Abstract

Dynamic (time) characteristics of adaptive systems are analyzed. A common adaptive system with a finite frequency band (or a finite response time) is described as a dynamic constant time-delay system, where time delay is to be much shorter than the time of coherence radius transfer through an optical beam by a mean wind speed. The questions of coherent beam formation are considered with use of the reference source. The analytical calculation of the Strehl parameter is made on basis of the generalized Huygens–Kirchhoff principle. An adaptive system is considered where the correcting phase is calculated with the use of both its derivatives and the signal, as well as adaptive systems using different time-predicting algorithms of the correcting signal for future time points. The use of a predicted phase front of the correcting wave allows much longer time delays. The stronger the phase distortions in the optical wave, the higher the time gain in comparison with common (with constant time delay) adaptive systems.

© 2010 Optical Society of America

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References

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  1. V. P. Lukin and V. L. Mironov, “Dynamic characteristics of adaptive optical systems,” Kvantovaya Electronika 12, 1959–1962 (1985).
  2. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  3. D. L. Greenwood, “Bandwidth specification for adaptive optics system,” J. Opt. Soc. Am. 67, 390–393 (1977).
    [CrossRef]
  4. A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in Turbulent Atmosphere (Nauka, Moscow, 1976).
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  6. V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).
  7. V. P. Lukin and V. E. Zuev, “Dynamic characteristics of adaptive systems,” Appl. Opt. 26, 139–144 (1987).
    [CrossRef] [PubMed]
  8. D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensative systems,” J. Opt. Soc. Am. 66193–206 (1976).
    [CrossRef]
  9. I. Y. Wang and L. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–88 (1978).
    [CrossRef]
  10. V. P. Lukin and V. V. Pokasov, “Optical wave phase fluctuations,” Appl. Opt. 20, 121–135 (1981).
    [CrossRef] [PubMed]
  11. L. A. Bolbasova and V. P. Lukin, “Modal anisoplanatism of phase fluctuations,” Atmos. Oceanic Opt. 21, 1070–1075 (2008).
  12. V. V. Voitsekhovich, Influence of Atmospheric Turbulence on the Accuracy of Determination of Wavefront Parameter, Preprint N. 862 (Institute of Space Researches, Moscow, 1984).

2008 (1)

L. A. Bolbasova and V. P. Lukin, “Modal anisoplanatism of phase fluctuations,” Atmos. Oceanic Opt. 21, 1070–1075 (2008).

1987 (1)

1986 (1)

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).

1985 (1)

V. P. Lukin and V. L. Mironov, “Dynamic characteristics of adaptive optical systems,” Kvantovaya Electronika 12, 1959–1962 (1985).

1984 (1)

V. V. Voitsekhovich, Influence of Atmospheric Turbulence on the Accuracy of Determination of Wavefront Parameter, Preprint N. 862 (Institute of Space Researches, Moscow, 1984).

1981 (1)

1978 (2)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

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

1977 (1)

1976 (3)

Bolbasova, L. A.

L. A. Bolbasova and V. P. Lukin, “Modal anisoplanatism of phase fluctuations,” Atmos. Oceanic Opt. 21, 1070–1075 (2008).

Fried, D. L.

Greenwood, D. L.

Greenwood, D. P.

Gurvich, A. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in Turbulent Atmosphere (Nauka, Moscow, 1976).

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Khmelevtsov, S. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in Turbulent Atmosphere (Nauka, Moscow, 1976).

Kon, A. I.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in Turbulent Atmosphere (Nauka, Moscow, 1976).

Lukin, V. P.

L. A. Bolbasova and V. P. Lukin, “Modal anisoplanatism of phase fluctuations,” Atmos. Oceanic Opt. 21, 1070–1075 (2008).

V. P. Lukin and V. E. Zuev, “Dynamic characteristics of adaptive systems,” Appl. Opt. 26, 139–144 (1987).
[CrossRef] [PubMed]

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).

V. P. Lukin and V. L. Mironov, “Dynamic characteristics of adaptive optical systems,” Kvantovaya Electronika 12, 1959–1962 (1985).

V. P. Lukin and V. V. Pokasov, “Optical wave phase fluctuations,” Appl. Opt. 20, 121–135 (1981).
[CrossRef] [PubMed]

Markey, L. K.

Mironov, V. L.

V. P. Lukin and V. L. Mironov, “Dynamic characteristics of adaptive optical systems,” Kvantovaya Electronika 12, 1959–1962 (1985).

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in Turbulent Atmosphere (Nauka, Moscow, 1976).

Noll, R. J.

Pokasov, V. V.

Voitsekhovich, V. V.

V. V. Voitsekhovich, Influence of Atmospheric Turbulence on the Accuracy of Determination of Wavefront Parameter, Preprint N. 862 (Institute of Space Researches, Moscow, 1984).

Wang, I. Y.

Zuev, V. E.

Appl. Opt. (2)

Atmos. Oceanic Opt. (1)

L. A. Bolbasova and V. P. Lukin, “Modal anisoplanatism of phase fluctuations,” Atmos. Oceanic Opt. 21, 1070–1075 (2008).

J. Opt. Soc. Am. (4)

Kvantovaya Electronika (1)

V. P. Lukin and V. L. Mironov, “Dynamic characteristics of adaptive optical systems,” Kvantovaya Electronika 12, 1959–1962 (1985).

Proc. IEEE (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other (3)

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in Turbulent Atmosphere (Nauka, Moscow, 1976).

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).

V. V. Voitsekhovich, Influence of Atmospheric Turbulence on the Accuracy of Determination of Wavefront Parameter, Preprint N. 862 (Institute of Space Researches, Moscow, 1984).

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Figures (2)

Fig. 1
Fig. 1

Comparison of the quality of prediction for the three different schemes: curve 1, prediction based on the average phase value; curve 2, adaptive correction with constant delay; curve 3, statistical prediction on the one measured value.

Fig. 2
Fig. 2

Time correlation function of the phase difference γ ( τ ) = b Δ S ( | v τ | ) . Separation of the observation points are for curve I, κ 0 | ρ 1 ρ 2 | = 0.2 ; curve II, κ 0 | ρ 1 ρ 2 | = 1.0 ; curve 3, κ 0 | ρ 1 ρ 2 | = 3.0 . The dashed curve shows the correlation function of phase fluctuations for comparison.

Equations (65)

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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 ( 0 , ρ 1 ; L , 0 ; t + τ ) S ( 0 , ρ 2 ; L , 0 ; t + τ ) ] i [ S corr ( ρ 1 ; t ) S corr ( ρ 2 ; t ) ] } .
= exp { 1 2 { [ S ( ρ 1 ; t + τ ) S ( ρ 2 ; t + τ ) ] [ S ( ρ 1 ; t ) S ( ρ 2 ; t ) ] } 2 } ,
S ( ρ 1 ; t + τ ) = S ( ρ 1 + υ τ ; t ) ,
S ( ρ ; t ) = j = 1 a j ( t ) F j ( ρ R ) ,
a j ( t ) = 1 R 2 d 2 ρ F j ( ρ R ) S ( ρ , t ) W ( ρ ) ,
S ( ρ 1 , t + τ ) = S ( ρ 1 ; t ) + τ S ( ρ 1 ; t ) τ = S ( ρ 1 ; t ) + υ τ | S ( ρ 1 ; t ) | τ = 0 υ τ .
υ τ | S ( ρ 1 + v τ ; t ) | τ = 0 = ρ 1 S ( ρ 1 ; t ) ,
= exp { τ 2 υ 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 1 S ( ρ 1 ; t ) = j = 2 a ( t ) ρ 1 F j ( ρ 1 R ) ,
D S ( | ρ 1 ρ 2 | ) = 6.88 [ | ρ 1 ρ 2 | r 0 ] 5 3 ,
x 1 S ( ρ 1 ; t ) = a 2 R + 4 x 1 a 4 R 2 + 4 x 1 a 5 R 2 + 4 y 1 a 6 R 2 ,
y 1 S ( ρ 1 ; t ) = a 3 R + 4 y 1 a 4 R 2 4 y 1 a 5 R 2 + 4 x 1 a 6 R 2 .
[ ρ 1 S ( ρ 1 ; t ) ] 2 = a 2 2 + a 3 2 R 2 + 16 ( x 1 2 + y 1 2 ) a 4 2 R 4 + 4 x 1 a 5 2 R 2 + 4 y 1 a 6 2 R 2 + 4 ( x 1 2 + y 1 2 ) a 5 2 R 4 + 4 ( x 1 2 + y 1 2 ) a 6 2 R 4 ,
ρ 1 S ( ρ 1 ; t ) ρ 2 S ( ρ 2 ; t ) = a 2 2 + a 3 2 R 2 + 16 ( x 1 x 2 + y 1 y 2 ) a 4 2 R 4 + 4 ( x 1 x 2 + y 1 y 2 ) a 5 2 R 4 + 4 ( x 1 x 2 + y 1 y 2 ) a 6 2 R 4 .
= exp { 12 a 4 2 R 4 v 2 τ 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 k ( ρ 1 2 ρ 2 2 ) 2 L } × exp [ 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 ρ d ρ exp [ ρ 2 4 a 2 ( 1 + Ω 2 ) 0.88 τ 2 v 2 ρ 2 r 0 5 3 R 7 3 ]
τ 3.48 ( r 0 v ) ( a Ω r 0 ) 1 6 ,
τ 0.53 ( r 0 v ) ( a r 0 ) 1 6 .
S corr ( ρ ; t + τ ) = S ̂ ( ρ + υ τ ; t ) S ( ρ ; t ) + ρ S ( ρ ; t ) v τ ,
| S τ ( ρ ; t + τ ) | τ = 0 = 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 + υ τ ) S ( ρ 1 ) ρ 1 S ( ρ 1 ) υ τ ] i [ S ( ρ 2 + υ τ ) S ( ρ 2 ) ρ 2 S ( ρ 2 ) υ τ ] } .
= exp { τ 4 υ 4 8 ρ 1 2 S ( ρ 1 ; t ) ρ 2 2 [ S ( ρ 2 ; t ) ] 2 } .
= exp ( 8.48 τ 4 υ 4 R 13 3 r 0 5 3 ( ρ 1 ρ 2 ) 2 ) .
τ s < ( r 0 v ) ( a r 0 ) 7 12
τ c τ ( a r 0 ) 5 12 .
S ̂ ( ρ , t + τ ) = S ( ρ + v τ , t ) .
v = v 0 + δ v ,
S corr ( ρ , t + τ ) = S ̂ ( ρ , t + τ ) = S ( ρ + v 0 τ , t ) ,
S ( ρ + v τ , t ) = S ( ρ + v 0 τ , t ) + ρ S ( ρ + v τ , t ) v τ .
S ( ρ , t + τ ) S ̂ ( ρ , t + τ ) = v τ | S ( ρ + v τ , t ) | δ v τ = 0 δ v τ .
δ v = exp { i δ v τ [ v τ S ( r 1 + v 0 τ , t ) v τ S ( r 2 + v 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 ] δ v ,
δ v y , z = 1 2 π σ exp ( δ v 2 y , z 2 σ 2 ) ,
δ v = ( 1 + 1.76 τ 2 σ 2 ( ρ 1 ρ 2 ) 2 r 0 5 3 R 7 3 ) 1 .
I ( 0 ) = 2 π 2 a 2 λ 2 L 2 0 ρ d ρ exp ( ρ 2 4 a 2 ) [ 1 + 1.76 τ 2 σ 2 ( ρ 1 ρ 2 ) 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 4 λ 2 L 2 ( 1 7.04 τ 2 σ 2 r 0 5 3 a 1 3 + ) ,
τ w 0.38 ( a r 0 ) 7 12 ( r 0 σ ) ,
S ̂ ( ρ , t + τ ) = S ( ρ , t ) = M .
S ̑ ( ρ , t + τ ) = S ( ρ , t ) .
S ̑ ( ρ , t + τ ) = b S ( τ ) S ( ρ , t ) .
e 2 = [ S ( ρ , t + τ ) S ̑ ( ρ , t + τ ) ] 2
for Scheme 1 :     e 2 1 = σ S 2 ,
for Scheme 2 :     e 2 2 = 2 σ S 2 [ 1 b S ( τ ) ] = D S ( | v τ | ) ,
for Scheme 3 :     e 2 3 = σ S 2 [ 1 b S 2 ( τ ) ] ,
1 . σ S 2 1 ; 2 . D S ( | v τ | ) 1 ; 3 . σ S 2 ( 1 b S 2 ( τ ) ) 1 .
[ S ( ρ 1 ) S ( ρ 2 ) ] 2 = D S ( | ρ 1 ρ 2 | ) = [ S 1 ( ρ 1 ) S 1 ( ρ 2 ) ] 2 = D S 1 ( | ρ 1 ρ 2 | ) ,
S 1 2 ( ρ ) = S 2 ( ρ ) a 1 2 ,
β = [ S ( ρ 1 , t + τ ) S ( ρ 2 , t + τ ) ] [ S corr ( ρ 1 , t + τ ) S corr ( ρ 2 , t + τ ) ] ,
β 2 1 = D S ( | ρ 1 ρ 2 | ) , β 2 2 = 2 D S ( | ρ 1 ρ 2 | ) 2 B Δ S ( | v τ | ) ,
β 2 3 = { [ S ( ρ 1 , t + τ ) b S ( τ ) S ( ρ 1 , t ) ] [ S ( ρ 2 , t + τ ) b S ( τ ) S ( ρ 1 , t ) ] } 2 = D S ( | ρ 1 ρ 2 | ) { 1 + b S 2 ( τ ) } 2 b S ( τ ) B Δ S ( | v τ | ) .
B Δ S ( | v τ | ) = 2 B S ( | v τ | ) B S ( | ( ρ 1 ρ 2 ) v τ | ) B S ( | ( ρ 1 ρ 2 ) + v τ | ) .
1 . β 2 1 D S ( | ρ 1 ρ 2 | ) 1 ,
2 . β 2 2 D S ( | ρ 1 ρ 2 | ) = 2 ( 1 b Δ S ( | v τ | ) ) ,
3 . β 2 3 D S ( | ρ 1 ρ 2 | ) = ( 1 + b S 2 ( τ ) ) 2 b S ( τ ) b Δ S ( | v τ | ) .
S ( ρ ; t ) = j = 2 a j ( t ) F j ( ρ R )
S ̂ 4 ( ρ ; t + τ ) = j = 2 a j ( t ) b j ( τ ) F j ( ρ R ) .
β 2 4 = { [ S ( ρ 1 , t + τ ) S ̂ 1 ( ρ 1 , t + τ ) ] [ S ( ρ 2 , t + τ ) S ̂ 1 ( ρ 2 , t + τ ) ] } 2 = { j = 2 [ a j ( t + τ ) b j ( τ ) a j ( t ) ] [ F j ( ρ 1 R ) F j ( ρ 2 R ) ] } 2 = i = 2 j = 2 [ a i a j + b j ( τ ) b i ( τ ) a i a j b j ( τ ) a i ( t + τ ) a j ( t ) b i ( τ ) a j ( t + τ ) a i ( t ) ] [ F j ( ρ 1 R ) F j ( ρ 2 R ) ] [ F i ( ρ 1 R ) F i ( ρ 2 R ) ] .
β 2 i i = [ 1 b i 2 ( τ ) ] a i 2 [ F i ( ρ 1 R ) F i ( ρ 2 R ) ] .
β 2 28 = a 2 a 8 { 1 b 28 ( τ ) [ b 2 ( τ ) + b 8 ( τ ) b 2 ( τ ) b 8 ( τ ) ] } .
a 2 2 R 2 [ 1 b 2 2 ( τ ) ] ( ρ 1 ρ 2 ) 2
2 a 2 2 R 2 [ 1 b 2 ( τ ) ] ( ρ 1 ρ 2 ) 2
Δ = [ 1 b j 2 ( τ ) 2 ( 1 b j ( τ ) ) ] 1 = 2 ( 1 + b j ( τ ) ) .

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