Abstract

A new propagation model is developed for the intensity fluctuations of a laser beam propagating through extended clear-air turbulence. The field of the optical wave is modeled as the sum of a coherent (deterministic) component and a random component, the intensity of which is assumed governed by the generalized n distribution of Nakagami. We further assume that the statistics are inherently nonstationary by treating the average intensity of the random portion of the field as a fluctuating quantity. The resulting unconditional IK distribution for the intensity fluctuations is a generalized form of the K distribution that is applicable to all conditions of atmospheric turbulence for which data have been obtained, including weak turbulence for which the K distribution is not theoretically applicable.

© 1985 Optical Society of America

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References

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  1. V. T. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961).
  2. D. A. DeWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
    [CrossRef]
  3. J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
    [CrossRef]
  4. P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,”J. Phys. A 11, 403–422 (1978).
    [CrossRef]
  5. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  6. 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]
  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. G. C. Valley, W. P. Brown, “Intensity statistics for propagation through a turbulent layer,” Appl. Opt. 21, 3002–3004 (1982).
    [CrossRef] [PubMed]
  9. L. R. Bissonnette, “Propagation model of laser beams in turbulence,”J. Opt. Soc. Am. 73, 262–268 (1983).
    [CrossRef]
  10. M. Tur, M. J. Beran, “Wave propagation in random media: a comparison of two theories,”J. Opt. Soc. Am. 73, 1343–1349 (1983).
    [CrossRef]
  11. K. S. Gochelashvili, V. I. Shishov, “Strong fluctuations of laser radiation intensity in a turbulent atmosphere—the distribution function,” Sov. Phys. JETP 47(6), 1028–1030 (1978).
  12. M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” reprint from Statistical Methods of Radio Wave Propagation (Pergamon, Oxford, 1960).
  13. D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
    [CrossRef]
  14. R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,”J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  15. 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, 675–686 (1981).
    [CrossRef]
  16. K. S. Miller, Multidimensional Gaussian Distributions (Wiley, New York, 1964).
  17. J. I. Marcum, P. Swerling, “Studies of target detection by pulsed radar,”IRE Trans. Inform. Theory IT-6, 59–308 (1960).
    [CrossRef]
  18. I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).
  19. R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 10, 110–112 (1984).
    [CrossRef]

1984 (1)

1983 (3)

1982 (3)

1981 (2)

1979 (1)

1978 (4)

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,”J. Phys. A 11, 403–422 (1978).
[CrossRef]

K. S. Gochelashvili, V. I. Shishov, “Strong fluctuations of laser radiation intensity in a turbulent atmosphere—the distribution function,” Sov. Phys. JETP 47(6), 1028–1030 (1978).

I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).

1974 (1)

D. A. DeWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
[CrossRef]

1960 (1)

J. I. Marcum, P. Swerling, “Studies of target detection by pulsed radar,”IRE Trans. Inform. Theory IT-6, 59–308 (1960).
[CrossRef]

Andrews, L. C.

Beran, M. J.

Bissonnette, L. R.

Brown, W. P.

Clifford, S. F.

Dashen, R.

DeWolf, D. A.

D. A. DeWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
[CrossRef]

Elbaum, M.

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

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(6), 1028–1030 (1978).

Hill, R. J.

Ito, S.

Jao, J. K.

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[CrossRef]

Lewinski, D. J.

D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
[CrossRef]

Marcum, J. I.

J. I. Marcum, P. Swerling, “Studies of target detection by pulsed radar,”IRE Trans. Inform. Theory IT-6, 59–308 (1960).
[CrossRef]

Miller, K. S.

K. S. Miller, Multidimensional Gaussian Distributions (Wiley, New York, 1964).

Miller, P. F.

P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,”J. Phys. A 11, 403–422 (1978).
[CrossRef]

Nakagami, M.

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” reprint from Statistical Methods of Radio Wave Propagation (Pergamon, Oxford, 1960).

Parry, G.

Phillips, R. L.

Pusey, P. N.

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(6), 1028–1030 (1978).

Swerling, P.

J. I. Marcum, P. Swerling, “Studies of target detection by pulsed radar,”IRE Trans. Inform. Theory IT-6, 59–308 (1960).
[CrossRef]

Tatarski, V. T.

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

Tur, M.

Valley, G. C.

Yakushkin, I. G.

I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
[CrossRef]

IRE Trans. Inform. Theory (1)

J. I. Marcum, P. Swerling, “Studies of target detection by pulsed radar,”IRE Trans. Inform. Theory IT-6, 59–308 (1960).
[CrossRef]

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

I. G. Yakushkin, “Moments of field propagating in randomly inhomogeneous medium in the limit of saturated fluctuations,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1194–1201 (1978).

J. Opt. Soc. Am. (7)

J. Phys. A (1)

P. F. Miller, “The probability distribution of a wave at a very large depth within an extended region,”J. Phys. A 11, 403–422 (1978).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

D. A. DeWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
[CrossRef]

J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
[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(6), 1028–1030 (1978).

Other (3)

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” reprint from Statistical Methods of Radio Wave Propagation (Pergamon, Oxford, 1960).

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

K. S. Miller, Multidimensional Gaussian Distributions (Wiley, New York, 1964).

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Equations (17)

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U ( t ) = e i ω t k = 1 N v k ( t ) exp [ i ψ k ( t ) ] ,
p 1 ( I ) = N b ( I A ) N - 1 exp [ - N ( A 2 + I ) / b ] × I N - 1 ( 2 N A b I ) ,             I > 0 ,
p 2 ( b ) = 1 b 0 exp ( - b / b 0 ) ,             b > 0 ,
p 1 ( I b ) = α b ( I A ) α - 1 exp [ - α ( A 2 + I ) / b ] × I α - 1 ( 2 α A b I ) ,             I > 0.
p ( I ) = 0 p 1 ( I b ) p 2 ( b ) d b ,
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 ,
I n = 0 I n 0 p 1 ( I b ) p 2 ( b ) d b d I = 0 p 2 ( b ) 0 I n p 1 ( I b ) d I d b ,             n = 1 , 2 , 3 , ,
0 I n p 1 ( I b ) d I = ( b α ) n n ! L n ( α - 1 ) ( - α A 2 b ) ,
I n = ( b 0 α ) n n ! k = 0 n Γ ( α + n ) Γ ( α + k ) ( α ρ ) k k ! ,
ρ = A 2 / b 0
I n I n = n ! α n ( 1 + ρ ) n k = 0 n Γ ( α + n ) Γ ( α + k ) ( α ρ ) k k ! ,             n = 1 , 2 , 3 , .
p ( y ) = { 2 α ( 1 + ρ ) [ ( 1 + ρ ) y ρ ] ( α - 1 ) / 2 K α - 1 ( 2 α ρ ) × I α - 1 { 2 [ α y ( 1 + ρ ) ] 1 / 2 } , y < ρ 1 + ρ 2 α ( 1 + ρ ) [ ( 1 + ρ ) y ρ ] ( α - 1 ) / 2 I α - 1 ( 2 α ρ ) × K α - 1 { 2 [ α y ( 1 + ρ ) ] 1 / 2 } , y > ρ 1 + ρ
F ( x ) = 0 x p ( y ) d y ,
F ( x ) = { 2 α ρ [ ( 1 + ρ ) x ρ ] α / 2 K α - 1 ( 2 α ρ ) × I α { 2 [ α x ( 1 + ρ ) ] 1 / 2 } , x < ρ 1 + ρ . 1 - 2 α ρ [ ( 1 + ρ ) x ρ ] α / 2 I α - 1 ( 2 α ρ ) × K α { 2 [ α x ( 1 + ρ ) ] 1 / 2 } , x > ρ 1 + ρ
lim α ρ I n I n = 1
lim α ρ 0 I n I n = n ! .
I n I n ~ n ! Γ ( α + n ) α n Γ ( α ) ( 1 + ρ + ) ,

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