Abstract

The probability density function (PDF) of the scattered intensity is expressed in terms of the corresponding lower-order moments, whether measured or derived theoretically, assuming that the probability density is unimodal. The sought-for PDF is given by a gamma PDF modulated by a series of generalized Laguerre polynomials; the expansion coefficients are expressed in terms of the scattered-intensity moments with the nth coefficient containing only the first n moments. The Laguerre PDF is not a statistical fit to the data; there are no free parameters in this expansion, and the answer depends directly on the first n moments. The log-intensity PDF is also obtained from the lower-order intensity moments. Laboratory scattering experiments that employed light scattering from a photorefractive crystal (a problem formally similar to the turbulent-atmosphere scattering problem) were performed. The first five scattered-intensity moments were measured as well as the intensity histogram; with the exception of the behavior in the immediate vicinity of the maximum, the Laguerre PDF is in good agreement with the experimental data. The Dashen–Flatté approach to the theoretically derived moments by means of path integrals is also discussed in the context of the Laguerre PDF.

© 1999 Optical Society of America

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References

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  1. S. Karp, R. Gagliardi, S. Moran, Optical Channels (Plenum, New York, 1988). Chapters 5–7 contain many pertinent references.
  2. E. Jakeman, P. Pusey, “Significance of the K-distribution in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  3. G. Parry, P. Pusey, “K-distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  4. R. Barakat, “Sums of independent lognormally distributed random variables,” J. Opt. Soc. Am. 66, 211–216 (1976).
    [CrossRef]
  5. R. Barakat, “Weak-scatter generalization of the K-density function with applications to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401–409 (1986).
    [CrossRef]
  6. E. Jakeman, R. Tough, “Generalized K-distribution: a statistical model of weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
    [CrossRef]
  7. V. Gudimetla, J. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213–1218 (1982).
    [CrossRef]
  8. R. Phillips, L. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
    [CrossRef]
  9. L. Bissonnette, P. Wizinowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979).
    [CrossRef] [PubMed]
  10. L. Andrews, R. 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]
  11. W. Strohbein, T. Wang, J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
    [CrossRef]
  12. J. Churnside, R. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  13. V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  14. D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
    [CrossRef]
  15. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  16. S. Flatté, D. Bernstein, R. Dashen, “Intensity moments by path integral techniques for wave propagation through random media with applications to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
    [CrossRef]
  17. R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 10, 110–112 (1984).
    [CrossRef]
  18. R. Dashen, “Asymptotic scheme for waves in random media,” Opt. Lett. 17, 91–93 (1991).
    [CrossRef]
  19. M. Charnotskii, J. Gozani, V. Tatarskii, V. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32.
  20. K. Furutsu, Random Media and Boundaries (Springer-Verlag, Berlin, 1993). See Chapter 7 for an excellent summary of Furutsu’s many papers.
  21. K. Furutsu, “On the probability distribution of irradiance in a turbulent medium,” J. Phys. Soc. Jap. 57, 1167–1172 (1988).
    [CrossRef]
  22. C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
    [CrossRef]
  23. R. Barakat, “Second-order statistics of integrated intensities and detected photons, the exact analysis,” J. Mod. Opt. 43, 1237–1252 (1996). Contains references to previous work and an overview of the expansion.
    [CrossRef]
  24. V. Romanovsky, “Generalization of some types of frequency curves of Professor Pearson,” Biometrika 16, 106–117 (1924).
    [CrossRef]
  25. M. Tiku, “Laguerre series forms of non-central χ2 and F distributions,” Biometrika 52, 415–427 (1965).
    [PubMed]
  26. I. Selin, Detection Theory (Princeton U. Press, Princeton, 1965), Chap. 2.
  27. A. Whalen, Detection of Signals in Noise (Academic, New York, 1971).
  28. P. Huber, “Robust statistics, a review,” Ann. Math. Stat. 43, 1043–1067 (1972).
  29. S. Clifford, R. 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]
  30. Z. Kopal, Numerical Analysis (Wiley, New York, 1955).
  31. P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1960).
  32. S. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975), Chaps. 7–9.
  33. J. Shohat, J. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1943).

1996 (2)

D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
[CrossRef]

R. Barakat, “Second-order statistics of integrated intensities and detected photons, the exact analysis,” J. Mod. Opt. 43, 1237–1252 (1996). Contains references to previous work and an overview of the expansion.
[CrossRef]

1991 (1)

1988 (1)

K. Furutsu, “On the probability distribution of irradiance in a turbulent medium,” J. Phys. Soc. Jap. 57, 1167–1172 (1988).
[CrossRef]

1987 (2)

1986 (1)

1985 (1)

1984 (2)

R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 10, 110–112 (1984).
[CrossRef]

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

1983 (1)

S. Flatté, D. Bernstein, R. Dashen, “Intensity moments by path integral techniques for wave propagation through random media with applications to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

1982 (2)

1981 (1)

1979 (3)

1978 (1)

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

1976 (1)

1975 (1)

W. Strohbein, T. Wang, J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

1972 (1)

P. Huber, “Robust statistics, a review,” Ann. Math. Stat. 43, 1043–1067 (1972).

1965 (1)

M. Tiku, “Laguerre series forms of non-central χ2 and F distributions,” Biometrika 52, 415–427 (1965).
[PubMed]

1924 (1)

V. Romanovsky, “Generalization of some types of frequency curves of Professor Pearson,” Biometrika 16, 106–117 (1924).
[CrossRef]

Andrews, L.

Barakat, R.

R. Barakat, “Second-order statistics of integrated intensities and detected photons, the exact analysis,” J. Mod. Opt. 43, 1237–1252 (1996). Contains references to previous work and an overview of the expansion.
[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, 401–409 (1986).
[CrossRef]

R. Barakat, “Sums of independent lognormally distributed random variables,” J. Opt. Soc. Am. 66, 211–216 (1976).
[CrossRef]

Bernstein, D.

S. Flatté, D. Bernstein, R. Dashen, “Intensity moments by path integral techniques for wave propagation through random media with applications to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

Bevington, P.

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1960).

Bissonnette, L.

Charnotskii, M.

M. Charnotskii, J. Gozani, V. Tatarskii, V. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32.

Churnside, J.

Clifford, S.

Craig, J.

D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
[CrossRef]

Dashen, R.

R. Dashen, “Asymptotic scheme for waves in random media,” Opt. Lett. 17, 91–93 (1991).
[CrossRef]

R. Dashen, “Distribution of intensity in a multiply scattering medium,” Opt. Lett. 10, 110–112 (1984).
[CrossRef]

S. Flatté, D. Bernstein, R. Dashen, “Intensity moments by path integral techniques for wave propagation through random media with applications to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

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

Ewart, T.

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

Flatté, S.

S. Flatté, D. Bernstein, R. Dashen, “Intensity moments by path integral techniques for wave propagation through random media with applications to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

Furutsu, K.

K. Furutsu, “On the probability distribution of irradiance in a turbulent medium,” J. Phys. Soc. Jap. 57, 1167–1172 (1988).
[CrossRef]

K. Furutsu, Random Media and Boundaries (Springer-Verlag, Berlin, 1993). See Chapter 7 for an excellent summary of Furutsu’s many papers.

Gagliardi, R.

S. Karp, R. Gagliardi, S. Moran, Optical Channels (Plenum, New York, 1988). Chapters 5–7 contain many pertinent references.

Gozani, J.

M. Charnotskii, J. Gozani, V. Tatarskii, V. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32.

Gudimetla, V.

Hill, R.

Holmes, J.

Huber, P.

P. Huber, “Robust statistics, a review,” Ann. Math. Stat. 43, 1043–1067 (1972).

Jakeman, E.

E. Jakeman, R. Tough, “Generalized K-distribution: a statistical model of weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
[CrossRef]

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

Karp, S.

S. Karp, R. Gagliardi, S. Moran, Optical Channels (Plenum, New York, 1988). Chapters 5–7 contain many pertinent references.

Kopal, Z.

Z. Kopal, Numerical Analysis (Wiley, New York, 1955).

Macaskill, C.

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

Meyer, S.

S. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975), Chaps. 7–9.

Moran, S.

S. Karp, R. Gagliardi, S. Moran, Optical Channels (Plenum, New York, 1988). Chapters 5–7 contain many pertinent references.

Mudge, D.

D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
[CrossRef]

Parry, G.

Phillips, R.

Pusey, P.

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

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

Romanovsky, V.

V. Romanovsky, “Generalization of some types of frequency curves of Professor Pearson,” Biometrika 16, 106–117 (1924).
[CrossRef]

Selin, I.

I. Selin, Detection Theory (Princeton U. Press, Princeton, 1965), Chap. 2.

Shohat, J.

J. Shohat, J. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1943).

Speck, J.

W. Strohbein, T. Wang, J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Strohbein, W.

W. Strohbein, T. Wang, J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Tamarkin, J.

J. Shohat, J. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1943).

Tatarskii, V.

M. Charnotskii, J. Gozani, V. Tatarskii, V. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32.

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Thomas, J.

D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
[CrossRef]

Tiku, M.

M. Tiku, “Laguerre series forms of non-central χ2 and F distributions,” Biometrika 52, 415–427 (1965).
[PubMed]

Tough, R.

Wang, T.

W. Strohbein, T. Wang, J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Wedd, A.

D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
[CrossRef]

Whalen, A.

A. Whalen, Detection of Signals in Noise (Academic, New York, 1971).

Wizinowich, P.

Zavorotny, V.

M. Charnotskii, J. Gozani, V. Tatarskii, V. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32.

Ann. Math. Stat. (1)

P. Huber, “Robust statistics, a review,” Ann. Math. Stat. 43, 1043–1067 (1972).

Appl. Opt. (1)

Biometrika (2)

V. Romanovsky, “Generalization of some types of frequency curves of Professor Pearson,” Biometrika 16, 106–117 (1924).
[CrossRef]

M. Tiku, “Laguerre series forms of non-central χ2 and F distributions,” Biometrika 52, 415–427 (1965).
[PubMed]

J. Acoust. Soc. Am. (1)

C. Macaskill, T. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,” J. Acoust. Soc. Am. 76, 1466–1473 (1984).
[CrossRef]

J. Math. Phys. (1)

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

J. Mod. Opt. (1)

R. Barakat, “Second-order statistics of integrated intensities and detected photons, the exact analysis,” J. Mod. Opt. 43, 1237–1252 (1996). Contains references to previous work and an overview of the expansion.
[CrossRef]

J. Opt. Soc. Am. (5)

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

J. Phys. Soc. Jap. (1)

K. Furutsu, “On the probability distribution of irradiance in a turbulent medium,” J. Phys. Soc. Jap. 57, 1167–1172 (1988).
[CrossRef]

Opt. Laser Technol. (1)

D. Mudge, A. Wedd, J. Craig, J. Thomas, “Statistical measurements of irradiance fluctuations produced by a reflective membrane optical scintillator,” Opt. Laser Technol. 28, 381–387 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Fluids (1)

S. Flatté, D. Bernstein, R. Dashen, “Intensity moments by path integral techniques for wave propagation through random media with applications to sound in the ocean,” Phys. Fluids 26, 1701–1713 (1983).
[CrossRef]

Phys. Rev. Lett. (1)

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

Radio Sci. (1)

W. Strohbein, T. Wang, J. Speck, “On the probability distribution of line-of-sight fluctuations for optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Other (10)

M. Charnotskii, J. Gozani, V. Tatarskii, V. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32.

K. Furutsu, Random Media and Boundaries (Springer-Verlag, Berlin, 1993). See Chapter 7 for an excellent summary of Furutsu’s many papers.

S. Karp, R. Gagliardi, S. Moran, Optical Channels (Plenum, New York, 1988). Chapters 5–7 contain many pertinent references.

I. Selin, Detection Theory (Princeton U. Press, Princeton, 1965), Chap. 2.

A. Whalen, Detection of Signals in Noise (Academic, New York, 1971).

Z. Kopal, Numerical Analysis (Wiley, New York, 1955).

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1960).

S. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975), Chaps. 7–9.

J. Shohat, J. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1943).

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

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Figures (2)

Fig. 1
Fig. 1

Circles, experimental values of W(I); curve, W(I) as determined from the first five measured intensity moments.

Fig. 2
Fig. 2

W(χ) as determined from the same intensity moments as used in Fig. 1.

Tables (1)

Tables Icon

Table 1 Values of the Measured Intensity Moments Il a

Equations (36)

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

2<h2h2<,
1<h2h2<.
W(h)=Wg(h)n=0Wn Ln(β-1)βhμ,
Wg(h)=1Γ(β) βμβhβ-1 exp(-βh/μ),
μ=h
β=h2h2-h2.
Ln(β-1)(x)=l=0nn+β-1n-l (-x)ll!.
Wn=n!Γ(β)l=0n (-β/μ)lhll!(n-l)!Γ(β+l),
W0=1,W1=W2=0,
W(h)=Wg(h)1+n=3WnLn(β-1)βhμ.
W(I)=Wg(I)1+n=3WnLn(β-1)(βI),
Wg(I)=1Γ(β) ββIβ-1 exp(-βI),
Wn=n!Γ(β)l=0n (-β)lIll!(n-l)!Γ(β+l).
0W(I)dI1,
0Wg(I)dI=1,
0Wg(I)Ln(β-1)(βI)dI=0,forn1.
χ=loghhlog I
W(χ)=ββΓ(β) exp(βχ)exp[-β exp(χ)]×1+n=3WnLn(β-1)[β exp(χ)],
χt=0(log I)tW(I)dI,t=0, 1, 2,,
=ββΓ(β) n=0WnRn(t)(β),
Rn(t)=0(log I)tIβ-1 exp(-βI)Ln(β-1)(βI)dI.
Rn(t)(β)=1ββ 0fn(t)(x)exp(-x)dx,
fn(t)(x)log(x/β)txβLn(β-1)(x).
Rn(t)(β)=1ββ k=1Nwk fn(t)(xk).
Rn(t)(β)=1ββ 0fn(t)(x)exp(-x)dx+1ββ fn(t)(x)exp(-x)dx,
1ββ 0fn(t)(x)exp(-x)dx1ββ fn(t)2exp(-/2).
Rn(β)=1ββ fn(t)2exp(-/2)+1ββ exp(-)k=1Nwkfn(t)(xk+).
Ir=0Ws(I)ItdI,
0Ws(I)dI=1.
αΩ/Φ,
In=n!1+12 n(n-1)Cα+0(α2),
W(I)=exp(-I)
β=(1+2Cα)-11-Cα;
mk=0IkW(I)dI
Δp=m0m1mpm1m2mp+1mpmp+1m2p>0,p=0, 1, ,
Ωp=m1m1mp+1m2m3mp+2mp+1mp+2m2p+1>0,p=0, 1, .

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