Abstract

In a compensated imaging (CI) system the wave-front distortion is changing randomly in time and the data processing must follow the changes and even, to some extent, predict them. Furthermore, the system is not infinitely fast; it has time delays and bandwidth limitations which can be quite significant. Couple to this the substantial photon noise in the wave-front measurements, and it becomes desirable to find the optimal data processing to use for system control. This paper derives the optimal linear control algorithm assuming the aberration and noise statistics are known for a compensated imaging system, and discusses the properties of this algorithm.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Hardy, J. Lefebvre, and C. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977) (this issue).
    [CrossRef]
  2. J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wavefront sensor,” (unpublished).
  3. R. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977) (previous paper).
    [CrossRef]
  4. H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
    [CrossRef]
  5. “Restoration of Atmospherically Degraded Images,” Woods Hole Summer Study, July1966. .
  6. J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Correction of Optical Imaging Systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.
  7. J. C. Wyant, “White Light Extended Source Shearing Interferometer,” Appl. Opt. 13, 200 (1974).
    [CrossRef] [PubMed]
  8. Analog Data Processor, U. S. Patent, 3, 921, 080November18, 1975, Assigned to Itek Corporation, Lexington, Mass.
  9. J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
    [CrossRef]
  10. J. W. Hardy, C. L. Koliopoulos, and J. K. Bowker, “Radial Grating A. C. Interferometer,” (unpublished).
  11. Freeman J. Dyson, “Photon noise and atmospheric noise in active optical systems,” J. Opt. Soc. Am. 65, 551–558 (1975).
    [CrossRef]
  12. R. Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–395 (1977) (this issue).
    [CrossRef]
  13. E. Wallner, “Minimizing atmospheric dispersion effects in compensated imaging,” J. Opt. Soc. Am. 67, 407–409 (1977) (this issue).
    [CrossRef]

1977 (4)

1975 (1)

1974 (2)

J. C. Wyant, “White Light Extended Source Shearing Interferometer,” Appl. Opt. 13, 200 (1974).
[CrossRef] [PubMed]

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[CrossRef]

1953 (1)

H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
[CrossRef]

Babcock, H. W.

H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
[CrossRef]

Bowker, J. K.

J. W. Hardy, C. L. Koliopoulos, and J. K. Bowker, “Radial Grating A. C. Interferometer,” (unpublished).

Bowker, K.

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wavefront sensor,” (unpublished).

Cone, P. F.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[CrossRef]

Dyson, Freeman J.

Feinleib, J.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[CrossRef]

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Correction of Optical Imaging Systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

Hardy, J.

J. Hardy, J. Lefebvre, and C. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977) (this issue).
[CrossRef]

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wavefront sensor,” (unpublished).

Hardy, J. W.

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Correction of Optical Imaging Systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

J. W. Hardy, C. L. Koliopoulos, and J. K. Bowker, “Radial Grating A. C. Interferometer,” (unpublished).

Hudgin, R.

Koliopoulos, C.

J. Hardy, J. Lefebvre, and C. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977) (this issue).
[CrossRef]

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wavefront sensor,” (unpublished).

Koliopoulos, C. L.

J. W. Hardy, C. L. Koliopoulos, and J. K. Bowker, “Radial Grating A. C. Interferometer,” (unpublished).

Lefebvre, J.

Lipson, S. G.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[CrossRef]

Wallner, E.

Wyant, J. C.

J. C. Wyant, “White Light Extended Source Shearing Interferometer,” Appl. Opt. 13, 200 (1974).
[CrossRef] [PubMed]

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Correction of Optical Imaging Systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[CrossRef]

J. Opt. Soc. Am. (5)

Pub Astr. Soc. Pac. (1)

H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
[CrossRef]

Other (5)

“Restoration of Atmospherically Degraded Images,” Woods Hole Summer Study, July1966. .

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-Time Correction of Optical Imaging Systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wavefront sensor,” (unpublished).

Analog Data Processor, U. S. Patent, 3, 921, 080November18, 1975, Assigned to Itek Corporation, Lexington, Mass.

J. W. Hardy, C. L. Koliopoulos, and J. K. Bowker, “Radial Grating A. C. Interferometer,” (unpublished).

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 (6)

FIG. 1
FIG. 1

Dependence of the estimation error on sensor noise and number of terms for temporal averaging.

FIG. 2
FIG. 2

Atmospheric turbulence distribution used to evaluate the estimator gives r0 = 10 c (equal turbulence in tropopause peak and the exponential).

FIG. 3
FIG. 3

Dependence of estimation error (100 terms) on system speed and sensor noise.

FIG. 4
FIG. 4

Predictor error variance vs relative sensor noise and number of terms (K0 = 2).

FIG. 5
FIG. 5

Term ordering by weight for the optimal estimator.

FIG. 6
FIG. 6

Estimation error as a function of position in the aperture.

Equations (18)

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

ϕ ˆ ( x i , t ) = j k A i j k ϕ ˜ 0 ( x j , t - k δ ) .
D ( x , t ) = [ ϕ a ( y + x , t + t ) - ϕ a ( y , t ) ] 2 ,
C j k l m = n ( x j , t - k δ ) n ( x l , t - m δ ) .
C j k l m C 0 e - s x j - x l / D 0 δ k m ,
E i = [ ϕ a ( x i , t ) - ϕ ˆ ( x i , t ) ] 2
= [ ϕ a ( x i , t ) - j = 1 N k = k 0 A i j k ϕ ˜ 0 ( x j , t - k δ ) ] 2 ,
E i = [ ϕ a ( x i , t ) - j = 1 N k = k 0 A i j k ( ϕ a ( x j , t - k δ ) + n ( x j , t - k δ ) ] 2 .
E i = { ϕ a ( x i , t ) - 1 N l = 1 N ϕ a ( x i , t ) - j = 1 N k = k 0 A i j k [ ϕ a ( x j , t - k δ ) + n ( x j , t - k δ ) ] + 1 N j , l = 1 N k = k 0 A i j k [ ϕ a ( x l , t - k δ ) + n ( x l , t - k δ ) ] } 2
= E i 0 + j = 1 N k = k 0 A i j k B ¯ j k + j , l = 1 N k , m = k 0 A i j k A i l m C ¯ j k l m ,
E i 0 = 1 N j = 1 N { D ( x i - x j , 0 ) + C i 0 j 0 } - 1 2 N 2 j , l = 1 N { D ( x j - x l , 0 ) + C j 0 l 0 } ,
B ¯ j k = D ( x i - x j , k δ ) + C i k j 0 - 1 N l = 1 N { D ( x i - x l , k δ ) + D ( x j - x l , - k δ ) + C i k l 0 + C j 0 l k } + 1 N 2 l , m = 1 N { D ( x l - x m , k δ ) + C l k m 0 } ,
C ¯ j k l m = - 1 2 D [ x j - x l , ( m - k ) δ ] - 1 2 C j k l m + 1 2 N n = 1 N { D [ x n - x l , ( m - k ) δ ] + D [ x j - x n , ( m - k ) δ ] + C n k l m + C j k n m } - 1 2 N 2 n , r = 1 N { D [ x n - x r , ( m - k ) δ ] + C n k r m } .
2 l m A i l m C j k l m + B j k = 0.
A i j k = 0             for             j i .
D ( O , t ) = d 0 t 5 / 3             ( waves 2 ) ,
C j k j m = C 0 δ k m             ( waves 2 ) ,
C 0 d 0 δ 5 / 3 .
D ( x , t ) = 2.91 λ 2 d h C n 2 ( h ) x - h ω t 5 / 3             ( waves 2 ) ,