Abstract

An analysis of the variance and covariance of irradiance of a finite beam wave in strong turbulence with effects of small residual log-amplitude perturbations of a spherical wave is presented. The method is based upon phenomenological elaboration of the extended Huygens-Fresnel formulation that succeeds quantitatively in reproducing the variance and covariance of observed scintillation irradiance in strong turbulence. It is found that the normalized variance reaches unity and the covariance approaches that of a spherical wave in the limit of very high turbulence.

© 1978 Optical Society of America

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References

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  1. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  4. M. H. Lee, R. A. Elliott, J. F. Holmes, and J. R. Kerr, J. Opt. Soc. Am. 66, 1389 (1976).
    [Crossref]
  5. R. S. Lawrence and J. W. Strohbehn, IEEE Proc. 58, 1523 (1970).
    [Crossref]
  6. M. H. Lee, J. F. Holmes, and J. R. Kerr, J. Opt. Soc. Am. (to be published).
  7. J. R. Dunphy and J. R. Kerr, J. Opt. Soc. Am. 63, 981 (1973).
    [Crossref]
  8. S. F. Clifford and H. T. Yura, J. Opt. Soc. Am. 64, 1641 (1974).
    [Crossref]

1976 (1)

1974 (2)

1973 (1)

1972 (1)

1971 (1)

1970 (1)

R. S. Lawrence and J. W. Strohbehn, IEEE Proc. 58, 1523 (1970).
[Crossref]

Banakh, V. A.

Clifford, S. F.

Dunphy, J. R.

Elliott, R. A.

Holmes, J. F.

M. H. Lee, R. A. Elliott, J. F. Holmes, and J. R. Kerr, J. Opt. Soc. Am. 66, 1389 (1976).
[Crossref]

M. H. Lee, J. F. Holmes, and J. R. Kerr, J. Opt. Soc. Am. (to be published).

Kerr, J. R.

Khemelevtsov, S. S.

Krekov, G. M.

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, IEEE Proc. 58, 1523 (1970).
[Crossref]

Lee, M. H.

M. H. Lee, R. A. Elliott, J. F. Holmes, and J. R. Kerr, J. Opt. Soc. Am. 66, 1389 (1976).
[Crossref]

M. H. Lee, J. F. Holmes, and J. R. Kerr, J. Opt. Soc. Am. (to be published).

Lutomirski, R. F.

Mironov, V. L.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, IEEE Proc. 58, 1523 (1970).
[Crossref]

Tsvik, R. Sh.

Yura, H. T.

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Equations (21)

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I 2 ( ρ ¯ ) = ( 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 α 0 2 + i k 2 L ( 1 - L / F ) r 1 2 - r 2 2 + r 3 2 - r 4 2 ) - i k L ρ ¯ ( r ¯ 1 - r ¯ 2 + r ¯ 3 - r ¯ 4 ) ) × exp { l 1 ( r ¯ 1 , ρ ¯ ) + l 2 ( r ¯ 2 , ρ ¯ ) + l 3 ( r ¯ 3 , ρ ¯ ) + l 4 ( r ¯ 4 , ρ ¯ ) + i [ ϕ 1 ( r ¯ 1 , ρ ¯ ) - ϕ 2 ( r ¯ 2 , ρ ¯ ) r + ϕ 3 ( r ¯ 3 , ρ ¯ ) - ϕ 4 ( r ¯ 4 , ρ ¯ ) ] } ,
M ( r ¯ 1 , r ¯ 2 , r ¯ 3 , r ¯ 4 , ρ ¯ ) = exp { l ( r ¯ 1 , ρ ¯ ) + l ( r ¯ 2 , ρ ¯ ) + l ( r ¯ 3 , ρ ¯ ) + l ( r ¯ 4 , ρ ¯ ) + i [ ϕ ( r ¯ 1 , ρ ¯ ) - ϕ ( r ¯ 2 , ρ ¯ ) + ϕ ( r ¯ 3 , ρ ¯ ) - ϕ ( r ¯ 4 , ρ ¯ ) ] } = exp { - ½ [ D ( r ¯ 2 - r ¯ 1 - D ( r ¯ 3 - r ¯ 1 ) + D ( r ¯ 4 - r ¯ 1 ) c + D ( r ¯ 3 - r ¯ 2 ) - D ( r ¯ 4 - r ¯ 2 ) + D ( r ¯ 4 - r ¯ 3 ) ] + 2 C l ( r ¯ 3 - r ¯ 1 ) + 2 C l ( r ¯ 4 - r ¯ 2 ) - ( i / 2 ) [ D l ϕ ( r ¯ 3 - r ¯ 1 ) - D l ϕ ( r ¯ 2 - r ¯ 4 ) ] } ,
e - ( 1 / 2 ) D ( r ) = e - ( 1 / 2 ) [ D l ( r ) + D ϕ ( r ) ] e - ( 1 / 2 ) D ϕ ( r ) = exp ( - r 5 / 3 / ρ 0 5 / 3 ) ,
exp [ - ½ ( D 12 - D 13 + D 14 + D 23 - D 24 + D 34 ) ] { κ δ ( r ¯ 1 - r ¯ 2 ) exp [ - ½ ( - D 13 + D 14 + D 23 - D 24 + D 34 ) ] + κ δ ( r ¯ 1 - r ¯ 4 ) exp [ - ½ ( D 12 - D 13 + D 23 - D 24 + D 34 ) ] - } ,
κ = 6 π 5 Γ ( 6 5 ) ρ 0 2 ( L k ) 2 4 π U 0 2 α 0 2 I
exp ( - r ¯ i - r ¯ j 5 / 3 / ρ 0 5 / 3 )
σ I N = I 2 - I 2 I 2 = [ 2 0 r exp ( - r 2 2 α 0 2 + 4 C l ( r ) ) d r ] / α 0 2 - 1.
e 4 C l ( r ) 1 + 4 C l ( r ) .
C l ( r ) = 0.18 0 1 d u 0 d q e - q J 0 ( 5.56 u q 3 / 5 r ρ 0 ) .
I 2 I 2 2 α 0 2 0 r exp ( - r 2 / 2 α 0 2 ) [ 1 + 4 C l ( r ) ] d r = 2 + 1.44 α 0 2 0 r d r exp ( - r 2 / 2 α 0 2 ) 0 1 d u 0 d q e - q × J 0 { 5.56 u q - 3 / 5 ( r ρ 0 ) } .
σ i N 2 = I 2 I 2 - 1 = 1 + 0.324 D 0 q 3 / 5 erf ( 3.93 D q - 3 / 5 ) e - q d q ,
erf ( z ) = 2 π 0 z e - t 2 d t .
σ I N 2 1 + 0.29 / D .
B I ( ρ ¯ 1 , ρ ¯ 2 ) = I ( ρ ¯ 1 ) I ( ρ ¯ 2 ) = ( 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 α 0 2 + i k 2 L ( 1 - L / F ) ( r 1 2 - r 2 2 c + r 3 2 - r 4 2 ) - i k L [ ρ ¯ 1 ( r ¯ 1 - r ¯ 2 ) + ρ ¯ 2 ( r ¯ 3 - r ¯ 4 ) ] ) × H ( r ¯ 1 , r ¯ 2 , r ¯ 3 , r ¯ 4 ; ρ ¯ 1 , ρ ¯ 2 ) ,
H ( r ¯ 1 , r ¯ 2 , r ¯ 3 , r ¯ 4 ; ρ ¯ 1 , ρ ¯ 2 ) = exp { l ( r ¯ 1 , ρ ¯ 1 ) + l ( r ¯ 2 , ρ ¯ 1 ) + l ( r ¯ 3 , ρ ¯ 2 ) + l ( r ¯ 4 , ρ ¯ 2 ) + i [ ϕ ( r ¯ 1 , ρ ¯ 1 ) - ϕ ( r ¯ 2 , ρ ¯ 1 ) + ϕ ( r ¯ 3 , ρ ¯ 2 ) - ϕ ( r ¯ 4 , ρ ¯ 2 ) ] } = exp { - ½ [ D ( r ¯ 2 - r ¯ 1 ) - D ( r ¯ 3 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 4 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 3 - r ¯ 2 ; ρ ¯ 2 - ρ ¯ 1 ) - D ( r ¯ 4 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 4 - r ¯ 3 ) ] + 2 C l ( r ¯ 3 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + 2 C l ( r ¯ 4 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) - ( i / 2 ) [ D l ϕ ( r ¯ 3 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) - D l ϕ ( r ¯ 4 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) ] } .
exp { - ½ [ D ( r ¯ 2 - r ¯ 1 ) - D ( r ¯ 3 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 4 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 3 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) - D ( r ¯ 4 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 4 - r ¯ 3 ) } κ 2 δ ( r ¯ 1 - r ¯ 2 ) δ ( r ¯ 3 - r ¯ 4 ) exp { - ½ [ - D ( r ¯ 3 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 4 - r ¯ 1 , ρ ¯ 2 - ρ ¯ 1 ) + D ( r ¯ 3 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) - D ( r ¯ 4 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) ] } .
B I ( ρ ¯ 1 , ρ ¯ 2 ) = κ 2 U 0 4 ( k 2 π L ) 4 d r ¯ 2 d r ¯ 4 × exp ( - r 2 2 + r 4 2 α 0 2 + 4 C l ( r ¯ 4 - r ¯ 2 , ρ ¯ 2 - ρ ¯ 1 ) ) .
B I ( ρ ¯ ) = I 2 2 π α 0 2 d r ¯ exp ( - r 2 2 α 0 2 + 4 C l ( r ¯ , ρ ¯ ) ) .
C l ( r ¯ , ρ ¯ ) C l ( ρ ¯ )
C I ( ρ ¯ ) = I 2 ( e 4 C l ( ρ ) - 1 ) .
C I N ( ρ ) = e 4 C l ( ρ ) - 1 1 + 0.29 / D [ e 4 C l ( ρ ) - 1 ] - 0.29 D [ e 4 C l ( ρ ) - 1 ] .