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 I_{0} 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.

Stanley M. Flatté, Charles Bracher, and Guang-Yu Wang J. Opt. Soc. Am. A 11(7) 2080-2092 (1994)

References

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Results of the Chi-Square (χ^{2}) and Kolmogorov-Smirnov (D) Tests and Standard Deviation Errors of Probability Density (_{f}E) and Distribution (_{F}E) Functions for the Exponential-Bessel (Subscript eb) and Log-Normal (subcript ln) Models; runs 37–45 and 50–51^{a}

EXPERIMENTAL CONDITIONS

CHI-SQUARE TEST

KOLMOGOROV-SMIRNOV TEST

STANDARD DEVIATION ERROR OF DENSITY FUNCTION

STANDARD DEVIATION ERROR OF DISTRIBUTION FUNCTION

Run #

z/z_{A}

<η^{2}>

m−1

χ^{2}_{eb}

χ^{2}_{ln}

D_{eb}

D_{ln}

_{f}E_{eb}

_{f}E_{ln}

_{F}E_{eb}

_{F}E_{ln}

37

4.02

2.33

49

1419

689

0.065

0.170

0.137

0.094

0.024

0.038

38

4.82

2.28

49

370

385

0.060

0.183

0.072

0.062

0.021

0.041

39

5.36

2.29

49

447

457

0.086

0.179

0.078

0.070

0.027

0.042

40

5.90

2.29

49

352

225

0.079

0.177

0.060

0.040

0.027

0.039

41

6.46

2.23

49

565

582

0.118

0.226

0.090

0.084

0.036

0.056

42

6.93

2.27

49

308

121

0.074

0.161

0.056

0.028

0.026

0.035

43

7.43

2.26

49

541

232

0.121

0.210

0.075

0.048

0.038

0.051

45

7.68

2.25

49

305

278

0.091

0.196

0.063

0.050

0.031

0.048

44

8.11

2.26

49

269

148

0.078

0.180

0.057

0.033

0.028

0.042

51

9.93

2.23

90

147

139

0.032

0.080

0.048

0.037

0.009

0.024

50

10.3

2.24

93

201

152

0.051

0.129

0.048

0.032

0.018

0.034

z/z_{A} is the normalized propagation distance; 〈η^{2}〉, the second-order moment of the irradiance normalized to unit mean; and (m − 1), the number of degrees of freedom for the χ^{2}-test.

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}

EXPERIMENTAL CONDITIONS

CHI-SQUARE TEST

KOLMOGOROV-SMIRNOV TEST

STANDARD DEVIATION ERROR OF DENSITY FUNCTION

STANDARD DEVIATION ERROR OF DISTRIBUTION FUNCTION

Run #

z/z_{A}

<η^{2}>

m−1

χ^{2}_{eb}

χ^{2}_{ln}

D_{eb}

D_{ln}

_{f}E_{eb}

_{f}E_{ln}

_{F}E_{eb}

_{F}E_{ln}

22

4.15

2.30

49

574

217

0.072

0.194

0.080

0.043

0.025

0.041

29

5.05

2.22

49

188

66

0.032

0.134

0.046

0.021

0.014

0.027

26

5.88

2.21

49

207

66

0.046

0.145

0.050

0.025

0.020

0.036

25

6.61

2.19

49

172

70

0.037

0.135

0.048

0.025

0.017

0.034

179

7.10

2.16

80

214

165

0.057

0.153

0.058

0.052

0.033

0.049

178

7.49

2.15

77

228

247

0.057

0.147

0.054

0.060

0.031

0.049

180

7.92

2.16

96

214

254

0.041

0.132

0.053

0.058

0.023

0.043

181

8.58

2.13

84

260

408

0.081

0.179

0.058

0.079

0.040

0.060

182

9.95

2.11

78

244

231

0.057

0.140

0.055

0.058

0.032

0.047

183

10.6

2.09

82

249

265

0.069

0.155

0.058

0.065

0.036

0.052

z/z_{A}, 〈η^{2}〉,(m − 1), χ^{2}, D, and subscripts eb and ln, as in Table I.

Table III

Comparative Values of Chi-Square(χ^{2}), Kolmogorov-Smirnov Statistic (D), and Standard Deviation Errors of Irradiance Density (_{f}E) and Distribution(_{F}E) 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.

Hypothesis

Normal complex-Amplitude Probability Distribution

Log-normal complex-Amplitude Probability Distribution

Test

χ^{2}

356

257

D

0.067

0.162

_{f}E

0.064

0.051

_{F}E

0.026

0.042

Tables (3)

Table I

Results of the Chi-Square (χ^{2}) and Kolmogorov-Smirnov (D) Tests and Standard Deviation Errors of Probability Density (_{f}E) and Distribution (_{F}E) Functions for the Exponential-Bessel (Subscript eb) and Log-Normal (subcript ln) Models; runs 37–45 and 50–51^{a}

EXPERIMENTAL CONDITIONS

CHI-SQUARE TEST

KOLMOGOROV-SMIRNOV TEST

STANDARD DEVIATION ERROR OF DENSITY FUNCTION

STANDARD DEVIATION ERROR OF DISTRIBUTION FUNCTION

Run #

z/z_{A}

<η^{2}>

m−1

χ^{2}_{eb}

χ^{2}_{ln}

D_{eb}

D_{ln}

_{f}E_{eb}

_{f}E_{ln}

_{F}E_{eb}

_{F}E_{ln}

37

4.02

2.33

49

1419

689

0.065

0.170

0.137

0.094

0.024

0.038

38

4.82

2.28

49

370

385

0.060

0.183

0.072

0.062

0.021

0.041

39

5.36

2.29

49

447

457

0.086

0.179

0.078

0.070

0.027

0.042

40

5.90

2.29

49

352

225

0.079

0.177

0.060

0.040

0.027

0.039

41

6.46

2.23

49

565

582

0.118

0.226

0.090

0.084

0.036

0.056

42

6.93

2.27

49

308

121

0.074

0.161

0.056

0.028

0.026

0.035

43

7.43

2.26

49

541

232

0.121

0.210

0.075

0.048

0.038

0.051

45

7.68

2.25

49

305

278

0.091

0.196

0.063

0.050

0.031

0.048

44

8.11

2.26

49

269

148

0.078

0.180

0.057

0.033

0.028

0.042

51

9.93

2.23

90

147

139

0.032

0.080

0.048

0.037

0.009

0.024

50

10.3

2.24

93

201

152

0.051

0.129

0.048

0.032

0.018

0.034

z/z_{A} is the normalized propagation distance; 〈η^{2}〉, the second-order moment of the irradiance normalized to unit mean; and (m − 1), the number of degrees of freedom for the χ^{2}-test.

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}

EXPERIMENTAL CONDITIONS

CHI-SQUARE TEST

KOLMOGOROV-SMIRNOV TEST

STANDARD DEVIATION ERROR OF DENSITY FUNCTION

STANDARD DEVIATION ERROR OF DISTRIBUTION FUNCTION

Run #

z/z_{A}

<η^{2}>

m−1

χ^{2}_{eb}

χ^{2}_{ln}

D_{eb}

D_{ln}

_{f}E_{eb}

_{f}E_{ln}

_{F}E_{eb}

_{F}E_{ln}

22

4.15

2.30

49

574

217

0.072

0.194

0.080

0.043

0.025

0.041

29

5.05

2.22

49

188

66

0.032

0.134

0.046

0.021

0.014

0.027

26

5.88

2.21

49

207

66

0.046

0.145

0.050

0.025

0.020

0.036

25

6.61

2.19

49

172

70

0.037

0.135

0.048

0.025

0.017

0.034

179

7.10

2.16

80

214

165

0.057

0.153

0.058

0.052

0.033

0.049

178

7.49

2.15

77

228

247

0.057

0.147

0.054

0.060

0.031

0.049

180

7.92

2.16

96

214

254

0.041

0.132

0.053

0.058

0.023

0.043

181

8.58

2.13

84

260

408

0.081

0.179

0.058

0.079

0.040

0.060

182

9.95

2.11

78

244

231

0.057

0.140

0.055

0.058

0.032

0.047

183

10.6

2.09

82

249

265

0.069

0.155

0.058

0.065

0.036

0.052

z/z_{A}, 〈η^{2}〉,(m − 1), χ^{2}, D, and subscripts eb and ln, as in Table I.

Table III

Comparative Values of Chi-Square(χ^{2}), Kolmogorov-Smirnov Statistic (D), and Standard Deviation Errors of Irradiance Density (_{f}E) and Distribution(_{F}E) 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.

Hypothesis

Normal complex-Amplitude Probability Distribution

Log-normal complex-Amplitude Probability Distribution