Abstract

The high-order irradiance moments recently measured in turbulent air up to the fifth order in connection with the K distribution are compared with previous theory based on a cluster expansion of the irradiance moments mν = 〈Iν〉/〈Iν, ν = 1, 2, 3, …. Good agreement is confirmed over the entire observed range 0.1 ≲ σI2/〈I2 ≲ 6 of the normalized variance of irradiance σI2/〈I2, including the regions of both unsaturated and saturated scintillations. Also shown are the theoretical curves of the mean and the variance of the log-amplitude, and good consistency is found with values of previous experiments made exclusively in terms of the log-amplitude and also of recent experiments made simultaneously on both log-amplitude and irradiance. The K distribution fits the experimental results of the mν values well within the range σI2/〈I2 ≳ 2 but definitely fails to fit those of the mean and the variance of the log-amplitude.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of laser radiation,” Trans. LRG-73-T-28 (Aerospace Corporation, Los Angeles, Calif.1973).
  2. M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
    [Crossref]
  3. G. Parry and P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [Crossref]
  4. G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [Crossref]
  5. K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
    [Crossref]
  6. K. Furutsu, “Review of the theory of the irradiance distribution in turbulent media with a particular emphasis on analytical methods,” Radio Sci. 14, 287–299 (1979).
    [Crossref]
  7. V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).
  8. I. G. Yakushikin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Radiophys. Quantum Electron. 21, 835–840 (1978).
    [Crossref]
  9. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [Crossref]
  10. D. A. de Wolf, “Saturation of irradiance fluctuations due to turbulent atmosphere,” J. Opt. Soc. Am. 58, 461–466 (1968).
    [Crossref]
  11. G. R. Ochs and T.-i Wang, NOAA/ERL, U.S. Department of Commerce, Boulder, Colo. 80303 (personal communication).
  12. M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurement of the average amplitude of a light wave propagating in a turbulent atmosphere,” Radiophys. Quantum Electron. 13, 36–39 (1970).
    [Crossref]
  13. S. F. Clifford and R. J. Hill, “Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution,” J. Opt. Soc. Am. 71, 112–114 (1981).
    [Crossref]
  14. R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1140–1445 (1981).
    [Crossref]

1981 (3)

1979 (3)

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

K. Furutsu, “Review of the theory of the irradiance distribution in turbulent media with a particular emphasis on analytical methods,” Radio Sci. 14, 287–299 (1979).
[Crossref]

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

1978 (1)

I. G. Yakushikin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Radiophys. Quantum Electron. 21, 835–840 (1978).
[Crossref]

1977 (1)

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

1976 (1)

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
[Crossref]

1974 (1)

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

1970 (1)

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurement of the average amplitude of a light wave propagating in a turbulent atmosphere,” Radiophys. Quantum Electron. 13, 36–39 (1970).
[Crossref]

1968 (1)

Andrews, L. C.

Clifford, S. F.

Dashen, R.

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

de Wolf, D. A.

Furutsu, K.

K. Furutsu, “Review of the theory of the irradiance distribution in turbulent media with a particular emphasis on analytical methods,” Radio Sci. 14, 287–299 (1979).
[Crossref]

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
[Crossref]

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurement of the average amplitude of a light wave propagating in a turbulent atmosphere,” Radiophys. Quantum Electron. 13, 36–39 (1970).
[Crossref]

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of laser radiation,” Trans. LRG-73-T-28 (Aerospace Corporation, Los Angeles, Calif.1973).

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurement of the average amplitude of a light wave propagating in a turbulent atmosphere,” Radiophys. Quantum Electron. 13, 36–39 (1970).
[Crossref]

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of laser radiation,” Trans. LRG-73-T-28 (Aerospace Corporation, Los Angeles, Calif.1973).

Hill, R. J.

Kallistratova, M. A.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurement of the average amplitude of a light wave propagating in a turbulent atmosphere,” Radiophys. Quantum Electron. 13, 36–39 (1970).
[Crossref]

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of laser radiation,” Trans. LRG-73-T-28 (Aerospace Corporation, Los Angeles, Calif.1973).

Khrupin, A. S.

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

Klyatskin, V. I.

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Lomadze, S. O.

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

Ochs, G. R.

G. R. Ochs and T.-i Wang, NOAA/ERL, U.S. Department of Commerce, Boulder, Colo. 80303 (personal communication).

Parry, G.

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

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

Phillips, R. L.

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of laser radiation,” Trans. LRG-73-T-28 (Aerospace Corporation, Los Angeles, Calif.1973).

Pusey, P. N.

Tatarskii, V. I.

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Wang, T.-i

G. R. Ochs and T.-i Wang, NOAA/ERL, U.S. Department of Commerce, Boulder, Colo. 80303 (personal communication).

Yakushikin, I. G.

I. G. Yakushikin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Radiophys. Quantum Electron. 21, 835–840 (1978).
[Crossref]

Zavorotnyi, V. U.

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

J. Math. Phys. (2)

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,” J. Math. Phys. 17, 1252–1263 (1976).
[Crossref]

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

J. Opt. Soc. Am. (4)

Opt. Acta (1)

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

Radio Sci. (1)

K. Furutsu, “Review of the theory of the irradiance distribution in turbulent media with a particular emphasis on analytical methods,” Radio Sci. 14, 287–299 (1979).
[Crossref]

Radiophys. Quantum Electron. (3)

M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, V. V. Pokasov, and A. S. Khrupin, “Probability distribution of ‘strong’ fluctuations of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[Crossref]

I. G. Yakushikin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Radiophys. Quantum Electron. 21, 835–840 (1978).
[Crossref]

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, “Measurement of the average amplitude of a light wave propagating in a turbulent atmosphere,” Radiophys. Quantum Electron. 13, 36–39 (1970).
[Crossref]

Sov. Phys. JETP (1)

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Other (2)

G. R. Ochs and T.-i Wang, NOAA/ERL, U.S. Department of Commerce, Boulder, Colo. 80303 (personal communication).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of laser radiation,” Trans. LRG-73-T-28 (Aerospace Corporation, Los Angeles, Calif.1973).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Comparison of the experimental values of mν = 〈Iν〉/〈Iν, ν = 2, 3, 4, 5, observed by Parry and Pusey3 with the corresponding theoretical values given by expressions (3) and (6). The curve of Δ′ shown in the upper part of the figure was obtained according to expressions (4) and (5), with the m4 values given by the smoothed curve of the experimental m4. The open circles and crosses indicate the theoretical values given by expressions (3) and (6), respectively, with the above values of Δ′. The dotted–dashed and broken curves indicate the moments expected from the lognormal and K distributions.

Fig. 2
Fig. 2

Normalized moments mν, ν = 2, 3, 4, 5, versus β0 = (0.5Cn2k7/6L11/6)1/2 observed in Ref. 4 and the theoretical results. Curves a and b in the lower part of the figure represent the theory given by expressions (3) and (6) with values of the smoothed curves of Δ′ and m2, respectively. The corresponding values are also shown for each of some chosen data by open circles for Δ′ and by open circles and crosses for the mν values given by expressions (3) and (6), respectively. The meanings of the dotted–dashed and broken curves are the same as those for Fig. 1.

Fig. 3
Fig. 3

Same comparison as in Fig. 1 for the experimental values observed by Ochs and Wang,11 with 1-km path using He–Ne laser as the source. Solid curves a and b show the theoretical values given by expressions (3) and (6), respectively.

Fig. 4
Fig. 4

(a) Theoretical dependence of - χ on σx. Solid curves a and b show the dependencies given by expressions (13) and (14) with the values of Δ′ in Fig. 1. The dotted–dashed and broken curves show the corresponding relations given by the lognormal and K distributions, respectively. (b) Experimental dependence of - χ on σχ: crosses, L = 250 m; filled circles, L = 1750 m (from Ref. 12).

Fig. 5
Fig. 5

Log-amplitude variance σχ2 versus normalized irradiance variance σI2/〈I2. The experimental values11 were observed simultaneously with the irradiance moments in Fig. 3. Solid curves a and b show the values predicted by Eqs. (13) and (14) for the same Δ′ values as employed in Fig. 3. The dotted–dashed and broken curves show the corresponding values for the lognormal and K distributions.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

m ν I ν / I ν ,             ν = 1 , 2 , 3 ,
m ν = exp [ ( ν 2 ) ln m 2 + ( ν 3 ) ln ( m 3 / m 2 3 ) + ( ν 4 ) ln ( m 4 m 2 6 / m 3 4 ) + ] ,
m ν ~ exp { ( ½ ) ν ( ν - 1 ) K 1 [ 1 - ( ν - 2 ) Δ ] } ,
K 1 = ln m 2 ,             Δ = 1 - ( ) ln m 3 / ln m 2 .
m 4 ~ m 3 4 / m 2 6 ,
m ν ~ exp { ( ½ ) ν ( ν - 1 ) K 1 [ 1 + ( ν - 2 ) Δ ] - 1 } ,
Δ Δ ( 1 - Δ ) - 1 = 3 ln m 2 / ln m 3 - 1 > 0 ,
P ( E ) = ( 2 π i ) - 1 - i - i - d ν exp ( - ν E ) m ν ,             = + 0
P ( E ) = exp ( - a 2 / 2 ) δ ( E - E c ) + { ν 0 a b - 1 exp [ - ( a 2 + b 2 ) / 2 ] I 1 ( a b ) , E < E c 0 , E > E c ,
ν 0 = Δ - 1 - 2 > 0 , a 2 = ( 1 + ν 0 ) K 1 Δ - 1 , E c = K 1 ( 2 Δ ) - 1 , b 2 = 2 ν 0 ( E c - E ) .
P ( E ) ~ ( 2 π K 1 ) - 1 / 2 exp ( - F 2 / 2 ) ,             a ~ b 1 , F = a - b ~ K 1 1 / 2 [ E / K 1 + 1 / 2 + ( Δ / 2 ) ( E / K 1 - ½ ) ( E / K 1 - ³ / ) ] ,
E = ν ln m ν | ν = 0 ,             ( E - E ) n = n ν n ln m ν | ν = 0 ,             n = 2 , 3.
E = - ½ ( 1 + 2 Δ ) K 1 < 0 ( E - E ) 2 = ( 1 + 3 Δ ) K 1 , ( E - E ) 3 = - 3 Δ K 1 < 0 ,
E = - ½ ( 1 - 2 Δ ) - 1 K 1 < 0 , ( E - E ) 2 = ( 1 - Δ ) ( 1 - 2 Δ ) - 2 K 1 , ( E - E ) 3 = - 3 Δ ( 1 - Δ ) ( 1 - 2 Δ ) - 3 K 1 < 0 ,
m ν = exp [ 2 χ ν + 2 σ χ 2 ν 2 + ]
m ν , K = ν ! Γ ( ν + y ) / y ν Γ ( y ) = exp ( 2 χ K ν + 2 σ χ , K 2 ν 2 + ) ,