Abstract

The recently developed IK distribution and its connection with other distributions is examined in the context of optical waves scattered by a turbulent medium. In particular, it is shown how the IK distribution, K distribution, and homodyned K distribution all evolve as marginal density functions from compound or doubly stochastic Gaussian optical fields. In this setting the genesis of the IK distribution is very similar to that of the homodyned K distribution, and there is one special case (α = 1) in which the two distributions are identical. Also, both the homodyned K and the IK distribution are shown to reduce to the K distribution in strong-turbulence regimes. The IK distribution is further examined as a model for the irradiance when the distribution parameter α is restricted to half-integer values, leading to simpler forms of the distribution. The special case of the IK distribution corresponding to α = 1/2 shows a particularly good fit with experimental data over a wide range of conditions of atmospheric turbulence.

© 1986 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman (McGraw-Hill, New York, 1961).
  2. J. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
    [Crossref]
  3. D. A. DeWolf, “Waves in turbulent air: a phenomenological model,” Proc. IEEE 62, 1523–1529 (1974).
    [Crossref]
  4. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [Crossref]
  5. J. K. Jao, M. Elbaum, “First-order statistics of a non-Rayleigh fading signal and its detection,” Proc. IEEE 66, 781–789 (1978).
    [Crossref]
  6. 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]
  7. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [Crossref]
  8. 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]
  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, 760–764 (1982).
    [Crossref]
  10. L. R. Bissonnette, “Propagation model of laser beams in turbulence,” J. Opt. Soc. Am. 73, 262–268 (1983).
    [Crossref]
  11. M. Tur, M. J. Beran, “Wave propagation in random media: a comparison of two theories,” J. Opt. Soc. Am. 73, 1343–1349 (1983).
    [Crossref]
  12. 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]
  13. 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).
  14. R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 10, 110–112 (1984).
    [Crossref]
  15. R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [Crossref]
  16. E. Jakeman, P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [Crossref]
  17. G. Parry, “Measurements of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [Crossref]
  18. E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [Crossref]
  19. A. K. Majumdar, “Higher-order statistics of laser-irradiance fluctuations due to turbulence,” J. Opt. Soc. Am. A 1, 1067–1074 (1984).
    [Crossref]
  20. 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).
  21. D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,” IEEE Trans. Antennas Prop. AP-31, 490–498 (1983).
    [Crossref]
  22. E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980).
    [Crossref]
  23. D. Link, “Phase statistics for a laser beam propagating through atmospheric turbulence,” Ph.D. dissertation (University of Central Florida, Orlando, Florida, 1985).
  24. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–210 (1977).
    [Crossref]
  25. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  26. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians (Macmillan, New York, 1985).

1985 (1)

1984 (2)

1983 (3)

1982 (2)

1981 (2)

R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
[Crossref]

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

1980 (1)

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

1979 (1)

1978 (4)

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).

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]

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

1977 (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–210 (1977).
[Crossref]

1976 (1)

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

1974 (1)

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

Andrews, L. C.

Beran, M. J.

Bissonnette, L. R.

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).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Ito, S.

Jakeman, E.

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

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

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–210 (1977).
[Crossref]

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

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980).
[Crossref]

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 Prop. AP-31, 490–498 (1983).
[Crossref]

Link, D.

D. Link, “Phase statistics for a laser beam propagating through atmospheric turbulence,” Ph.D. dissertation (University of Central Florida, Orlando, Florida, 1985).

Majumdar, A. K.

McWhirter, J. G.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–210 (1977).
[Crossref]

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.

G. Parry, “Measurements 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, 195–210 (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]

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

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–210 (1977).
[Crossref]

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

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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).

Tatarski, V. I.

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

Tur, M.

IEEE Trans. Antennas Prop. (1)

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

IEEE Trans. Antennas Propag. (1)

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

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (2)

J. Phys. A (2)

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]

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

Opt. Acta (1)

G. Parry, “Measurements 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, 195–210 (1977).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

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

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 (7)

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

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

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).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980).
[Crossref]

D. Link, “Phase statistics for a laser beam propagating through atmospheric turbulence,” Ph.D. dissertation (University of Central Florida, Orlando, Florida, 1985).

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Figures (3)

Fig. 1
Fig. 1

Plots of the normalized IK distribution for the cases α = 1 and ρ = 0, 1, 4.

Fig. 2
Fig. 2

Normalized moments of the IK distribution for the cases α = 1/2, 3/2, 9/2. The dashed curves represent the theoretical curves of the K distribution.

Fig. 3
Fig. 3

Comparison of experimental values of the normalized moments of irradiance fluctuations observed by Parry and Pusey7 with corresponding theoretical vlaues of the IK distribution with α = 1/2.

Equations (31)

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U ( t ) = e i ω t [ A e i θ ( t ) + R ( t ) e i Φ ( t ) ] ,
R ( t ) e i Φ ( t ) = j = 1 N r j ( t ) e i ϕ j ( t ) ,
p 1 ( I ) = 1 b exp [ ( A 2 + I ) / b ] I 0 ( 2 A b I ) , I > 0 ,
b = R 2 = j = 1 N r j 2 .
p 2 ( I | b ) = 1 b exp [ ( A 2 + I ) / b ] I 0 ( 2 A b I ) , I > 0
p 4 ( I ) = 0 p 2 ( I | b ) p 3 ( b ) d b , I > 0 ,
p 3 ( b ) = α ( α b / b 0 ) α 1 Γ ( α ) b 0 exp [ α b / b 0 ] , b > 0 ,
p 4 ( I ) = α ( α / b 0 ) α 1 Γ ( α ) b 0 × 0 b α 2 exp [ α b / b 0 ( A 2 + I ) / b ] I 0 ( 2 A b I ) d b ,
p 5 ( I ) = 2 Γ ( α ) b 0 ( α I b 0 ) ( α 1 ) / 2 K α 1 ( 2 α I b 0 ) , I > 0 ,
p 6 ( I ) = { 2 b 0 K 0 ( 2 A / b 0 ) I 0 ( 2 I / b 0 ) , I < A 2 2 b 0 I 0 ( 2 A / b 0 ) K 0 ( 2 I / b 0 ) , I > A 2 ,
I = | U ( t ) | 2 = A 2 + R 2 + 2 A R cos ( Φ θ ) .
C ( u ) = E [ e i u I ] = 0 π π e i u I p 8 ( R , Φ ) d Φ d R ,
p 7 ( R , Φ | b ) = R π b e R 2 / b ,
p 8 ( R , Φ ) = 0 p 7 ( R , Φ | b ) p 3 ( b ) d b = α R ( α / b 0 ) α 1 Γ ( α ) π b 0 0 b α 2 exp [ α b / b 0 R 2 / b ] d b ,
p 8 ( R , Φ ) = R 2 α 1 Γ ( α ) π b 0 0 ( α x ) α exp [ x / b 0 α R 2 / x ] d x .
C ( u ) = 1 b 0 0 ( α x ) α exp [ x / b 0 α A 2 / x ] ( α x i u ) α × exp ( α 2 A 2 / x 2 α x i u ) F 1 ( 1 α ; 1 ; A 2 u 2 α x i u ) d x ,
p 9 ( I ) = 1 2 π e i u I C ( u ) d u = α b 0 ( I A ) α 1 0 exp [ x / b 0 α ( A 2 + I ) / x ] × I α 1 ( 2 α A x I ) d x x ,
p 9 ( 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 .
p 9 ( I ) = 0 p 10 ( I | b ) p 11 ( b ) d b ,
p 10 ( 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 11 ( b ) = 1 b 0 e b / b 0 , 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 , ,
I n I n = n ! α n ( 1 + ρ ) n k = 0 n ( n k ) Γ ( α + n k ) Γ ( α ) ( α ρ ) k k ! , n = 1 , 2 , 3 , .
p 12 ( I ) = { 2 b 0 I exp ( A 2 b 0 ) cosh 2 I b 0 , I < A 2 2 b 0 I cosh ( A 2 b 0 ) exp ( 2 I b 0 ) , I > A 2 ,
I 1 2 ( x ) = 2 π x cosh x
K 1 2 ( x ) = π 2 x e x .
α = M + 1 2 , M = 1 , 2 , 3 ,
K M 1 2 ( x ) = π 2 x e x j = 0 M 1 ( M + j ) ! ( 2 x ) j j ! ( M j ) !
I M 1 2 ( x ) = 1 2 π x [ e x k = 0 M 1 ( 1 ) k ( M + k ) ! ( 2 x ) k k ! ( M k ) ! + ( 1 ) M e x k = 0 M 1 ( M + k ) ! ( 2 x ) k k ! ( M k ) ! ] ,
I < A 2 , p 12 ( I ) = I ( M 1 ) / 2 2 ( M + 1 2 ) b 0 exp ( 2 A M + 1 2 b 0 ) × j = 0 M 1 ( M + j ) ! j ! ( M j ) ! ( 4 A M + 1 2 b 0 ) j × [ exp ( 2 ( M + 1 2 ) I b 0 ) × k = 0 M 1 ( 1 ) k ( M + k ) ! k ! ( M k ) ! ( 4 ( M + 1 2 ) I b 0 ) k + ( 1 ) M exp ( 2 ( M + 1 2 ) I b 0 ) × k = 0 M 1 ( M + k ) ! k ! ( M k ) ! ( 4 ( M + 1 2 ) I b 0 ) k ]
I < A 2 , p 12 ( I ) = I ( M 1 ) / 2 2 ( M + 1 2 ) b 0 exp ( 2 ( M + 1 2 ) I b 0 ) × j = 0 M 1 ( M + j ) ! j ! ( M j ) ! ( 4 ( M + 1 2 ) I b 0 ) j × [ exp ( 2 A M + 1 2 b 0 ) × k = 0 M 1 ( 1 ) k ( M + k ) ! k ! ( M k ) ! ( 4 A M + 1 2 b 0 ) k + ( 1 ) M exp ( 2 A M + 1 2 b 0 ) × k = 0 M 1 ( M + k ) ! k ! ( M k ) ! ( 4 A M + 1 2 b 0 ) k ] .

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