Abstract

This Letter presents a rigorous mathematical proof of the applicability of phase approximation in the Huygens–Kirchhoff method for the problems of laser-beam propagation in a turbulent atmosphere. The calculated results are in good agreement with experimental data.

© 1977 Optical Society of America

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References

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  1. A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe Izluchenie υ Turbulentnoi Atmosfere (English transl.: Laser Radiation in the Turbulent Atmosphere), V. I. Tatarski, ed. (Nauka, Moscow, 1976).
  2. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, R. Sh. Tsvyk, J. Opt. Soc. Am. 64, 516 (1974).
    [Crossref]
  3. V. I. Tatarski, Rasprostraneiye Korotkikh Voln υ Srede so Sluchainymi Neodnorodnostyama υ Priblizhenii Markovskogo Sluchainogo Protsessa (English transl.: Short Wave Propagation in a Medium with Random Inhomogeneities in the Approximation of Markovian Process), preprint (Dept. of Oceanology of Atmospheric Physics and Geography of the U.S.S.R. Academy of Sciences, Moscow, 1970).
  4. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
    [Crossref]
  5. L. A. Chernov, Volny υ Sluchaino-Neodnorodnykh Sredakh (English transl.: Waves in Random Inhomogeneous Media) (Nauka, Moscow, 1975).
  6. V. I. Klyatskin, Statisticheskoye Opisaniye Dinamicheskikh Sistem s Fluktuiruyushchimi Parametrami (English transl.: Statistical Description of Dynamic Systems with Fluctuating Parameters) (Nauka, Moscow, 1975).
  7. V. I. Tatarski, Rasprostraneniye Voln υ Turbulentnoy Atmosfere (English transl.: Wave Propagation in the Turbulent Atmosphere) (Nauka, Moscow, 1967).
  8. I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18 (11), 1660 (1975).
  9. R. L. Fante, Proc. IEEE 63 (12), 1669 (1975).
    [Crossref]

1975 (3)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
[Crossref]

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18 (11), 1660 (1975).

R. L. Fante, Proc. IEEE 63 (12), 1669 (1975).
[Crossref]

1974 (1)

Banakh, V. A.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
[Crossref]

Chernov, L. A.

L. A. Chernov, Volny υ Sluchaino-Neodnorodnykh Sredakh (English transl.: Waves in Random Inhomogeneous Media) (Nauka, Moscow, 1975).

Fante, R. L.

R. L. Fante, Proc. IEEE 63 (12), 1669 (1975).
[Crossref]

Gochelashvili, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
[Crossref]

Gurvich, A. S.

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

Khmelevtsov, S. S.

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, R. Sh. Tsvyk, J. Opt. Soc. Am. 64, 516 (1974).
[Crossref]

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

Klyatskin, V. I.

V. I. Klyatskin, Statisticheskoye Opisaniye Dinamicheskikh Sistem s Fluktuiruyushchimi Parametrami (English transl.: Statistical Description of Dynamic Systems with Fluctuating Parameters) (Nauka, Moscow, 1975).

Kon, A. I.

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

Krekov, G. M.

Mironov, V. L.

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, R. Sh. Tsvyk, J. Opt. Soc. Am. 64, 516 (1974).
[Crossref]

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

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
[Crossref]

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Rasprostraneiye Korotkikh Voln υ Srede so Sluchainymi Neodnorodnostyama υ Priblizhenii Markovskogo Sluchainogo Protsessa (English transl.: Short Wave Propagation in a Medium with Random Inhomogeneities in the Approximation of Markovian Process), preprint (Dept. of Oceanology of Atmospheric Physics and Geography of the U.S.S.R. Academy of Sciences, Moscow, 1970).

V. I. Tatarski, Rasprostraneniye Voln υ Turbulentnoy Atmosfere (English transl.: Wave Propagation in the Turbulent Atmosphere) (Nauka, Moscow, 1967).

Tsvyk, R. Sh.

Yakushkin, I. G.

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18 (11), 1660 (1975).

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

I. G. Yakushkin, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18 (11), 1660 (1975).

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

R. L. Fante, Proc. IEEE 63 (12), 1669 (1975).
[Crossref]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, Proc. IEEE 63 (3), 790 (1975).
[Crossref]

Other (5)

L. A. Chernov, Volny υ Sluchaino-Neodnorodnykh Sredakh (English transl.: Waves in Random Inhomogeneous Media) (Nauka, Moscow, 1975).

V. I. Klyatskin, Statisticheskoye Opisaniye Dinamicheskikh Sistem s Fluktuiruyushchimi Parametrami (English transl.: Statistical Description of Dynamic Systems with Fluctuating Parameters) (Nauka, Moscow, 1975).

V. I. Tatarski, Rasprostraneniye Voln υ Turbulentnoy Atmosfere (English transl.: Wave Propagation in the Turbulent Atmosphere) (Nauka, Moscow, 1967).

V. I. Tatarski, Rasprostraneiye Korotkikh Voln υ Srede so Sluchainymi Neodnorodnostyama υ Priblizhenii Markovskogo Sluchainogo Protsessa (English transl.: Short Wave Propagation in a Medium with Random Inhomogeneities in the Approximation of Markovian Process), preprint (Dept. of Oceanology of Atmospheric Physics and Geography of the U.S.S.R. Academy of Sciences, Moscow, 1970).

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

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Equations (21)

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2 i k u ( x , ρ ) x + Δ u ( x , ρ ) + k 2 ξ 1 ( x , ρ ) u ( x , ρ ) = 0 .
G ( x , x ; ρ , ρ ) = δ ( ρ ρ ) ,
u ( x , ρ ) = d 2 ρ u 0 ( ρ ) G ( x , x 0 ; ρ , ρ ) .
G ( x , x ; ρ , ρ ) G 0 ( x , x ; ρ , ρ ) = G 0 ( x , x ; ρ , ρ ) exp [ iS ( x , x ; ρ , ρ ) ] ,
S ( x , x ; ρ , ρ ) = i k 2 x x d ζ ξ 1 ( ζ , ρ ζ x x x + ρ x ζ x x )
2 i k G ˜ x + Δ G ˜ + k 2 ξ 1 G ˜ = G 0 exp ( iS )
G ( x , x ; ρ , ρ ) = G 0 ( x , x ; ρ , ρ ) + i 2 k x x d x d 2 ρ G 1 ( x , x ; ρ , ρ ) × G 0 ( x , x ; ρ , ρ ) Δ exp [ iS ( x , x ; ρ , ρ ) ] .
G 1 ( x , x ; ρ , ρ ) = G ( x , x ; ρ , ρ ) .
G ( x , x ; ρ , ρ ) = G 0 ( x , x ; ρ , ρ ) + i 2 k x x d x d 2 ρ G ( x , x ; ρ , ρ ) × G 0 ( x , x ; ρ , ρ ) Δ exp [ iS ( x , x ; ρ , ρ ) ] .
u ( x , ρ ) = d 2 ρ u 0 ( ρ ) G 0 ( x , x 0 ; ρ , ρ ) + i 2 k x 0 x d x d 2 ρ u ( x , ρ ) G 0 ( x , x ; ρ , ρ ) × Δ exp [ iS ( x , x ; ρ , ρ ) ] ,
u 0 ( x , ρ ) = d 2 ρ u 0 ( ρ ) G 0 ( x , x 0 ; ρ , ρ ) .
u ( x , ρ ) = u 0 ( x , ρ ) + i 2 k x 0 x d x d 2 ρ u ( x , ρ ) × G 0 ( x , x ; ρ , ρ ) Δ exp [ iS ( x , x ; ρ , ρ ) ] .
γ 2 n ( x , ρ 1 , , ρ 2 n ) = u ( x , ρ 1 ) u * ( x , ρ 2 ) u ( x , ρ 2 n 1 ) u * ( x , ρ 2 n ) ,
Γ 2 n ( x , ρ 1 , , ρ 2 n ) = γ 2 n ( x , ρ 1 , , ρ 2 n ) .
Γ 2 n ( x , ρ 1 , , ρ 2 n ) = Γ 2 n 0 ( x , ρ 1 , , ρ 2 n ) + i 2 k x 0 x d x d 4 n ρ 1 , , ρ 2 n × Γ 2 n ( x , ρ 1 , , ρ 2 n ) × G 2 n , 0 ( x , x ; ρ 1 , , ρ 2 n ; ρ 1 , , ρ 2 n ) × j = 1 2 n ( 1 ) j + 1 Δ j × exp [ i j = 1 2 n ( 1 ) j + 1 S ( x , x ; ρ j , ρ j ) ] ,
G 2 n , 0 ( x , x ; ρ 1 , , ρ 2 n ; ρ 1 , , ρ 2 n ) = G 0 ( x , x ; ρ 1 , ρ 1 ) × G 0 * ( x , x ; ρ 2 , ρ 2 ) G 0 ( x , x ; ρ 2 n 1 , ρ 2 n 1 ) × G 0 * ( x , x ; ρ 2 n , ρ 2 n ) .
Γ 2 n 0 ( x , ρ 1 , , ρ 2 n ) = u 0 ( x , ρ 1 ) u 0 * ( x , ρ 2 ) u 0 ( x , ρ 2 n 1 ) u 0 * ( x , ρ 2 n ) .
Γ 2 n = Γ 2 n 0 + j = 1 Γ 2 n , j ,
Γ 2 n , j ( x , ρ 1 , , ρ 2 n ) = i 2 k x 0 x d x × d 2 n ρ 1 , , ρ 2 n Γ 2 n , j 1 ( x , ρ 1 , , ρ 2 n ) × G 2 n , 0 ( x , x ; ρ 1 , , ρ 2 n ; ρ 1 , , ρ 2 n ) × l = 1 2 n ( 1 ) l + 1 Δ l × exp [ i l = 1 2 n ( 1 ) l + 1 S ( x , x ; ρ l , ρ l ) ] , Γ 2 n , 0 ( x , ρ 1 , , ρ 2 n ) = Γ 2 n 0 ( x , ρ 1 , , ρ 2 n ) .
u ( x , ρ ) = u 0 ( x , ρ ) , Γ 2 ( x , ρ 1 , ρ 2 ) = Γ 2 0 ( x , ρ 1 , ρ 2 ) .
σ I 2 = 0.7 β 0 2 + 0.3 β 0 2 + 0 ( β 0 4 ) , β 0 1 , σ I 2 = ( 1 + 0.5 β 0 4 / 5 ) + 0.36 β 0 4 / 5 + 0 ( β 0 8 / 5 ) , β 0 1 ,

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