Abstract

In this paper, we try to clarify some of the issues involved in the log-normal paradox for irradiance scintillations of an optical wave propagating through the turbulent atmosphere. Quite arbitrarily, a perturbation has been introduced in the log-normal distribution of the irradiance fluctuations. The results show that the log-irradiance variance and covariance functions are extremely sensitive to the log-normal assumption. We find that the deviation from the log-normal distribution, in the saturation region, may be detectable experimentally. This appears to be confirmed by recent Russian experimental results. We also show that our results lend insight into other distributions as well. In particular, there are certain regions where the Rice–Nakagami probability distribution cannot exist.

© 1974 Optical Society of America

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References

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  1. M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 8, 717 (1965).
  2. M. E. Gracheva, Izv. Vuz. Radiofizika (Russian) 10, 775 (1967).
  3. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [Crossref]
  4. G. R. Ochs and R. S. Lawrence, J. Opt. Soc. Am. 59, 226 (1969).
    [Crossref]
  5. A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 11, 1360 (1968).
  6. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and Vl. V. Pokasov, Preprint (USSR Academy of Science, Dept. of Oceanology, Physics, Atmospheres, and Geography, Moscow, 18April1973), p. 39.
  7. D. A. de Wolf, J. Opt. Soc. Am. 58, 461 (1968).
    [Crossref]
  8. D. A. de Wolf, J. Opt. Soc. Am. 59, 1455 (1969).
    [Crossref]
  9. D. A. de Wolf, J. Opt. Soc. Am. 73, 171 (1973).
    [Crossref]
  10. K. Furutsu, J. Opt. Soc. Am. 62, 240 (1972).
    [Crossref]
  11. V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.-JETP 33, 703 (1971)].
  12. J. W. Strohbehn and T.-i. Wang, J. Opt. Soc. Am. 62, 1061 (1972).
    [Crossref]
  13. T.-i. Wang and J. W. Strohbehn, J. Opt. Soc. Am. 64, 583 (1974).
    [Crossref]
  14. V. I. Klyatskin, Izv. Vuz. Radiofizika (Russian) 15, 540 (1972).
  15. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 5319 (National Technical Information Service, Springfield, Va., 1971).

1974 (1)

1973 (1)

D. A. de Wolf, J. Opt. Soc. Am. 73, 171 (1973).
[Crossref]

1972 (3)

1971 (1)

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.-JETP 33, 703 (1971)].

1969 (2)

1968 (2)

A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 11, 1360 (1968).

D. A. de Wolf, J. Opt. Soc. Am. 58, 461 (1968).
[Crossref]

1967 (2)

M. E. Gracheva, Izv. Vuz. Radiofizika (Russian) 10, 775 (1967).

D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
[Crossref]

1965 (1)

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 8, 717 (1965).

de Wolf, D. A.

Fried, D. L.

Furutsu, K.

Gracheva, M. E.

M. E. Gracheva, Izv. Vuz. Radiofizika (Russian) 10, 775 (1967).

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 8, 717 (1965).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and Vl. V. Pokasov, Preprint (USSR Academy of Science, Dept. of Oceanology, Physics, Atmospheres, and Geography, Moscow, 18April1973), p. 39.

Gurvich, A. S.

A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 11, 1360 (1968).

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 8, 717 (1965).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and Vl. V. Pokasov, Preprint (USSR Academy of Science, Dept. of Oceanology, Physics, Atmospheres, and Geography, Moscow, 18April1973), p. 39.

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and Vl. V. Pokasov, Preprint (USSR Academy of Science, Dept. of Oceanology, Physics, Atmospheres, and Geography, Moscow, 18April1973), p. 39.

Keister, M. P.

Klyatskin, V. I.

V. I. Klyatskin, Izv. Vuz. Radiofizika (Russian) 15, 540 (1972).

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.-JETP 33, 703 (1971)].

Lawrence, R. S.

Mevers, G. E.

Ochs, G. R.

Pokasov, Vl. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and Vl. V. Pokasov, Preprint (USSR Academy of Science, Dept. of Oceanology, Physics, Atmospheres, and Geography, Moscow, 18April1973), p. 39.

Strohbehn, J. W.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 5319 (National Technical Information Service, Springfield, Va., 1971).

Wang, T.-i.

Izv. Vuz. Radiofizika (Russian) (4)

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 8, 717 (1965).

M. E. Gracheva, Izv. Vuz. Radiofizika (Russian) 10, 775 (1967).

A. S. Gurvich, Izv. Vuz. Radiofizika (Russian) 11, 1360 (1968).

V. I. Klyatskin, Izv. Vuz. Radiofizika (Russian) 15, 540 (1972).

J. Opt. Soc. Am. (8)

Zh. Eksp. Teor. Fiz. (1)

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.-JETP 33, 703 (1971)].

Other (2)

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and Vl. V. Pokasov, Preprint (USSR Academy of Science, Dept. of Oceanology, Physics, Atmospheres, and Geography, Moscow, 18April1973), p. 39.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 5319 (National Technical Information Service, Springfield, Va., 1971).

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Figures (5)

Fig. 1
Fig. 1

Probability-density function of log-irradiance vs (lnI − 〈lnI〉)/σlnI for σχa = 0.5.

Fig. 2
Fig. 2

Cumulative-probability distribution function of log-irradiance for different values of δ (σχa = 0.5).

Fig. 3
Fig. 3

Cumulative-probability distribution function of log-irradiance for different values of σχ2 (σχa = 0.5).

Fig. 4
Fig. 4

Experimental measurements of the cumulative-probability distribution function of log-irradiance from Gracheva et al. (Ref. 6). 1 : β02 < 1; 2 : 1 < β02 < 4; 3 : 4 < β02; 4 : 25 < β02. β02 = 0.31 × C2k7/6L11/6.

Fig. 5
Fig. 5

Numerical solution for σ12 vs normalized path length ξ.

Equations (50)

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u ( v d + v ) exp ( Ψ ¯ + Ψ 1 ) .
u = v d exp ( Ψ ¯ + Ψ 1 ) + ( v r + i v i ) exp ( Ψ ¯ + Ψ 1 ) ,
u u * = v d v d * exp ( 2 χ a ) + v d v * exp ( 2 χ a ) + v d * v exp ( 2 χ a ) + v v * exp ( 2 χ a ) ,
u u = v d 2 exp ( 2 Ψ ¯ + 2 Ψ 1 ) + 2 v d v exp ( 2 Ψ ¯ + 2 Ψ 1 ) + v 2 exp ( 2 Ψ ¯ + 2 Ψ 1 ) .
e y = exp ( y 2 / 2 ) ,
x e y = x y exp ( y 2 / 2 ) ,
x 1 x 2 e y = [ x 1 x 2 + x 1 y x 2 y ] exp ( y 2 / 2 ) .
u = exp [ χ ¯ a + i S ¯ a + σ χ a 2 / 2 - σ S a 2 / 2 + i χ 1 a S 1 a ] × [ v d + v r χ 1 a - v i S 1 a + i v i χ 1 a + i v r S 1 a ] ,
u u * = exp [ 2 χ ¯ a + 2 σ χ a 2 ] [ v d 2 + 4 v d v r χ 1 a + v r 2 + v i 2 + 4 v r χ 1 a 2 + 4 v i χ 1 a 2 ] ,
u u = exp [ 2 χ ¯ a + 2 i S ¯ a + 2 σ χ a 2 - 2 σ S a 2 + 4 i χ 1 a S 1 a ] × [ v d 2 + 4 v d ( v r χ 1 a - v i S 1 a + i v i χ 1 a + i v r S 1 a ) + v r 2 + 4 v r χ 1 a 2 - 4 v r S 1 a 2 + 8 i v r χ 1 a v r S 1 a - v i 2 - 4 v i χ 1 a 2 + 4 v i S 1 a 2 - 8 i v i χ 1 a v i S 1 a + 2 i v , v i + 8 i v r χ 1 a v i χ 1 a - 8 i v r S 1 a v i S 1 a - 8 v r S 1 a v i χ 1 a - 8 v r χ 1 a v i S 1 a ] ,
u = exp [ χ ¯ a + i S ¯ a - 1 2 A ] [ v d - b A ] ,
u u * = exp [ 2 χ ¯ a + 2 σ χ a 2 ] [ v d 2 + 4 b v d σ χ a 2 + b 2 σ χ a 2 + b 2 σ S a 2 + 4 b 2 σ χ a 4 + 4 b 2 χ 1 a S 1 a 2 ] ,
u u = exp [ 2 χ ¯ a + 2 i S ¯ a - 2 A ] [ ( v d - 2 b A ) 2 - b 2 A ] ,
A - ψ 1 2 = σ S a 2 - σ χ a 2 - 2 i χ 1 a S 1 a .
σ 1 2 = 1 2 Re [ ln u u - 4 ln u ]
σ 2 2 = - 1 2 Re [ ln u u ] .
σ 1 2 = σ χ a 2 + 1 2 ln | ( 1 - 2 b v d A ) 2 - b 2 v d 2 A | - 2 ln | 1 - b v d A | + 1 2 ln | 1 + 4 b v d σ χ a 2 + b 2 v d 2 ( σ χ a 2 + σ S a 2 ) + 4 b 2 v d 2 σ χ a 4 + 4 b v d 2 χ 1 a S 1 a 2 | ,
σ 2 2 = σ S a 2 - 1 2 ln | ( 1 - 2 b v d A ) 2 - b 2 v d 2 A | + 1 2 ln | 1 + 4 b v d σ χ a 2 + b 2 v d 2 σ χ a 2 + b 2 v d 2 σ S a 2 + 4 b 2 v d 2 σ χ a 4 + 4 b 2 v d 2 χ 1 a S 1 a 2 | .
σ 1 2 = ( 1 + b v d ) 2 σ χ a 2 - b 2 A 2 v d 2 ( 1 + 2 b A v d + 7 2 b 2 A 2 v d 2 + ) ,             2 b A v d < 1.
σ 2 2 σ S a 2 ( 1 + 2 b v d + b 2 v d 2 - + 2 b 2 A v d 2 - ) .
σ χ t 2 [ Re ( ln u ) ] 2 - Re ( ln u ) 2 = σ χ a 2 + ( ln w ) 2 - ln w 2 + 2 χ 1 a ln w
σ S t 2 [ Im ( ln u ) ] 2 - Im ( ln u ) 2 = σ S a 2 + 2 S 1 a Φ + Φ 2 - Φ 2 ,
w = [ ( v d + v r ) 2 + v i 2 ] 1 2
Φ = tan - 1 v i v d + v r .
σ χ t 2 ( 1 + b v d ) 2 σ χ a 2 + 0 ( b 3 )
σ S t 2 ( 1 + b v d ) 2 σ S a 2 + 0 ( b 3 ) .
σ χ t 2 σ 1 2 + b 2 A v d 2 + 0 ( b 3 )
σ S t 2 σ 2 2 - 2 b 2 v d 2 σ 2 4 + 0 ( b 3 ) .
ln I ln ( u u * ) = 2 χ ¯ a + 2 χ 1 a + 2 ln v d + ln ( 1 + 2 b v d χ 1 a + b 2 v d 2 χ 1 a 2 + b 2 v d 2 S 1 a 2 ) ,
ln I - 2 χ ¯ a 2 χ 1 a + 2 ln v d + ln ( 1 + δ 1 ) ,
δ 1 = b 2 S 1 a 2 v d 2 .
ln I 2 χ ¯ a + 2 ln v d + ln ( 1 + δ 1 )
σ ln I 2 4 σ χ a 2 + [ ln ( 1 + δ 1 ) ] 2 - ln ( 1 + δ 1 ) 2 .
ln ( 1 + δ 1 ) = - ln ( 1 + b 2 S 1 a 2 v d 2 ) p S ( S 1 a ) d S 1 a
[ ln ( 1 + δ 1 ) ] 2 = - [ ln ( 1 + b 2 S 1 a 2 v d 2 ) ] 2 × p S ( S 1 a ) d S 1 a .
Z = ln I - ln I ;
Z = 2 χ 1 a + ln ( 1 + δ 1 ) - ln ( 1 + δ 1 ) .
p Z ( Z ) = 1 4 π σ χ a δ - exp [ - ( Z - ln y ) 2 / 8 σ χ a 2 ] × e - ( y - 1 ) / 2 δ ( y - 1 ) U ( y - 1 ) d y ,
P Z ( Z ) = 1 ( 2 π δ ) 0 exp [ - y 2 / 2 δ ] F [ ln ( 1 + y 2 ) - Z 2 2 σ χ a ] d y             if             Z 0 , = 1 π [ 0 y 0 exp [ - y 2 / 2 δ ] { 2 - F [ Z - ln ( 1 + y 2 ) 2 2 σ χ a ] } d y + y 0 exp [ - y 2 / 2 δ ] F [ ln ( 1 + y 2 ) - Z 2 2 σ χ a ] d y             if             Z > 0 ,
F ( u ) 1 - erf ( u ) 1 - 2 π 0 u exp ( - t 2 ) d t .
β 0 2 = 0.31 C 2 k 7 / 6 L 11 / 6 > 1.
u = ū + A + i B ,
σ 1 2 1 2 Re [ ln u u - 4 ln u ] ,
u = ū ,
u u * = ū 2 + A 2 + B 2 ,
u u = ū 2 + A 2 - B 2 - 2 i A B .
σ 1 2 = 1 2 Re [ ln ( 1 - 2 B 2 + 2 i A B ) - 2 ln ( 1 - A 2 - B 2 ) ]
1 2 ln ( 1 - 2 B 2 ) - ln ( 1 - A 2 - B 2 ) .
σ 1 2 = - 1 2 ln ( 1 - 2 B 2 ) ,
σ 1 2 = 1 2 ln 1 - 2 B 2 ( 1 - B 2 ) 2 ,