Abstract

Approximating the probability-density function of optical irradiance fluctuations in the turbulent atmosphere under all propagation conditions requires a model with at least two parameters. Two phenomenological two-parameter models that have been proposed, the IK and the log-normally modulated Rician probability-density functions, are compared with measured probability-density functions of the irradiance of laser light in a turbulent atmosphere under a variety of propagation conditions. The parameters for each model are obtained from measured second and third moments. It is concluded that the log-normally modulated Rician model is the better approximation to the data. However, the effects of the intermittency of turbulence must be included in the model for short propagation paths.

© 1989 Optical Society of America

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).
  2. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 45–106.
    [CrossRef]
  3. C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astr. Soc. 111, 289–302 (1951).
  4. K. Gochelashvili, V. S. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
    [CrossRef]
  5. V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Sov. Phys. JETP 48, 27–31 (1978).
  6. R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  7. D. A. DeWolf, “Are strong irradiance fluctuations log normal or Rayleigh distributed?”J. Opt. Soc. Am. 59, 1455–1460 (1969).
    [CrossRef]
  8. L. R. Bissonnette, “Propagation model of laser beams in turbulence,”J. Opt. Soc. Am. 73, 262–268 (1983).
    [CrossRef]
  9. E. J. Fremouw, R. C. Livingston, D. A. Miller, “On the statistics of scintillating signals,”J. Atmos. Terr. Phys. 42, 717–731 (1980).
    [CrossRef]
  10. S. Ito, K. Furutsu, “Theoretical analysis of the high-order irradiance moments of light waves observed in turbulent air,”J. Opt. Soc. Am. 72, 760–764 (1982).
    [CrossRef]
  11. R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 9, 110–112 (1984).
    [CrossRef] [PubMed]
  12. N. Ben-Yosef, E. Goldner, “Splitting-source model for the statistics of irradiance scintillations,” J. Opt. Soc. Am. A 5, 126–131 (1988).
    [CrossRef]
  13. L. C. Andrews, R. L. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]
  14. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  15. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
    [CrossRef]
  16. G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [CrossRef]
  17. T. Wang, J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,”J. Opt. Soc. Am. 64, 994–999 (1974).
    [CrossRef]
  18. J. W. Strohbehn, T. Wang, J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
    [CrossRef]
  19. J. H. Churnside, S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  20. J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  21. L. C. Andrews, R. L. Phillips, B. K. Shivamoggi, “Relations of the parameters of the I–K distribution for irradiance fluctuations to physical parameters of the turbulence,” Appl. Opt. 27, 2150–2156 (1988).
    [CrossRef] [PubMed]
  22. R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” Appl. Opt. 27, 2194–2198 (1988).
    [CrossRef] [PubMed]
  23. J. M. Martin, S. Flatte, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  24. N. Ben-Yosef, E. Goldner, “Sample size influence on optical scintillation analysis. 1: Analytical treatment of the higher-order irradiance moments,” Appl. Opt. 27, 2167–2171 (1988).
    [CrossRef] [PubMed]
  25. E. Goldner, N. Ben-Yosef, “Sample size influence on optical scintillation analysis. 2: Simulation approach,” Appl. Opt. 27, 2172–2177 (1988).
    [CrossRef] [PubMed]
  26. R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” Mod. Opt. (to be published).
  27. V. I. Tatarskii, “Some new aspects in the problem of waves and turbulence,” Radio Sci. 22, 859–865 (1987).
    [CrossRef]

1988 (6)

1987 (3)

1985 (1)

1984 (1)

1983 (1)

1982 (1)

1981 (1)

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

1980 (1)

E. J. Fremouw, R. C. Livingston, D. A. Miller, “On the statistics of scintillating signals,”J. Atmos. Terr. Phys. 42, 717–731 (1980).
[CrossRef]

1979 (2)

1978 (1)

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Sov. Phys. JETP 48, 27–31 (1978).

1977 (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

1975 (1)

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

1974 (1)

1971 (1)

K. Gochelashvili, V. S. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

1969 (1)

1951 (1)

C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astr. Soc. 111, 289–302 (1951).

Andrews, L. C.

Ben-Yosef, N.

Bissonnette, L. R.

Churnside, J. H.

Clifford, S. F.

Dashen, R.

R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 9, 110–112 (1984).
[CrossRef] [PubMed]

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

DeWolf, D. A.

Flatte, S.

Frehlich, R. G.

R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” Appl. Opt. 27, 2194–2198 (1988).
[CrossRef] [PubMed]

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” Mod. Opt. (to be published).

Fremouw, E. J.

E. J. Fremouw, R. C. Livingston, D. A. Miller, “On the statistics of scintillating signals,”J. Atmos. Terr. Phys. 42, 717–731 (1980).
[CrossRef]

Furutsu, K.

Gochelashvili, K.

K. Gochelashvili, V. S. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

Goldner, E.

Hill, R. J.

Ito, S.

Jakeman, E.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Little, C. G.

C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astr. Soc. 111, 289–302 (1951).

Livingston, R. C.

E. J. Fremouw, R. C. Livingston, D. A. Miller, “On the statistics of scintillating signals,”J. Atmos. Terr. Phys. 42, 717–731 (1980).
[CrossRef]

Martin, J. M.

McWhirter, J. G.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Miller, D. A.

E. J. Fremouw, R. C. Livingston, D. A. Miller, “On the statistics of scintillating signals,”J. Atmos. Terr. Phys. 42, 717–731 (1980).
[CrossRef]

Parry, G.

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Phillips, R. L.

Pusey, P. N.

G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Shishov, V. S.

K. Gochelashvili, V. S. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

Shivamoggi, B. K.

Speck, J. P.

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

T. Wang, J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,”J. Opt. Soc. Am. 64, 994–999 (1974).
[CrossRef]

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 45–106.
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, “Some new aspects in the problem of waves and turbulence,” Radio Sci. 22, 859–865 (1987).
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Wang, T.

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

T. Wang, J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,”J. Opt. Soc. Am. 64, 994–999 (1974).
[CrossRef]

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Sov. Phys. JETP 48, 27–31 (1978).

Appl. Opt. (5)

J. Atmos. Terr. Phys. (1)

E. J. Fremouw, R. C. Livingston, D. A. Miller, “On the statistics of scintillating signals,”J. Atmos. Terr. Phys. 42, 717–731 (1980).
[CrossRef]

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

Mon. Not. R. Astr. Soc. (1)

C. G. Little, “A diffraction theory of the scintillation of stars on optical and radio wavelengths,” Mon. Not. R. Astr. Soc. 111, 289–302 (1951).

Opt. Acta (2)

K. Gochelashvili, V. S. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

Opt. Commun. (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Opt. Lett. (1)

Radio Sci. (2)

V. I. Tatarskii, “Some new aspects in the problem of waves and turbulence,” Radio Sci. 22, 859–865 (1987).
[CrossRef]

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

Sov. Phys. JETP (1)

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Sov. Phys. JETP 48, 27–31 (1978).

Other (3)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 45–106.
[CrossRef]

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” Mod. Opt. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Apparatus used for PDF measurements.

Fig. 2
Fig. 2

Measured (circles), LR, and IK PDF’s of normalized irradiance for a variance of 2.6 × 10−3. (a) Linear representation, (b) logarithmic representation.

Fig. 3
Fig. 3

Measured PDF (circles) of the 0.5-sec variances of the data from Fig. 2 and a log-normal PDF (curve) with the same first and second moments.

Fig. 4
Fig. 4

Measured (symbols) and log-normal (curves) PDF’s of normalized irradiance for those data segments with variances between 3 × 10−4 and 5 × 10−4 (squares), between 1.1 × 10−4 and 1.3 × 10−4 (circles), and between 6 × 10−3 and 9 ×10−3 (diamonds). (a) Linear representation, (b) logarithmic representation.

Fig. 5
Fig. 5

Measured (circles) and conditionally log-normal (curves) PDF’s of normalized irradiance for the data of Fig. 2. (a) Linear representation, (b) logarithmic representation.

Fig. 6
Fig. 6

Measured (circles), LR, and IK PDF’s of normalized irradiance for a variance of 0.12. (a) Linear representation, (b) logarithmic representation.

Fig. 7
Fig. 7

Measured (circles), LR, and IK PDF’s of normalized irradiance for a variance of 1.3. (a) Linear representation, (b) logarithmic representation.

Fig. 8
Fig. 8

Measured (circles), LR, and IK PDF’s of normalized irradiance for a variance of 5.2.

Equations (25)

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p ( I ) = 2 a ( 1 + ρ ) [ ( 1 + ρ ) I ρ ] ( a - 1 ) / 2 K a - 1 [ 2 ( a ρ ) 1 / 2 ] × I a - 1 { 2 [ a ( 1 + ρ ) I ] 1 / 2 } ,             I < ρ 1 + ρ = 2 a ( 1 + ρ ) [ ( 1 + ρ ) I ρ ] ( a - 1 ) / 2 I a - 1 [ 2 ( a ρ ) 1 / 2 ] × K a - 1 { 2 [ a ( 1 + ρ ) I ] 1 / 2 } ,             I < ρ 1 + ρ ,
I n = n ! a n ( 1 + ρ ) n k = 0 n Γ ( a + n ) Γ ( a + k ) a k ρ k k ! ,
I 2 = ρ 2 ( 1 + ρ ) 2 + 2 1 + a - 1 1 + ρ ,
I 3 = ρ 3 ( 1 + ρ ) 3 + 3 ( 1 + 2 a - 1 ) ρ 2 ( 1 + ρ ) 3 + 6 ( 1 + α - 1 ) ( 1 + 2 a - 1 ) ( 1 + ρ ) 2 ,
( I 2 2 - I 2 - 1 3 I 3 + 1 3 ) ρ 3 + ( 3 I 2 2 - 2 I 2 - I 3 ) × ( ρ 2 + ρ ) + I 2 2 - I 2 - 1 3 I 3 = 0
α - 1 = 1 2 ( 1 + ρ ) I 2 - 1 2 ρ 2 1 + ρ - 1
I n = 1 + 1 2 n ( n - 1 ) σ 2
I n = 1 + n ( n - 1 ) ( a ρ ) - 1 ,
p ( I ) = 1 2 σ I - 3 / 4 exp ( - 2 3 / 2 σ 1 - I ) ,
p ( I ) = 2 Γ ( a ) a ( a + 1 ) / 2 I ( a - 1 ) / 2 K a - 1 [ 2 ( a I ) 1 / 2 ] ,
I n = n ! a n Γ ( a + n ) Γ ( a ) .
a - 1 = 1 2 I 2 - 1.
p ( I ) = ( 1 + ρ ) e - ρ 2 π σ z 0 d z z 2 I 0 { 2 [ ( 1 + ρ ) ρ z I ] 1 / 2 } × exp [ - 1 + ρ z I - 1 2 σ z 2 ( ln z + 1 2 σ z 2 ) 2 ] ,
I n = ( n ! ) ( 1 + ρ ) n exp [ 1 2 n ( n - 1 ) σ z 2 ] k = 0 n ρ k ( n - k ) ! ( k ! ) 2 .
I 2 = ρ 2 + 4 ρ + 2 ( 1 + ρ ) 2 exp ( σ z 2 ) ,
I 3 = ρ 3 + 9 ρ 2 + 18 ρ + 6 ( 1 + ρ ) 3 exp ( 3 σ z 2 ) ,
( ρ 2 + 4 ρ + 2 ) 3 ( 1 + ρ ) 3 ( ρ 3 + 9 ρ 2 + 18 ρ + 6 ) = I 2 3 I 3 .
σ z 2 = ln [ ( 1 + ρ ) 2 ρ 2 + 4 ρ + 2 I 2 ] .
p ( I ) = 1 2 π σ z I exp [ - 1 2 σ z 2 ( ln I + 1 2 σ z 2 ) 2 ] ,
I n = exp [ 1 2 n ( n - 1 ) σ z 2 ] ,
σ z 2 = ln I 2 .
p ( I ) = 1 2 π σ z 0 d z z 2 exp [ - I z - 1 2 σ z 2 ( ln z + 1 2 σ z 2 ) 2 ] ,
I n = n ! exp [ 1 2 n ( n - 1 ) σ z 2 ] ,
σ z 2 = ln ( 1 2 I 2 ) .
p ( I ) = 0 p ( I σ 2 ) p ( σ 2 ) d σ 2 ,

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