Abstract

A closed-loop adaptive-optics servo system is an iterative optical–digital processor that can be characterized by a first-order difference equation. This identification leads to a new class of wave-front estimators, some of which are extremely sparse and permit real-time subaperture intensity weighting.

© 1995 Optical Society of America

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References

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  1. G. Gardner, B. Welsh, L. Thompson, Proc. IEEE 78, 1721 (1990).
    [CrossRef]
  2. D. Gavel, J. Morris, R. Vernon, J. Opt. Soc. Am. A 11, 914 (1994).
    [CrossRef]
  3. R. Parenti, R. Sasiela, J. Opt. Soc. Am. A 11, 288 (1994).
    [CrossRef]
  4. F. Roddier, M. Northcott, J. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991).
    [CrossRef]
  5. D. Sandler, S. Stahl, R. Angel, M. Lloyd-Hart, D. McCarthy, J. Opt. Soc. Am. A 11, 925 (1994).
    [CrossRef]
  6. G. Cochran, Reps. TR-688 and TR-766 (Optical Sciences Company, Placentia, Calif., 1986).
  7. J. R. P. Angel, Nature (London) 368, 203 (1994).
    [CrossRef]
  8. R. Hudgin, J. Opt. Soc. Am. 67, 375 (1977).
    [CrossRef]
  9. D. Fried, J. Opt. Soc. Am. 56, 1373 (1966).
  10. B. Ellerbroek, J. Opt. Soc. Am. A 11,783 (1994).
    [CrossRef]
  11. E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).
  12. D. Fried, J. Opt. Soc. Am. 67, 370 (1977).
    [CrossRef]
  13. W. Wild, E. Kibblewhite, V. Scor, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 726 (1994).
  14. G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).
  15. R. Sasiela, J. Mooney, Proc. Soc. Photo-Opt. Instrum. Eng. 551, 170 (1985).
  16. C. Coulman, Annu. Rev. Astron. Astrophys. 23, 19 (1985).
    [CrossRef]
  17. W. Wild, B. Carter, E. Kibblewhite, Proc. Soc. Photo-Opt. Instrum. Eng. 1920, 263 (1993).
  18. W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

1994 (8)

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

R. Parenti, R. Sasiela, J. Opt. Soc. Am. A 11, 288 (1994).
[CrossRef]

B. Ellerbroek, J. Opt. Soc. Am. A 11,783 (1994).
[CrossRef]

D. Gavel, J. Morris, R. Vernon, J. Opt. Soc. Am. A 11, 914 (1994).
[CrossRef]

D. Sandler, S. Stahl, R. Angel, M. Lloyd-Hart, D. McCarthy, J. Opt. Soc. Am. A 11, 925 (1994).
[CrossRef]

J. R. P. Angel, Nature (London) 368, 203 (1994).
[CrossRef]

W. Wild, E. Kibblewhite, V. Scor, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 726 (1994).

1993 (1)

W. Wild, B. Carter, E. Kibblewhite, Proc. Soc. Photo-Opt. Instrum. Eng. 1920, 263 (1993).

1991 (1)

F. Roddier, M. Northcott, J. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991).
[CrossRef]

1990 (1)

G. Gardner, B. Welsh, L. Thompson, Proc. IEEE 78, 1721 (1990).
[CrossRef]

1985 (2)

R. Sasiela, J. Mooney, Proc. Soc. Photo-Opt. Instrum. Eng. 551, 170 (1985).

C. Coulman, Annu. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

1977 (2)

1966 (1)

D. Fried, J. Opt. Soc. Am. 56, 1373 (1966).

Angel, J. R. P.

J. R. P. Angel, Nature (London) 368, 203 (1994).
[CrossRef]

Angel, R.

Carter, B.

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

W. Wild, B. Carter, E. Kibblewhite, Proc. Soc. Photo-Opt. Instrum. Eng. 1920, 263 (1993).

Cochran, G.

G. Cochran, Reps. TR-688 and TR-766 (Optical Sciences Company, Placentia, Calif., 1986).

Coulman, C.

C. Coulman, Annu. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

Ellerbroek, B.

Fried, D.

D. Fried, J. Opt. Soc. Am. 67, 370 (1977).
[CrossRef]

D. Fried, J. Opt. Soc. Am. 56, 1373 (1966).

Gardner, G.

G. Gardner, B. Welsh, L. Thompson, Proc. IEEE 78, 1721 (1990).
[CrossRef]

Gavel, D.

Golub, G.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Graves, J.

F. Roddier, M. Northcott, J. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991).
[CrossRef]

Hudgin, R.

Jeys, T.

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

Kelder-house, G.

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

Kibblewhite, E.

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

W. Wild, E. Kibblewhite, V. Scor, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 726 (1994).

W. Wild, B. Carter, E. Kibblewhite, Proc. Soc. Photo-Opt. Instrum. Eng. 1920, 263 (1993).

Lloyd-Hart, M.

Manning, H.

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

McCarthy, D.

Mooney, J.

R. Sasiela, J. Mooney, Proc. Soc. Photo-Opt. Instrum. Eng. 551, 170 (1985).

Morris, J.

Northcott, M.

F. Roddier, M. Northcott, J. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991).
[CrossRef]

Parenti, R.

Roddier, F.

F. Roddier, M. Northcott, J. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991).
[CrossRef]

Sandler, D.

Sasiela, R.

R. Parenti, R. Sasiela, J. Opt. Soc. Am. A 11, 288 (1994).
[CrossRef]

R. Sasiela, J. Mooney, Proc. Soc. Photo-Opt. Instrum. Eng. 551, 170 (1985).

Scor, V.

W. Wild, E. Kibblewhite, V. Scor, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 726 (1994).

Shi, F.

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

Stahl, S.

Thompson, L.

G. Gardner, B. Welsh, L. Thompson, Proc. IEEE 78, 1721 (1990).
[CrossRef]

Van Loan, C.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Vernon, R.

Vuilleumier, R.

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

Welsh, B.

G. Gardner, B. Welsh, L. Thompson, Proc. IEEE 78, 1721 (1990).
[CrossRef]

Wild, W.

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

W. Wild, E. Kibblewhite, V. Scor, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 726 (1994).

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

W. Wild, B. Carter, E. Kibblewhite, Proc. Soc. Photo-Opt. Instrum. Eng. 1920, 263 (1993).

Annu. Rev. Astron. Astrophys. (1)

C. Coulman, Annu. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Nature (1)

J. R. P. Angel, Nature (London) 368, 203 (1994).
[CrossRef]

Proc. IEEE (1)

G. Gardner, B. Welsh, L. Thompson, Proc. IEEE 78, 1721 (1990).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (5)

W. Wild, E. Kibblewhite, V. Scor, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 726 (1994).

W. Wild, B. Carter, E. Kibblewhite, Proc. Soc. Photo-Opt. Instrum. Eng. 1920, 263 (1993).

W. Wild, E. Kibblewhite, F. Shi, B. Carter, G. Kelder-house, R. Vuilleumier, H. Manning, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 1121 (1994).

E. Kibblewhite, R. Vuilleumier, B. Carter, W. Wild, T. Jeys, Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 272 (1994).

R. Sasiela, J. Mooney, Proc. Soc. Photo-Opt. Instrum. Eng. 551, 170 (1985).

Publ. Astron. Soc. Pac. (1)

F. Roddier, M. Northcott, J. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991).
[CrossRef]

Other (2)

G. Cochran, Reps. TR-688 and TR-766 (Optical Sciences Company, Placentia, Calif., 1986).

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

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Figures (3)

Fig. 1
Fig. 1

Adaptive-optics servo loop. The residual wave-front phase from reflection off the deformable mirror is φ(ti+1) – ϕ(ti), where φ(ti+1) is the phase of the incident wave front and ϕ (ti) is the reconstructed phase on the mirror’s surface. This residual is converted to subaperture tilt information by the wave-front sensor and subsequently reconstructed by the matrix M and then scaled by k. An updated signal is then sent to the mirror, thus completing the loop.

Fig. 2
Fig. 2

Noise propagator for a 6 × 6 actuator array as a function of k. Solid curve, least-squares estimator; dashed curve, Richardson estimator; dotted–dashed curve, Jacobi estimator.

Fig. 3
Fig. 3

Time-delay error normalized to (D/r0)5/3 as a function of k. The tip/tilt removed normalized variance at k = 0 converges to σ td 2 / ( D / r 0 ) 5 / 3 = 0.134 in the limit of large N. Solid curve, M = A+; dashed curve, Jacobi estimator.

Equations (8)

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σ φ 2 = g n σ ph 2 + σ td 2 + σ fe 2 + ,
ϕ i + 1 = ϕ i + k M 1 Γ t i t i + Γ s ( t ) d t ,
ϕ i + 1 = ϕ i + k M ( s i - A ϕ i ) .
g n σ ph 2 = σ n 2 = σ ph 2 k 2 N Tr ( HM M T ) ,
H = t = 0 Ω - 1 ( I - k A T M T ) t ( I - k MA ) t ,
[ X t ] i j = - 3.44 ( L D ) 5 / 3 u i - u j - v t 5 / 3 ,
σ td 2 = 1 N Tr [ X ˜ 0 - 2 ( I - k MA ) - 1 C ] + k 2 HMA X ˜ 0 A T M T + 2 k H C A T M T ] ( D r 0 ) 5 / 3 ,
C t = 1 Ω - 1 ( I - k MA ) t k MA X ˜ t .

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