Abstract

Application of the central limit theorem to the stochastic equation of propagation suggests that the probability distribution of the complex wave amplitude defined on the geometrical phase front is approximately normal. The resulting irradiance probability density function, valid in the strong scintillation regime, is an exponential multiplied by the modified Bessel function I0 both of argument proportional to the irradiance; it is not the Rice-Nakagami density function. Quantitative tests show that this exponential-Bessel function constitutes as good a fit as the log-normal to the irradiance probability data reported in this paper. Since the normal distribution hypothesis is consistent with the stochastic wave equation, the model proposed here should be a simple substitute to the often used but theoretically incorrect log-normal irradiance probability distribution model.

© 1979 Optical Society of America

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References

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  1. T.-i. Wang, J. W. Strohbehn, J. Opt. Soc. Am. 64, 583 (1974).
    [CrossRef]
  2. L. R. Bissonnette, “Log-Normal Probability Distribution of Strong Irradiance Fluctuations: an Asymptotic Analysis,” NATO-AGARD Conference Proceedings 183 on Optical Propagation in the Atmosphere (National Technical Information Service, Springfield, Va., 1975), paper 19.
  3. L. R. Bissonnette, “Modelling of Laser Beam Propagation in Atmospheric Turbulence,” presented at the Second International Symposium on Gas-Flow and Chemical Lasers, von Karman Institute, Rhode-Saint-Genèse, Belgium, 11–15 September 1978; Proceedings to appear in 1979. (Hemisphere Publishing Corporation, Washington, D.C.)
  4. J. W. Strohbehn, T.-i. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
    [CrossRef]
  5. R. A. Elliot, J. R. Dunphy, J. R. Kerr, in Digest of Topical Meeting on Optical Propagation Through Turbulence, Rain and Fog (Optical Society of America, Washington, D.C., 1977), paper WA5.
  6. L. R. Bissonnette, Appl. Opt. 16, 2242 (1977).
    [CrossRef] [PubMed]
  7. B. W. Lindgren, G. W. McElrath, Introduction to Probability and Statistics (Macmillan, New York, 1963).
  8. T.-i. Wang, J. W. Strohbehn, J. Opt. Soc. Am. 64, 994 (1974).
    [CrossRef]
  9. F. Davidson, A. Gonzalez-del-Valle, J. Opt. Soc. Am. 65, 655 (1975).
    [CrossRef]

1977 (1)

1975 (2)

J. W. Strohbehn, T.-i. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

F. Davidson, A. Gonzalez-del-Valle, J. Opt. Soc. Am. 65, 655 (1975).
[CrossRef]

1974 (2)

Bissonnette, L. R.

L. R. Bissonnette, Appl. Opt. 16, 2242 (1977).
[CrossRef] [PubMed]

L. R. Bissonnette, “Log-Normal Probability Distribution of Strong Irradiance Fluctuations: an Asymptotic Analysis,” NATO-AGARD Conference Proceedings 183 on Optical Propagation in the Atmosphere (National Technical Information Service, Springfield, Va., 1975), paper 19.

L. R. Bissonnette, “Modelling of Laser Beam Propagation in Atmospheric Turbulence,” presented at the Second International Symposium on Gas-Flow and Chemical Lasers, von Karman Institute, Rhode-Saint-Genèse, Belgium, 11–15 September 1978; Proceedings to appear in 1979. (Hemisphere Publishing Corporation, Washington, D.C.)

Davidson, F.

Dunphy, J. R.

R. A. Elliot, J. R. Dunphy, J. R. Kerr, in Digest of Topical Meeting on Optical Propagation Through Turbulence, Rain and Fog (Optical Society of America, Washington, D.C., 1977), paper WA5.

Elliot, R. A.

R. A. Elliot, J. R. Dunphy, J. R. Kerr, in Digest of Topical Meeting on Optical Propagation Through Turbulence, Rain and Fog (Optical Society of America, Washington, D.C., 1977), paper WA5.

Gonzalez-del-Valle, A.

Kerr, J. R.

R. A. Elliot, J. R. Dunphy, J. R. Kerr, in Digest of Topical Meeting on Optical Propagation Through Turbulence, Rain and Fog (Optical Society of America, Washington, D.C., 1977), paper WA5.

Lindgren, B. W.

B. W. Lindgren, G. W. McElrath, Introduction to Probability and Statistics (Macmillan, New York, 1963).

McElrath, G. W.

B. W. Lindgren, G. W. McElrath, Introduction to Probability and Statistics (Macmillan, New York, 1963).

Speck, J. P.

J. W. Strohbehn, T.-i. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

Strohbehn, J. W.

Wang, T.-i.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Radio Sci. (1)

J. W. Strohbehn, T.-i. Wang, J. P. Speck, Radio Sci. 10, 59 (1975).
[CrossRef]

Other (4)

R. A. Elliot, J. R. Dunphy, J. R. Kerr, in Digest of Topical Meeting on Optical Propagation Through Turbulence, Rain and Fog (Optical Society of America, Washington, D.C., 1977), paper WA5.

B. W. Lindgren, G. W. McElrath, Introduction to Probability and Statistics (Macmillan, New York, 1963).

L. R. Bissonnette, “Log-Normal Probability Distribution of Strong Irradiance Fluctuations: an Asymptotic Analysis,” NATO-AGARD Conference Proceedings 183 on Optical Propagation in the Atmosphere (National Technical Information Service, Springfield, Va., 1975), paper 19.

L. R. Bissonnette, “Modelling of Laser Beam Propagation in Atmospheric Turbulence,” presented at the Second International Symposium on Gas-Flow and Chemical Lasers, von Karman Institute, Rhode-Saint-Genèse, Belgium, 11–15 September 1978; Proceedings to appear in 1979. (Hemisphere Publishing Corporation, Washington, D.C.)

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Figures (4)

Fig. 1
Fig. 1

Comparison between the experimental ○, the exponential-Bessel —, and the log-normal - - - irradiance probability density functions; runs 37–41. η is the normalized irradiance and z/zA, the normalized propagation distance.

Fig. 2
Fig. 2

Comparison between the experimental ○, the exponential-Bessel —, and the log-normal - - - irradiance probability density functions; runs 42–45 and 50–51. η and z/zA, as in Fig. 1.

Fig. 3
Fig. 3

Comparison between the experimental ○, the exponential-Bessel —, and the log-normal - - - irradiance probability density functions; runs 22, 25, 26, and 29. η and z/zA, as in Fig. 1.

Fig. 4
Fig. 4

Comparison between the experimental ○, the exponential-Bessel —, and the log-normal - - - irradiance probability density functions; runs 178–183. η and z/zA, as in Fig. 1.

Tables (3)

Tables Icon

Table I Results of the Chi-Square (χ2) and Kolmogorov-Smirnov (D) Tests and Standard Deviation Errors of Probability Density (fE) and Distribution (FE) Functions for the Exponential-Bessel (Subscript eb) and Log-Normal (subcript ln) Models; runs 37–45 and 50–51a

Tables Icon

Table II Results of the Chi-Square and Kolmogorov-Smirnov Tests and Standard Deviation Errors of Probability Density and Distribution Functions for the Exponential-Bessel and Log-Normal Models; runs 22, 25, 26, 29, and 178–183 a

Tables Icon

Table III Comparative Values of Chi-Square(χ2), Kolmogorov-Smirnov Statistic (D), and Standard Deviation Errors of Irradiance Density (fE) and Distribution(FE) Functions for the Tested Hypotheses of Normal and Log-Normal Probability Distribution of the Complex wave Amplitude Defined on the Geometric Phase Front; the Numbers Listed are Averages over All Runs.

Equations (39)

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E = A exp [ i k ( z + ϕ ) - i ω t ] ,
( z + V · ) V = ( N - n 0 ) / n 0 ,
( z + V · ) A + ½ A · V - i 2 k 2 A = 0 ,
V = ϕ .
V = V + v ; v = 0 ,
A = A + a ; a = 0 ,
V = r / ( z - F ) .
a z + r ( z - F ) · a + a ( z - F ) - i 2 k 2 a = g ( z , r ) ,
g ( z , r ) = - v · A - ½ A · v - ½ ( a · v - a · v ) - ( v · a - v · a ) .
a ( 0 , r ) = lim r a ( z , r ) = 0
a ( z , r ) = i k 2 π 0 z d u ( u - z ) - d 2 s g ( u , s ) × exp [ i k 2 ( z - u ) F - u F - z | r - F - z F - u s | 2 ] .
a ( z , r ) 0 z d u ( - v · A - ½ A · v ) ,
ln a ( z , r ) ~ - ½ 0 z d u · v .
A = x 0 + i y 0 ,
a = x + i y ,
f ( x , y ) = 1 2 π σ x σ y ( 1 - ρ 2 ) 1 / 2 × exp [ - 1 2 ( 1 - ρ 2 ) ( x 2 σ x 2 - 2 ρ x y σ x σ y + y 2 σ y 2 ) ] ,
σ x 2 = x 2 ,
σ y 2 = y 2 ,
ρ = x y / σ x σ y .
f ( η ) = 1 2 π γ δ exp ( - γ δ 2 η - η 0 ) × 0 2 π d θ exp [ - γ δ ( δ 2 - 1 ) 1 / 2 η cos ( 2 θ + θ 1 ) + C η 1 / 2 sin ( θ + θ 2 ) ] ,
η = I / I = A A * / A A * .
γ = I σ x 2 + σ y 2 ,
δ = σ x 2 + σ y 2 2 σ x σ y ( 1 - ρ 2 ) 1 / 2 ,
η 0 = 1 2 ( 1 - ρ 2 ) ( x 0 2 σ x 2 - 2 ρ x 0 y 0 σ x σ y + y 0 2 σ y 2 ) ,
C 2 = I σ x 2 σ y 2 ( 1 - ρ 2 ) 2 [ ( ρ 2 + σ y 2 σ x 2 ) x 0 2 + ( ρ 2 + σ x 2 σ y 2 ) y 0 2 - 2 ρ x 0 y 0 ( σ x 2 + σ y 2 ) σ x σ y ] ,
tan θ 1 = 2 ρ σ x σ y ( σ y 2 - σ x 2 ) ,
tan θ 2 = σ y 2 x 0 - ρ σ x σ y y 0 σ x 2 y 0 - ρ σ x σ y x 0 .
f r ( η ) = I σ 2 exp [ - I σ 2 η - ( x 0 2 + y 0 2 ) σ 2 ] × I 0 [ 2 I 1 / 2 ( x 0 2 + y 0 2 ) 1 / 2 η 1 / 2 σ 2 ] ,
f e b ( η ) = δ exp ( - δ 2 η ) × I 0 [ δ ( δ 2 - 1 ) 1 / 2 η ] .
f ln ( η ) = [ η σ ln I ( 2 π ) 1 / 2 ] - 1 exp [ - ( ln η - ln η ) 2 2 σ ln I 2 ] ,
f i = ( F i + 1 - F i ) / Δ U ,
δ = 1 / ( 3 - η 2 ) 1 / 2 ,
ln η = - 1 2 σ ln I 2 ,
σ ln I 2 = ln η 2 .
χ t 2 = i = 1 m M [ p i - p t ( η i ) ] 2 p t ( η i )
p t ( η i ) = η i η i + 1 d η f t ( η )
D t = max - < η i < F i - F t ( η i ) ,
E f t = { 1 m i = 1 m [ f i - f t ( η i ) ] 2 } 1 / 2 ,
E F t = { 1 m i = 1 m [ F i - F t ( η i ) ] 2 } 1 / 2 .

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