Abstract

A new probability-density function (PDF) is proposed for irradiance scintillations in the case of strong scintillation (i.e., irradiance variance decreases with further increases in path-averaged refractive-index turbulence). This new PDF is named the log-normally modulated exponential PDF. This PDF is compared with experimental PDF’s of both irradiance and photon counts obtained from atmospheric laser propagation; the agreement is excellent (superior to that of the K PDF). Receiver–aperture-averaged irradiance is shown to be log normal for sufficiently large apertures. Comparison of moments of measured irradiance with theoretical moments shows that the moment comparison method of testing irradiance statistics can be very misleading because of the limitations of receivers and the sensitivity of high-order moments to very large irradiances.

© 1987 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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1986

1985

1984

1982

1981

1979

1978

1977

A. S. Gurvich, M. A. Kallistratova, F. E. Martvell, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 20, 1020–1031 (1977).

1975

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

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvilli, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

1974

K. S. Gochelashvilli, V. I. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

T. Wang, J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,”J. Opt. Soc. Am. 64, 994–999 (1974).
[CrossRef]

Andrews, L. C.

Applebaum, G.

Azar, Z.

Z. Azar, H. M. Loebenstein, G. Applebaum, E. Azoulay, U. Halavee, M. Tamir, M. Tur, “Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence,” Appl. Opt. 24, 2401–2407 (1985).
[CrossRef] [PubMed]

E. Azoulay, Z. Azar, M. Tur, “Aperture-averaged statistics of laser intensity fluctuations in strong turbulence,” in Proceedings of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, Italy, 1986), pp. 211–214.

Azoulay, E.

Z. Azar, H. M. Loebenstein, G. Applebaum, E. Azoulay, U. Halavee, M. Tamir, M. Tur, “Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence,” Appl. Opt. 24, 2401–2407 (1985).
[CrossRef] [PubMed]

E. Azoulay, Z. Azar, M. Tur, “Aperture-averaged statistics of laser intensity fluctuations in strong turbulence,” in Proceedings of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, Italy, 1986), pp. 211–214.

Barakat, R.

Bissonnette, L. R.

Briccolani, E.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvilli, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Churnside, J. H.

R. J. Hill, J. H. Churnside, “Measured statistics of optical scintillation in strong refractive turbulence relevant to laser eye safety,” submitted to Health Phys.

Clifford, S. F.

Conforti, G.

Consortini, A.

Dashen, R.

Elliott, R. A.

Ewart, T. E.

C. Macaskill, T. E. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,”J. Acoust. Soc. Am. 76, 1466–1773 (1984).
[CrossRef]

Furutsu, K.

Gochelashvilli, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvilli, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

K. S. Gochelashvilli, V. I. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, 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. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

Gurvich, A. S.

A. S. Gurvich, M. A. Kallistratova, F. E. Martvell, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 20, 1020–1031 (1977).

M. E. Gracheva, A. S. Gurvich, 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. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

Halavee, U.

Hill, R. J.

Ito, S.

Kallistratova, M. A.

A. S. Gurvich, M. A. Kallistratova, F. E. Martvell, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 20, 1020–1031 (1977).

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, 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. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

Kerr, J. R.

Loebenstein, H. M.

Macaskill, C.

C. Macaskill, T. E. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,”J. Acoust. Soc. Am. 76, 1466–1773 (1984).
[CrossRef]

Majumdar, A. K.

Martvell, F. E.

A. S. Gurvich, M. A. Kallistratova, F. E. Martvell, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 20, 1020–1031 (1977).

Ochs, G. R.

Parry, G.

G. Parry, “Measurement 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]

Phillips, R. L.

Pincus, P. A.

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, 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. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvilli, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Pusey, P. N.

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvilli, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

K. S. Gochelashvilli, V. I. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Speck, J. P.

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

Strohbehn, J. W.

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

T. Wang, J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,”J. Opt. Soc. Am. 64, 994–999 (1974).
[CrossRef]

Tamir, M.

Tur, M.

Z. Azar, H. M. Loebenstein, G. Applebaum, E. Azoulay, U. Halavee, M. Tamir, M. Tur, “Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence,” Appl. Opt. 24, 2401–2407 (1985).
[CrossRef] [PubMed]

E. Azoulay, Z. Azar, M. Tur, “Aperture-averaged statistics of laser intensity fluctuations in strong turbulence,” in Proceedings of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, Italy, 1986), pp. 211–214.

Wang, T.

Wizinowich, P. L.

Appl. Opt.

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

A. S. Gurvich, M. A. Kallistratova, F. E. Martvell, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 20, 1020–1031 (1977).

J. Acoust. Soc. Am.

C. Macaskill, T. E. Ewart, “The probability distribution of intensity for acoustic propagation in a randomly varying ocean,”J. Acoust. Soc. Am. 76, 1466–1773 (1984).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Lett.

Proc. IEEE

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvilli, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Radio Sci.

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

Sov. Phys. JETP

K. S. Gochelashvilli, V. I. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Other

R. J. Hill, J. H. Churnside, “Measured statistics of optical scintillation in strong refractive turbulence relevant to laser eye safety,” submitted to Health Phys.

E. Azoulay, Z. Azar, M. Tur, “Aperture-averaged statistics of laser intensity fluctuations in strong turbulence,” in Proceedings of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, Italy, 1986), pp. 211–214.

M. E. Gracheva, A. S. Gurvich, 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. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Configuration of the experiment.

Fig. 2
Fig. 2

PDF’s of normalized irradiance for σI2 = 2.83 (β02 = 36): K (— — —), log-normal (- - -), and log-normally modulated exponential (—) functions and data (○). (a) Logarithmic representation; (b) linear representation for I < 2.

Fig. 3
Fig. 3

PDF’s of normalized irradiance for σI2 = 4.13 (β02 = 23): K (— — —), log-normal (- - -), and log-normally modulated exponential (—) functions and data (○). (a) Logarithmic representation; (b) linear representation for I < 2.

Fig. 4
Fig. 4

PDF’s of normalized irradiance through a 25-mm aperture: log normal (—) and data (○). σI2 = 2.45.

Fig. 5
Fig. 5

PDF’s of photon counts: K (— — —), dead-time-corrected K (- - -), and log-normally modulated exponential (—) functions and data from Fig. 7a of Ref. 17 (○).

Fig. 6
Fig. 6

PDF’s of photon counts: K (— — —), dead-time-corrected K (- - -), and log-normally modulated exponential (—) functions and data from Fig. 7b of Ref. 17 (○).

Fig. 7
Fig. 7

Third, fourth, and fifth normalized moments of irradiance I3, I4, and I5, respectively, as functions of second moment I2: K (— — —), log-normal(- - -) and log-normally modulated exponential (—) functions and data (○).

Fig. 8
Fig. 8

Second-, third-, fourth-, and fifth-moment integrands (—) and the irradiance PDF (- - -) for a log-normally modulated exponential function with σI2 = 3.5.

Fig. 9
Fig. 9

Third, fourth, and fifth normalized moments of irradiance (I3, I4, I5) as functions of the second moment I2 for a PDF truncated at I = 40: K (— — —), log-normal (- - -), and log-normally modulated exponential (—) functions and data (○).

Equations (18)

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

p ( I ) = 0 p ( I z ) p ( z ) d z ,
p ( I ) = 1 2 π σ z 0 d z z 2 exp [ - I z - ( ln z + ½ σ z 2 ) 2 2 σ z 2 ]
p ( I ) z 0 ( 1 + σ z 2 I z 0 ) - 1 / 2 exp [ - I z 0 ( 1 + σ z 2 2 I z 0 ) ] ,
ln z 0 = σ z 2 2 - σ z 2 I z 0 .
σ I 2 = 2 exp ( σ z 2 ) - 1 ,
p ( n ) = 0 d I ( γ I ) n exp ( - γ I ) n ! p ( I ) ,
γ = A T h ν ,
p ( n ) = 1 2 π σ z n ¯ 0 d z z 2 ( 1 + 1 n ¯ z ) - n - 1 exp [ - ( ln z + ½ σ z 2 ) 2 2 σ z 2 ] ,
p ( n ) = Γ ( M + n ) Γ ( M ) ( M / n ¯ ) M U ( M + n , M , M / n ¯ ) ,
I n = I n .
I n = n ! exp [ ½ n ( n - 1 ) σ z 2 ] .
I n = n ! ( I 2 / 2 ) n ( n - 1 ) / 2 ,
I n = n ! Γ ( 2 I 2 - 2 + n ) ( 2 I 2 - 2 ) n Γ ( 2 I 2 - 2 ) ,
I n = I 2 n ( n - 1 ) / 2 .
I n = 0 I n p ( I ) d I .
σ χ 2 = ¼ ln I 2 ,
σ χ 2 = π 2 24 + 1 4 n = 0 ( n + 2 I 2 - 2 ) - 2 .
σ χ 2 = π 2 24 + 1 4 ln ( I 2 / 2 ) .

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