Abstract

A direct calculation of the normalized variance of irradiance for a beam wave of arbitrary aperture and focus in strong saturation employs a generalized function interpretation of the spherical-wave coherence function with the extended Huygens-Fresnel principle. The asymptotic limit is unity, and the limit of the probability distribution for the irradiance is exponential when the amplitude perturbation terms are neglected. Higher-order terms in the asymptotic series for a focused beam wave extend previously published results.

© 1976 Optical Society of America

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References

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  1. S. F. Clifford and H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
    [Crossref]
  2. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [Crossref]
  3. A. M. Whitman and M. J. Beran, “Asymptotic theory of irradiance fluctuations in a beam propagating in a random medium,” J. Opt. Soc. Am. 65, 765–768 (1975).
    [Crossref]
  4. R. L. Fante, “Irradiance scintillations: Comparison of theory with experiment,” J. Opt. Soc. Am. 65, 548–550 (1975).
    [Crossref]
  5. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khemelevtsov, and R. Sh. Tsvik, “Focused-laser beam scintillations in the turbulent atomosphere,” J. Opt. Soc. Am. 64, 516–518 (1974).
    [Crossref]
  6. J. R. Dunphy and J. R. Kerr, “Turbulence effects on target illumination by laser sources: Unified analysis and experimental verification,” Paper No. 21, NATO/AGARD Conference on Optical Propagation in the Atmosphere, Lyngby, Denmark, October 27–31, 1975.
  7. H. T. Yura, “The mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [Crossref] [PubMed]
  8. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [Crossref]
  9. M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Cambridge U. P., Cambridge, England, 1958).
    [Crossref]
  10. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  11. R. L. Fante, “Two-source spherical-wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 66, 74 (1976).
    [Crossref]
  12. J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).
  13. F. P. Carlson and A. Ishimaru, “Propagation of spherical waves in locally homogeneous random media,” J. Opt. Soc. Am. 59, 319–327 (1969).
    [Crossref]
  14. R. F. Lutomirski and H. T. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. 61, 482–487 (1971).
    [Crossref]
  15. J. W. Strohbehn, Ting-i Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
    [Crossref]
  16. Ting-i Wang and J. W. Strohbehn, “Log-normal paradox in atmospheric scintillations,” J. Opt. Soc. Am. 64, 583–591 (1974).
    [Crossref]

1976 (1)

1975 (5)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[Crossref]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[Crossref]

A. M. Whitman and M. J. Beran, “Asymptotic theory of irradiance fluctuations in a beam propagating in a random medium,” J. Opt. Soc. Am. 65, 765–768 (1975).
[Crossref]

R. L. Fante, “Irradiance scintillations: Comparison of theory with experiment,” J. Opt. Soc. Am. 65, 548–550 (1975).
[Crossref]

J. W. Strohbehn, Ting-i Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[Crossref]

1974 (3)

1972 (1)

1971 (1)

1969 (1)

Banakh, V. A.

Beran, M. J.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[Crossref]

Carlson, F. P.

Clifford, S. F.

Dunphy, J. R.

J. R. Dunphy and J. R. Kerr, “Turbulence effects on target illumination by laser sources: Unified analysis and experimental verification,” Paper No. 21, NATO/AGARD Conference on Optical Propagation in the Atmosphere, Lyngby, Denmark, October 27–31, 1975.

Elliott, R. A.

J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).

Fante, R. L.

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[Crossref]

Holmes, J. F.

J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).

Ishimaru, A.

Kerr, J. R.

J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).

J. R. Dunphy and J. R. Kerr, “Turbulence effects on target illumination by laser sources: Unified analysis and experimental verification,” Paper No. 21, NATO/AGARD Conference on Optical Propagation in the Atmosphere, Lyngby, Denmark, October 27–31, 1975.

Khemelevtsov, S. S.

Krekov, G. M.

Lee, M. H.

J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).

Lighthill, M. J.

M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Cambridge U. P., Cambridge, England, 1958).
[Crossref]

Lutomirski, R. F.

Mironov, V. L.

Pincus, P. A.

J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[Crossref]

Sishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[Crossref]

Speck, J. P.

J. W. Strohbehn, Ting-i Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[Crossref]

Strohbehn, J. W.

J. W. Strohbehn, Ting-i Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[Crossref]

Ting-i Wang and J. W. Strohbehn, “Log-normal paradox in atmospheric scintillations,” J. Opt. Soc. Am. 64, 583–591 (1974).
[Crossref]

Tsvik, R. Sh.

Wang, Ting-i

J. W. Strohbehn, Ting-i Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[Crossref]

Ting-i Wang and J. W. Strohbehn, “Log-normal paradox in atmospheric scintillations,” J. Opt. Soc. Am. 64, 583–591 (1974).
[Crossref]

Whitman, A. M.

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. (8)

Proc. IEEE (2)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[Crossref]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Sishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[Crossref]

Radio Sci. (1)

J. W. Strohbehn, Ting-i Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[Crossref]

Other (4)

J. R. Kerr, R. A. Elliott, J. F. Holmes, M. H. Lee, and P. A. Pincus, “Propagation of multiwavelength laser radiation through atmospheric turbulence,” , available from National Technical Information Service (1976).

M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Cambridge U. P., Cambridge, England, 1958).
[Crossref]

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

J. R. Dunphy and J. R. Kerr, “Turbulence effects on target illumination by laser sources: Unified analysis and experimental verification,” Paper No. 21, NATO/AGARD Conference on Optical Propagation in the Atmosphere, Lyngby, Denmark, October 27–31, 1975.

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Figures (1)

FIG. 1
FIG. 1

Comparison of first- and second-order expansions with experimental (open circle) and Monte Carlo (lower curve) results reproduced from Ref. 5. The abscissa term Ds is equivalent to 2 α 5 / 6 in the present paper. The expansions are

Equations (32)

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U ( r ) = U 0 exp ( - r 2 / 2 a 2 - i k r 2 / 2 F ) ,
U ( p ) = U 0 k e i k ( L + p 2 / 2 L ) 2 π i L - d r exp [ - r 2 2 a 2 + i k 2 L × ( 1 - L F ) r 2 - i k L p · r + ψ ( p , r ) ] ,
I ( p ) = U 0 2 k 2 ( 2 π L ) 2 - d r 1 d r 2 exp [ - r 1 2 + r 2 2 2 a 2 + i k 2 L ( 1 - L F ) ( r 1 2 - r 2 2 ) - i k L p · ( r 1 - r 2 ) ] × exp [ ψ ( p , r 1 ) + ψ * ( p , r 2 ) ] .
exp [ ψ ( p , r 1 ) + ψ * ( p , r 2 ) ] = exp [ - ( r 1 - r 2 / ρ 0 ) 5 / 3 ] ,
ρ 0 = ( 0.546 C n 2 L k 2 ) - 3 / 5 .
I ( p ) = U 0 2 k 2 ( 2 π L ) 2 - d r d R exp [ - R 2 + 1 4 r 2 a 2 i k L × ( 1 - L F ) R · r - i k L p · r - ( r ρ 0 ) 5 / 3 ] .
I ( p ) = U 0 2 a 2 2 ( k L ) 2 0 r d r J 0 ( k L p r ) × exp { - [ a 2 k L ( 1 - L F ) ] 2 r 2 - r 2 4 a 2 - ( r ρ 0 ) 5 / 3 } .
I ( p ; α ) = D 4 8 ( k L ) 2 U 0 2 0 d x x J 0 ( k L D p x ) × exp { - [ D 4 k L ( 1 - L F ) ] 2 D 2 x 2 - x 2 } e - ( α x ) 5 / 3 .
lim α α 2 I ( p ; α ) = 1 8 D 4 ( k / L ) 2 U 0 2 3 5 Γ ( 6 5 ) .
I ( p ; α ) ~ 1 8 D 4 ( k / L ) 2 U 0 2 3 5 Γ ( 6 5 ) α - 2             ( α ) ,
I ( p ; α ) = D 4 8 ( k L ) 2 U 0 2 3 5 Γ ( 6 5 ) α - 2 - D 4 8 ( k L ) 2 U 0 2 3 5 Γ ( 12 5 ) × [ D 2 k 2 p 2 4 L 2 + D 4 k 2 16 L 2 ( 1 - L F ) 2 + 1 ] α - 4 + O ( α - 6 ) .
I 2 ( p ) = U 0 4 ( k 2 π L ) 4 - d r 1 d r 2 d r 3 d r 4 × exp [ - r 1 2 + r 2 2 + r 3 2 + r 4 2 2 a 2 + i k 2 L ( 1 - L F ) ( r 1 2 - r 2 2 + r 3 2 - r 4 2 ) - i k p L ( r 1 - r 2 + r 3 - r 4 ) ] H ( p ; r 1 , r 2 , r 3 , r 4 ) ,
H = exp [ - ( 1 / ρ 0 5 / 3 ) ( r 1 - r 2 5 / 3 + r 3 - r 4 5 / 3 + r 1 - r 4 5 / 3 + r 2 - r 3 5 / 3 - r 1 - r 3 5 / 3 - r 2 - r 4 5 / 3 ) + 2 C χ ( r 1 - r 3 ) + 2 C χ ( r 2 - r 4 ) ] .
I 2 ( p , α ) = D 8 U 0 4 ( k 2 π L ) 4 - d x 1 d x 2 d x 3 d x 4 × exp [ - 2 ( x 1 2 + x 2 2 + x 3 2 + x 4 2 ) + i k 2 L D 2 ( 1 - L F ) ( x 1 2 - x 2 2 + x 3 2 - x 4 2 ) - i k D L p ( x 1 - x 2 + x 3 - x 4 ) ] × exp [ - α 5 / 3 ( x 1 - x 2 5 / 3 + x 3 - x 4 5 / 3 + x 1 - x 4 5 / 3 + x 2 - x 3 5 / 3 - x 2 - x 4 5 / 3 - x 1 - x 3 5 / 3 ) ] .
lim α α 2 exp [ - α 5 / 3 ( x 1 - x 2 5 / 3 + x 1 - x 4 5 / 3 ) ]
6 5 π Γ ( 6 5 ) exp ( - α 5 / 3 x 2 - x 4 5 / 3 ) [ δ ( x 1 - x 2 ) + δ ( x 1 - x 4 ) ]
lim α α 4 I 2 ( p ; α ) = 2 D 8 64 U 0 4 ( k L ) 4 [ 3 5 Γ ( 6 5 ) ] 2
I 2 ( p ; α ) ~ 2 [ D 4 8 ( k L ) 2 U 0 2 3 5 Γ ( 6 5 ) ] 2 α - 4             ( α ) .
lim α σ I n 2 = lim α I 2 ( p ; α ) - I ( p ; α ) 2 I ( p ; α ) 2 = 1.
σ I F 2 = I F 2 ( 0 ; α ) - I F ( 0 ; α ) 2 I F ( 0 ; α ) 2 = 1 + 13 18 ( 5 3 ) 2 2 - 2 / 3 Γ ( 2 3 ) ( Γ ( 12 5 ) Γ ( 6 5 ) ) 2 α - 2 / 3 + 121 1728 ( 5 3 ) 4 2 - 4 / 3 Γ ( 1 3 ) ( Γ ( 18 5 ) Γ ( 6 5 ) ) 2 α - 4 / 3 + O ( α - 2 ) = 1 + 3.132 α - 2 / 3 + 9.414 α - 4 / 3 + O ( α - 2 ) .
I 3 ( p ; α ) ~ 3 ! [ D 4 8 U 0 2 ( k L ) 2 3 5 Γ ( 6 5 ) ] 3 α - 6 ,
I 3 ( p ; α ) / I ( p ; α ) 3 ~ 3 !
I n ( p ; α ) / I ( p ; α ) n ~ n !
σ I = ( 1 + 3.132 α - 2 / 3 ) 1 / 2 ,
σ I = ( 1 + 3.132 α - 2 / 3 + 9.414 α - 4 / 3 ) 1 / 2 .
lim t - f t ( x ) F ( x ) d x = lim v - f v ( x ) F ( x ) d x ,
lim α α 2 e - α 5 / 3 r 5 / 3 = 6 5 Γ ( 6 5 ) r - 1 δ ( r )
lim α | α 2 0 d r r e - α 5 / 3 r 5 / 3 F ( r ) - 3 5 Γ ( 6 5 ) F ( o ) | = 0
0 d r r e - α 5 / 3 r 5 / 3 = 3 5 Γ ( 6 5 ) α - 2 ,
lim α α 2 0 d r r e - ( α r ) 5 / 3 F ( r ) - F ( o ) lim α α 2 max F ( r ) 0 d r r 2 e - ( α r ) 5 / 3 = lim α α 2 max F ( r ) 3 5 Γ ( 9 4 ) α - 3 = 0 ,
lim α α 2 e - α 5 / 3 r - r 0 5 / 3 = 6 5 π Γ ( 6 5 ) δ ( r - r 0 ) .
lim α | α 2 d r e - α 5 / 3 r - r 0 5 / 3 F ( r ) - 6 5 π Γ ( 6 5 ) F ( r 0 ) | = lim α α 2 0 2 π d θ 0 d r r e - α 5 / 2 r 5 / 3 × F ( r + r 0 ) - F ( r 0 ) = 0