James H. Churnside and R. J. Hill. NOAA/ERL/Wave Propagation Laboratory, Boulder, Colorado 80303. Received July 26, 1986; accepted December 4, 1986.
J. H. Churnside and R. J. Hill
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A
727
Probability density of irradiance scintillations for strong
path-integrated refractive turbulence James H. Churnside and R. J. Hill NOAA/ERL/Wave Propagation Laboratory, Boulder, Colorado 80303 Received July 26, 1986; accepted December 4, 1986
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-normallymodulated 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.
INTRODUCTION Many probability-density functions (PDF's) have been proposed for irradiance in strong scintillation,'-9 and one must pause when considering introducing yet another candidate PDF. The PDF introduced here, the log-normally modulated exponential PDF, is justified by its excellent agreement with data. Like the K PDF,' the log-normally modulated exponential is a single-parameter PDF describing the approach to the exponential PDF in the asymptotic limit of saturated scintillation. Our PDF does not describe the weak-turbulence regime or the initial onset of saturation. For sufficiently large, spatially incoherent transmitting and receiving apertures, it has been shown that the predictions of weak-scintillation theory hold well into the regime where irradiance from a coherent transmitter,
observed by a
varies depending on the log-normally distributed large-scale patches of irradiance. LOG-NORMALLY MODULATED EXPONENTIAL FUNCTION The model for the irradiance used in this paper assumes that the irradiance fluctuations arise from two multiplicative random processes; one of the processes obeys negative expo-
nential statistics, and the other obeys log-normal statistics. The PDF of irradiance can then be expressed as PW = fo p(I1z)p(z)dz,
(1)
where the conditional density p(Ilz) is a negative exponen-
point receiver, shows the saturation-of-scintillation effect.'0
tial function whose mean value z is a log-normal variate.
An aperture with a diameter exceeding XL/po, where X is the
Thus
electromagnetic wavelength, L is the propagation path length, and po is the wave coherence length, is sufficiently large for application of weak-turbulence theory. For saturated scintillation, spatial scales in the scintillation pattern between p0 and XL/po contribute little to the variance of irradiance.1"-13 Thus it is plausible that the irradiance has a log-normal PDF well within the saturation regime when measured through apertures as small as X;Lin diameter. Indeed, data are presented that verify this. Even irradiance measured by using apertures considerably smaller,than JXL
=
2J1
I dz
I
(In z + 1/2uz2)2
L___
z
ex[
0z2 JeU,
(2)
is the density function for the normalized irradiance I (this is the irradiance divided by the mean irradiance), where xz2 is the variance of the logarithm of the modulation. For large values of I a saddle-point integration gives p(I)
zO(1 + 0u2 iz0) 1/2 exp[-
Izo(1
+ 2'
iZo)0
(3)
but larger than pa has been shown to be closely log normal.14
Other observations show the approach to log-normal statistics with increasing aperture size.15'16 In the saturation regime, the small-scale patches of irradi-
where z0 is the solution of 2 az=
ance have the size of the small-scale width of the irradiance
covariance; this width is proportional to pa. The remainder of the irradiance pattern resides in distinctly larger patches of size XL/po. A heuristic picture arises of the small-scale
spikes of irradiance riding upon (i.e., being modulated by) the large irradiance patches. The log-normally modulated exponential PDF is obtained by assuming that the smallscale spikes are exponentially distributed with a mean that 0740-3232/87/040727-07$02.00
(4)
To obtain some idea of the accuracy of this approximation, the PDF was calculated at I = 5. With an intensity variance of 2, the saddle-point approximation was 4.7% below a numerical evaluation of Eq. (2). Increasing cr2 to 4 increases
this error to 6.4%. The saddle-point method is therefore useful for intensities greater than about 5 in most cases. © 1987 Optical Society of America
728
J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987
J. H. Churnside and R. J. Hill
COMPARISON OF THE PROBABILITY-DENSITY FUNCTION WITH DATA By means of the experimental
10
configuration of Fig. 1, the
irradiance was measured during periods of high turbulence
1
in August of 1985. A 4-mW He-Ne laser operating on the 633-nm line was used as the radiation source. The natural beam divergence was about a milliradian. Since changes in
10-1-
vertical temperature gradient along the path could conceivably move the beam up or down by some reasonable fraction of a milliradian, a negative cylindrical lens was used to ex-
pand the vertical beam width by about a factor of 2. The
10-2
propagation range was 1 km at a height of about 2 m. The
surface was flat, uniform grassland. At the receiver, the signal was detected by a photomultiplier tube after passing through a 1-mm aperture and a 1-nm optical bandpass filter. After amplification, the irradiance signal was fed into the computer. In the computer, the irradiance signal was digitized with
cA 10-3
10-4_
12 bits of resolution at a rate of 5,000 samples per second.
After collecting 10,833 samples, the system sampled the output of an incoherent-light optical scintillometer. This instrument provided a measure of C,2 averaged over a 250-m path parallel to the main propagation path. After C,2 was sampled, the data record was stored on the system disk for later processing. Each data run consisted of 50 such records, or a little over half a million intensity samples. Ten
l
Gu
10 X0-
l
10
20 I
data runs were obtained. After each data run, a single record was collected with the laser blocked to permit estima-
(a)
tion of offsets and noise in the receiver electronics. Measured C,2 values for the data sets ranged from 5.2 X 13 10-13 to 1.9 X 10-12 m-2 . The Rytov spherical-wave, zeroinner-scale, log-intensity variance flo2 is defined as 0.5 k716 L 1 116 Cn2 , where k = 27r/A. For our experiment,
from 12 to 43.
l
/o 2 ranged
The regime of strong turbulence,
40
l
2 -
where
spherical-wave irradiance variance decreases with increasing turbulence, corresponds to flo2 values exceeding about 2.5, and our data are clearly well into this regime.
The normalized irradiance data were sorted into histogram bins. For normalized irradiance values less than 5, the bin width was 0.1. For values from 5 to 10, from 10 to 20, and from 20 to 40, the corresponding bin widths were 0.3, 1.0,
AX
and 3.0. These data are represented by a circle at the center
'\
\ \
of each bin in Figs. 2 and 3. The uncertainty in the measure-
ment was calculated for each data point, with the assumption that successivesamples of irradiance were independent. The resulting error bars were smaller than the data symbols for probability densities greater than about 10-5. We should note that the error bars in Fig. 2 may underestimate the actual uncertainty, since successivesamples are not coimpletely independent.
1
\ \
z
9
k
Lens Turbulent Atmosphere
\ \X
// ~~~~~ 12 Lu
n Filter
|
'.o
,
Detector
0
(b)
K (--Computer Configuration of the experiment.
\
X
Fig. 2.
Fig. 1.
A
l 4
Aperture
Laser
I
Amplifier
PDF's of normalized irradiance for
2 1
=
2.83 (Qo2
=
36):
), log-normal (----), and log-normally modulated exponential (-) functions and data (0). (a) Logarithmic representation; (b) linear representation
for I < 2.
J. H. Churnside and R. J. Hill
0.
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A
729
10
20
0.
0
5
15
I (a) Fig. 4.
PDF's of normalized irradiance through a 25-mm aperture:
log normal (-) and data (0). or2= 2.45.
The variance of normalized irradiance was also calculated from the data. This was compared with the theoretical variance al2 = 2 exp(o-f2) - 1,
(5)
calculated by using Eq; (2) to get an estimate of a-z2.In each figure, the solid line is a plot of the log-normally modulated exponential function whose parameter oZ2was inferred in
I,
0 1
1
\
'2
(b) Fig. 3. PDF's of normalized irradiance for au2 = 4.13 (,B02 = 23): K (---), log-normal --- -), and log-normally modulated exponential (-) functions and data (0). (a) Logarithmic representation; (b) linear representation for I < 2.
this fashion. The dashed lines represent the log-normal PDF's and K PDF's with the same intensity variance as the data. Figure 2 represents the data run with the lowest irradiance variance of the 10 data runs (0rI2 = 2.83). Comparing the theoretical density functions, one finds that the log-normal density function is above the k PDF for irradiance values from some small value to about 2 and for values greater than about 17. Elsewhere, the K is above the log normal. The log-normally modulated exponential function is between the log-normal and the K functions at all irradiance values. The data points also tend to lie between the K and the log-normal functions. Figure 3 represents the data run with the highest irradiance variance of the 10 data runs (af2 = 4.13). In this case, the points at which the log-normal and the K PDF's cross are at larger irradiance values, occurring at about I = 2.5 and 23. Although the three density functions are fairly close together over the range of available data values, the data favor the log-normally modulated exponential PDF over the log-normal or the K PDF. The propagation model that leads to the log-normally modulated exponential PDF suggests that the fluctuations that are due to the small-scale spikes of irradiance should be
J. H. Churnside and R. J. Hill
J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987
730 reduced
whose diameter
by an aperture
is about
a Fresnel
1
I
I
I
I
I
l
l
I
24
28
I
zone size. The fluctuations observed through such an aperture would therefore
be expected to be very nearly log-
normally distributed. For this reason, a limited amount of data was taken through a 25-mm aperture. These data
10-1
were, in fact, very nearly log normal. Typical results are presented in Fig. 4. The solid line represents the log-normal PDF with the same variance of irradiance (a/ 2 = 2.45) as the
data. With the large aperture, successive samples tended to
10-2
be correlated, and no error bars were calculated for this case.
PHOTON-COUNT STATISTICS Since some of the previous work has dealt with the statistics of photon-counting experiments, 1 7 it is interesting to exam-
'10-'
ine the photon-count statistics produced by log-normally modulated exponential irradiance statistics. Photon counting permits the use of much smaller apertures than does use
10-4
-
of irradiance measurements. If the spatial spikes of irradiance are progressively smaller in size for increasing irradiance (as suggested by Prokhorov et al.12), then we have
under-estimated our measured irradiance at large irradiance
1o
values because of aperture averaging by our 1-mm-diameter
io-_
receiver aperture. Thus photon-count PDF's might be more trustworthy in the extreme tails of the PDF. For counting times much less than the coherence time of the fluctuations, the PDF of the photon count is a conditional Poisson process related to the irradiance PDF by the formula
10-
6
) 4
8
12
16
20
32
3
n
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 (0). 1
I?
=IdI' (,y )n exp(-yI') p(T) v ~~~~~~~~~~~~~~~~~~~~~p(n)
\ _
10-1-\
(6)
where y is a constant related to the experimental configuration and I' is the actual (unnormalized) irradiance. For apertures much smaller than the transverse coherence length of the irradiance fluctuations, ly is given by
AT
10-2
aE. 10-3-
\_Assuming
10-4-
\
(7
where A is the detector area, T is the count time, and hp is the photon energy. that the irradiance obeys log-normally modulated exponential statistics, the photon-count statistics can be described by the density function p~n) =
-
4
1_
dz
I
Z2
zn
1+1 ) nz /
1(in
L
z+
/20, 2)2
2az
(8) 10-5 -
x
\
40 and are therefore not plotted in Figs. 2 and 3 remain in agreement with the log-normally modulated exponential PDF. Extreme caution must be exercised in considering measured moments. Consortini and Conforti2 ' and Consortini et al.22 showed that the effects of detector saturation on the higher-order moments can be severe. In their work the log-normal distribution and Furutsu distribution were considered. The sensitivity of moments to large values of irradiance is easily seen from the integrand of the moment integral.23 The moments are given by (In) =
J
p(I)dL
(15)
= '/4 in
I2,
2
where axe is the log-amplitude variance.
(16)
Clifford and Hill24
evaluated the log-amplitude variance of the K PDF to obtain
aX
+ 4(n
+ I22)
(17)
For the log-normally modulated exponential PDF the corresponding relationship is 2=
en
24
+l
4
ln(I2/2).
(18)
As for the untruncated higher moments, the log-normally modulated exponential PDF predicts log-amplitude variance values that are between those of the log-normal func-
tion and the K function. DISCUSSION The log-normally modulated exponential PDF has been
The integrand, Inp(I), is plotted in Fig. 8 for several values of n. These curves assume a log-normally modulated expo-
nential density function with a second moment of 4.5, which is typical of our data. Because the amplifier gain was optimized for each run of the experiment, our detector saturation voltage corresponds to values of I varying from about 40
to 100, depending on the run. All runs appear to be affected by saturation to some extent. Thus Fig. 8 shows that the fourth and fifth measured moments are greatly underestimated.
Figure 8 also shows the need for extremely large
sample spaces to reduce scatter in the measured moments. To obtain accurate estimates of the third moment, Fig. 8 shows that several events near I = 100 should be observed.
At I = 100, p(I) is a little over 10-7, and several tens of millions of independent samples are needed. The required number of samples to resolve the fourth and fifth moments is much greater. Because of these uncertainties in the measured moments, we truncated the theoretical and experimental PDF's at a
shown to agree with measured irradiance PDF's as well as
with measured photon-count PDF's. It is shown to be superior to the K PDF. Both the log-normally modulated exponential function and the K PDF apply to the approach to exponential statistics in the saturation regime, not to the regime of increasing irradiance variance with increases in path-averaged refractive turbulence strength. The moments of the log-normally modulated exponential PDF, the K PDF, and the log-normal PDF are compared with moments of measured irradiance. The disagreement of the moments of the log-normally modulated exponential PDF with the data is dramatic, considering the close agreement of the PDF's. As emphasized by others, the reason is the combination of the sensitivity of the moments to extremely large and improbable values of irradiance and the limitations of the receiver and the number of independent samples attainable. The truncated moments, on the other hand, show good agreement with the log-normally modulated exponential, as expected on the basis of the agreement of
J. H. Churnside and R. J. Hill
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A
the PDF's. We conclude that irradiance moments are a misleading statistic and that deviations from log-normal statistics are not so extreme as had been supposed. Comparison of theoretical PDF's with measured PDF's is recommended over comparison of the moments.
It is also recom-
mended that plots of In p(I) be given to demonstrate the sufficiency of the sample space for given moments and that, either moments from truncated PDF's be compared or the detector saturation effect be calculated for the theoretical PDF's. The log normality of aperture-averaged scintillations for strong path-integrated refractive turbulence has been demonstrated. It may be that early experiments reported lognormal PDF's of irradiance in strong scintillation because of receiver apertures with diameters that exceeded the smallscale width of the irradiance covariance. Further experimentation is needed to reveal the effect of finite aperture size on the shape of the irradiance PDF. Irradiance statistics of collimated beams have been shown to be log normal in some experiments 23 ' 25 ' 26 but not in oth-
ers.18 The experiments demonstrating log normality have in common that the beam diameter was of the order of XL/po. Consequently, beam wander might be producing the observed log normality of collimated beams. The laboratory experimentsl8'25' 26 are particularly well suited to investigating the effects of the spatial limitation of a beam wave on the irradiance statistics because the beam diameter can be easily
varied, provided that there exists sufficient laser power to use a small receiver aperture,
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Shishov, "Laser irradiance propagation in turbulent media," Proc. IEEE 63, 790-811 (1975). 13. R. J. Hill and S. F. Clifford, "Theory of saturation
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scintillation by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am. 71, 675-686 (1981).
14. R. J. Hill and J. H. Churnside, "Measured statistics of optical scintillation in strong refractive turbulence relevant to laser eye safety," submitted to Health Phys. 15. Z. Azar, H. M. Loebenstein, G. Applebaum, E. Azoulay, U. Halavee, M. Tamir, and M. Tur, "Aperture averaging of the
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17. G. Parry, "Measurement of atmospheric turbulence induced intensity fluctuations (1981).
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19. R. Dashen, "The distribution of intensity in a multiply scattering medium," Opt. Lett. 10, 110-112 (1984). 20. R. L. Phillips and L. C. Andrews, "Measured statistics of laser-
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