5 ) .
w i + 1 2 w i + w i 1 = a 2 2 W y 2 i + a 4 12 4 W y 4 i + O ( a 6 ) ,
2 W y 2 i = w i + 1 2 w i + w i 1 a 2 a 2 12 4 W y 4 i + O ( a 4 ) .
W y i + 1 W y i = a 2 ( w i + 2 2 w i + 1 + w i a 2 + w i + 1 2 w i + w i 1 a 2 ) a 3 24 ( 4 W y 4 i + 4 W y 4 i + 1 ) a 3 12 4 W y 4 i + 1 2 + O ( a 5 )
W y i + 1 W y i = 1 2 a ( w i + 2 w i + 1 w i + w i 1 ) + O ( a 5 ) .
s y i + 2 s y i + 1 = 1 2 a ( w i + 3 w i + 2 w i + 1 + w i ) ,
s z i + t s z i + 2 t = 1 2 a ( w i w i + t w i + 2 t + w i + 3 t ) ,
i = 1 , 2 , t , t + 1 , t + 2 , 2 t , , m 3 t .

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