Abstract

In adaptive optical systems that compensate for random wave-front disturbances, a wave front is measured and corrections are made to bring it to the desired shape. For most systems of this type, the local wave-front slope is first measured, the wave front is next reconstructed from the slope, and a correction is then fitted to the reconstructed wave front. Here a more realistic model of the wave-front measurements is used than in the previous literature, and wave-front estimation and correction are analyzed as a unified process rather than being treated as separate and independent processes. The optimum control law is derived for an arbitrary array of slope sensors and an arbitrary array of correctors. Application of this law is shown to produce improved results with noisy measurements. The residual error is shown to depend directly on the density of the slope measurements, but the sensitivity to the precise location of the measurements that was indicated in the earlier literature is not observed.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Hardy, J. E. Lefebvre, and C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [Crossref]
  2. M. P. Rimmer, “Methods for evaluating lateral shear interferograms,” Appl. Opt. 13, 623–629 (1974).
    [Crossref] [PubMed]
  3. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–374 (1977).
    [Crossref]
  4. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–377 (1977).
    [Crossref]
  5. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–398 (1979).
    [Crossref]
  6. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139 (1978).
    [Crossref]
  7. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [Crossref]
  8. J. Herrmann, “Least squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [Crossref]
  9. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1005 (1980).
    [Crossref]
  10. J. Herrmann, “Cross-coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [Crossref]
  11. D. L. Fried, “Statistics for a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [Crossref]
  12. B. L. McGlamery and P. E. Silva (Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California 92152), “A preliminary comparison of wave-front error measurement devices for use in compensated imaging systems” (unpublished report, March1975).
  13. D. L. Fried, “Required number of degrees-of-freedom for an adaptive optics system,” (October1975).
  14. R. H. Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–395 (1977).
    [Crossref]
  15. J. Y. Wang and J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–88 (1978).
    [Crossref]
  16. E. P. Wallner, “Comparison of wave-front sensor configurations using optimal reconstruction and correction,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 42–53 (1982).

1982 (1)

E. P. Wallner, “Comparison of wave-front sensor configurations using optimal reconstruction and correction,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 42–53 (1982).

1981 (1)

1980 (2)

1979 (2)

1978 (2)

1977 (4)

1974 (1)

1965 (1)

Cubalchini, R.

Fried, D. L.

Hardy, J. W.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Koliopoulos, C. L.

Lefebvre, J. E.

Markey, J. K.

McGlamery, B. L.

B. L. McGlamery and P. E. Silva (Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California 92152), “A preliminary comparison of wave-front error measurement devices for use in compensated imaging systems” (unpublished report, March1975).

Noll, R. J.

Rimmer, M. P.

Silva, P. E.

B. L. McGlamery and P. E. Silva (Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California 92152), “A preliminary comparison of wave-front error measurement devices for use in compensated imaging systems” (unpublished report, March1975).

Southwell, W. H.

Wallner, E. P.

E. P. Wallner, “Comparison of wave-front sensor configurations using optimal reconstruction and correction,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 42–53 (1982).

Wang, J. Y.

Appl. Opt. (1)

J. Opt. Soc. Am. (12)

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–374 (1977).
[Crossref]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–377 (1977).
[Crossref]

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–398 (1979).
[Crossref]

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

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

J. Herrmann, “Least squares wave-front errors of 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, 998–1005 (1980).
[Crossref]

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

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

R. H. Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–395 (1977).
[Crossref]

J. Y. Wang and J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–88 (1978).
[Crossref]

J. W. Hardy, J. E. Lefebvre, and C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

E. P. Wallner, “Comparison of wave-front sensor configurations using optimal reconstruction and correction,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 42–53 (1982).

Other (2)

B. L. McGlamery and P. E. Silva (Visibility Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California 92152), “A preliminary comparison of wave-front error measurement devices for use in compensated imaging systems” (unpublished report, March1975).

D. L. Fried, “Required number of degrees-of-freedom for an adaptive optics system,” (October1975).

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 (5)

Fig. 1
Fig. 1

Displaced subaperture wave-front sensor configuration.

Fig. 2
Fig. 2

Performance of displaced subaperture sensor configuration.

Fig. 3
Fig. 3

Comparison of optimal and closed-loop reconstructors.

Fig. 4
Fig. 4

Common subaperture wave-front sensor configuration.

Fig. 5
Fig. 5

Comparison of sensor configurations.

Equations (41)

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

- d x W A ( x ) = 1 ,
- d x W A ( x ) ϕ ( x ) = 0 ,
ϕ ( x ) = ψ ( x ) - - d x W A ( x ) ψ ( x )
s n = - d x W s n ( x ) [ ϕ n s ( x ) + v ( x ) ] ,
s n = - d x [ - W s n s ( x ) ϕ ( x ) + W s n ( x ) v ( x ) ] ,
v ( x ) = v ( x ) ϕ ( x ) = 0.
c j = n M j n s n ,
ϕ ^ ( x ) = j c j r j ( x ) ,
( x ) = ϕ ^ ( x ) - ϕ ( x ) = j r j ( x ) n M j n s n - ϕ ( x ) .
2 ( x ) = j j n n r j ( x ) r j ( x ) M j n M j n s n s n - 2 j n r j ( x ) M j n s n ϕ ( x ) + ϕ 2 ( x ) ,
c 2 = - d x W A ( x ) 2 ( x )
= j j n n M j n M j n s n s n × - d x W A ( x ) r j ( x ) r j ( x ) - 2 j n M j n - d x W A ( x ) r j ( x ) s n ϕ ( x ) + - d x W A ( x ) ϕ 2 ( x ) .
S n n = s n s n = - d x - d x [ W s n s ( x ) W s n s ( x ) ϕ ( x ) ϕ ( x ) + W s n ( x ) W s n ( x ) v ( x ) v ( x ) ] .
R j j = - d x W A ( x ) r j ( x ) r j ( x ) .
A j n = - d x W A ( x ) r j ( x ) s n ϕ ( x ) = - d x - d x W A ( x ) r j ( x ) W s n s ( x ) ϕ ( x ) ϕ ( x ) .
0 2 = - d x W A ( x ) ϕ 2 ( x ) .
c 2 = j j n n M j n M j n S n n R j j - 2 j n M j n A j n + 0 2 .
d c 2 d M j n = 2 j n M j n S n n R j j - 2 A j n = 0.
R j j M j n S n n = A j n .
M j n * = R j j - 1 A j n S n n - 1 .
* 2 = 0 2 - M j n * A j n = 0 2 - R j j - 1 A j n S n n - 1 A j n .
D ( x , x ) = [ ψ ( x ) - ψ ( x ) ] 2 .
ϕ ( x ) = ψ ( x ) - - d x W A ( x ) ψ ( x ) = - d x W A ( x ) [ ψ ( x ) - ψ ( x ) ] .
ϕ ( x ) ϕ ( x ) = - d x - d x W A ( x ) W A ( x ) × [ ψ ( x ) - ψ ( x ) ] [ ψ ( x ) - ψ ( x ) ] = - ½ - d x - d x W A ( x ) W A ( x ) × [ D ( x , x ) - D ( x , x ) - D ( x , x ) + D ( x , x ) ] = - ½ D ( x , x ) + g ( x ) + g ( x ) - a ,
g ( x ) = ½ - d x W A ( x ) D ( x , x ) ,
a = ½ - d x - d x W A ( x ) W A ( x ) D ( x , x ) .
0 2 = - d x W A ( x ) ϕ 2 ( x ) = a .
S n n = - d x - d x [ - ½ W s n s ( x ) W s n s ( x ) D ( x , x ) + W s n ( x ) W s n ( x ) v ( x ) v ( x ) ] ,
A j n = - ½ - d x - d x W A ( x ) r j ( x ) W s n s ( x ) D ( x , x ) .
v ( x ) v ( x ) = k n n σ n 2 δ ( x - x ) ,
- d x - d x W s n ( x ) W s n ( x ) v ( x ) v ( x ) = k n n σ n 2 - d x W s n ( x ) W s n ( x ) .
ϕ ^ ( x ) = c j r j ( x ) .
s n = s n - - d x [ - W s n s ( x ) ϕ ^ ( x ) ] = s n - - d x [ - W s n s ( x ) c j r j ( x ) ] .
P n j = - d x [ - W s n ( x ) r j ( x ) ] .
s n = s n - P n j c j .
c j = P n j + s n ,
c 2 = 0 2 - 2 P n j + A j n + P n j + P n j + S n n R j j .
D ( x ¯ , x ¯ ) = 6.8839 ( x ¯ - x ¯ ) / r 0 5 / 3 ,
0 2 = 1.3103 ( A / r 0 ) 5 / 3 ,
σ s 2 = 6.4051 L - 1 / 3 r 0 - 5 / 3 ,
σ n 2 = K M = K ρ A s ,