Abstract

Modal wave-front reconstruction by use of Zernike polynomials and Karhunen–Loève functions from average slope measurements with circular and annular apertures is discussed because of its practical applications in astronomy. A new error source, referred to as the remaining error, is formulated theoretically and evaluated numerically. The total reconstruction error is found to be the sum of the uncompensated wave-front residual error, the measurement error, and the remaining error. Numerical calculation shows that modal wave-front reconstruction with atmospheric Karhunen–Loève functions results in a smaller residual error than with Zernike polynomials.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  3. R. J. Noll, “Phase estimation from slope-type wave-front sensors,”J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  4. J. Herrmann, “Least-squares wave front errors with minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  5. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,”J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  6. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,”J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  7. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  8. R. H. Hudgin, “Optimal wave-front estimation,”J. Opt. Soc. Am. 67, 378–382 (1977).
    [CrossRef]
  9. E. P. Wallner, “Optimal wave-front correction using slope measurements,”J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  10. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [CrossRef]
  11. M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
    [CrossRef]
  12. G.-m. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 562–573 (1994).
    [CrossRef]
  13. P. Wintoft, G.-m. Dai, “Neural network for modal compensation of atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 103–109 (1994).
    [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  15. G.-m. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Lovève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995).
    [CrossRef]
  16. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,”J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  17. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,”J. Opt. Soc. Am. 68, 78–86 (1978).
    [CrossRef]
  18. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  19. D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,”J. Opt. Soc. Am. 68, 1651–1658 (1978).
    [CrossRef]
  20. G.-m. Dai, “Wavefront simulation for atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 62–72 (1994).
    [CrossRef]

1995 (1)

1992 (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
[CrossRef]

1990 (1)

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

1989 (1)

1986 (1)

1983 (1)

1981 (1)

1980 (2)

1979 (1)

1978 (3)

1977 (3)

1976 (1)

Cubalchini, R.

Dai, G.-m.

G.-m. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Lovève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995).
[CrossRef]

G.-m. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 562–573 (1994).
[CrossRef]

G.-m. Dai, “Wavefront simulation for atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 62–72 (1994).
[CrossRef]

P. Wintoft, G.-m. Dai, “Neural network for modal compensation of atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 103–109 (1994).
[CrossRef]

Freischlad, K. R.

Fried, D. L.

Gardner, C. S.

Herrmann, J.

Hudgin, R. H.

Koliopoulos, C. L.

Markey, J. K.

Noll, R. J.

Roddier, N.

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

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
[CrossRef]

Southwell, W. H.

Wallner, E. P.

Wang, J. Y.

Welsh, B. M.

Wintoft, P.

P. Wintoft, G.-m. Dai, “Neural network for modal compensation of atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 103–109 (1994).
[CrossRef]

Comput. Electron. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
[CrossRef]

J. Opt. Soc. Am. (12)

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,”J. Opt. Soc. Am. 69, 972–977 (1979).
[CrossRef]

R. J. Noll, “Phase estimation from slope-type wave-front sensors,”J. Opt. Soc. Am. 68, 139–140 (1978).
[CrossRef]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

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]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–378 (1977).
[CrossRef]

R. H. Hudgin, “Optimal wave-front estimation,”J. Opt. Soc. Am. 67, 378–382 (1977).
[CrossRef]

J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,”J. Opt. Soc. Am. 68, 78–86 (1978).
[CrossRef]

D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,”J. Opt. Soc. Am. 68, 1651–1658 (1978).
[CrossRef]

J. Herrmann, “Least-squares wave front errors with minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
[CrossRef]

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,”J. Opt. Soc. Am. 70, 998–1006 (1980).
[CrossRef]

J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,”J. Opt. Soc. Am. 71, 989–992 (1981).
[CrossRef]

E. P. Wallner, “Optimal wave-front correction using slope measurements,”J. Opt. Soc. Am. 73, 1771–1776 (1983).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

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

Other (3)

G.-m. Dai, “Wavefront simulation for atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 62–72 (1994).
[CrossRef]

G.-m. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 562–573 (1994).
[CrossRef]

P. Wintoft, G.-m. Dai, “Neural network for modal compensation of atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 103–109 (1994).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Normalized (D = r0) residual error after Zernike or K–L compensation. Solid curve, K–L compensation; dotted curve, Zernike compensation.

Fig. 2
Fig. 2

Wave-front sensing configurations relevant to astronomical applications. (a) Circular pupil with 32 subapertures, (b) annular pupil with 28 subapertures and an obscuration ratio of 0.28. Square and circular subapertures are considered. The arrow bars indicate average slope measurements within given subapertures.

Fig. 3
Fig. 3

Remaining errors versus the number of actual wave-front modes M for the circular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 4
Fig. 4

Remaining errors versus the number of actual wave-front modes M for the annular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 5
Fig. 5

Remaining errors versus the number of reconstructed modes N for the circular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 6
Fig. 6

Remaining errors versus the number of reconstructed modes N for the circular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 7
Fig. 7

Measurement errors versus the number of reconstructed modes N for the circular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 8
Fig. 8

Measurement errors versus the number of reconstructed modes N for the annular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 9
Fig. 9

Total reconstruction errors versus the number of reconstructed modes N for the circular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Fig. 10
Fig. 10

Total reconstruction errors versus the number of reconstructed modes N for the annular aperture. Solid and dotted curves, K–L reconstruction with square and circular subapertures, respectively; dashed and long-dashed curves, Zernike reconstruction with square and circular subapertures, respectively.

Tables (1)

Tables Icon

Table 1 Minimum Reconstruction Errors and Maximum Strehl Ratios

Equations (28)

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

ϕ ( R r ) = i = 2 a i F i ( r ) ,
1 π W ( r ) F i ( r ) F j ( r ) d r = δ i j ,
W ( r ) = { 1 if | r | 1 0 otherwise .
a i = 1 π W ( r ) F i ( r ) ϕ ( R r ) d r .
ϕ ( R r ) x | l = i = 2 a i F i ( r ) x | l ,
ϕ ( R r ) y | l = i = 2 a i F i ( r ) y | l ,
S = G A = G [ A f A r ] ,
G = [ F 2 ( r , θ ) 1 x F 3 ( r , θ ) 1 x F M ( r , θ ) 1 x F 2 ( r , θ ) 1 y F 3 ( r , θ ) 1 y F M ( r , θ ) 1 y F 2 ( r , θ ) k x F 3 ( r , θ ) k x F M ( r , θ ) k x F 2 ( r , θ ) k y F 3 ( r , θ ) k y F M ( r , θ ) k y ] .
ϕ ˆ ( R r ) = i = 2 N b i F i ( r ) ,
S ˜ = HB ,
B = H + S ,
B = H + G A = H + [ H G r ] [ A f A r ] = H + ( H A f + G r A r ) = A f + C A r ,
C = H + G r = ( H T H ) 1 H T G r ,
σ r 2 = i = 2 N | b i a i | 2 = i = 2 N j = N + 1 j = N + 1 c i j a j a j c j i t ,
σ ϕ 2 = 1 π W ( r ) [ ϕ ( R r ) ϕ ˆ ( R r ) ] 2 d r = 1 π W ( r ) [ i = 2 a i F i ( r ) i = 2 N b i ( r ) F i ( r ) ] 2 d r .
S ˆ = S + E ,
B = H + S ˆ = H + ( H A f + G r A r ) + H + E = A f + C A r + H + E .
b i = a i + j = N + 1 c i j a j + j = 1 2 k h i j + e j ,
σ ϕ 2 = 1 π W ( r ) [ i = 2 a i F i ( r ) i = 2 N a i F i ( r ) i = 2 N j = N + 1 c i j a j F i ( r ) + i = 2 N j = 1 2 k h i j + e j F i ( r ) ] 2 d r .
σ ϕ 2 = i = N + 1 a i 2 + i = 2 N j = N + 1 j = N + 1 c i j a j a j c j i t + i = 2 N j = 1 2 k j = 1 2 k h i j + e j e j ( h + ) j i t .
e j e j = σ 0 2 δ j j ,
i = 2 N j = 1 2 k j = 1 2 k h i j + e j e j ( h + ) j i t = σ 0 2 i = 2 N j = 1 2 k [ ( H T H ) 1 H T ] i j [ H ( H T H ) 1 ] j i = σ 0 2 tr [ ( H T H ) 1 ] ,
σ ϕ 2 = i = N + 1 a i 2 + i = 2 N j = N + 1 j = N + 1 c i j a j a j c j i t + σ 0 2 tr [ ( H t H ) 1 ] . = Δ N + σ r 2 + σ M 2 .
Z i ( r ) = R n m ( r ) Θ m ( θ ) ,
R n m ( r ) = s = 0 ( n m ) / 2 ( 1 ) s n + 1 ( n s ) ! r n 2 s s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! ,
Θ m ( θ ) = { 2 cos m θ , m 0 even mode 2 sin m θ , m 0 odd mode 1 , m = 0 .
K i ( r ) = S p q ( r ) Θ q ( θ ) ,
σ r 2 = i = 2 N j = N + 1 M j = N + 1 M c i j a j a j c j i t .

Metrics