Abstract

A new class of optimal wave-front estimators for adaptive-optics systems that minimize time-delay errors that are due to atmospheric winds is developed. These matrices utilize the past time history of the servo system by means of temporal Kolmogorov cross-covariance matrices. Except in the simplest case, a nonlinear matrix equation must be iteratively solved. Strehl ratio performance for selected estimators is given.

© 1996 Optical Society of America

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References

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  1. W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, Opt. Lett. 20, 955 (1995).
    [CrossRef] [PubMed]
  2. G. A. Tyler, J. Opt. Soc. Am. A 1, 251 (1984).
    [CrossRef]
  3. E. P. Wallner, Proc. SPIE 351, 42 (1982).
  4. E. P. Wallner, J. Opt. Soc. Am. A 73, 1171 (1983).
  5. W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
    [CrossRef]
  6. W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
    [CrossRef]
  7. W. Wild, E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994).
    [CrossRef]
  8. C. Shelton, Mt. Wilson Observatory, Mt. Wilson, Calif. 91023 (personal communication, 1996).

1995 (2)

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, Opt. Lett. 20, 955 (1995).
[CrossRef] [PubMed]

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

1994 (2)

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

W. Wild, E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

1984 (1)

1983 (1)

E. P. Wallner, J. Opt. Soc. Am. A 73, 1171 (1983).

1982 (1)

E. P. Wallner, Proc. SPIE 351, 42 (1982).

Carter, B.

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

Farmiga, N.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

Kelderhouse, G.

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

Kibblewhite, E.

W. Wild, E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Kibblewhite, E. J.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, Opt. Lett. 20, 955 (1995).
[CrossRef] [PubMed]

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

Manning, H.

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

Scor, V.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. Wild, E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Shelton, C.

C. Shelton, Mt. Wilson Observatory, Mt. Wilson, Calif. 91023 (personal communication, 1996).

Shi, F.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

Tyler, G. A.

Vuilleumier, R.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, Opt. Lett. 20, 955 (1995).
[CrossRef] [PubMed]

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

Wallner, E. P.

E. P. Wallner, J. Opt. Soc. Am. A 73, 1171 (1983).

E. P. Wallner, Proc. SPIE 351, 42 (1982).

Wild, W.

W. Wild, E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Wild, W. J.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, Opt. Lett. 20, 955 (1995).
[CrossRef] [PubMed]

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

E. P. Wallner, J. Opt. Soc. Am. A 73, 1171 (1983).

G. A. Tyler, J. Opt. Soc. Am. A 1, 251 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (4)

E. P. Wallner, Proc. SPIE 351, 42 (1982).

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994).
[CrossRef]

W. Wild, E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Other (1)

C. Shelton, Mt. Wilson Observatory, Mt. Wilson, Calif. 91023 (personal communication, 1996).

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Figures (1)

Fig. 1
Fig. 1

Strehl ratio versus Δt plot; on the left axis the curves from top to bottom are for the least-squares estimator (dotted), the Wiener-Kolmogorov optimal estimator (dotted-dashed), the optimal estimator in Eq. (8) (dashed), and the η = 1 POE (solid). The η = 0 and η = 1 POE's (the lowest and next-to-lowest curves on the axis, respectively) were computed with the assumption that Δt = 0.4. For Δt > 0.4 the η = 1 POE maximizes the relative Strehl ratio over the other estimators.

Equations (12)

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Γ = [ φ ( t i + 1 ) - ϕ dm ( t i ) ] [ φ ( t i + 1 ) - ϕ dm ( t i ) ] T ,
ϕ ( t i + 1 ) = a 0 ϕ ( t i ) + k M [ s ( t i + 1 ) - A ϕ ( t i ) ] ,
s ( t i ) = A φ ( t i ) + n ( t i )
ϕ ( t η ) = k i = 0 η ( a 0 I - k MA ) η - i Ms ( t i ) .
X φ i i = φ ( t i ) φ ( t i ) T .
[ X φ i i ] j j = - 3.44 | x j - x j - v τ ( i - i ) r 0 | 5 / 3
Tr ( Γ ) = Tr [ ϕ dm ( t i ) - ϕ ( t i ) ] [ ϕ dm ( t i ) - ϕ ( t i ) ] T + Tr [ ϕ ( t i + 1 ) - ϕ ( t i ) ] [ ϕ ( t i + 1 ) - ϕ ( t i ) ] T + Tr [ ϕ ( t i ) - φ ( t i ) ] [ ϕ ( t i ) - φ ( t i ) ] T + cross terms .
M = X ˜ φ i i A T { A X ˜ φ i i A T + X n i i } - 1 ,
[ X ˜ φ i i ] j j = - 3.44 ( I - P ) × | x j - x j - v τ ( i - i ) r 0 | 5 / 3 ( I - P ) ,
M = X ˜ φ i i A T { A ( 3 X ˜ φ i i - X ˜ φ i , i - 1 - X ˜ φ i - 1 , i ) A T + σ n 2 I } - 1 ,
M T = { k a 0 ψ i i A T β T - k 2 AMA ψ i i A T β T + k 2 A ψ i i A T + k 2 a 0 A ψ i - 1 , i A T - k 3 A MA ψ i - 1 , i A T + k A ψ i , i - 1 A T β T + k a 0 X n i i β T - k 2 AMX n i i β T + k 2 X n i i + k 2 a 0 X n i i - k 3 AMX n i i + k X n i i β T } - 1 { k 2 A ψ i i A T β T M T MA + k A X ˜ φ i i + k 3 A ψ i - 1 , i A T M T MA + k a 0 A X ˜ φ i - 1 , i - k 2 AMA X ˜ φ i - 1 , i - k 2 A X ˜ φ i - 1 , i MA + k 2 X n i i β T M T MA + k 3 X n i i M T MA } ,
ψ m , m = 3 X ˜ φ m , m - X ˜ φ m , m - 1 - X ˜ φ m - 1 , m

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