Abstract

The effect of time delay on characteristics of an optical adaptive system is considered. Classification of the optical adaptive systems as dynamic feedback systems has been made. Systems with constant delay, a high-speed adaptive system as well as a new class of systems, i.e., predicting adaptive systems, are investigated.

© 1987 Optical Society of America

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References

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  1. V. P. Lukin, Kvantovaya Elektron. 8, 2145 (1981).
  2. R. J. Noll, “Zernike Polynomials and Atmospheric Turbulence,” J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  3. V. P. Lukin, Kvantovaya Elektron. 10, 993 (1983).
  4. J. Y. Wang, J. K. Markey, “Modal Compensation of Atmospheric Turbulence Phase Distortion,” J. Opt. Soc. Am. 68, 78 (1978).
    [CrossRef]
  5. G. C. Valley, S. M. Wandzura, “Spatial Correlation of Phase-Expansion Coefficients for Propagation Through Atmospheric Turbulence,” J. Opt. Soc. Am. 69, 712 (1979).
    [CrossRef]
  6. V. E. Zuev, P. A. Konyaev, V. P. Lukin, Izv. Vuzov. Fiz. 28, No. 11, 6 (1985).
  7. A. N. Efimov, Prediction of Random Processes (Znanie, Moscow, 1978).
  8. V. P. Lukin, V. V. Pokasov, “Optical Wave Phase Fluctuations,” Appl. Opt. 20, 121 (1981).
    [CrossRef] [PubMed]
  9. D. L. Fried, “Least-Squares Fitting a Wave-Front Distortion Estimate to an Array of Phase-Difference Measurements,” J. Opt. Soc. Am. 67, 370 (1977).
    [CrossRef]
  10. V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.
  11. V. V. Voitsekhovich, “Temporal Characteristics of an Adaptive Astronomical System,” preprint IKI AN SSSR, PR-873 (1984), p. 24.

1985 (1)

V. E. Zuev, P. A. Konyaev, V. P. Lukin, Izv. Vuzov. Fiz. 28, No. 11, 6 (1985).

1984 (1)

V. V. Voitsekhovich, “Temporal Characteristics of an Adaptive Astronomical System,” preprint IKI AN SSSR, PR-873 (1984), p. 24.

1983 (2)

V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.

V. P. Lukin, Kvantovaya Elektron. 10, 993 (1983).

1981 (2)

1979 (1)

1978 (1)

1977 (1)

1976 (1)

Efimov, A. N.

A. N. Efimov, Prediction of Random Processes (Znanie, Moscow, 1978).

Fried, D. L.

Konyaev, P. A.

V. E. Zuev, P. A. Konyaev, V. P. Lukin, Izv. Vuzov. Fiz. 28, No. 11, 6 (1985).

Lukin, V. P.

V. E. Zuev, P. A. Konyaev, V. P. Lukin, Izv. Vuzov. Fiz. 28, No. 11, 6 (1985).

V. P. Lukin, Kvantovaya Elektron. 10, 993 (1983).

V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.

V. P. Lukin, Kvantovaya Elektron. 8, 2145 (1981).

V. P. Lukin, V. V. Pokasov, “Optical Wave Phase Fluctuations,” Appl. Opt. 20, 121 (1981).
[CrossRef] [PubMed]

Markey, J. K.

Mironov, V. L.

V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.

Noll, R. J.

Pokasov, V. V.

Tartakovskii, V. A.

V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.

Valley, G. C.

Voitsekhovich, V. V.

V. V. Voitsekhovich, “Temporal Characteristics of an Adaptive Astronomical System,” preprint IKI AN SSSR, PR-873 (1984), p. 24.

Wandzura, S. M.

Wang, J. Y.

Zuev, V. E.

V. E. Zuev, P. A. Konyaev, V. P. Lukin, Izv. Vuzov. Fiz. 28, No. 11, 6 (1985).

V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.

Appl. Opt. (1)

Izv. Vuzov. Fiz. (1)

V. E. Zuev, P. A. Konyaev, V. P. Lukin, Izv. Vuzov. Fiz. 28, No. 11, 6 (1985).

J. Opt. Soc. Am. (4)

Kvantovaya Elektron. (2)

V. P. Lukin, Kvantovaya Elektron. 10, 993 (1983).

V. P. Lukin, Kvantovaya Elektron. 8, 2145 (1981).

preprint IKI AN SSSR, PR-873 (1)

V. V. Voitsekhovich, “Temporal Characteristics of an Adaptive Astronomical System,” preprint IKI AN SSSR, PR-873 (1984), p. 24.

Third All-Union Conference on Atmospheric Optics (1)

V. E. Zuev, V. P. Lukin, V. L. Mironov, V. A. Tartakovskii, in Third All-Union Conference on Atmospheric Optics, Part 2,” Tomsk Izd. Inst. Atm. Optics (1983), pp. 5–13.

Other (1)

A. N. Efimov, Prediction of Random Processes (Znanie, Moscow, 1978).

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Equations (56)

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I ( 0 ) = λ - 2 L - 2 d 4 ρ 1 , 2 A ( ρ 1 ) A ( ρ 2 ) exp [ - i k ( ρ 1 2 - ρ 2 2 ) 2 L ] × exp { i [ S ( O , ρ 1 ; L , O ; t + τ ) - S ( O , ρ 2 ; L , O ; t + τ ) ] - i [ S ( O , ρ 1 ; L , O ; t ) - S ( O , ρ 2 ; L , O ; t ) ] } ,
= exp ( - ½ { [ S ( ρ 1 , t + τ ) - S ( ρ 2 , t + τ ) ] - [ S ( ρ 1 , t ) - S ( ρ 2 , t ) ] } 2 ) ,
S ( ρ , t + τ ) = S ( ρ + v τ , t ) ,
S ( ρ , t ) = j = 1 a j ( t ) F j ( ρ / R ) ,
a j = 1 R 2 d 2 ρ F j ( ρ / R ) S ( ρ , t ) W ( ρ ) ,
W ( ρ ) = { 1 , ρ R 0 , ρ > R
S ( ρ + v τ , t ) = S ( ρ , t ) + v τ S ( ρ + v τ , t ) τ = 0 · v τ ,
v τ S ( ρ + v τ , t ) τ = 0 = ρ S ( ρ , t ) ,
= exp ( - τ 2 v 2 2 { [ ρ 1 S ( ρ 1 , t ) ] 2 + [ ρ 2 S ( ρ 2 , t ) ] 2 - 2 ρ 1 S ( ρ 1 , t ) ρ 2 S ( ρ 2 , t ) } ) ,
ρ S ( ρ , t ) = j = 2 a j ( t ) ρ F j ( ρ / R ) ,
D S ( ρ 1 - ρ 2 ) = 6.88 ( ρ 1 - ρ 2 / r 0 ) 5 / 3 ,
= exp [ - 12 a 4 2 R 4 τ 2 v 2 ( ρ 1 - ρ 2 ) 2 ] ,
a 4 2 = 0.0736 ( R / r 0 ) 5 / 3 .
I ( 0 ) = λ - 2 L - 2 d 4 ρ 1 , 2 A ( ρ 1 ) A * ( ρ 2 ) exp [ - i κ ( ρ 1 2 - ρ 2 2 ) 2 L - 0.88 τ 2 v 2 r 0 5 / 3 R 7 / 3 ( ρ 1 - ρ 2 ) 2 ] .
I ( 0 ) = 2 π 2 a 2 λ 2 L 2 0 d ρ ρ exp [ - ρ 2 4 a 2 ( 1 + Ω 2 ) - 0.88 τ 2 v 2 r 0 5 / 3 R 7 / 3 ρ 2 ] .
τ 3.48 ( r 0 v ) · ( a / Ω r 0 ) 1 / 6 .
τ 0.53 ( r 0 v ) · ( a / r 0 ) 1 / 6 .
S ^ ( ρ , t + τ ) = S ( ρ , t ) + S τ ( ρ , t + τ ) τ = 0 · τ ,
S τ ( ρ , t + τ ) τ = 0 = Δ ρ S ( ρ , t ) · v .
I ( 0 ) = λ - 2 L - 2 d 4 ρ 1 , 2 A ( ρ 1 ) A * ( ρ 2 ) exp [ - i k ( ρ 1 2 - ρ 2 2 ) 2 L ] × exp { i [ S ( ρ 1 + v τ , t ) - S ( ρ 1 , t ) - ρ 1 S ( ρ 1 , t ) v τ ] - i [ S ( ρ 2 + v τ , t ) - S ( ρ 2 , t ) - ρ 2 S ( ρ 2 , t ) v τ ] } .
= exp { - τ 4 v 4 8 [ ρ 1 2 S ( ρ 1 , t ) - ρ 2 2 S ( ρ 2 , t ) ] 2 } .
= exp [ - 8.48 τ 4 v 4 r 0 5 / 3 R 13 / 3 ( ρ 1 - ρ 2 ) 2 ] .
τ c ( r 0 v ) · ( a / r 0 ) 7 / 12 .
S τ = S y , v y + S z v z = ρ S ( ρ , t ) v ,
τ c τ = O [ ( a / r 0 ) 5 / 12 ] .
S ^ ( r , t + τ ) = S ( r + v τ , t ) .
v = v 0 + δ v ,
v = v 0 ,             δ v 2 = δ v y 2 + δ v z 2 .
S ^ ( r , t + τ ) = S ( r + v 0 , t )
S ( r + v τ , t ) = S ( r + v 0 τ , t ) + r + v τ S ( r + v τ , t ) v = v 0 · δ v τ .
S ( r , t + τ ) - S ^ ( r , t + τ ) = r + v 0 τ S ( r + v 0 τ , t ) δ v τ ;
δ v = exp { i δ v τ [ r 1 + v 0 τ S ( r 1 + v 0 τ , t ) - r 2 + v 0 τ S ( r 2 + r 0 τ , t ) ] } δ v .
δ v = exp ( - 0.88 τ 2 δ v 2 r 0 5 / 3 R 7 / 3 ρ 1 - ρ 2 2 ) ,
p ( δ v z ) = ( 2 π σ ) - 1 exp ( - δ v z 2 / 2 σ 2 ) , σ y 2 = σ z 2 = σ 2 .
= [ 1 + 1.76 τ 2 ( ρ 1 - ρ 2 ) 2 σ 2 r 0 5 / 3 R 7 / 3 ] - 1 .
I ( 0 ) = 2 π 2 a 2 λ 2 L 2 0 d ρ ρ exp ( - ρ 2 / 4 a 2 ) ( 1 + 1.76 τ 2 ρ 2 σ 2 / r 0 5 / 3 a 7 / 3 ) = 2 π 2 a 2 λ 2 L 2 ( r 0 5 / 3 a 7 / 3 3.52 τ 2 σ 2 ) [ - exp ( r 0 5 / 3 a 1 / 3 7.04 τ 2 σ 2 ) ] × E i ( - r 0 5 / 3 a 1 / 3 7.04 τ 2 σ 2 ) .
E i ( - z ) = exp ( - z ) ( - z ) ( 1 - 1 ! z + 2 ! z 2 - 3 ! z 3 + ) .
I ( 0 ) = 4 π 2 a 2 λ 2 L 2 ( 1 - 7.04 τ 2 σ 2 r 0 5 / 3 a 1 / 3 + ) ,
τ ( r 0 σ ) ( a / r 0 ) 1 / 6 .
S ^ ( r , t + τ ) = S ( r , t ) = m .
S ^ ( r , t + τ ) = S ( r , t ) .
S ^ ( r , t + τ ) = f S ( τ ) S ( r , t ) ;
e 2 = [ S ( r , t + τ ) - S ^ ( r , t + τ ) ] 2
e 2 1 = σ S 2 ,             e 2 2 = 2 σ S 2 [ 1 - f s ( τ ) ] = D S ( τ ) , e 2 3 = σ S 2 [ 1 - f S 2 ( τ ) ] ,
( 1 ) σ S 2 1 ; ( 2 ) D S ( τ ) 1 ; ( 3 ) σ S 2 [ 1 - f S 2 ( τ ) ] 1.
β = [ S ( r 1 , t + τ ) - S ^ ( r 1 , t + τ ) ] - [ S ( r 2 , t + τ ) - S ^ ( r 2 , t + τ ) ] .
β 2 1 = D S ( r 1 - r 2 ) ;             β 2 2 = 2 D S ( r 1 - r 2 ) - 2 B Δ S ( τ ) ; β 2 3 = D S ( r 1 - r 2 ) [ 1 + f S 2 ( τ ) ] - 2 f S ( τ ) B Δ S ( τ ) .
B Δ S ( τ ) = 2 B S ( v τ ) - B S [ ( r 1 - r 2 ) + v τ ] - B S [ ( r 1 - r 2 ) - v τ ]
S ˜ ( r , t ) = j = 2 a j ( t ) F j ( r / R )
S ^ ( r , t + τ ) = j = 2 a j ( t ) f j ( τ ) F j ( r / R ) ,
β 2 = { [ S ( r 1 , t + τ ) - S ( r 2 , t + τ ) ] - [ S ^ ( r 1 , t + τ ) - S ^ ( r 2 , t + τ ) ] } 2 = j = 2 i = 2 [ a i a j + f i ( τ ) f j ( τ ) × a i a j - f j ( τ ) a i ( t + τ ) a j ( t ) - f i ( τ ) f j ( τ ) a i a j - f i ( τ ) a i ( t ) a j ( t + τ ) ] [ F j ( r 1 / R ) - F j ( r 2 / R ) ] [ F i ( r 1 / R ) - F i ( r 2 / R ) ] .
β 2 22 = [ 1 - f 2 2 ( τ ) ] a 2 2 ( x 1 - x 2 R ) 2 ,             F 2 ( r / R ) = x / R ;
β 2 28 = a 2 a 8 { 1 - f 28 ( τ ) [ f 2 ( τ ) + f 8 ( τ ) - f 2 ( τ ) f 8 ( τ ) ] } .
β 2 22 + β 2 33 = a 2 2 R 2 [ 1 - f 2 2 ( τ ) ] ( ρ 1 - ρ 2 ) 2 ,
2 a 2 2 / R 2 [ 1 - f 2 ( τ ) ] ( ρ 1 - ρ 2 ) 2 .
Δ = [ 1 + f j ( τ ) ] / 2.

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