Abstract

The log-normally modulated Rician model for the probability-density function of optical irradiance fluctuations caused by refractive turbulence can be obtained by applying central-limit theorem arguments to various aspects of the scattered fields. Each of the resulting Gaussianrandom variables is assumed to be also jointly Gaussian when it is sampled simultaneously at two points in space, thus extending the model to predict the joint probability-density function of the irradiance fluctuations at two points in space. Good agreement with experimental results is provided by this model.

© 1989 Optical Society of America

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References

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  1. T. Wang, J. W. Strohbehn, “Perturbed log-normal distribution of irradiance fluctuations,”J. Opt. Soc. Am. 64, 994–999 (1974).
    [Crossref]
  2. 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]
  3. J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [Crossref]
  4. J. H. Churnside, S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [Crossref]
  5. J. H. Churnside, R. G. Frehlich, “Comparison of two two-parameter models of optical scintillation in the turbulent atmosphere,” submitted to J. Opt. Soc. Am. A.
  6. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), p. 15.
  7. J. Ohtsubo, “Joint probability density function of partially developed speckle patterns,” Appl. Opt. 27, 1290–1292 (1988).
    [Crossref] [PubMed]
  8. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [Crossref]
  9. R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,”J. Opt. Soc. Am. 68, 892–899 (1978).
    [Crossref]

1988 (1)

1987 (2)

1978 (1)

1975 (1)

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]

1974 (1)

1970 (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Churnside, J. H.

Clifford, S. F.

Frehlich, R. G.

J. H. Churnside, R. G. Frehlich, “Comparison of two two-parameter models of optical scintillation in the turbulent atmosphere,” submitted to J. Opt. Soc. Am. A.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), p. 15.

Hill, R. J.

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Ohtsubo, J.

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]

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Wang, T.

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]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

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

Proc. IEEE (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Radio Sci. (1)

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]

Other (2)

J. H. Churnside, R. G. Frehlich, “Comparison of two two-parameter models of optical scintillation in the turbulent atmosphere,” submitted to J. Opt. Soc. Am. A.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), p. 15.

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Figures (15)

Fig. 1
Fig. 1

Block diagram of experimental configuration used for measurements.

Fig. 2
Fig. 2

Correlation function for 100-m-path data. The curve is the theoretical result and the circles are measured values.

Fig. 3
Fig. 3

Joint probability density p(I1, I2) for the 100-m-path data with a 0.77-mm separation. (a) Linear histogram, (b) logarithmic contour plot.

Fig. 4
Fig. 4

Theoretical joint probability density p(I1, I2) corresponding to the data of Fig. 3. (a) Linear surface plot, (b) logarithmic contour plot.

Fig. 5
Fig. 5

Ratio Rp of experimental to theoretical probabilities in 0.1 × 0.1 bins beginning at the designated I1 values and plotted against the I2 values for the case shown in Figs. 3 and 4.

Fig. 6
Fig. 6

Joint probability density p(I1, I2) for the 100-m-path data with a 1.5-mm separation. (a) Linear histogram, (b) logarithmic contour plot.

Fig. 7
Fig. 7

Theoretical joint probability density p(I1, I2) corresponding to the data of Fig. 6. (a) Linear surface plot, (b) logarithmic contour plot.

Fig. 8
Fig. 8

Ratio Rp of experimental to theoretical probabilities in 0.1 × 0.1 bins beginning at the designated I1 values and plotted against the I2 values for the case of Figs. 6 and 7.

Fig. 9
Fig. 9

Joint probability density p(I1, I2) for the 100-m-path data with a 6.1-mm separation. (a) Linear histogram, (b) logarithmic contour plot.

Fig. 10
Fig. 10

Theoretical joint probability density p(I1, I2) corresponding to the data of Fig. 9. (a) Linear surface plot, (b) logarithmic contour plot.

Fig. 11
Fig. 11

Ratio Rp of experimental to theoretical probabilities in 0.1 × 0.1 bins beginning at the designated I1 values and plotted against the I2 values for the case shown in Figs. 9 and 10.

Fig. 12
Fig. 12

Correlation function for 1200-m-path data.

Fig. 13
Fig. 13

Joint probability density p(I1, I2) for the 1200-m-path data with a 5-mm separation. (a) Linear histogram, (b) logarithmic contour plot.

Fig. 14
Fig. 14

Theoretical joint probability density p(I1, I2) corresponding to the data of Fig. 13. (a) Linear surface plot, (b) linear histogram, (c) logarithmic contour plot.

Fig. 15
Fig. 15

Ratio Rp of experimental to theoretical probabilities in 1.0 × 1.0 bins beginning at the designated I1 values and plotted against the I2 values for the case shown in Figs. 13 and 14.

Equations (25)

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U 1 = ( U r 1 + i U i 1 ) exp ( χ 1 + i S 1 ) , U 2 = ( U r 2 + i U i 2 ) exp ( χ 2 + i S 2 ) ,
p ( U r 1 , U r 2 , U i 1 , U i 2 ) = 1 4 π 2 σ 1 2 σ 2 2 ( 1 - μ 2 ) exp { - 1 2 ( 1 - μ 2 ) × [ ( U r 1 - U r 1 ¯ ) 2 + U i 1 2 σ 1 2 + ( U r 2 - U r 2 ¯ ) 2 + U i 2 2 σ 2 2 - 2 μ ( U r 1 - U r 1 ¯ ) ( U r 2 - U r 2 ¯ ) + 2 μ U i 1 U i 2 σ 1 σ 2 ] } .
E 1 = U r 1 2 + U i 1 2 , θ 1 = tan - 1 ( U i 1 / U r 1 ) , E 2 = U r 2 2 + U i 2 2 , θ 2 = tan - 1 ( U i 2 / U r 2 ) ,
p ( E 1 , E 2 ) = 1 16 π 2 σ 1 2 σ 2 2 ( 1 - μ 2 ) 0 2 π d θ 1 0 2 π d θ 2 exp { - 1 2 ( 1 - μ 2 ) [ E 1 - 2 E 1 1 / 2 U r 1 ¯ cos θ 1 + U r 1 ¯ 2 σ 1 2 + E 2 - 2 E 2 1 / 2 U r 2 ¯ cos θ 2 + U r 2 ¯ 2 σ 2 2 - 2 μ ( E 1 1 / 2 cos θ 1 - U r 1 ¯ ) ( E 2 1 / 2 cos θ 2 - U r 2 ¯ ) + 2 μ ( E 1 E 2 ) 1 / 2 sin θ 1 sin θ 2 σ 1 σ 2 ] } .
p ( E 1 , E 2 ) = ( 1 + r 1 ) ( 1 + r 2 ) 4 π 2 ( 1 - μ 2 ) 0 2 π d θ 1 0 2 π d θ 2 × exp { - 1 1 - μ 2 [ ( 1 + r 1 ) E 1 E 1 + ( 1 + r 2 ) E 2 E 2 + r 1 + r 2 - 2 μ r 1 r 2 - 2 ( 1 + r 1 ) 1 / 2 ( r 1 1 / 2 - μ r 2 1 / 2 ) ( E 1 E 2 ) 1 / 2 cos θ 1 - 2 ( 1 + r 2 ) 1 / 2 ( r 2 1 / 2 - μ r 1 1 / 2 ) ( E 2 E 2 ) 1 / 2 cos θ 2 - 2 μ ( 1 + r 1 ) 1 / 2 ( 1 + r 2 ) 1 / 2 ( E 1 E 1 E 2 E 2 ) 1 / 2 cos ( θ 1 - θ 2 ) ] } .
p ( E 1 , E 2 ) = ( 1 + r 1 ) ( 1 + r 2 ) 4 π 2 ( 1 - μ 2 ) E 1 E 2 exp { - 1 1 - μ 2 [ ( 1 + r 1 ) E 1 E 1 + ( 1 + r 2 ) E 2 E 2 + r 1 + r 2 - 2 μ ( r 1 r 2 ) 1 / 2 ] } × 0 2 π d θ 1 exp ( 2 1 - μ 2 { [ r 1 ( 1 + r 1 ) E 1 E 1 ] 1 / 2 - μ ( r 1 r 2 ) 1 / 2 } ) cos θ 1 × I 0 { 2 1 - μ 2 [ ( 1 + r 2 ) ( r 1 1 / 2 - μ r 2 1 / 2 ) 2 E 2 E 2 + μ 2 ( 1 + r 1 ) ( 1 + r 2 ) E 1 E 1 E 2 E 2 + 2 μ ( 1 + r 1 ) 1 / 2 ( 1 + r 2 ) ( r 2 1 / 2 - μ r 1 1 / 2 ) ( E 1 E 1 ) 1 / 2 E 2 E 2 cos θ 1 ] 1 / 2 } .
p ( E 1 , E 2 ) = ( 1 + r 1 ) ( 1 + r 2 ) ( 1 - μ 2 ) E 1 E 2 exp { - 1 1 - μ 2 [ ( 1 + r 1 ) E 1 E 1 + ( 1 + r 2 ) E 2 E 2 + r 1 + r 2 - 2 μ ( r 1 r 2 ) 1 / 2 ] } × n = - I n [ 2 μ ( 1 + r 1 ) 1 / 2 ( 1 + r 2 ) 1 / 2 1 - μ 2 ( E 1 E 1 E 2 E 2 ) 1 / 2 ] I n [ 2 ( 1 + r 1 ) 1 / 2 1 - μ 2 ( r 1 1 / 2 - μ r 2 1 / 2 ) ( E 1 E 1 ) 1 / 2 ] × I n [ 2 ( 1 + r 2 ) 1 / 2 1 - μ 2 ( r 2 1 / 2 - μ r 1 1 / 2 ) ( E 2 E 2 ) 1 / 2 ] .
I 1 = z 1 E 1 E 1 , I 2 = z 2 E 2 E 2 ,
p ( I 1 , I 2 z 1 , z 2 ) = ( 1 + r ) 2 2 π ( 1 - μ 2 ) z 1 z 2 × exp { - 1 1 - μ 2 [ ( 1 + r ) ( I 2 z 1 + I 2 z 2 ) + 2 ( 1 - μ ) r ] } × 0 2 π d θ exp ( 2 1 - μ { [ r ( 1 + r ) I 1 z 1 ] 1 / 2 - μ r } cos θ ) × I 0 { 2 1 - μ 2 [ r ( 1 + r ) ( 1 - μ ) 2 I 2 z 2 + μ 2 ( 1 + r ) 2 I 1 z 1 I 2 z 2 + 2 μ ( 1 + r ) 3 / 2 r 1 / 2 ( 1 - μ ) ( I 1 z 1 ) 1 / 2 I 2 z 2 cos θ ] 1 / 2 } ,
p ( I 1 , I 2 z 1 , z 2 ) = ( 1 + r ) 2 ( 1 - μ 2 ) z 1 z 2 × exp { - 1 1 - μ 2 [ ( 1 + r ) ( I 1 z 1 + I 2 z 2 ) + 2 ( 1 - μ ) r ] } × n = - I n [ 2 μ ( 1 + r ) 1 + μ ( I 1 I 2 z 1 z 2 ) 1 / 2 ] × I n [ 2 r 1 / 2 ( 1 + r ) 1 / 2 1 + μ ( I 1 z 1 ) 1 / 2 ] I n [ 2 r 1 / 2 ( 1 + r ) 1 / 2 1 + μ ( I 2 z 2 ) 1 / 2 ] .
p ( I 1 , I 2 ) = 0 d z 1 0 d z 2 p ( I 1 I 2 z 1 , z 2 ) p ( z 1 , z 2 ) ,
p ( z 1 , z 2 ) = 1 2 π σ z 2 ( 1 - ρ 2 ) 1 / 2 z 1 z 2 exp { - 1 2 ( 1 - ρ 2 ) σ z 2 × [ ( ln z 1 + 1 2 σ z 2 ) 2 + ( ln z 2 + 1 2 σ z 2 ) 2 - 2 ρ ( ln z 1 + 1 2 σ z 2 ) ( ln z 2 + 1 2 σ z 2 ) ] } .
I 1 = 1 , I 1 2 = r 2 + 4 r + 2 ( r + 1 ) exp ( σ z 2 ) , I 1 I 2 = r 2 + 2 ( 1 + μ ) r + 1 + μ 2 ( r + 1 ) 2 exp ( ρ σ z 2 ) .
p ( I 1 , I 2 ) = 1 2 π ( 1 - ρ 2 ) 1 / 2 σ z 2 I 1 I 2 × exp { - 1 2 ( 1 - ρ 2 ) σ z 2 [ ( ln I 1 + 1 2 σ z 2 ) 2 + ( ln I 2 + 1 2 σ z 2 ) 2 - 2 ρ ( ln I 1 + 1 2 σ z 2 ) ( ln I 2 + 1 2 σ z 2 ) ] } .
I 1 u I 2 2 = exp { ½ [ u ( u - 1 ) + v ( v - 1 ) + 2 ρ u v ] σ z 2 } .
B ( d ) = 16 π 2 k 2 0 d K K 0 L d z Φ n ( K , z ) J 0 ( d K z L ) × sin 2 [ K 2 z ( L - z ) 2 k L ] ,
ρ = B ( d ) / B ( 0 ) , σ z 2 = B ( 0 ) .
σ z 2 = 0.5 k 7 / 6 L 11 / 6 C n 2 ,
p ( I 1 , I 2 z 1 , z 2 ) = 1 ( 1 - μ 2 ) z 1 z 2 exp [ - 1 1 - μ 2 ( I 1 z 1 + I 2 z 2 ) ] × I 0 [ 2 μ 1 - μ 2 ( I 1 I 2 z 1 z 2 ) 1 / 2 ] .
I 1 u I 2 v = u ! v ! 2 F 1 ( - u , - v ; 1 ; μ 2 ) × exp { ( 1 / 2 ) [ u ( u - 1 ) + v ( v - 1 ) + 2 ρ u v ] σ z 2 } ,
I 1 u = u ! exp [ ( 1 / 2 ) u ( u - 1 ) σ z 2 ] I 1 I 2 = ( 1 + μ 2 ) exp ( ρ σ z 2 ) .
R = ( 1 + μ 2 ) exp ( ρ σ z 2 ) - 1 2 exp ( σ z 2 ) - 1 .
R = σ I 2 - 1 2 σ I 2 ,
p ( I 1 , I 2 ) = 1 ( 2 π ) 1 / 2 σ z 0 d z 1 z 3 × exp [ - I 1 + I 2 z - ( ln z + ½ σ z 2 ) 2 2 σ z 2 ] ,
σ z 2 = ln [ ½ ( σ I 2 + 1 ) ] .

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