Abstract

By using results from perturbation techniques for weak turbulence and the asymptotic theory for strong turbulence, we develop expressions for the parameters α and ρ of the IK distribution in terms of the Rytov variance for plane waves σ12=1.23Cn2k7/6L11/6 and the inner scale of turbulence parameter l0. Comparisons of the resulting scintillation index to experimental data and numerical results from the solution of the fourth-moment equation for 3-D propagation show good agreement.

© 1988 Optical Society of America

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References

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  1. J. Strohbehn, Ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
    [CrossRef]
  2. K. S. Gochelashvili, V. I. Shishov, “Strong Fluctuations of Laser Radiation Intensity in a Turbulent Atmosphere—the Distribution Function,” Sov. Phys. JETP 47, 1028 (1978).
  3. E. Jakeman, P. N. Pusey, “Significance of K-Distributions in Scattering Experiments,” Phys. Rev. Lett. 40, 546 (1978).
    [CrossRef]
  4. G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
    [CrossRef]
  5. E. Jakeman, “On the Statistics of K-Distributed Noise,” J. Phys. A 13, 31 (1980).
    [CrossRef]
  6. G. Parry, “Measurements of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
    [CrossRef]
  7. R. J. Hill, S. F. Clifford, “Theory of Saturation of Optical Scintillation by Strong Turbulence for Arbitrary Refractive-Index Spectra,” J. Opt. Soc. Am. 71, 675 (1981).
    [CrossRef]
  8. R. L. Phillips, L. C. Andrews, “Measured Statistics of Laser-Light Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. 71, 1440 (1981).
    [CrossRef]
  9. 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 (1982).
    [CrossRef]
  10. W. A. Coles, R. G. Frehlich, “Simultaneous Measurements of Angular Scattering and Intensity Scintillation in the Atmosphere,” J. Opt. Soc. Am. 72, 1042 (1982).
    [CrossRef]
  11. R. L. Fante, “Inner-Scale Size Effect on the Scintillations of Light in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 277 (1983).
    [CrossRef]
  12. L. R. Bissonnette, “Propagation Model of Laser Beams in Turbulence,’ J. Opt. Soc. Am. 73, 262 (1983).
    [CrossRef]
  13. R. Dashen, “Distribution of Intensity in a Multiply Scattering Medium,” Opt. Lett. 10, 110 (1984).
    [CrossRef]
  14. A. Consortini, G. Conforti, “Detector Saturation Effect on Higher-Order Moments of Intensity Fluctuations in Atmospheric Laser Propagation Measurement,” J. Opt. Soc. Am. A 1, 1075 (1984).
    [CrossRef]
  15. A. M. Whitman, M. J. Beran, “Two-Scale Solution for Atmospheric Scintillation,” J. Opt. Soc. Am. A 2, 2133 (1985).
    [CrossRef]
  16. A. Consortini, E. Briccolani, G. Conforti, “Strong-Scintillation-Statistics Deterioration due to Detector Saturation,” J. Opt. Soc. Am. A 3, 101 (1986).
    [CrossRef]
  17. R. Barakat, “Weak-Scatter Generalization of the K-Density Function with Applications to Laser Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. A 3, 401 (1986).
    [CrossRef]
  18. L. C. Andrews, R. L. Phillips, “Mathematical Genesis of the I–K Distribution for Random Optical Fields,” J. Opt. Soc. Am. A 3, 1912 (1986).
    [CrossRef]
  19. R. G. Frehlich, “Intensity Covariance of a Point Source in a Random Medium with a Kolmogorov Spectrum and an Inner Scale of Turbulence,” J. Opt. Soc. Am. A 4, 360 (1987).
    [CrossRef]
  20. D. Link, R. L. Phillips, L. C. Andrews, “Theoretical Model for Optical-Wave Phase Fluctuations,” J. Opt. Soc. Am. A 4, 374 (1987).
    [CrossRef]
  21. J. H. Churnside, R. J. Hill, “Probability Density of Irradiance Scintillations for Strong Path-Integrated Refractive Turbulence,” J. Opt. Soc. Am. A 4, 727 (1987).
    [CrossRef]
  22. 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 (1987).
    [CrossRef]
  23. V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961).
  24. M. E. Gracheva, A. S. Gurvich, M. A. Kallistratova, “Measurements of the Variance of ‘Strong’ Intensity Fluctuations of Laser Radiation in the Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 55 (1970).
  25. J. W. Strohbehn, T.-I. Wang, J. P. Speck, “On the Probability Distribution of Line-of-Sight Fluctuations of Optical Signals,” Radio Sci. 10, 59 (1975).
    [CrossRef]
  26. H. Cramer, Mathematical Methods of Statistics (Princeton U.P., Princeton, NJ, 1966), p. 364.
  27. V. I. Tatarski, V. U. Zavorontnyi, “Wave Propagation in Random Media with Fluctuating Turbulent Parameters,” J. Opt. Soc. Am. A 2, 2069 (1985).
    [CrossRef]
  28. R. Fante, “Detection of Multiscatter Targets in K-Distributed Clutter,” IEEE Trans. Antennas Propag. AP-32, 1358(1984).
    [CrossRef]
  29. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
    [CrossRef]
  30. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians (Macmillan, New York, 1985).

1987 (4)

1986 (3)

1985 (2)

1984 (3)

1983 (2)

1982 (2)

1981 (3)

1980 (1)

E. Jakeman, “On the Statistics of K-Distributed Noise,” J. Phys. A 13, 31 (1980).
[CrossRef]

1979 (1)

1978 (2)

K. S. Gochelashvili, V. I. Shishov, “Strong Fluctuations of Laser Radiation Intensity in a Turbulent Atmosphere—the Distribution Function,” Sov. Phys. JETP 47, 1028 (1978).

E. Jakeman, P. N. Pusey, “Significance of K-Distributions in Scattering Experiments,” Phys. Rev. Lett. 40, 546 (1978).
[CrossRef]

1975 (2)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
[CrossRef]

J. W. Strohbehn, T.-I. Wang, J. P. Speck, “On the Probability Distribution of Line-of-Sight Fluctuations of Optical Signals,” Radio Sci. 10, 59 (1975).
[CrossRef]

1970 (1)

M. E. Gracheva, A. S. Gurvich, M. A. Kallistratova, “Measurements of the Variance of ‘Strong’ Intensity Fluctuations of Laser Radiation in the Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 55 (1970).

Andrews, L. C.

Barakat, R.

Beran, M. J.

Bissonnette, L. R.

Briccolani, E.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
[CrossRef]

Churnside, J. H.

Clifford, S. F.

Coles, W. A.

Conforti, G.

Consortini, A.

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton U.P., Princeton, NJ, 1966), p. 364.

Dashen, R.

Fante, R.

R. Fante, “Detection of Multiscatter Targets in K-Distributed Clutter,” IEEE Trans. Antennas Propag. AP-32, 1358(1984).
[CrossRef]

Fante, R. L.

Frehlich, R. G.

Furutsu, K.

Gochelashvili, K. S.

K. S. Gochelashvili, V. I. Shishov, “Strong Fluctuations of Laser Radiation Intensity in a Turbulent Atmosphere—the Distribution Function,” Sov. Phys. JETP 47, 1028 (1978).

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
[CrossRef]

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, M. A. Kallistratova, “Measurements of the Variance of ‘Strong’ Intensity Fluctuations of Laser Radiation in the Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 55 (1970).

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, M. A. Kallistratova, “Measurements of the Variance of ‘Strong’ Intensity Fluctuations of Laser Radiation in the Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 55 (1970).

Hill, R. J.

Ito, S.

Jakeman, E.

E. Jakeman, “On the Statistics of K-Distributed Noise,” J. Phys. A 13, 31 (1980).
[CrossRef]

E. Jakeman, P. N. Pusey, “Significance of K-Distributions in Scattering Experiments,” Phys. Rev. Lett. 40, 546 (1978).
[CrossRef]

Kallistratova, M. A.

M. E. Gracheva, A. S. Gurvich, M. A. Kallistratova, “Measurements of the Variance of ‘Strong’ Intensity Fluctuations of Laser Radiation in the Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 55 (1970).

Link, D.

Parry, G.

G. Parry, “Measurements of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
[CrossRef]

G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
[CrossRef]

Phillips, R. L.

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
[CrossRef]

Pusey, P. N.

G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
[CrossRef]

E. Jakeman, P. N. Pusey, “Significance of K-Distributions in Scattering Experiments,” Phys. Rev. Lett. 40, 546 (1978).
[CrossRef]

Shishov, V. I.

K. S. Gochelashvili, V. I. Shishov, “Strong Fluctuations of Laser Radiation Intensity in a Turbulent Atmosphere—the Distribution Function,” Sov. Phys. JETP 47, 1028 (1978).

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
[CrossRef]

Speck, J. P.

J. W. Strohbehn, T.-I. Wang, J. P. Speck, “On the Probability Distribution of Line-of-Sight Fluctuations of Optical Signals,” Radio Sci. 10, 59 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T.-I. Wang, J. P. Speck, “On the Probability Distribution of Line-of-Sight Fluctuations of Optical Signals,” Radio Sci. 10, 59 (1975).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, V. U. Zavorontnyi, “Wave Propagation in Random Media with Fluctuating Turbulent Parameters,” J. Opt. Soc. Am. A 2, 2069 (1985).
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961).

Wang, T.-I.

J. W. Strohbehn, T.-I. Wang, J. P. Speck, “On the Probability Distribution of Line-of-Sight Fluctuations of Optical Signals,” Radio Sci. 10, 59 (1975).
[CrossRef]

Whitman, A. M.

Zavorontnyi, V. U.

IEEE Trans. Antennas Propag. (1)

R. Fante, “Detection of Multiscatter Targets in K-Distributed Clutter,” IEEE Trans. Antennas Propag. AP-32, 1358(1984).
[CrossRef]

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

M. E. Gracheva, A. S. Gurvich, M. A. Kallistratova, “Measurements of the Variance of ‘Strong’ Intensity Fluctuations of Laser Radiation in the Atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 55 (1970).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (10)

A. Consortini, G. Conforti, “Detector Saturation Effect on Higher-Order Moments of Intensity Fluctuations in Atmospheric Laser Propagation Measurement,” J. Opt. Soc. Am. A 1, 1075 (1984).
[CrossRef]

A. M. Whitman, M. J. Beran, “Two-Scale Solution for Atmospheric Scintillation,” J. Opt. Soc. Am. A 2, 2133 (1985).
[CrossRef]

A. Consortini, E. Briccolani, G. Conforti, “Strong-Scintillation-Statistics Deterioration due to Detector Saturation,” J. Opt. Soc. Am. A 3, 101 (1986).
[CrossRef]

R. Barakat, “Weak-Scatter Generalization of the K-Density Function with Applications to Laser Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. A 3, 401 (1986).
[CrossRef]

L. C. Andrews, R. L. Phillips, “Mathematical Genesis of the I–K Distribution for Random Optical Fields,” J. Opt. Soc. Am. A 3, 1912 (1986).
[CrossRef]

R. G. Frehlich, “Intensity Covariance of a Point Source in a Random Medium with a Kolmogorov Spectrum and an Inner Scale of Turbulence,” J. Opt. Soc. Am. A 4, 360 (1987).
[CrossRef]

D. Link, R. L. Phillips, L. C. Andrews, “Theoretical Model for Optical-Wave Phase Fluctuations,” J. Opt. Soc. Am. A 4, 374 (1987).
[CrossRef]

J. H. Churnside, R. J. Hill, “Probability Density of Irradiance Scintillations for Strong Path-Integrated Refractive Turbulence,” J. Opt. Soc. Am. A 4, 727 (1987).
[CrossRef]

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 (1987).
[CrossRef]

V. I. Tatarski, V. U. Zavorontnyi, “Wave Propagation in Random Media with Fluctuating Turbulent Parameters,” J. Opt. Soc. Am. A 2, 2069 (1985).
[CrossRef]

J. Phys. A (1)

E. Jakeman, “On the Statistics of K-Distributed Noise,” J. Phys. A 13, 31 (1980).
[CrossRef]

Opt. Acta (1)

G. Parry, “Measurements of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Jakeman, P. N. Pusey, “Significance of K-Distributions in Scattering Experiments,” Phys. Rev. Lett. 40, 546 (1978).
[CrossRef]

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser Irradiance Propagation in a Turbulent Media,” Proc. IEEE 63, 790 (1975).
[CrossRef]

Radio Sci. (1)

J. W. Strohbehn, T.-I. Wang, J. P. Speck, “On the Probability Distribution of Line-of-Sight Fluctuations of Optical Signals,” Radio Sci. 10, 59 (1975).
[CrossRef]

Sov. Phys. JETP (1)

K. S. Gochelashvili, V. I. Shishov, “Strong Fluctuations of Laser Radiation Intensity in a Turbulent Atmosphere—the Distribution Function,” Sov. Phys. JETP 47, 1028 (1978).

Other (4)

J. Strohbehn, Ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961).

H. Cramer, Mathematical Methods of Statistics (Princeton U.P., Princeton, NJ, 1966), p. 364.

L. C. Andrews, Special Functions for Engineers and Applied Mathematicians (Macmillan, New York, 1985).

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Figures (5)

Fig. 1
Fig. 1

Variation of the scintillation index with σ1 for both plane waves and spherical waves with zero inner scale.

Fig. 2
Fig. 2

Variations of the square root of the scintillation index with σ1 for plane waves and zero inner scale. The solid curve is that predicted by the IK distribution, and the experimental data are from Ref. 24. The dots represent measured data at a fixed propagation distance of 1750 m, while the pluses are the same at 250 m.

Fig. 3
Fig. 3

Variation of the square root of the scintillation index with σ1 for spherical waves and zero inner scale. The solid curve is that predicted by the IK distribution, and the experimental data are from Refs. 6 and 8. The dots are Ref. 8 data with variable propagation path lengths up to 1500 m, while the pluses are Ref. 6 data taken at 1250 m.

Fig. 4
Fig. 4

Comparison of the square root of the scintillation index for the IK distribution with numerical results from Ref. 15 based on fourth-moment equations for 3-D propagation.

Fig. 5
Fig. 5

Variation of the square root of the scintillation index with σ1 and inner scale of turbulence l0 and comparison with the experimental data of Ref. 24. The theoretical curves correspond to a propagation path length of 1750 m.

Equations (42)

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U ( t ) = exp ( i ω t ) [ A exp ( i θ ) + R exp ( i ϕ ) ] .
p ( I ) = { 2 α b 0 ( I A ) α - 1 K α - 1 ( 2 A α b 0 ) I α - 1 ( 2 α I b 0 ) , I < A 2 , 2 α b 0 ( I A ) α - 1 I α - 1 ( 2 A α b 0 ) K α - 1 ( 2 α I b 0 ) , I > A 2 ,
σ I 2 = I 2 I 2 - 1 ,
σ L n I 2 = 16 π 2 k 2 0 L 0 κ ϕ n ( κ ) sin 2 [ κ 2 ( L - η ) 2 k ] d κ d η ,
σ L n I 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 = σ 1 2 .
σ I 2 = exp ( σ 1 2 ) - 1 σ 1 2 ( 1 + 0.5 σ 1 2 ) ,             σ 1 2 1             ( plane wave ) ,
σ I 2 = exp ( 0.41 σ 1 2 ) - 1 0.41 σ 1 2 ( 1 + 0.5 σ 1 2 ) , σ 1 2 1 ( spherical wave ) .
σ I 2 1 + 0.86 σ 1 4 / 5 , σ 1 2 1 ( plane wave ) ,
σ I 2 1 + 2.8 σ 1 4 / 5 , σ 1 2 1 ( spherical wave ) .
I n I n = n ! a n ( 1 + ρ ) n k = 1 n Γ ( α + n ) Γ ( α + k ) ( α ρ ) k k ! ,             n = 1 , 2 , 3 , ,
σ I 2 = 2 ( 1 + ρ ) 2 ( 1 2 + 1 + ρ α ) .
σ I 2 2 α ρ ,             ρ 1.
α ρ = 2 σ 1 2 ( 1 + 0.5 σ 1 2 ) ,             σ 1 2 1 ( plane wave ) ,
α ρ = 4.88 σ 1 2 ( 1 + 0.2 σ 1 2 ) , σ 1 2 1 ( spherical wave ) .
σ I 2 1 + 2 α ,             ρ 1.
α = 2.33 σ 1 4 / 5 , σ 1 2 1 ( plane wave ) ,
α = 0.71 σ 1 4 / 5 , σ 1 2 1 ( spherical wave ) .
α = 2.33 α 1 4 / 5 , ρ = 2 α σ 1 2 ( 1 + 0.5 σ 1 2 ) ,
α = 0.71 σ 1 4 / 5 , ρ = 4.88 α σ 1 2 ( 1 + 0.2 σ 1 2 ) .
σ I 2 1 + 2.03 σ 0 1 / 3 ,             σ 0 2 1 ,
σ 0 2 = 6.94 σ 1 2 β 7 / 6 .
β = λ L l 0 2 ,
α = 0.985 σ 0 1 / 3 = 1.36 ( α 1 2 β 7 / 6 ) 1 / 6 .
ϕ n ( κ ) = 0.033 C n 2 exp ( - κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6 ,
σ L n I 2 = 1.23 ( c n 2 σ ˜ 2 ) k 7 / 6 L 11 / 6 = σ 1 2 σ ˜ 2 ,
σ ˜ 2 3.864 ( 1 + 1 31.14 β 2 ) 11 / 12 × sin ( 11 6 tan - 1 ( 5.58 β ) ) - 1.69 β 5 / 6 .
σ I 2 = σ I 2 σ ˜ 2 ( 1 + 0.5 σ 1 2 σ ˜ 2 ) ,             σ 1 2 1 ,
ρ = 2 α σ 1 2 σ ˜ 2 ( 1 + 0.5 σ 1 2 σ ˜ 2 ) .
σ L N I 2 = 0.264 π 2 k 2 C n 2 ( I 1 - I 2 ) ,
I 1 = 0 L 0 κ exp ( - κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6 d κ d η ,
I 2 = 0 L 0 k exp ( - κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6 cos [ κ 2 ( L - η ) k ] d κ d η .
I 1 = ½ L κ 0 - 5 / 3 U ( 1 ; 1 / 6 ; κ 0 2 / κ m 2 ) ,
I 2 = ¼ κ 0 - 5 / 3 0 L [ U ( 1 ; 1 / 6 ; X - i Y ) + U ( 1 ; 1 / 6 ; X + i Y ) ] d η ,
X ± i Y = κ 0 2 [ 1 κ m 2 ± i ( L - η ) k ] .
I 2 = k 4 i κ 0 - 11 / 3 a - i b a + i b U ( 1 ; 1 / 6 ; t ) d t ,
a ± i b = κ 0 2 ( 1 κ m 2 ± i L k ) .
U ( 1 ; 1 / 6 ; t ) = 6 5 F 1 1 ( 1 ; 1 / 6 ; t ) + Γ ( - 5 / 6 ) t 5 / 6 e t ,
I 2 = 3 10 i κ 0 - 11 / 3 [ ( a + i b ) F 2 2 ( 1 , 1 ; 2 , 1 / 6 ; a + i b ) - ( a - i b ) F 2 2 ( 1 , 1 ; 2 , 1 / 6 ; a - i b ) ] + 3 Γ ( - 5 / 6 ) 22 i k κ 0 - 11 / 3 [ ( a + i b ) 11 / 6 F 1 1 ( 11 / 6 ; 17 / 6 ; a + i b ) - ( a - i b ) 11 / 6 F 1 1 ( 11 / 6 ; 17 / 6 ; a - i b ) ] ,
σ L n I 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 σ ˜ 2 ,
σ ˜ 2 = 1.06 ( k L κ 0 2 ) 5 / 5 U ( 1 ; 1 / 6 ; κ 0 2 / κ m 2 ) - 1.272 ( k L κ 0 2 ) 5 / 6 × n = 0 ( 1 ) n ( L κ 0 2 / k ) n ( 2 ) n ( 1 / 6 ) n ( 1 + k 2 L 2 κ m 4 ) ( n + 1 ) / 2 × sin [ ( n + 1 ) tan - 1 ( L κ m 2 k ) ] + 3.864 n = 0 ( 11 / 6 ) n ( L κ 0 2 / k ) n ( 17 / 6 ) n n ! ( 1 + k 2 L 2 κ m 2 ) 11 / 12 + n / 2 × sin [ ( n + 11 6 ) tan - 1 ( L κ m 2 k ) ] .
( a ) n = Γ ( a + n ) Γ ( a ) ,             n = 0 , 1 , 2 , .
1.06 ( k L κ 0 2 ) 5 / 6 U ( 1 ; 1 / 6 ; κ 0 2 / κ m 2 ) 1.272 ( k L κ 0 2 ) 5 / 6 + 1.06 Γ ( - 5 / 6 ) ( k L κ m 2 ) 5 / 6 , κ 0 2 / κ m 2 1 ,

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