Transformation of statistical characteristics of picosecond

laser pulses by multiple-scattering media

Arun K. Majumdar

Higher-order (up to eighth) temporal moments to characterize non-Gaussian picosecond laser pulses were

measured in a laboratory-simulation experiment for laser communication through multiple-scattering media.

A technique employing deconvolution of higher-order moments (taking into account the impulse response of

the detector and sampling system) is described to determine the true moments from the observed sampling

scope traces of the pulse shapes. Measuring the broadening of the original pulses (-20-ps pulse width)

generated with a single spatial mode GaAlAs laser diode by current modulation gain switching method, the

temporal moments of order n = 2-8 were analyzed as a function of optical depth - of the multiple-scattering

medium (consisting of latex spheres suspended in water) (5.5 < r < 10.31) and receiver's field of views (FOV)

ranging from 0.503 to 2.10. Higher-order temporal moments curves are presented as a function of both

optical depth and FOV showing the general trend of the moments initially to rise quickly and finally to attain

saturation or limiting values. A simple empirical relationship of a polynomial function of the order of 3 is

obtained to express the broadened pulse width (n = 2) in the range of optical depth and FOV considered here.

The results described in this paper clearly indicate the significance of the contribution of the longer part of the

non-Gaussian tails of the broadened pulses to practical design problems in all-weather atmospheric optical

communications.

1. Introduction

The recent advances in picosecond lasers have re-ceived increasing attention in connection with the de-velopments of very high digital data rate with atmo-spheric optical communication, precise ranging, andthe investigation of remote sensing of planetary atmo-sphere. The received laser pulses after propagatingthrough a temporally dispersive atmospheric channelsuffer distortion, and a delayed tail appears as a resultof multiple scattering. Thus in a pulse position modu-lation (PPM) format for short-pulsed laser communi-cation, for example, the received stretched pulse shapecauses pulse spreading into adjacent slots (interslotinterference) and, if severely spread, can overlap intoadjacent frames (intersymbol interference).1

The present paper reports new results on the trans-formation of statistical characteristics of picosecondlaser pulses due to multiple-scattering effects in a lab-oratory-simulation experiment in terms of higher-or-der (up to eighth-order) temporal moments. The idea

When this work was done the author was with Lockheed-Califor-

nia Company, Kelly Johnson Research & Development Center at

Rye Canyon, P.O. Box 551, Burbank, California 91520; he is now

with University of Colorado at Denver, Electrical Engineering De-

partment, Denver, Colorado 80202.

Received 9 June 1986.0003-6935/86/244649-07$02.00/0.© 1986 Optical Society of America.

of temporal moments to describe the pulse signals wasproposed by various workers, for example, Baird,2 An-derson and Askne,3 Yeh and Liu,4 and Ito and Fur-utsu.5 But their papers did not contain experimentaldata of moments higher than the second order, which isreported for the first time in this present paper. High-er-order moments contain the information about theskewness and other higher-order properties of thepulse. These temporal moments can be treated in thesame way as other statistical parameters previouslyreported: for example, higher-order statistics of laser-irradiance fluctuations due to turbulence.6 7 Experi-mental results of transformation of temporal charac-teristics as a function of optical depth of the multiple-scattering medium and the receiver field of view (FOV)are presented in the present work. Because of theimperfections of the acquisition device and measure-ment system, the available data do not represent theoriginal waveform.

Therefore, to determine the correct higher-ordertemporal moments from the observed stretchedpulses, deconvolution of the time domain waveforms isnecessary. Equations relating the true higher-ordermoments from the observed sampling scope traces ofthe pulse shapes are derived taking into account theeffective impulse responses of both the detector andsampling head. The technique can be useful for ana-lyzing and deconvolving the higher-order temporalcharacteristics of non-Gaussian pulses to gain detailedinformation about broadening mechanism.

15 December 1986 / Vol. 25, No. 24 / APPLIED OPTICS 4649

SAMPLING SCOPEL

OUTPUTJ ) SCATTERING -1 N DETECTOR S SAMPLING L ) t

MEDIUM i- SYSTE M I

h(tl-

_-Fig. 1. Various pulse shape waveforms in the experimental ar-

rangement.

The importance of evaluating higher-order tempo-ral moments lies in the fact that the information can bevery useful in many practical problems, e.g., to solvethe boundary-value problem necessary for modelingthe complex problem of ultrashort laser pulse propaga-tion in a multiple-scattering medium and also to recon-struct the original pulse back to increase the SNR inall-weather atmospheric optical communications.

11. Higher-Order Temporal Moments in Optical PulsesThe concepts of temporal moments have been used

for describing quantum mechanical packets,2' 8 wavedispersion,3 waves in random medium,4 and diffusionequation.9 The temporal characteristics of picosec-ond optical pulses (after propagating in a multiple-scattering medium) can be described by the nth tem-poral moment defined by

(tn) = J tnf(t)dt/f f(t)dt,

the sampling scope, and h(t) is the effective impulseresponse of the combination of the detector and sam-pling head. Note that two conditions need to be im-posed so that Eq. (1) can be interpreted physically:(1) the normalization condition, normalized with re-spect to the area under the curve of the temporal pulse;(2) the time origin condition so that the pulses start attime t = 0, providing a convenient time reference.Thus (t) can be interpreted as a mean arrival time andthe mean-square pulse width (which may be used as ameasure of the pulse broadening) is expressed as

(At2 ) = (t 2) - (t)2 .

Note that the observed higher-order moments com-puted directly from the sampling scope pulse shapesg(t) are different from the moments of f(t), which arethe true moments of the multiple-scattered pulses.To derive the relationship between the observed andtrue temporal moments, let m, pn, and C be the nthmoments (not normalized) of true pulse shape, f(t), theeffective impulse response h(t) of both the detectorand the sampling head and of the observed pulse shapeg(t), respectively.

The various temporal moments are then given by

(1)

where f(t) is the pulse shape under study which isobtained by broadening of an initial short laser pulseM0(t) due to multiple-scattering process. The mo-ments defined by Eq. (1) can also be expressed in termsof the Fourier transform F(v) of f(t) as follows:

(tp) = (/aV)nF(v)j.= 0 /F()I_0. (2)

The expansion coefficients defined by9

Xn = (JO/l/)0 logF(v) =o (3)

are related to the moments (tn) by the simple relationXn= ((t- (t))n) (4)

so that F(v) can be expressed as

F(v) = exp[x + xi(-iv) + IX2(jv)2

+ 1! X3(-iV) + * * * ]} (5)

and F(v) can thus be evaluated from the temporalmoments using Eq. (4). Therefore, evaluating thehigher-order (for n up to 8 for practical purposes)temporal moments can be reduced to solve the bound-ary-value problem for this complex multiple-scatter-ing situation and to determine the Fourier transformF(v).

Figure 1 shows the various pulse shape functions atthe various points of the experimental arrangement.The pulse f(t) is detected by a high-speed photodiodeand observed with a sampling scope with a fast sam-pling head. Let g(t) be the observed pulse shape on

Jn = ftnf(t)dt,

Pn = f.tn h(t)dt,J

C0 = J tng(t)dt.

(7)

(8)

(9)

Note that g(t) = f(t) h(t), where denotes convolu-tion. Taking nth-order moments of both sides of theconvolution and changing the variables,

Jo. tng(t)dt = | (x + T)nfXh(T)d-dx.

Expanding (x + r)n binomially and integrating sepa-rately with respect to the variables x and , from Eq.(10),

Cn = mnh(r)dT + nn-1 J Th(T)dr

+ n -2) Th(T)d2! Jo

+. . . +... + o nh(r)dT.

Thus the observed moments (Cn values) can be ex-pressed in terms of the true moments (n values) andthe moments of the impulse response of the instru-ments (Pn values) as follows:

Cn = mnP + nn-lPl

+ n (n 1) n-2P2 +* + . mopn- (12)

Note that in the experimental setup the values C, C1,C2, . . , are known from the measured moments of theobserved pulse shapes directly from the samplingscope; pO, p1, P2, . . , are also known or estimated from

4650 APPLIED OPTICS / Vol. 25, No. 24 / 15 December 1986

(6)

(10)

(11)

the impulse response of the photodetector and thesampling head from independent measurements; mo,Ml, M2, . . , the true moments of the scattered pulse,are determined from Eq. (12). When the moments areall normalized with respect to their respective zeroth-order moment, i.e., let

m n = Mn/Mo, (13)

Pn P0/Po' (14)

Cn = Cn/C0 , (15)

then the following relation is obtained (note that C0 =

mop0 ):

-- (n -1)- C m = M + nmn-l1pl + n2! mn-2P2 ... + ... + Pn- (16)

Equation (16) is the relationship between the observedand true temporal moments. This equation will beuseful in determining the true moments up to eighthorder from the experiment to be described in the nextsection.

For a detailed comparison of the different experi-mental pulses scattered through the medium in vari-ous conditions of the optical depth, the normalizedratio of the temporal moment [Mn] was then computedwhere

[mn] = (mn)/(m )n. (17)

Note that the calculation of higher-order momentsfrom a finite length sample leads to errors, which in-crease as the order of the moment n increases. Thecontribution to the moment of order n of longer part ofthe delayed tails of the distorted temporal waveformincreases as n increases. Therefore, care must be tak-en to measure the broadened pulse as accurately aspossible.

The impulse response (effective) h(t) of both thedetector and the sampling head was determined fromthe pulse response when the photodiode was illuminat-ed by a picosecond mode-locked dye laser input. Thefunctional form of h(t) was found to hold the followingempirical relationship:

h(t) = A1 exp[-c(t - to)2] sinbt, (18)

where Al, c, to, and b are constants. The numericalvalues of c, to, and b were determined by comparing themodeled output with the observed output using a dyelaser input and is described in detail somewhere else.'0

In the present work, the moments of the combinedimpulse response h(t) of both the detector and sam-pling head were computed directly from Eq. (8) bynumerical integration with a computer.

Since the higher-order moments are related to tem-poral distortions of the pulses, the transformation oftemporal characteristics of the picosecond laser pulseswere investigated as a function of optical depth of themultiple-scattering medium and also of the receiverFOV. The experimental results will be described inthe next section.

111. Experiments

A. Picosecond Optical Pulse Generation with a

Semiconductor Diode Laser

Since to produce extremely short pulses was not arequirement for the experiment, the generation ofbroader pulses (-10-30 ps) was acceptable. The cur-rent modulated gain switchingll1" 2 method was used inthe present experiment. A very low threshold (-18mA), highly efficient, single spatial mode GaAlAs laserdiode (Ortel Corp. LDS 10-OMF, X = 0.837 !rm) wasdriven by a step recovery diode (Hewlett Packard33002A). The laser diode was mounted in an imped-ance matched high frequency package, which includedan integral optical power-monitoring photodiode.The laser emitted up to 10 mW of optical power andhad a direct modulation bandwidth of up to 6 GHz.The input to the step recovery diode was a 100-MHzoscillator operating at -1 W, and the output was atrain of narrow, high amplitude negative pulses at thesame frequency. The laser was driven with a very low(finally without a) dc bias below the threshold and wasultimately excited only by the step recovery diode.This process finally generated a continuous train ofoptical pulses at the repetition rate of 100 MHz. Vary-ing the dc bias, the intensity of these pulses could becontrolled.

B. Scattering System

A scattering cell (5 cm in diameter, 7.4 cm in length,with optically flat entrance and exit windows) was usedto simulate a low visibility atmosphere. The construc-tion and design of the cell were similar to that reportedearlier.13 Uniform latex microspheres (Dow Chemi-cal) were used as scatterers in the experiment. Thediameters of the microspheres were 0.176 Am (lot2M4V), 0.312 ,m (lot 1A72), 1.091,um (lot 5866), 2.99Jim (1A37), and 6.4 ,m (lot 1A93). The solutions wereprepared in the same way as described in Ref. 13.Each solution consisted of monodisperse particles.Desired densities and optical depth were prepared bydiluting concentrated solutions in the latex spheres(10% by weight) with distilled and deionized water.From the empty cell and the scattering cell measure-ments of the received signal of the laser as a function ofFOV, the optical thickness was determined by extrap-olating to a zero FOV value of the ratio of Io (intensitywithout medium) and I (intensity with medium), usingthe relation X = [ln(IoII)]Fov- o-

C. Experimental Setup

Figure 2 shows the photograph of the experimentalsetup for measuring the distorted and broadened opti-cal pulses after propagating through a multiple-scat-tering medium. Picosecond optical pulses (-10-25ps) generated by impulse train current modulation of asemiconductor laser (as described earlier) were colli-mated by a microscope objective (20 X, N.A. = 0.5).The collimated beam, 1 cm in diameter, was transmit-ted through the scattering cell containing the uniformlatex microspheres suspended in water. At the exit

15 December 1986 / Vol. 25, No. 24 / APPLIED OPTICS 4651

window of the scattering cell, there was a variable iris.This variable iris provided the receiver FOV to varyfrom 0.215 to 2.10, so that the temporal characteristicsof the multiple-scattered optical pulses could be re-corded with increasing FOV. The scattered light wasfinally collected by an IR lens system (consisting of acombination of collimating and focusing lenses) andwas focused on the detector. The detector was a highspeed GaAlAs/GaAs PIN photodiode (Ortel Corp. PD050-OM). The photodiode had a typical rise time of50 ps, a 3-dB bandwidth of >7 GHz, and quantumefficiency of 65%. It also featured an extremely lowcapacitance and dark current. The photodiode wasoperated in an ac coupled mode using a circuit wherethe dc and ac photocurrents were decoupled with amicrowave bias T. The amount of incident opticalpower could be determined by measuring the currentalong the dc bias path. The ac coupled signal ap-peared at the rf output of the bias T.

The temporal intensity profiles of the multiple-scat-tered pulses, obtained from the detector system, wererecorded with a sampling oscilloscope (Tektronix7904) with the 7S11 and 7T11 sampling sweep unitsand a S-4 sampling head. This sampling head had arise time of 25 ps. The optical pulse waveforms wererecorded, and photographs were taken for the -timehistory of the optical pulse for each FOV and each setof optical depth of the scattering medium. The outputdata curves from the photographs were later placedinto the CADAM computer graphics system via the digi-tizer PAD. This computer system allowed the auto-matic scalings of all the curves and provided the gener-ation of a set of points along those data curves. Theinformation was then stored on computer tape andtransferred to the Xerox Computer System for furtherprocessing and analysis. Moment programs werewritten in the VAX computer system and were utilizedto compute the higher-order temporal moments of thebroadened pulses from the data points.

IV. Results and Discussion

The effective impulse response h(t) of the combina-tion of the sampling system and the photodetector wasdetermined from the deconvolution of the temporalwaveform when the photodiode was illuminated by apicosecond mode-locked dye laser.'0 The result wasthe following:

h(t) = Al exp[-2.6363 X 10- 4(t - 94.412)2]

X sin(1.6244 X 10-2

t), (19)

where A is a constant, depends on the input power,and t is expressed in picoseconds. Temporal momentsof orders n = 1-8 were evaluated for the instrumentimpulse response, and the true moments were thencomputed from the observed moments following themethod described earlier in the text. Figure 3 shows atypical sampling scope recording of the time history ofa set of laser pulses after propagating through variousscattering media of optical depths ranging from - =5.35 to 10.31 for a fixed FOV of 2.10. The ringing of

Fig. 2. (a) Photograph of experimental setup; (b) diode laser mountwith step recovery diode and microscope objective; and (c) detector

mount with collimating and focusing IR lens system.

-.1 I-sopS FOV = 2.10

T= 535

_ 2 mVTT = 7.05

T T= 7 31

~~~71T r= 7.5 5

111= 8.19

T = 10.31

Fig. 3. Sampling scope traces of the time history of a set of laserpulses for optical depths: T = 5.35, 7.05, 7.31, 7.55, 8.19, and 10.31

and field of view of 2.10.

the pulses, which is the oscillatory transient, is mostlydue to the characteristics of the sampling oscillo-scope.'4"15 In general, the pulses propagating throughthe multiple-scattering medium are attenuated andbroadened simultaneously with increasing opticaldepth. The scattered pulse shapes are generallyasymmetrical in nature, and the degree of asymmetryincreases with the increasing optical depth of the me-dium. Note also that the displays of Fig. 3 do notrepresent the true scattered pulse shapes but ratherqualitatively the pulse shapes convolved with the in-strument (sampling scope and photodetector). Thetrue higher-order temporal moments of the pulsesbroadened by the multiple-scattering medium wereobtained from the observed moments by deconvolvingthe latter using the method described earlier.

Figures 4-6 show the measured higher-order tempo-ral moments for n = 3-8 as a function of optical depthfor three FOVs of 0.503, 1.09, and 2.10, respectively.

4652 APPLIED OPTICS / Vol. 25, No. 24 / 15 December 1986

(b)

106

10

103

07 /

10/

I? 31m3

6 7 9 9 10 11OPTICAL DEPTH. T

Fig. 4. Higher-order temporal moments (of order n = 3-8) of scat-

tered pulses vs optical depth for FOV = 0.503.

108

FFOV 1 a0cc9 td [n n g

mE* 106[r7I E

z 0~~~~~~~~~~~~~060

io4~~~~~~~~~U~

0.

0~~~~~100

Fo W ie FO102 tmorlmmnsinraewt

fg 5.hige-rertmoa moments vs optical depth forcte that =h

For a fixued fiedove, the tep ralmoment ineset withsthe morerapdl ofth momrents.g otia meanetha the-contribution fc tht longtai of the lse saee becomesigneifinl imortant.e For acfixed. FOVithe increase

causoe of the uthtlesftedh scatere ucetid

1E 108 |[-71

16 107

z15 1060

04

03

102

OPTICAL DEPTH, T

Fig.6. Higher-order temporal moments vs optical depth for FOV =2.10.

with increasing optical thickness with the result thatincreasing broadening and hence increasing temporalmoments were observed. Furthermore, it is also ob-served from Figs. 4-6 that the temporal moments atfirst increase with optical depth quickly and then in-crease later slowly tending to reach finally to satura-tion values for large optical depths. Increasing thereceiver FOV to 2.10, the moments tend to attain satu-ration values which occur at the optical depth around r= 7.6. As far as the shape of a pulse is concerned, thismeans that a limiting asymmetry (i.e., non-Gaussiannature) of the pulse broadening was finally obtained.If the receiver FOV is kept on increasing, the contribu-tion of the scattering component of the received signalpulse intensity will finally reach limiting values, butthe noise component will gradually increase with in-creasing optical depth, reaching to a point where thelimited dynamic range of the detection system canmake it difficult to measure small scattered intensity.Therefore, the received broadened pulse is actually theresultant convolution including the contributions ofthe noise component. In the discussion, the possibleeffects of noise on the temporal moments are ignored.Effects of noise in a noisy environment on the nth-order temporal moments can, however, be evaluatedassuming the received random signal to be statisticallyindependent of the noise and are not considered here.

A series of pulse broadening measurements was alsomade by varying the FOV of the receiver. Figures 7and 8 show the results for the temporal moments as afunction of FOV. When the optical depth, r = 5.01,the moments increases at first quickly and then fallsoff slowly after attaining maximum values at a FOV -1.260. For a larger optical depth of -X = 10.31, the

15 December 1986 / Vol. 25, No. 24 / APPLIED OPTICS 4653

u u./ U.4 0.6 0.8 1.0 1.2 1.4 1.6FIELD OF VIEW FOVI. DEGREE

FOV

0 0.321°

0 0.503° Ad \CURVEA 2.1° / FITTING

30/

/ UP' CURVE,/ / FITTING -

0 -' o'/

FITTINGCURVEORIGINAL FITTINGPULSE

0 WIDTH

0 - A _ _ _

A

V

-

0u

3

2

1.8 2.0 2.2 2.4

Fig. 7. Higher-order temporal moments of the scattered pulses vsFOV of reception for optical depth, r = 5.01.

1011,= 10.31

100

106 0. 0. 0.1.510125.76182. .

SE

C 106

0 1o 5 1 c 1

2 0 3

0 0.2 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.8 2.0 2.2FIELD OF VIEW IFOVI. DEGREE

Fig. 8. Higher-order moments vs FOV for optical depth,,r= 10.31.

temporal moments curves initially rise quickly, butlater the rate of increase is slower and finally the curvesare saturated. The saturation values start at a smallerFOV of P This means that by increasing the FOV,the contribution of the multiple scattering to the longtails of the scattered laser pulses are more pronouncedat the beginning, after which a limiting value of FOV isattained; increasing FOV further, photon historieswith larger path lengths do not contribute very muchto the longer tails of the asymmetric pulses. LargerFOV receivers will give larger signals but will alsoadmit relatively more background noise which can besignificantly important beyond a certain limit due to asaturated spatio-temporal spread. This is importantfrom the design point of view because of the fact thatnot much is to be gained by increasing FOV muchbeyond the limiting value of the FOV, if one wants to

1 2 3 4 5 6 7

OPTICAL DEPTH, T

Fig. 9. Scattered laser pulse width vs optical depth T for three fieldsof view: FOV = 0.321, 0.503, and 2.1°.

reconstruct the complete pulse shape at the receiver toincrease the SNR. Furthermore, beyond a limitingFOV of the receiver, the received stretched pulse shapewill not cause any further pulse spreading into adja-cent slots in a pulse position modulation format tocause interslot or intersymbol interfaces.

One other important parameter for design purposesis the pulse width which is obtained from Eq. (6) usingthe deconvolved temporal moments of order n = 2.Figure 9 shows the pulse width vs optical depth, forfixed values of the FOV, indicating that the pulsewidth increases at a faster rate initially but later at aslower rate and finally attains limiting or saturatedvalues. As the FOV is increased from 0.321 to 2.1°, thepulse width increases with FOV, and the curves satu-rate at AT = 10.3. When these pulse width values aredivided into three subsets depending on the FOV, asimple polynomial function of order 3 fits all the re-sults. The pulse width can then be expressed in thefollowing form (FOV = 0):

8 9 10 11

(t2)1/ = a0 (O) + a 1(O)T + a2 (O)T2 + a 3(0)T3.

The polynomial coefficients for 5.5 < r < 10.31 aregiven in Table I. The curve fittings with the experi-mental values are quite close.

V. Conclusions

New experimental results on the transformation ofstatistical quantities of propagation of picosecond la-

Table 1. Polynomial Coefficients ao(O) ... Ca3(0) as a function of FOV =e for 5.5 < r 10.31

FOV

edegree ao(O) al(O) a2(0) a3(0)

0.321 195.7 -61.6 6.5 -0.21a

0.503 68.8 -29.3 4.8 -0.23

2.1 67.0 -33.6 6.2 -0.31

a For 7.0 < r < 10.31.

4654 APPLIED OPTICS / Vol. 25, No. 24 / 15 December 1986

SE

2

2

|T= 5.01|

: o103 / ^ t O I8

i>/ f ~~~~~~~~Vlmi71102

0[5-

A I515

o -0 -/0 ~0Iff31

100 Cl I I I I I I I I I

-50E

4

1

(20)

ser pulses by multiple-scattering media are reported.Higher-order (up to eighth) temporal moments wereintroduced to characterize the asymmetric broadeningof the pulses, e.g., the pulses with sharp rise but withlonger tail which typically occur in a pulse propagationthrough low visibility atmosphere. Temporal mo-ments of order n = 3-8 were analyzed as a function ofoptical depths (5.5 • - < 10.31) and FOVs rangingfrom 0.503 to 2.1°. The increase of temporal momentswith both the order n and the optical depth r indicatesthat the contributions of the longer parts of the tails ofthe broadened pulses can be significant. The experi-mental curves show the general trend that the mo-ments initially increase with optical depth quickly, butlater the rate of increase is slower and finally attainssaturation or limiting values. Temporal moments as aFOV function show similar trends, indicating thatphoton histories with larger path lengths (obtaineddue to the larger angle of acceptance) do not contributevery much to the longer tails of the asymmetric pulses.

From the results of pulse width broadening (fromsecond-order temporal moments) with optical depth, asimple empirical relationship of a polynomial functionof third order was shown to fit all the results in therange of the optical depth and field of view consideredhere.

Evaluating the higher-order (for n up to 8 for practi-cal purposes) temporal moments can be reduced tosolve the boundary-valve problem for analyzing thecomplicated problem of propagation of ultrashort la-ser pulses in a multiple-scattering medium. From thecommunication point of view, it is of great interest toreconstruct the original pulse back so that the receivedSNR can be greatly increased even if the pulse haspropagated through a multiple-scattering low visibili-ty atmosphere. The results described in this papercan also be useful in adaptive modeling of a multipathcommunication channel and in designing adaptive fil-

ters for improving all-weather optical communicationperformance.References1. G. Prati and R. M. Gagliardi, "Decoding with Stretched Pulses

in Laser PPM Communications," IEEE Trans. Commun.COM-31, 1037 (1983).

2. L. C. Baird, "Moments of a Wave Packet," Am. J. Phys. 40, 327

(1972).3. D. G. Anderson and J. I. H. Askne, "Wave Packets in Strongly

Dispersive Media," Proc. IEEE 62, 1518 (1974).

4. K. C. Yeh and C. H. Liu, "An Investigation of Temporal Mo-

ments of Stochastic Waves," Radio Sci. 12, 671 (1977).

5. S. Ito and K. Furutsu, "Theory of Light Pulse Propagationthrough Thick Clouds," J. Opt. Soc. Am. 70, 366 (1980).

6. A. K. Majumdar, "Higher-Order Statistics of Laser-Irradiance

Fluctuations due to Turbulence," J. Opt. Soc. Am. A 1, 1067

(1984).7. A. K. Majumdar, "Higher-Order Skewness and Excess Coeffi-

cients of some Probability Distributions Applicable to Optical

Propagation Phenomena," J. Opt. Soc. Am. 69, 199 (1979).

8. F. De Martini, C. H. Townes, T. K. Gustafson, and P. L. Kelley,

"Self-steepening of Light Pulses," Phys. Rev. 164, 312 (1967).

9. K. Furutsu, "Diffusion Equation Derived from Space-Time

Transport Equation," J. Opt. Soc. Am. 70, 360 (1980).

10. A. K. Majumdar, "Deconvolution of Non-Gaussian Temporal

Characteristics in Picosecond Laser Pulse Measurements," sub-mitted for publication (1986).

11. H. Ito, H. Yokoyama, S. Murata, and H. Inaba, "Picosecond

Optical Pulse Generation from an r.f. Modulated AlGaAs d.h.

Diode Laser," Electron. Lett. 15, 738 (1979).12. G. J. Aspin, J. E. Carroll, and R. G. Plumb, "The Effect of Cavity

Length on Picosecond Pulse Generation with Highly r.f. Modu-

lated AlGaAs d.h. Lasers," Appl. Phys. Lett. 39, 860 (1981).

13. A. K. Majumdar, "Laboratory Simulation Experiment for Opti-cal Communication Through Low-Visibility Atmosphere Using

a Diode Laser," IEEE. J. Quantum Electron. QE-20,919 (1984).14. G. Cowan, Tektronix, Inc., Beaverton, OR; private communica-

tion (1985).15. D. Bloom, Stanford U.; private communication from a short

course, "Picosecond Electronics," U. California, Santa Barbara(1985).

The author would like to thank K. Y. Lau of OrtelCorp. for helpful discussions regarding the picosecondpulse generation from the diode laser: R. F. Luto-mirski of Pacific Sierra Research Corp., Santa Monica;R. G. Bradley of Sandia National Laboratories, Albu-querque; and W. L. Gans of NBS, Boulder for technicaldiscussions; and Sharon La Traille, Dan Martin, andCharles Franklin for data processing, programming,and computation.

15 December 1986 / Vol. 25, No. 24 / APPLIED OPTICS 4655