Vol. 1, No. 11/November 1984/J. Opt. Soc. Am. A 1067

Higher-order statistics of laser-irradiance fluctuations due toturbulence

Arun K. Majumdar

Lockheed-California Company, Kelly Johnson Research & Development Center at Rye Canyon, P.O. Box 551,

Burbank, California 91520

Received January 8, 1984; accepted July 10, 1984

Higher-order coefficients (up to eighth order) that give information concerning the contribution from the tails ofnon-Gaussian distributions are utilized to compare model distributions (K, log-normal, Rice-Nakagami, and a uni-

versal statistical model distribution) with the measured data of 0.6328-Atm laser-irradiance fluctuations propagatedthrough a laboratory-simulated turbulence. Expressions for the coefficients of the K distribution are derived inthis paper. The comparison is made using a plot of the coefficients as a function of u- = (1.23CQ2 k716

L 11/6)1/2, C,,2 being the refractive-index structure constant. Different values of the exponent a (or p) in r,,-oa

(or r,, LP) computed from the measured data show that the exponent parameters increase with the order of nof the coefficients rn. The parameters a or p can be useful for accurate modeling of the complex process.

1. INTRODUCTION

The subject of the statistics of the irradiance fluctuations inturbulent media is still, in my opinion, unsettled and in need

of additional developments. The random spatial and tem-poral fluctuations in the refractive index of the medium giverise to irradiance fluctuations of an optical beam propagatingthrough it. The irradiance fluctuations resulting from tur-bulence studied here can be considered as examples of sto-chastic processes of physical origin. A statistical descriptionof such processes contributes to the understanding of thenature, the origins, and the time histories. In this paper, I

present a technique to analyze non-Gaussian statistics byusing more-advanced statistical characteristics, such as thehigher-order skewness and the excess coefficients. The ap-plication of this technique is discussed, using data from alaboratory-simulated experiment for studying irradiancefluctuations that result from turbulence.

Higher-order skewness and excess coefficients of someprobability distributions that are applicable to optical prop-agation phenomena were studied previously.' In Ref. 1 it was

pointed out that the higher-order moments can give moreinformation concerning the contributions from the tails of theprobability density functions. In order to determine suffi-ciently well the physical nature of a complicated randomprocess, such as laser fluctuations through turbulence, it isnecessary to consider fluctuations beyond the fourth-ordercorrelation. For practical purposes, moments up to eighthorder are usually sufficient for the statistical characterizationof the random process, although the moments for some sta-tistical distributions do not define the process uniquely.2 Tounderstand better the statistical characteristics of irra-diance-fluctuation phenomena, data from a laboratory-sim-ulation experiment 3 were used; measurements were takenunder controlled experimental conditions. In Ref. 3, therelevance of the experiment to the problem of defining aproper irradiance probability model was pointed out bycomparing the higher-order (up to sixth order) cumulants andthe normalized statistical parameters of proposed distribu-

tions with the experimental data as the order n was increased.In Ref. 3, however, measurements from all the higher-ordercoefficients (for example, up to eighth order) were not in-cluded; they are included in this paper. A comparison of themeasured coefficients up to eighth order with those of themodel distributions shows the usefulness and the applicabilityof the technique described here. The significance of thesehigher-order parameters in relation to measureable quantitiesand the means of comparing model distributions with ex-perimental data are also pointed out.

From a practical point of view, it is necessary to know theform of the probability density functions and the variance ofirradiance fluctuations, for example, for designing laser sys-tems to operate under atmospheric conditions, particularlyfor an atmospheric optical communication system.4 It isexpected that some of the methods of analysis and of mea-surement discussed here in connection with the study of tur-bulence-induced fluctuations will be applicable to otherprocesses of physical origin.

2. HIGHER-ORDER COEFFICIENTS INSTATISTICAL ANALYSIS

Let I be the received irradiance after an optical beam haspropagated through a turbulent medium, which is charac-terized by the following parameters: propagation path lengthL, turbulence-strength parameter (refractive-index structureconstant) C,, 2, and the smallest and the largest scale sizes ofturbulence lo and Lo, respectively. The nth-order centralmoments (n = 2, 3,.. ., 8) of I can be written as

(1)

where ( ) indicates an ensemble average. The higher-ordernondimensional coefficients are defined as'

skewness: r 3 = A31A2 ,

excess: r 4 = M4//A22 - 3,

superskewness: r5 = IA5/I.L3/l.2 - 10,

(2)

(3)

(4)

0740-3232/84/111067-08$02.00 © 1984 Optical Society of America

Arun K. Majumdar

it. = ((I - (I) )n ) ,

1068 J. Opt. Soc. Am. A/Vol. 1, No. 11/November 1984

superexcess: r 6 = JU6//123 - 15, (5)

hyperskewness: r 7 = /17/,u3 1222- 105, (6)

hyperexcess: rs = u8/IL24 - 105. (7)

Note that, for a Gaussian probability distribution, the odd-order coefficients r 5 and r7 do not exist and become inde-terminate, whereas the even-order coefficients r 4 = r6 = r8= 0. Any nonzero value of the even-order parameters indi-cates that the random process deviates from a Gaussian dis-tribution.

In order to relate the above higher-order coefficients to theparameters of irradiance fluctuations resulting from turbu-lence, let us consider the expression for the variance of thenormalized irradiance fluctuation given by5

a12 = 1.23C, 2k 7 /16L 1 1 /1 6 , (8)

where k = 27r/X is the wave number of the optical beam. Theparameter a1

2 is calculated by the method of smooth pertur-bations (MSP), also called Rytov's method. The procedurefor applying the higher-order analysis to fluctuating opticalsignals is outlined below.

From a propagation experiment, the coefficients F3 -r8 canbe measured. Note that the usual notations for the skewnessand the excess coefficients yl and 72 are the same as ournotations r 3 = y, and r 4 = 72. Measured higher-ordercoefficients were compared with those of the model distri-butions by first setting the measured value of F3 equal to thoseof the model distributions. The parameters of each modeldistribution were thus determined from comparisons of r3 .Next the predicted values of the higher-order coefficients forthe model distributions were computed using these parame-ters and compared with the experimental data. The detailsof this method using laboratory-simulated data are discussedin Section 3. For different values of the MSP parameter ul(obtained by varying the propagation path length L), thestatistics of received irradiance may be different; this corre-sponds to different values for F3 . By adjusting these r3 val-ues, different parameters of the model distributions arecomputed, and finally the coefficients r 4 -r8 are calculated.The coefficients r 4-r 8 can now be plotted as a function of a,to compare with the measured coefficients, and the subtledifferences among statistical distributions can be observed.

First, three model distributions were considered for com-parison: the K, the log-normal, and the Rice-Nakagamidistributions. The higher-order coefficients were calculatedpreviously in Ref. 1 for the log-normal and the Rice-Nakagamidistributions. The higher-order coefficients for the morerecently proposed K distribution6 are derived here, and theexpressions are given in Appendix A.

Next, a recently proposed7 universal statistical model(USM) for irradiance fluctuations in a turbulent medium wascompared with the experimental data. This universal modelis a three-parameter distribution and may have potentialapplications under all conditions of turbulence.

Recently some researchers studied the higher-order sta-tistics of laser-light fluctuations in atmospheric turbulence. 7-"The difference between their research and that reported inthe present paper is that they considered the normalizedmoments (In)/(I) n for values of it up to 4, as did Grachevaet al., 8 and for n up to only 5, as other researchers did.7' 9-'1

The difficulty in analyzing the normalized-moments data

arises because the data are affected by the instrumental cal-ibration, since the average received irradiance (I) can be af-fected by any unwanted change in the gains of the instrumentsduring the experiment. Even and odd higher-order coeffi-cients can provide different information, which can be ofsignificance in understanding complex phenomena. For nodd, any deviation of Yn from zero indicates that the randomprocess deviates from a Gaussian distribution. Also, thevalues of Fn for n even can be of interest. The higher-ordercoefficients, if derived from the cumulants, can be indepen-dent of instrumental calibration, and any change in instru-mental gains will not affect the Fn values. The higher-orderskewness and the excess coefficients, which usually come froma small part of the signal representing the random process, candescribe the most asymmetric part more significantly thanthe normalized-moments data, as described by the other re-searchers. 7 -"1 Any departure from the Gaussian statistics canbe determined from these coefficients. Also, the coefficientsr5 -r 8 can indicate the occurrence of sharp jumps' 2 or occa-sional spikes in a random signal. I believe that the higher-order coefficients r 5 -F8 can provide more-detailed infor-mation in a complex process, such as optical irradiance fluc-tuations through turbulence. This will open new avenues ofapproach; for example, a more-accurate statistical model canbe developed to fit the experimental results, and this will beuseful in solving such problems as the origin of turbulence bydetermining the physical mechanism associated with thenon-Gaussian effects.

3. COMPARISON WITH EXPERIMENTALRESULTS

A. Comparison with Log-Normal, Rice-Nakagami, andK DistributionsA laboratory-simulation approach was chosen to assess theapplicability and the utility of the technique of analyzinghigher-order coefficients. The details of the experimentalsetup are described elsewhere3 and therefore will not be re-peated here. Essentially, a collimated He-Ne laser beam (X= 0.6328-,um) was propagated through heat-generated at-mospheric turbulence of strength C, 2 = 2.476 X 10-11 m- 213

with the smallest scale size 1l = 2.74 mm. The effectivepropagation path length was increased by transmitting overmultiple-folded paths (obtained by a series of equally spacedplane mirrors). The measurements were performed for op-tical paths from a single path of 2.5 m in length up to 13 pathsof 32.5 m in length transmitted through independent regionsof the laboratory-generated homogeneous and isotropic tur-bulent medium.

Note that, although the coefficients of skewness and excessof irradiance fluctuations were considered in Ref. 3, no de-tailed study of the higher-order coefficients (as described here)was made. The notation r,, in Ref. 3 was for higher-ordernormalized cumulants rn = Xn/(VX)n, Xn being the nthcumulant, which is different from the present notation for thenondimensional higher-order coefficients.

For each path length, the value of al was calculated usingEq. (8). In the experiment, the output of the photomultipliertube after amplification by a differential amplifier was digi-tized by an analog-to-digital converter (14 bits) interfaced to

Arun K. Majumdar

Vol. 1, No. 11/November 1984/J. Opt. Soc. Am. A 1069

Table 1. Parameters of the Three Probability Density Functions

Parametera of Distributionr3 K Log-Normal Rice-Nakagami

from Distribution Distribution DistributionO'l Experiment () (co) (/)

0.154 0.259 1.0074 66.248

0.291 0.634 1.0434 10.350

0.422 0.754 1.0607 7.070

0.550 1.215 1.2146 2.180

0.674 1.762 1.2874 0.510C

0.797 2.962 0.2042b 1.6563

0.918 3.090 0.2390 1.6983

1.037 3.662 0.4178 1.8875

1.156 5.571 1.3150 2.5195

1.273 5.504 1.2760 2.4979

1.389 5.871 1.4975 2.6170

1.504 4.949 0.9717 2.3152

1.619 4.707 0.8515 2.2353

a Parameters are obtained by setting r3 (measured) equal to r3 (model distribution).b r3 > 2 for K distribution since 0 ' , • 3/2.c r3 < 2 and 0 S r4 • 6 for Rice-Nakagami distribution.

the PDP 15/20 data-acquisition system. The data-processingsystem constructed real-time frequency histograms from atime series at a rate determined by the sampling criteria. Inorder to determine the sampling rate, the temporal-frequencyspectra of irradiance fluctuations were obtained by applyingthe fast Fourier transform method (available in the PDP 15/20computer) to the photocurrent produced by laser light afterpropagating through turbulence. The highest frequencydetermined from such power spectra 3 was about 160 Hz. Thiswas also verified from the higher-frequency break point vI/1o(vI is the transverse component of the average wind velocityand lo is the smallest scale size of turbulence) within the in-ertial subrange of generated turbulence. Thus the samplerate by the computer was determined from twice the highestobserved frequency. However, most of the data were obtainedat a faster rate of typically between 500 and 2000 samples/sec.The instantaneous signal after the photomultiplier and am-plifier system was converted to a 14-bit digital signal by theanalog-to-digital converter, and the 512 discrete channels wereused for storing the digitized data in the buffers of the PDP15/20 computer. In order to provide an interactive data-acquisition system for real-time signal processing and to havea high level of programming flexibility, the software IECS

(Interactive Experiment Control System), developed at theUniversity of California, Irvine, was used extensively. Thedata-processing system can construct a real-time frequencyhistogram from a time series at a rate determined by thesampling criteria discussed above. The raw data points forthe histograms were smoothed by using the Edgeworth series.Next, the higher-order moments, the central moments, andthe cumulants up to eighth order were computed by using theprograms already written in the computer. The signal waskept at the highest possible level without clipping or beingsaturated so that the quantization noise level was minimum.The higher-order coefficients r3 -r8 were computed from themeasurements of higher-order parameters. For a particularpath length (or a,), the measured value of the P3 coefficientwas set equal to that of any of the proposed model distribu-tions to obtain the parameter of the distribution. By usingthis parameter, higher-order coefficients of the model dis-

tribution were then predicted for this a1 and compared withthose obtained from measurements. Thus, essentially bymatching the r3 value identically with the measured data, theparameters of the model distributions were evaluated, and thetheoretical curves for the higher-order coefficients were thendrawn by using the parameters. The actual experimentalpoints were then compared with the curves predicted by themodel distributions. The parameters defining the threeprobability density functions (mentioned above) were com-puted for increasing values of a, by matching the r 3 coeffi-cients in each case to achieve the results in Table 1. Notethat, for the log-normal distribution, the parameter co =exp(a 2), where a2

= var(log I). In the case of a three-pa-rameter model (such as a USM, which is discussed below), theparameters were obtained graphically from simultaneousnonlinear equations containing the normalized moments.

Figure 1 shows the higher-order coefficients plotted as afunction of ai. For purposes of comparison with the theo-retical models, the K, the log-normal, and the Rice-Nakagamidistribution curves for the higher-order coefficients are alsoshown. The coefficients tend to show a similar trend ofvariation with al, i.e., they increase with a1 to a maximumvalue at a, 1.4 and then decrease at a slower rate with fur-ther increases in a1. Note that, for the K distribution, theparameter 3 [=(a2

- 1)/2, where ao2 = (I2)/(I)2 - 1, (I) beingthe mean and (I2) the second moment of irradiance fluctua-tions] has a range of values 3/2 Ž 2 Ž 0. These limits of valuesof # set the range of value of r 3 to be 2 < r 3 < 5.875. For theparticular experiment described in this paper, this corre-sponds to six to thirteen paths. Therefore, for up to fivepropagation paths no K distribution fit is possible. Also, forthe Rice-Nakagami distribution, r3 < 2 and 0 ' r 4 < 6.Therefore the Rice-Nakagami distribution can fit the dataonly up to four paths.

For a1 < 0.5, the data for r3 -r8 do not coincide with eitherthe log-normal or the Rice-Nakagami distribution, althoughthe log-normal distribution appears to be closer. This is thecase in which +\/EX < lo, lo being the smallest-scale size ofturbulence, and, for the inhomogeneities in that region, ageometrical-optics approximation can be applied. For a1 >

Arun K. Majumdar

1070 J. Opt. Soc. Am. A/Vol. 1, No. 11/November 1984

107

106

105

I

acn

LU

U_

uJ

CM

CD

I

104

103

102

101

100 1

10 -1L

Arun K. Majumdar

0.25 0.5 0.75 1.0 1.25 1.5

01 = (1.23 Cn2 k7 16 11116)112

1.75

Fig. 1. Plot of higher-order coefficients r3 -P8 as a function of the MSP parameter al. Comparison of experimental points with three modeldistributions.

0.5, the higher-order coefficients increase with a,. It is in-teresting to determine the slopes of the curves for variouscoefficients in different regions of a,. The slope or the ex-ponent a in r, ala appears to increase with the order of thecoefficients. After a value of approximately a, a11 = 0.8,

the slopes of the curves for the data points become less but stillincrease slightly with the order of the coefficients. The slopes(or the exponent a) were computed from a plot of log rP,versus log a1. Figure 2 shows some examples of this compu-tation for r 6, F7 , and r8. Different values of the exponent

Vol. 1, No. 11/November 1984/J. Opt. Soc. Am. A 1071

a (or p) for a, (or L) variation (i.e., Fn a ia or nP- LP) for

regions I and II (as shown in Fig. 2) are summarized in Table

2. From Fig. 1, again, in the region a, > 0.5 we have >XL>

lo. Compared with the predicted theoretical models, themeasured data points in the region 0.5 < al < 0.8 are close tothe log-normal distribution. This is still in the weak-turbu-lence regime. For a, > 0.8, the data appear to lie in between

the K distribution and the log-normal distribution, but moreclosely to the K distribution. This technique of analyzinghigher-order coefficients is useful in determining the depar-tures (or the closeness) of measured data from the proposedmodel distributions. However, there is a large amount of

scatter associated with the higher-order coefficients. Thereason that the amount of scatter increases with the order ofthe coefficients is because of the finite number of samples used

during the measurements to establish the value of thehigher-order coefficients, which depend strongly on the values

of the irradiance corresponding to the tails of the distribution.The tails of the distribution are noisy, and a large number of

samples is required to establish the higher-order coefficientsaccurately. By taking more sample points and by utilizingstatistical signal-processing schemes, the scatter may be

minimized. Also note that, for rF, some of the data points ofthe model distributions (the log-normal and the K distribu-tions) were quite large and were outside the range of the curves

shown in Fig. 1. After a, = 1.04, we approach the strong-turbulence regime. In this region, there are large departuresof the data points from the log-normal distribution, but thedata points are closer to the K distribution. For a1 > 1.39,the data tend to saturate, and the coefficients start to decreasebut at a slower rate. Compared with the log-normal distri-bution, the data still are closer to the K distribution. How-

715- Cr6orier 6

0 0~~~~~~~~~~~~~~0~~~~~~~~~~~~~~

b 0~~~~~

WEGION I ~REGION 1 1Sj I III*II1:III:I

= ~ ~~ 04060 0012 0 64 0656 0 46 0 40 0 32 0024 0 16 006 0 0 06 016 0024

11 ! f

Fig. 2. Plot of log I,, versus log a,. Examples show the computa-tions of exponent a (for P, -ala) for n = 6, 7, 8 for region I (0.50 <ao < 0.8) and for region II (0.8 • al • 1.37).

ever, neither the log-normal nor the K distribution representsthe measured data points exactly.

B. Comparison with a Universal Statistical ModelHigher-order coefficients also have been computed for a re-cently proposed USM7 applicable under all conditions ofturbulence for which data are available. The distribution wasderived under the assumption that the field of an optical beampropagating through a turbulent medium is composed of twoprincipal components: a specular (or line-of-sight) compo-nent and a diffuse component (from off-axis scattering terms),each of which has an amplitude that is m distributed. Theprobability density function of this USM is given by 7

=IMm) exp(-mI/b) F_ k! Lk(mI/b)bF (M) kok

k Fk\2 (M + j)(rm/M)i

j 0 (j r)m(m-k+j)(9)

where b = (R 2 ), R being the amplitude of the diffuse com-ponent; m and M are the parameters of the distributions de-scribing the diffuse and the specular components, respectively;and r is the power ratio of the specular-to-diffuse componentsof the field. Lk is the kth Laguerre polynomial, r is thegamma function, and (k) = k!/j!(k - j)! is the binomial coef-ficient.

The higher-order coefficients r3 -r 8 were calculated for thisUSM by using the normalized moments. In order to comparethe higher-order coefficients from this model with the threeparameters m, M, and r, the normalized second- and third-order moments were matched identically with the measureddata. To obtain the three parameters (m, M, and r), it isnecessary to solve simultaneous nonlinear equations. How-ever, we used a simpler graphical method to obtain m, M, andr. Note that the normalized moments are given by7

(In) 1 (fln)2 k(I) n (1 Yk=O Ik I (10)

where

r(m + k)mkr(m)

and

r(M + k)ak - Mkr(M)

By introducing the notation Nn = (In)/(I) n as the nor-malized moment, the second and third normalized-momentsequations from Eq. (10) were solved to obtain the followingsimultaneous equations:

Table 2. Summary of Exponents a (or p) for Various Higher-Order Coefficients

Slope of Exponent a for al Slope of Exponent p for Path L

Higher-Order (i.e., rn a-t) (i.e., rn - LP)

Coefficient Region Ia Region III Region I Region II

r3 1.983 1.772 1.817 1.624

r 4 5.233 2.96 4.80 2.71

r 5 5.427 3.11 4.97 2.85

r 6 7.71 5.43 7.07 4.98

r 7 8.49 6.14 7.78 5.63

r8 11.54 8.51 b 10.58 7.80

a As shown in Fig. 2.b Estimated from large scattered data.

Arun K. Majumdar

1072 J. Opt. Soc. Am. A/Vol. 1, No. 11/November 1984 Arun K. Majumdar

r2 + 2 -4)r+ ()

and

-(33 + 9r2) + I(3r3 + 9r2)2-8r3[r3 - 3(1 + r)3+ 9r2+ 9r + m+ + m) (, + 2)!I1 , (12)

where x = 1/M, M being the parameter of the USM.By using various values of the parameter m and by then

plotting x versus r from Eqs. (11) and (12), the values of the

_ a - 1.504

zol - 1.156

g el - 0.918

\\ ..-.o10614

parameters x (= 1/M) and r were determined from the inter-section of the two curves. Thus the parameters m, M, and rwere estimated for a particular value of al (as defined above).Various sets of the parameters m, M, and r were thus deter-mined graphically for various values of a1. Four sample ir-radiance histograms, which span the al range for which theparameters m, M, and r are determined, are shown in Fig. 3.Parameters of the USM as a function of al are shown in Fig.4. Using these parameter values, the theoretical curves forr 3 -r8 for this universal model were drawn as a function of al(Fig. 5). Note that the measured data are close to the theo-retical curves predicted by this USM. The values of the threeparameters for al below 0.67 were not easily matched by usingthe graphical method. A thorough computer method forsolving simultaneous nonlinear equations is necessary to de-termine the parameters for all ranges of al. Compared withFig. 1, we note from Fig. 5 that this universal model predictstheoretical curves that are closer to data points than thosepredicted by log-normal, Rich-Nakagami, and K distribu-tions. This USM seems to be a more versatile model withwider ranges of applicability, covering weak- to strong-tur-bulence regimes.

IRRADIANCE (ARBITRARY SCALE)

Fig. 3. Four sample histograms for various values of al.

102

101

, loo

lo-l

N _ aM

N=, ~:z: A

N

__ I \\...

\\.

I~~~~~4-."~~~~~~~

I

lo-2 L-

Fig. 4. Parameters m, M, and r of the USM as a function of al.

10

. r3. r

4

A "5So r6A r7So 18

01

,10

/

"' /0",d

r7103

101

loo0.25 0.5 0.75 1.0 1.25 1.5 1.75

di - 0.23C2 11161.ll6)II2

i`5

r4

0.

P 4 "3

I I I30,75 1.0 15

l - 1.23 C4

20

716L1116(112

I1. 1.5

5

iI

I

7

lo,

r'6

7-

Fig. 5. Plot of r3-r8 as a function of or, for the USM.

Vol. 1, No. 11/November 1984/J. Opt. Soc. Am. A 1073

4. CONCLUSIONS AND DISCUSSION

Higher-order coefficients (up to eighth order for practicalpurposes) have been utilized to analyze particularly thenon-Gaussian statistics. These coefficients are different fromthe normalized moments reported by other researchers. Thehigher-order coefficients r 3 -F8 , as defined in this paper, cangive more insight into the complex problem of non-Gaussianprocesses applicable to many realistic situations. An exampleof analysis of irradiance fluctuations through turbulence isconsidered here. We have shown that the theoretical analysisand the modeling extend in a straightforward way for in-creasing order of moments. Although the model theoreticallyyields a more exact fit of the random variable being estimated,practical limits arise in forming reasonable accurate samplemoments. The accurate measuring of moments as high as theeighth moment would require a large number of samples; this

means that the sampling rate has to be increased beyond theusual Nyquist rate. The accuracy of the measured higher-order moments will thus depend on the increased samplingrate and hence is limited by the duration of the experiment.The large scatter of the data points with increased higher-order coefficients can be minimized by increasing the samplingrate and by using better signal-processing schemes.

It is necessary to measure the higher-order coefficients,which depend strongly on the values of the irradiance corre-sponding to the tails of the distribution. For example, thespiking nature of a random field of irradiance at the receiverplane can -be detected by measuring the higher-order mo-ments. The spikes also observed' 3 in the time derivative ofturbulent velocity fluctuations constitute the principal non-Gaussian contribution when the spikes occur in clusters. Inthe design of a laser communication and lidar system oper-ating in the lower atmosphere, the spikes of the random ir-radiance function of the receiver aperture plane may be acritical factor in the performance of the receiver system. Thisis important when the mean level of irradiance after propa-gating through a turbulent atmosphere is below the receiveraction threshold.

The information obtained from the higher-order coeffi-cients defined in this paper can be used to define an accurateirradiance probability model, for example, for laser fluctua-tions through turbulence.

In order to assess the utility and the applicability of thishigher-order analysis, higher-order coefficients obtained frommeasured data of a laboratory-simulation experiment werefirst compared with those for three proposed model distri-butions: K, log-normal, and Rice-Nakagami distributions.The higher-order coefficient curves as a function of al showsimilar trends of variation. For a, < 0.5, for which a geo-metrical-optics approximation can be applied, the distributionfunction is not close to any of the three model distributions.For 0.5 < a1 < 0.8, still in the weak-turbulence regime, thedata points are closer to the log-normal distribution. For a1> 0.8, the data appear to lie between the K distribution andthe log-normal distribution but are closer to the K distribu-ltion. For a, > 1.04, we approach the strong-turbulence re-gime, and there is a large departure from the log-normal dis-tribution. For at > 1.39, the data tend to saturate but are stillcloser to the K distribution. The measured data points donot, however, represent the exact K distribution or the log-normal distribution for all values of ,1.

Values of the exponent a (or p) in I> - ala (or Pn - LP)

were obtained from the measured data for both regions I andII (defined above) of the turbulence. The slopes and the ex-ponents appear to increase with the order n of the coefficients.Furthermore, the rates of increase of the slopes in region II areslower than those in region I. These exponent parameters a(or p) in urc, (or LP) can be quite useful for accurate modelingof the complex process of optical scintillations under all con-ditions of turbulence and for predicting a universal model forall the data.

The higher-order coefficients obtained from measurementswere also compared with a USM containing the three pa-rameters m, M, and r. A graphical method was used to solvethe simultaneous nonlinear equations involving second andthird normalized moments to obtain the set of values of theparameters for each a1 . The theoretical curves were thendrawn for these higher-order coefficients defined by thisuniversal model. Compared with the other models (log-normal, Rice-Nakagami, and K distributions), this modelappears to fit the experimental data reasonably well. Thescatter of the data is much less for this model compared withthe other models. This model seems to be potentially appli-cable under all conditions of turbulence from weak to strongin the saturation regime. However, a set of simultaneousnonlinear equations needs to be solved to obtain the threeparameters exactly.

APPENDIX A: HIGHER-ORDER COEFFICIENTS

FOR THE K DISTRIBUTION

The coefficients F4-178 as defined in this paper are derivedbelow for the K distribution. The probability density func-tion of irradiance I can be written as6

P() = ( ) (Y+l)/2 I(Y-1)I2K yji(2v1 y7(7 ) (Al)

The parameter of the K distribution is given by y = 2/(U2- 1), where a 2

= (12)/(1)2 - 1. Note that the K distributionexists only9 for a2 > 1 and, from the definition of y and theobserved values of a2 at and beyond the maximum (1 < a 2 <

4), that y has a range'4 of values 2/3 < y < a. Although, inprinciple, y can theoretically have the range 0 < y < a, Ky_is the modified Bessel function of the order y - 1. The mo-ments of this distribution are given by the following expres-sion:

(I)k k! F(k + y)Yk F (y) (A2)

By using the relation between moments and central mo-ments

Mr = E ) Mr-j(_Ml)jj=0

(A3)

and by substituting 3 = 1/y (and calling 3 the parameter forK distribution), the expressions for r 3-r 8 have been derived.Note that 3/2 >: ' Ž 0. The coefficients are

I =2(6/2 + 60 + 1)(1 + 20)3/2

24(1 + 20)!(33) + (9 + 123 _)(I + 2/3)2

(A4)

(A5)

Arun K. Majumdar

1074 J. Opt. Soc. Am. A/Vol. 1, No. 11/November 1984 Arun K. Majumdar

where we define

(1 + Na)b! e (I + 0)(1 + 2i)(1 + 3ni ... (1 +dNO). (A6)With the above definition,

r 120(1 + 3fl)!(40) + 60(1 + 2)!-20(1 + a) + 4[6(1 + 1)(20) + 2](1 + 213)

- 10,

r 720(1 + 413)!(513) + 360(1 + 33! - [120(1 + 21)!] + 30(1 + 1) - 5(1 + 2)3 - 15,

- 5040(1 + 5/3)!(61) + [1680(1 + 31)!](1 + 61) + 210(1 + 21)! - 42(1 + 1) + 62(6132 + 613 + 1)(1 + 213)2 105,

r 40320(1 + 613)!(7/) + 20160(1 + 513)! - [6720(1 + 413)!] + [1680(1 + 313)!] - [336(1 + 213)!] + 56(1 + 3) - 7(1 + 2_)4

(A9)

105. (A10)

ACKNOWLEDGMENTS

I wish to thank Hideya Gamo of the University of California,Irvine, California, for his interest and valuable discussions,Glenn E. Bowie of Lockheed-California Company for en-couragement and helpful comments, and Loren Clare of theJet Propulsion Laboratory, California Institute of Technology,Pasadena, California, for technical discussions.

REFERENCES

1. A. K. Majumdar, J. Opt. Soc. Am. 69, 199 (1979).2. A. K. Majumdar, Opt. Commun. 50, 1 (1984).3. A. K. Majumdar and H. Gamo, Appl. Opt. 21, 2229 (1982).4. A. K. Majumdar and G. H. Fortescue, Appl. Opt. 22, 2495

(1983).

5. V. I. Tatarskii, "The effects of the turbulent atmosphere on wavepropagation," TT-68-50464 (National Science 'Foundation,Washington, D.C., 1971).

6. E. Jakeman and P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978).7. R. L. Phillips and L. C. Andrews, J. Opt. Soc. Am. 72, 864

(1982).8. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pok-

asov, in Laser Beam Propagation in the Atmosphere, J. W.Strohbehn, ed. (Springer-Verlag, Berlin, 1978).

9. G. Parry and P. N. Pusey, J. Opt. Soc. Am. 69, 796 (1979).10. G. Parry, Opt. Acta 28, 715 (1981).11. R. L. Phillips and L. C. Andrews, J. Opt. Soc. Am. 71, 1440

(1981).12. See, for example, R. Betchov and C. Lorenzen, Phys. Fluids 17,

1503 (1974).13. R. Betchov, Arch. Mech. Stosow. 28, 837 (1976).14. S. F. Clifford and R. J. Hill, J. Opt. Soc. Am. 71, 112 (1981).

(A7)

(A8)