Abstract

This Letter presents a rigorous mathematical proof of the applicability of a phase approximation in the Huygens–Kirchhoff method for the problems of space-limited optical-beam propagation in a turbulent atmosphere.

© 1979 Optical Society of America

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References

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  1. V. A. Banakh, V. L. Mironov, Opt. Lett. 1, 172 (1977).
    [CrossRef] [PubMed]
  2. A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie v Turbulentnoi Atmosfere (English translation, Laser Radiation in the Turbulent Atmosphere), V. I. Tatarskii, ed. (Nauka, Moscow, 1976).
  3. V. A. Banakh, V. L. Mironov, Kvantovaya Elektron. (Moscow) 5, 1535 (1978).
  4. V. I. Tatarskii, Rasprostraneniye Voin v Turbulentnoy Atmosfere (English translation, Wave Propagation in the Turbulent Atmosphere (Nauka, Moscow, 1967).
  5. V. I. Tatarskii, Rasprostraneniye Korothihh Voin v Srede so Sluchainymi Neodnorodnostyami v Priblizhenii Markovskogo Sluchainogo Protsessa (English translation, Short-Wave Propagation in a Medium with Random Inhomogeneities in the Approximation of Markovian Process), preprint (Department of Oceanology of Atmospheric Physics and Geography of the USSR Academy of Sciences, Moscow, 1970).
  6. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
    [CrossRef]
  7. I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19384 (1976).

1978

V. A. Banakh, V. L. Mironov, Kvantovaya Elektron. (Moscow) 5, 1535 (1978).

1977

1976

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19384 (1976).

1975

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Banakh, V. A.

V. A. Banakh, V. L. Mironov, Kvantovaya Elektron. (Moscow) 5, 1535 (1978).

V. A. Banakh, V. L. Mironov, Opt. Lett. 1, 172 (1977).
[CrossRef] [PubMed]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Gochelashvili, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Gurvich, A. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie v Turbulentnoi Atmosfere (English translation, Laser Radiation in the Turbulent Atmosphere), V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Khmelevtsov, S. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie v Turbulentnoi Atmosfere (English translation, Laser Radiation in the Turbulent Atmosphere), V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Kon, A. I.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie v Turbulentnoi Atmosfere (English translation, Laser Radiation in the Turbulent Atmosphere), V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Mironov, V. L.

V. A. Banakh, V. L. Mironov, Kvantovaya Elektron. (Moscow) 5, 1535 (1978).

V. A. Banakh, V. L. Mironov, Opt. Lett. 1, 172 (1977).
[CrossRef] [PubMed]

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie v Turbulentnoi Atmosfere (English translation, Laser Radiation in the Turbulent Atmosphere), V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Rasprostraneniye Voin v Turbulentnoy Atmosfere (English translation, Wave Propagation in the Turbulent Atmosphere (Nauka, Moscow, 1967).

V. I. Tatarskii, Rasprostraneniye Korothihh Voin v Srede so Sluchainymi Neodnorodnostyami v Priblizhenii Markovskogo Sluchainogo Protsessa (English translation, Short-Wave Propagation in a Medium with Random Inhomogeneities in the Approximation of Markovian Process), preprint (Department of Oceanology of Atmospheric Physics and Geography of the USSR Academy of Sciences, Moscow, 1970).

Yakushkin, I. G.

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19384 (1976).

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19384 (1976).

Kvantovaya Elektron. (Moscow)

V. A. Banakh, V. L. Mironov, Kvantovaya Elektron. (Moscow) 5, 1535 (1978).

Opt. Lett.

Proc. IEEE

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63, 790 (1975).
[CrossRef]

Other

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie v Turbulentnoi Atmosfere (English translation, Laser Radiation in the Turbulent Atmosphere), V. I. Tatarskii, ed. (Nauka, Moscow, 1976).

V. I. Tatarskii, Rasprostraneniye Voin v Turbulentnoy Atmosfere (English translation, Wave Propagation in the Turbulent Atmosphere (Nauka, Moscow, 1967).

V. I. Tatarskii, Rasprostraneniye Korothihh Voin v Srede so Sluchainymi Neodnorodnostyami v Priblizhenii Markovskogo Sluchainogo Protsessa (English translation, Short-Wave Propagation in a Medium with Random Inhomogeneities in the Approximation of Markovian Process), preprint (Department of Oceanology of Atmospheric Physics and Geography of the USSR Academy of Sciences, Moscow, 1970).

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Equations (19)

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σ I 2 = Γ 4 ( x , 0 ) / Γ 2 2 ( x , 0 ) - 1 ,
Γ 4 0 ( x , 0 ) = ( q 2 π ) 3 Ω d r 1 , 2 , 3 exp [ - q 2 Ω ( r 1 2 + r 2 2 + g 2 r 3 2 ) + i q ( 1 - x F ) r 1 · r 2 ] exp [ 3 8 ( r 1 + r 3 5 / 3 + r 1 - r 3 5 / 3 + r 2 + r 3 5 / 3 + r 2 - r 3 5 / 3 - r 1 + r 2 5 / 3 - r 1 - r 2 5 / 3 ) ] ,
σ I 0 2 = 2.84 Γ 11 / 6 ( g 2 ) - 5 / 6 [ Re ( 1 + i g 2 - 1 ) 5 / 6 - 1 ] + O ( β 0 4 ) ,
Γ 2 ( x , 0 ) = Ω 2 / ( g 2 + 1.52 β 0 12 / 5 )
σ I 0 2 = 1 + 2.7 Ω - 1 / 3 β 0 - 4 / 5 + O [ Ω 1 / 3 β 0 4 / 5 ) - 2 ] ,
Γ 4 , 1 ( x , 0 ) = - 1 2 k ( q 2 π ) 3 Ω x q 0 1 d ζ ζ - 4 d r 1 , 2 , 3 , 4 × exp [ - q Ω 2 ζ 2 ( r 3 12 / 4 ) + i q ζ - 1 ( r 1 · r 2 + r 3 r 4 ) ] × d r 1 , 2 , 3 exp ( - q 2 Ω ( r 1 2 + r 2 2 + g 2 ( ζ ) r 3 2 ) + i q ζ - 1 × ( 1 - ζ x F ) r 1 · r 2 - i q ζ - 1 { [ 2 r 4 + i Ω ζ ( 1 - ζ x F ) r 3 2 ] × r 3 + r 1 · r 2 + r 2 · r 2 } ) × exp { j = 1 , 2 D ¯ s [ x , 0 ; r 1 + ( - 1 ) j r 2 ; r i + ( - 1 ) j r 2 ] - m = 1 , 2 j = 1 , 2 D ¯ s [ x , 0 ; r 3 2 + ( - 1 ) m r j ; r 3 + ( - 1 ) m r j ] } × exp { j = 1 , 2 D ¯ s [ x , x ; 0 ; r 1 + ( - 1 ) j r 2 ] - m = 1 , 2 j = 1 , 2 D ¯ s [ x , x , 0 ; r 3 2 + ( - 1 ) m r j ] } × { Δ [ D ¯ s ( x , x ; 0 ; r 1 + r 2 ) - D ¯ s ( x , x ; 0 ; r 1 - r 2 ) ] + [ D ¯ s ( x , x ; 0 ; r 1 + r 2 ) + D ¯ s ( x , x ; 0 ; r 1 - r 2 ) - D ¯ s ( x , x ; 0 ; r 1 + r 3 2 ) - D ¯ s ( x , x ; 0 ; - r 1 + r 3 2 ) ] × [ D ¯ s ( x , x ; 0 ; r 1 + r 2 1 ) - D ¯ s ( x , x ; 0 ; r 1 - r 2 ) - D ¯ s ( x , x ; 0 ; r 2 + r 3 2 ) - D ¯ s ( x , x ; 0 ; r 3 2 - r 2 ) ] } ,
g 2 ( ζ ) = 1 + Ω 2 ζ 2 ( 1 - ζ x F ) 2 , D ¯ s ( x , 0 ; r , r ) = ζ 0 1 d ξ ξ r + ( 1 - ξ ) r 5 / 3 , D ¯ s ( x , x ; r , r ) = ( 1 - ζ ) 0 1 d ξ ξ r + ( 1 - ξ ) r 5 / 3 ,
σ I 1 2 = Γ 4 , 1 ( x , 0 ) / Γ 2 2 ( x , 0 ) = 2.22 g 2 Ω - 3 / 2 β 0 2 × 0 1 d ζ ζ 5 / 3 ( 1 - ζ ) - 1 g 5 / 3 ( ζ ) ( 1 + Z 2 ) - 17 / 12 × [ Z cos ( 5 6 arctan Z ) - sin ( 5 6 arctan Z ) ] ,
Z = ζ [ g 2 ( ζ ) - ( 1 - ζ ) Ω 2 ( 1 - ζ x F ) / ζ 2 ] / [ Ω ( 1 - ζ ) ] .
σ I 1 2 = 0.3 β 0 2 + O ( β 0 4 ) ,             Ω 1 ,
σ I 1 2 = 0.4 β 0 2 + O ( β 0 4 ) ,             Ω 1 ,
σ I 1 2 = 0.17 β 0 2 + O ( β 0 4 ) ,             Ω = 1.
exp { j = 1 , 2 D ¯ s [ x , 0 ; r 1 + ( - 1 ) j r 2 ; r 1 + ( - 1 ) j r 2 ] - m = 1 , 2 j = 1 , 2 D ¯ s [ x , 0 ; r 3 2 + ( - 1 ) m r j ; r 3 + ( - 1 ) m r j ] } × exp { j = 1 , 2 D ¯ s [ x , x ; 0 ; r 1 + ( - 1 ) j r 2 ] - m = 1 , 2 j = 1 , 2 D ¯ s [ x , x ; 0 ; r 3 2 + ( - 1 ) m r j ] } = ( exp { - j = 1 , 2 D ¯ s [ x , 0 ; r 3 2 + ( - 1 ) j r 1 ; r 3 + ( - 1 ) j r 1 ] } n = 0 2 - n n ! { j = 1 , 2 D ¯ s [ x , 0 ; r 1 + ( - 1 ) j r 2 ; r 1 + ( - 1 ) j r 2 ] - D ¯ s [ x , 0 ; r 3 2 + ( - 1 ) j r 2 ; r 3 + ( - 1 ) j r 2 ] } n + exp { - j = 1 , 2 D s [ x , 0 ; r 3 2 + ( - 1 ) j r 2 ; r 3 + ( - 1 ) j r 2 ] n = 0 2 - n n ! [ j = 1 , 2 D ¯ s [ x , 0 ; r 1 + ( - 1 ) j r 2 ; r 1 + ( - 1 ) j r 2 ] - D ¯ s x , 0 ; r 3 2 + ( - 1 ) j r 1 ; r 3 + ( - 1 ) j r 1 ] } n ) × ( exp { - j = 1 , 2 D ¯ s [ x , x ; 0 ; r 3 2 + ( - 1 ) j r 1 ] } × n = 0 2 - n n ! { j = 1 , 2 D ¯ s [ x , x ; 0 ; r 1 + ( - 1 ) j r 2 ] - D ¯ s [ x , x ; 0 ; r 3 2 + ( - 1 ) j r 2 ] } n + exp { - j = 1 , 2 D ¯ s [ x , x ; 0 ; r 3 2 + ( - 1 ) j r 2 ] } × n = 0 2 - n n ! { j = 1 , 2 D ¯ s [ x , x ; 0 ; r 1 + ( - 1 ) j r 2 ] - D ¯ s [ x , x ; 0 ; r 3 2 + ( - 1 ) j r 1 ] } n ) .
σ I 1 2 = - 1.53 × 10 - 3 q 1 / 3 0 1 d ζ ( 1 - ζ ) 2 × d κ κ - 11 / 3 d r 1 , 2 exp [ - 1 4 g 2 q Ω ( r 1 2 + r 2 2 ) - 1 2 q Ω ζ g 2 ( ζ ) κ 2 ] exp [ - 1 2 q Ω ζ × ( 1 + { ( g 2 - 1 ) [ g 2 ( ζ ) - 1 ] 1 / 2 } ) ( κ · r 1 + κ · r 2 ) ] × exp [ - 3 8 ( 1 - ζ ) ( r 1 5 / 3 + r 2 5 / 3 ) - ζ 0 1 d ξ j = 1 , 2 ξ ( 1 - ζ ) r j + ( 1 - ξ ) ( r j + κ ζ ) 5 / 3 ] × [ κ 2 ( κ · r 1 + κ · r 2 ) - 7.25 · 10 - 2 × ( 1 - ζ ) 8 / 3 ( κ · r 1 ) ( κ · r 2 ) d p ( κ · p ) p - 11 / 3 × ( sin p · r 1 + sin p · r 2 ) ] + O [ f ( Ω , q , x F ) ] .
ζ ( 1 + α ) 5 / 3 [ 8 5 - 3 5 ζ + ζ α ] - ( 1 - ζ ) 8 / 3 ( - α ) 2 / 3 ( 1 + 2 5 ζ α ) = 0.
σ I 1 2 = 6.04 β 0 - 4 / 5 + O ( Ω 1 / 3 β 0 - 4 / 5 ) ,
σ I 2 = σ I 0 2 + σ I 1 2 = 1 + 6.04 β 0 - 4 / 5 + 2.7 Ω - 1 / 3 β 0 - 4 / 5 + O ( Ω 1 / 3 β 0 - 4 / 5 ) .
σ I 1 2 = 1.4 Ω - 7 / 6 β 0 2 + O ( Ω - 13 / 6 β 0 - 2 / 5 ) .
Ω β 0 84 / 25

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