Abstract

A novel model describing the probability-density function (PDF) of the irradiance fluctuations in a turbulent atmosphere is presented. The change in the PDF as turbulence strength increases is explained by postulating that the reduction of the transverse coherence leads effectively to splitting of the source to a number of secondary sources. Consequently, the observer detects a number of independent light sources simultaneously. The statistics of this superposition are different from those of a single source. An exact calculation based on this stipulation leads to a PDF that is practically log normal at low-level turbulence, transforms continuously, and converges to the exponential distribution at very strong turbulence. The plots of PDF moments fit between the log-normal and the K-PDF plots.

© 1988 Optical Society of America

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References

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  1. See, for example, the review: L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  2. E. Jakeman, P. N. Pusey, “A model for the non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [CrossRef]
  3. L. R. Bissonnette, P. L. Wizinowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979).
    [CrossRef] [PubMed]
  4. J. H. Churnside, R. J. Hill, G. R. Ochs, “Yet another probability distribution for intensity fluctuations in strong turbulence,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986;“Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4,727–733 (1987).
  5. R. L. Phillips, L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
    [CrossRef]
  6. 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]
  7. 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]
  8. R. Barakat, “Weak scatterer generalization of the K-density function with application to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401–409 (1986).
    [CrossRef]
  9. E. Pinsky, “Turbulence generation and its effects on the optical transmission of the atmosphere,” Ph.D. dissertation (The Hebrew University, Jerusalem, 1984; in Hebrew).
  10. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt.Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  11. R. L. Phillips, L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  12. G. R. Ochs, T.-I. Wang, NOAA/ERL, U.S. Department of Commerce, Boulder, Colo. 80303 (personal communication).
  13. G. Parry, J. G. Walker, “Statistics of stellar scintillations,” J. Opt. Soc. Am. 70, 1157–1159 (1980).
    [CrossRef]
  14. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  15. E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  16. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968).
  17. H. T. Yura, “An elementary derivation of the saturation of optical scintillations,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986.
  18. E. Jakeman, “On the statistics of K distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  19. A. Consortini, G. Conforti, “Detector saturation effects on higher-order moments of intensity fluctuations in atmospheric laser propagation measurements,” J. Opt. Soc. Am. A 1, 1075–1077 (1984).
    [CrossRef]
  20. A. Consortini, E. Briccolani, G. Conforti, “Strong scintillation statistics detoriation due to detector saturation,” J. Opt. Soc. Am. A 3, 101–107 (1986).
    [CrossRef]
  21. A. Consortini, R. J. Hill, “Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the K and log-normally modulated exponential probability densities,” Opt. Lett. 12, 304–306 (1987).
    [CrossRef] [PubMed]
  22. M. E. Gracheva, A. S. Gurevich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strobehn, ed. (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  23. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

1987 (1)

1986 (2)

1985 (1)

1984 (1)

1982 (2)

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]

R. L. Phillips, L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
[CrossRef]

1981 (1)

1980 (2)

G. Parry, J. G. Walker, “Statistics of stellar scintillations,” J. Opt. Soc. Am. 70, 1157–1159 (1980).
[CrossRef]

E. Jakeman, “On the statistics of K distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

1979 (3)

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

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

L. R. Bissonnette, P. L. Wizinowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979).
[CrossRef] [PubMed]

1978 (1)

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

1976 (1)

E. Jakeman, P. N. Pusey, “A model for the non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

1975 (1)

See, for example, the review: L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Andrews, L. C.

Barakat, R.

Bissonnette, L. R.

Briccolani, E.

Churnside, J. H.

J. H. Churnside, R. J. Hill, G. R. Ochs, “Yet another probability distribution for intensity fluctuations in strong turbulence,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986;“Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4,727–733 (1987).

Conforti, G.

Consortini, A.

Dashen, R.

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

Fante, L.

See, for example, the review: L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968).

Furutsu, K.

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]

Gracheva, M. E.

M. E. Gracheva, A. S. Gurevich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strobehn, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

Gurevich, A. S.

M. E. Gracheva, A. S. Gurevich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strobehn, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

Hill, R. J.

A. Consortini, R. J. Hill, “Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the K and log-normally modulated exponential probability densities,” Opt. Lett. 12, 304–306 (1987).
[CrossRef] [PubMed]

J. H. Churnside, R. J. Hill, G. R. Ochs, “Yet another probability distribution for intensity fluctuations in strong turbulence,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986;“Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4,727–733 (1987).

Ito, S.

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]

Jakeman, E.

E. Jakeman, “On the statistics of K distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for the non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurevich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strobehn, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

Ochs, G. R.

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

J. H. Churnside, R. J. Hill, G. R. Ochs, “Yet another probability distribution for intensity fluctuations in strong turbulence,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986;“Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4,727–733 (1987).

Parry, G.

G. Parry, J. G. Walker, “Statistics of stellar scintillations,” J. Opt. Soc. Am. 70, 1157–1159 (1980).
[CrossRef]

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

Phillips, R. L.

Pinsky, E.

E. Pinsky, “Turbulence generation and its effects on the optical transmission of the atmosphere,” Ph.D. dissertation (The Hebrew University, Jerusalem, 1984; in Hebrew).

Pokasov, V. V.

M. E. Gracheva, A. S. Gurevich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strobehn, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

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]

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for the non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Walker, J. G.

Wang, T.-I.

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

Wizinowich, P. L.

Yura, H. T.

H. T. Yura, “An elementary derivation of the saturation of optical scintillations,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

E. Jakeman, P. N. Pusey, “A model for the non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

J Opt. Soc. Am. (1)

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]

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. (3)

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

J. Opt.Soc. Am. (1)

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

J. Phys. A (1)

E. Jakeman, “On the statistics of K distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

Proc. IEEE (1)

See, for example, the review: L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Other (7)

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968).

H. T. Yura, “An elementary derivation of the saturation of optical scintillations,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986.

E. Pinsky, “Turbulence generation and its effects on the optical transmission of the atmosphere,” Ph.D. dissertation (The Hebrew University, Jerusalem, 1984; in Hebrew).

J. H. Churnside, R. J. Hill, G. R. Ochs, “Yet another probability distribution for intensity fluctuations in strong turbulence,” presented at the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 1986;“Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4,727–733 (1987).

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

M. E. Gracheva, A. S. Gurevich, S. S. Kashkarov, V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strobehn, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (10)

Fig. 1
Fig. 1

The two paths in the μσI0 domain used in the discussion.

Fig. 2
Fig. 2

The normalized intensity variance as a function of the parameter μ (which increases monotonically with increasing turbulence strength). The saturation is clearly observed and the insensitivity to the specific path easily seen.

Fig. 3
Fig. 3

The intensity PDF for point a in Fig. 1. σI0 = 1.0, μ = 0.51. Dashed line, log-normal distribution; solid line, present model.

Fig. 4
Fig. 4

The intensity PDF for point b in Fig. 1. σI0 = 1.87, μ = 0.65. Dashed line, log-normal distribution; solid line, present model.

Fig. 5
Fig. 5

The intensity PDF for point c in Fig. 1. σI0 = 2.2, μ = 1.5. Dashed line, log-normal distribution; solid line, present model.

Fig. 6
Fig. 6

The intensity PDF for point d in Fig. 1. σI0 = 2.2, μ = 48. Dashed line, log-normal distribution; solid line, present model.

Fig. 7
Fig. 7

The intensity PDF for point e in Fig. 1. σI0 = 1.5, μ = 2.0. Dashed line, log-normal distribution; solid line, present model.

Fig. 8
Fig. 8

The intensity PDF for point f in Fig. 1. σI0 = 1.02, μ = 10. Dashed line, log-normal distribution; solid line, present model.

Fig. 9
Fig. 9

The intensity PDF for point g in Fig. 1. σI0 = 1.0, μ = 25. Dashed line, log-normal distribution; solid line, present model.

Fig. 10
Fig. 10

Plot of moments (third, fourth, and fifth moments) along path 1. Dotted line, log-normal distribution; dashed line, K PDF; solid line, present model.

Equations (25)

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P ( E ) = N = 1 α N P N ( E ) ,
α N = exp ( μ ) μ N 1 ( N 1 ) ! ,
N = 1 α N = 1 .
f ( E ) = def f ( E ) P ( E ) d E = N = 1 α N f ( E ) P N ( E ) d E = N = 1 α N f N ,
E N = K = 1 N E N K exp ( j ψ N K ) ,
I 2 = [ ( μ + 1 ) g + 2 μ ( μ + 2 ) ] / ( μ + 1 ) 2 ,
I 3 = [ ( μ + 1 ) g 3 + 9 μ ( μ + 2 ) g + 6 μ 2 ( μ + 3 ) ] / ( μ + 1 ) 3 ,
I 4 = [ ( μ + 1 ) g 6 + 16 μ ( μ + 2 ) g 3 + 18 μ ( μ + 2 ) g 2 + 72 μ 2 ( μ + 3 ) g + 24 μ 2 ( μ + 4 ) ] / ( μ + 1 ) 4 ,
I n = n ! [ 1 + n ( n 1 ) 2 g 2 1 μ + 0 ( 1 μ 2 ) ] .
σ I 0 2 = exp ( β C n 2 k 7 / 6 L 11 / 6 ) 1 ,
lim μ σ I 0 2 + 1 μ = lim μ g μ <
I n = E n E n * = k = 1 n m = 1 n E n k E n m exp [ j ( ψ n k ψ n m ) ] ,
I n = k = 1 n E n k 2 = n J ,
I = n = 1 α n I n = n = 1 n α n J = ( μ + 1 ) J .
J = 1 μ + 1 .
E n k 2 p = E n k 2 p ( σ i 0 2 + 1 ) p ( p 1 ) / 2 ,
I n k = E n k 2 ; g = σ i 0 2 + 1 , J = I n k ,
I n k p = I n k p g p ( p 1 ) / 2 = J p g p ( p 1 ) / 2 .
I n q = [ k = 1 n E n k exp ( j ψ n k ) ] q [ m = 1 n E n m exp ( j ψ n m ) ] q .
= { ( q ; k 1 , , k n ) [ E n 1 exp ( j ψ n 1 ) ] k 1 × [ E n n exp ( j ψ n n ) ] k n } × { ( q ; m 1 , , m n ) [ E n 1 exp ( j ψ n 1 ) ] m 1 × [ E n n exp ( j ψ n n ) ] m n } ;
= ( q ; k 1 , , k n ) 2 E n 1 2 k 1 E n n 2 k n , = ( q ; k 1 , , k n ) 2 I n 1 k 1 I n n k n .
G = g 1 / 2 = ( σ i 0 2 + 1 ) 1 / 2 ,
I n q = ( μ + 1 ) q G q ( q ; k 1 , , k n ) 2 G k 1 2 + + k n 2 ,
I q = n = 1 α n I n q = exp ( μ ) G q ( μ + 1 ) q n = 1 μ n 1 ( n 1 ) ! × ( q ; k 1 , , k n ) 2 G k 1 2 + + k n 2 ,
I n q = q ! { 1 + q ( q 1 ) 2 g 2 1 μ + 1 + O [ 1 ( μ + 1 ) 2 ] } .

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