Abstract

An analysis of the problem of wave-front reconstruction from Shack–Hartmann measurements is presented. The wave-front aberration is assumed to result from passage of the wave front through Kolmogorov turbulence. Limitations of using Zernike polynomials as an orthogonal basis for wave-front reconstruction are highlighted, and the advantage of using the Karhunen–Loeve functions for computing the higher-order modes of the wave front is shown.

© 1992 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 283–368.
    [CrossRef]
  2. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  3. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
    [CrossRef]
  4. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  5. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A. 5, 963–985 (1988).
    [CrossRef]
  6. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
    [CrossRef]
  7. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  8. 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]
  9. R. H. Hugdin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  10. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1977).
    [CrossRef]
  11. J. Hermann, “Least-squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  12. N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  13. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  14. D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  15. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  16. H. W. Sorenson, Parameter Estimation (Dekker, New York, 1980).
  17. D. L. Fried, “Probability of getting a lucky short-exposure through turbulence,” J. Opt. Soc. Am. 68, 1650–1658 (1965).

1990 (2)

1988 (1)

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A. 5, 963–985 (1988).
[CrossRef]

1983 (1)

1980 (2)

1978 (1)

1977 (3)

1976 (1)

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1965 (2)

D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
[CrossRef]

D. L. Fried, “Probability of getting a lucky short-exposure through turbulence,” J. Opt. Soc. Am. 68, 1650–1658 (1965).

Ayers, G. R.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A. 5, 963–985 (1988).
[CrossRef]

Dainty, J. C.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A. 5, 963–985 (1988).
[CrossRef]

Fontanella, J. C.

Fried, D. L.

Hermann, J.

Hugdin, R. H.

Hunt, B. R.

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lohmann, A. W.

Markey, J. K.

Noll, R. J.

Northcott, M. J.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A. 5, 963–985 (1988).
[CrossRef]

Primot, J.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 283–368.
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Rousset, G.

Sorenson, H. W.

H. W. Sorenson, Parameter Estimation (Dekker, New York, 1980).

Southwell, W. H.

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

Wang, J. Y.

Weigelt, G.

Wirnitzer, B.

Appl. Opt. (1)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Astron. J. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short exposure photographs,” Astron. J. 193, L45–L48 (1974).
[CrossRef]

J. Opt. Soc. Am. (9)

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

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

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A. 5, 963–985 (1988).
[CrossRef]

Opt. Eng. (1)

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Other (2)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 283–368.
[CrossRef]

H. W. Sorenson, Parameter Estimation (Dekker, New York, 1980).

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Figures (4)

Fig. 1
Fig. 1

8 × 8 Shack–Hartmann array of lenslets superimposed on a circular aperture. The shading is used to indicate those lenslets whose outputs are used in the calculation of the coefficients of the basis functions.

Fig. 2
Fig. 2

Ideal residual wave-front error as a function of the number of basis functions used: the upper solid curve represents Zernike basis functions; the lower dashed curve represents Karhunen–Loeve functions.

Fig. 3
Fig. 3

Contour plots of both the Zernike and Karhunen–Loeve functions: (a) Z8, (b) Z16, (c) Z46, (d) K10, (e) K21, (f) K38. Each curve is plotted with 15 equally spaced contours.

Fig. 4
Fig. 4

Simulated residual wave-front error obtained by using the Shack–Hartmann configuration shown in Fig. 1. The upper solid curve represents Zernike basis functions. The lower dashed curve represents Karhunen–Loeve functions.

Tables (1)

Tables Icon

Table 1 Number of Basis Functions Required to Reduce Δj Below a Given Level

Equations (21)

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ϕ ( x , y ) = i = 0 + a i Ψ i ( x , y ) ,
Δ = aperture [ ϕ ( x , y ) - ϕ ( x , y ) av ] 2 ,
Θ A = H ,
Z even i = ( n + 1 ) 1 / 2 R n m ( r ) 2 cos m θ , m 0 , Z odd i = ( n + 1 ) 1 / 2 R n m ( r ) 2 sin m θ , m 0 , Z i = ( n + 1 ) 1 / 2 R n 0 ( r ) , m = 0 ,
R n m = s = 0 ( n - m ) / 2 ( - 1 ) s ( n - s ) ! r n - 2 s s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] ! .
( 1 / π ) 0 2 π 0 1 Z i ( r , θ ) Z i ( r , θ ) r d θ = δ i i ,
δ i i = { 1 i = i 0 i i .
Δ J z 0.2944 J - 3 / 2 ( D r 0 ) 5 / 3
ϕ ( x , y ) = i = 2 + a i z Z i ( x , y ) ,
Z 1 ( x , y ) = 1 ,
C = E ( A · A T ) = [ E ( a 2 a 2 ) E ( a 2 a 3 ) E ( a 2 a p ) E ( a 3 a 2 ) E ( a 3 a 3 ) E ( a 3 a p ) · · · · · · · · · E ( a p a 2 ) E ( a p a 3 ) E ( a p a p ) ] .
C = X · D · X T ,
E ( a j k a j k ) = { 0 j j D j j = j ,
K = X T · Z .
Δ J k = Δ 1 z - j = 2 N D j .
A ˜ = ( Θ T Θ ) - 1 Θ T H .
Δ ˜ M z = 1 K k = 1 K [ l = 2 M + 1 ( A ˜ l - S l ) 2 + M + 2 861 ( S l ) 2 ] ,
Δ ˜ M k = 1 K k = 1 K [ l = 2 861 ( A ˜ l - S l ) 2 ] ,
Θ A = H + ν ,
N = E ( ν ν T ) ,
A ¯ = ( Θ T N - 1 Θ + C - 1 ) - 1 Θ T N - 1 H .

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