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.
Jason R. W. Mclaren, John C. Thomas, Jessica L. Mackintosh, Kerry A. Mudge, Kenneth J. Grant, Bradley A. Clare, and William G. Cowley Appl. Opt. 51(25) 5996-6002 (2012)
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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
EXPERIMENTAL CONDITIONS
CHI-SQUARE TEST
KOLMOGOROV-SMIRNOV TEST
STANDARD DEVIATION ERROR OF DENSITY FUNCTION
STANDARD DEVIATION ERROR OF DISTRIBUTION FUNCTION
Run #
z/zA
<η2>
m−1
χ2eb
χ2ln
Deb
Dln
fEeb
fEln
FEeb
FEln
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/zA is the normalized propagation distance; 〈η2〉, the second-order moment of the irradiance normalized to unit mean; and (m − 1), the number of degrees of freedom for the χ2-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/zA
<η2>
m−1
χ2eb
χ2ln
Deb
Dln
fEeb
fEln
FEeb
FEln
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/zA, 〈η2〉,(m − 1), χ2, D, and subscripts eb and ln, as in Table I.
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.
Hypothesis
Normal complex-Amplitude Probability Distribution
Log-normal complex-Amplitude Probability Distribution
Test
χ2
356
257
D
0.067
0.162
fE
0.064
0.051
FE
0.026
0.042
Tables (3)
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
EXPERIMENTAL CONDITIONS
CHI-SQUARE TEST
KOLMOGOROV-SMIRNOV TEST
STANDARD DEVIATION ERROR OF DENSITY FUNCTION
STANDARD DEVIATION ERROR OF DISTRIBUTION FUNCTION
Run #
z/zA
<η2>
m−1
χ2eb
χ2ln
Deb
Dln
fEeb
fEln
FEeb
FEln
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/zA is the normalized propagation distance; 〈η2〉, the second-order moment of the irradiance normalized to unit mean; and (m − 1), the number of degrees of freedom for the χ2-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/zA
<η2>
m−1
χ2eb
χ2ln
Deb
Dln
fEeb
fEln
FEeb
FEln
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/zA, 〈η2〉,(m − 1), χ2, D, and subscripts eb and ln, as in Table I.
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.
Hypothesis
Normal complex-Amplitude Probability Distribution
Log-normal complex-Amplitude Probability Distribution
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