Computer simulation of superposed coherent andchaotic radiation

Giovanni Vannucci and Malvin Carl Teich

A computer simulation technique useful for generating superposed coherent and chaotic radiation of arbi-trary spectral shape is described. Its advantages over other techniques include flexibility and ease of imple-mentation, as well as the capability of incorporating spectral characteristics that cannot be generated byother methods. We discuss the implementation of the technique and present results to demonstrate its va-lidity. The technique can be used to obtain numerical solutions to photon statistics problems through com-puter simulation. We furthermore argue that experiments involving photon statistics can be carried outusing a wideband source in place of an amplitude-stabilized source whenever the spectral characteristics ofthe source are not important. Experimental results that corroborate the argument are presented.

1. Introduction

The photon statistics generated by thermal andchaotic light sources have evoked substantial interestever since the well-known experiment by HanburyBrown and Twiss",2 demonstrated the occurrence ofphoton correlation in light generated by natural sources.With most natural sources, however, the time scale overwhich photon correlation is observable is typically veryshort and is often virtually negligible; indeed very sen-sitive and carefully designed experiments are usuallyrequired in order to observe such correlation. Corre-spondingly, the coherence time of the associated elec-tromagnetic field is very short for such sources, as thecoherence time provides a measure of the time scale overwhich variations occur in the intensity of the field.3

To circumvent this difficulty, a number of researchershave devised a variety of techniques to generate simu-lated chaotic light with an increased coherence time inthe laboratory. Most of these techniques require a laseras the primary source of light, since they make use of thenartow bandwidth and high degree of coherence of laserradiation. They generate chaotic light by simulating,to different degrees, the physical characteristics ofnatural sources of chaotic light. One such method 4 '5

When this work was done both authors were with Columbia Uni-versity, Columbia Radiation Laboratory, Department of ElectricalEngineering, New York, New York 10027; G. Vannucci is now withBell Laboratories, Holmdel, New Jersey 07733.

Received 14 July 1979.0003-6935/80/040548-06$00.50/0.© 1980 Optical Society of America.

consists of scattering laser light from particles sus-pended in a fluid, each particle acting as an independentradiator whose frequency is the laser frequency,Doppler-shifted by the Brownian motion of the particle.In this configuration, the spectral bandwidth and shapecan be adjusted to some extent by manipulating thestatistical distribution of the size of the particles.Another method consists of scattering laser light froma moving rough surface, such as a rotating ground-glassscreen6 7 or a rotating roughened whee8; here the dif-ferent surface elements act as independent radiators,and the bandwidth of the light can be adjusted byvarying the speed of motion of the surface. Reference9 provides a survey of the properties of scattered lightin general.

If only the intensity variations of chaotic light are ofinterest (as is the case for the statistics of photon ar-rivals), it is possible to use the method suggested byRuggieri et al. 10 to simulate a superposition of coherentand chaotic radiation. This method uses two inde-pendent generators of Gaussian random noise to sim-ulate the in-phase and quadrature components of theradiation field. The sum of the squares of these com-ponents yields the simulated light intensity, which isthen used to modulate the output of a laser source. Itshould be noted that this method is less physical thanthe others cited above, since only the intensity (ampli-tude) variations of a chaotic radiation field are repro-duced (the phase of the field depends solely on thecharacteristics of the primary laser source). Indeed, thenarrow bandwidth and high degree of coherence of laserradiation are not necessary in this type of simulation,since the laser is used only to generate photons with thestatistics of a Poisson point process whose rate can be

548 APPLIED OPTICS / Vol. 19, No. 4 / 15 February 1980

controlled by modulating the intensity of the radiation.An arbitrary source yielding photons with the statisticsof a Poisson point process with constant rate can be usedin place of the laser to obtain the same simulated photonstatistics (doubly stochastic Poisson process3) as dis-cussed in Sec. V. Furthermore, it should be noted thatwith independent generators, only a limited class ofpower spectra can be reproduced.

We show here how a small laboratory computer canbe used to generate the two random processes neededfor the particular type of simulation discussed in thispaper. The computer can then drive the modulatordirectly through a digital-to-analog converter. Theadvantages of using the computer include increasedflexibility since an arbitrary spectral shape can be easilysimulated. Increased accuracy and simplicity in theexperimental setup are other benefits since criticaladjustments, noise filtering, and matching of the twoindependent noise generators are not necessary.

II. Polarized Chaotic RadiationA source of chaotic radiation can be modeled as a

collection of a large number of independent radiators,each emitting radiation at a certain frequency, with acertain phase, a certain amplitude, and a certain planeof polarization. If we assume that the phase is uni-formly distributed between 0 and 2ir rad and restrictourselves to a well-defined state of polarization, thecontribution to the radiation field in that state of po-larization from all the radiators emitting at a given an-gular frequency Wk may be expressed as

Ak(t) = ak coszkt + bk sinWkt, (1)

where ah and bk are random variables. If the numberof independent radiators emitting at wk is large, ak andbk (by virtue of the central limit theorem and the uni-form distribution of phase) are independent identicallydistributed zero-mean Gaussian random variableswhose variance is proportional to the number of radia-tors at that frequency.

If we assume that radiators exist only for discretefrequencies, the over-all radiation field may be ex-pressed as

A(t) = L ak cosWkt + bk sinWkt, (2)k=-w

where

(Sk = OO + kAw. (3)

In the limit where Awo - 0 this is equivalent to consid-ering a continuum of frequencies. As indicated above,the ak and bk coefficients are independent zero-meanGaussian random variables whose variance is a functionOf Cvk; i.e.,

E[akbl] = for any kJ

E[akal] = E[bkblf = 2 f k I (4)lA'Cr2(Wk) fork = 1.

Here 2(W) is a continuous function corresponding tothe power spectrum of the radiation. By substitutingEq. (3) into Eq. (2) we obtain

A(t) = (ak coskAwt + bk sinkAcot)J coswotk=-

+ [kE (bk coskAwt - ak sinkAwt) sinwotk=-

or

A(t) = x (t) coswot + y(t) sinwot,

where

x(t) = E (ak coskAcot + b sinkAct)k=- c

(t = (bk cosk/cot - a sinksco)k=-X

(5)

(6)

(7)

are two jointly Gaussian random processes corre-sponding to the in-phase and quadrature componentsof the radiation field. The two random processes havethe same marginal statistics.

Their autocorrelation can be easily evaluated with thehelp of Eq. (4):

Ryy(r) R..(T) = E[x(t)x(t + )]

=i {E[a^] cos[kAwtl cos[kAw(t + T)

+ E[bl] sin[kAwt] sin[kAw(t + T)]}

(8)= A 2 (WO + kAw) coskAwor.k=--

In the limit as -o 0, Eq. (8) becomes

Ryy(T) = Rxx(T) =SJ a2(Co + w) coswTdw. (

Using similar algebraic manipulations, the cross-cor-relation between the two random processes may bewritten as

Ry(r) = E[x(t + r)y(t)] = a2(W0 + w) sinw-rdw. (10)

We see that the two random processes are independentif the function o2 (W) is symmetric around wo (since thesine function is antisymmetric), but in general, for anarbitrary nonsymmetric power spectrum, the twocomponents will not be independent.

For narrowband radiation, if o denotes the centerfrequency, x (t) and y (t) are slowly varying with respectto wo, and the intensity of the radiation will be givenby

I(t) = x 2(t) + y 2(t). (11)

Using standard expressions for the moments of jointlyGaussian random variables, we can evaluate the auto-correlation of I(t):

RHj(T) = E[(x2(t) + y 2(t))(x 2(t + T) + y 2(t + ))]

= 4[Rx2(0) + RX (R) + ,(T)]. (12)

From the foregoing discussion, it is clear that in orderto simulate the intensity variations of chaotic light ofa given power spectrum a

2(),it is sufficient to generatetwo Gaussian random processes with the appropriateautocorrelation and cross-correlation prescribed by Eqs.(9) and (10). The two are then individually squared and

15 February 1980 / Vol. 19, No. 4 / APPLIED OPTICS 549

(9)

added, and the result is used to modulate the output ofa suitable light source (see Sec. V). The statistics ofphoton arrivals from such a source will be indistin-guishable from chaotic light with the desired spec-trum.

The basic notion for this type of simulation was firstsuggested by Lachs et al., 0,"' who used it with a lasersource to simulate a superposition of chaotic and co-herent radiation; the chaotic component possessed aLorentzian spectrum and was superposed on a singlecentrally located coherent mode. The prescribed su-perposed power spectrum has the form

oj(w) = ( (t) {A _ + B(w -wo), (13)(co - WO)2 + F2

where A and B are constants, 2r is the half-power ra-dian bandwidth, and is the Dirac delta function.Since this spectrum is symmetric about wo, Eq. (10)provides that x(t) and y(t) are statistically independent.Accordingly, it was possible to simulate this light withtwo independent laboratory noise generators. Theexperimental results obtained by Ruggieri et al. 0 arein very good agreement with the theory.

In general, for an arbitrary (nonsymmetric) powerspectrum, it will not be possible to use two independentsources to generate x (t) and y (t). Computer simulationprovides a solution to this problem by generating x (t)and y(t) with the appropriate joint statistics for anyspecified power spectrum.

Ill. Computer Simulation

The well-known fast Fourier transform (FFT) algo-rithm'2"13 is a fast algorithm for the evaluation of thediscrete Fourier transform (DFT). If we consider asequence of N (where N is even) complex numbers Zk,k = - N/2,.. ., N/2 - 1, the DFT of this sequence willbe another sequence of N complex numbers Al definedas

N/2-1Al = E ZkWlk, (14)

k=-N/2

where W = exp(27ri/N). Equivalently,

Al = N21 Zk (cos2 + i sinkr1) (15)k=-N/2 N N

from which we obtainN12-1 2ir 2ir

ImA1 = (ImZ cosk -1 + ReZh sink -1)k=-N/2 N NN12-1 2ir 2ir (16)

ReAl = Z (ReZk cosk - I - ImZk sink -I) k=-N/2 N N

If we consider band-limited light, the infinite sum-mations in Eq. (7) can be replaced by summations overa finite number N of frequency terms, so that

N/2-Ix(t) = E (ak coskAwt + b sinkAwt)

k=-N/2 (17)

N12-1Iy(t) = EN (bk coskAwt - ak sinkAwt).k= -N/2

It is immediately evident that Eqs. (16) and (17) areequivalent when the following substitutions aremade:

ImAj = x(lAt) IZk = ak A = 2-/NAt (18)ReA = y(lAt) ReZ = bk t = lAt.

Stated differently, if the real and imaginary parts of theZk 's are independent Gaussian random variables whosevariance is an appropriate function of k [as given by Eq.(4)], the real and imaginary parts of the Al's will corre-spond to Nyquist samples of x(t) and y(t) (taken atregular intervals). If we denote the interval betweensamples as At, the frequency step appearing in Eq. (17)will be Acw = 2w/NAt. For implementation, the sim-ulation program makes use of a generator of Gaussianrandom numbers with zero mean and given variance,which it calls 2N times to generate 2N random valuesfor the real and imaginary parts of the Zh's with vari-ance AWr2(Wo + kAw) as given by Eq. (4). The simu-lation program then uses the FFT algorithm to evaluatethe DFT of the Zk sequence, thus obtaining the Al se-quence, whose real and imaginary parts will be Nyquistsamples of the simulated x(t) and y(t). Therefore theNyquist samples of the simulated light intensity aregiven by the sequence of absolute values squaredAI 2.

The simulation program is thus able to evaluate thesimulated intensity and output the Nyquist samplesthrough a digital-to-analog converter channel. Thevoltage output would be fed into a low-pass filter toobtain the correct waveform to modulate the lightsource.'4 In practice it is simpler to choose At rathersmaller than the minimum necessary (the minimum isfmax/2, where fmax is the largest frequency to be repro-duced, as prescribed by the sampling theorem' 4 ), so thatinstead of a low-pass filter a simple one-stage RC filterwill be sufficient, or even no filter at all, depending onhow small At is. In the case of an infinite spectrum,such as the Lorentzian spectrum, choosing a small Atmeans truncating the spectrum very conservatively,which also has the advantage of minimizing errors dueto spectral truncation.

To simulate coherent modes superposed on the cha-otic radiation it is sufficient to add a fixed (complex)constant with the appropriate value to the corre-sponding frequency term before evaluating the FFT. Arelated technique has been described in detail byRuggieri et al. 10

IV. Implementation

The simulation method described was implementedon a DEC PDP 11/03 laboratory computer in. FORTRAN.

The generator of Gaussian random numbers was takenfrom DEC's Scientific Subroutine Package; the qualityof the random numbers generated by it was tested byevaluating the histogram of the distribution and theautocorrelation of the sequence of numbers generated,and was found to be quite satisfactory. The FFT al-gorithm was provided by subroutine FFT2 from theInternational Mathematical and Statistical Libraries(IMSL). This subroutine accepts as input a complex

550 APPLIED OPTICS / Vol. 19, No. 4 / 15 February 1980

Zzw*L.w0

Z

0

00I-.(

1.0 I I I I I I I l 1

0.5

0.1

0.05 -

0 20 40 60SAMPLE SEPARATION

80 100

Fig. 1. Autocorrelation coefficients of the in-phase and quadraturecomponents of computer-simulated Gaussian-Lorentzian radiationas a function of Nyquist sample separation. As expected, the twocurves are very similar and appear as virtually straight lines on a

log-linear plot, corresponding to an exponential falloff.

1-D array whose size N is a power of 2 and returns itsDFT in the same array space. The N elements of thearray are numbered from 1 to N, and the correspon-dence with the sequence of Zk's appearing in Eq. (14)is as follows:

ZJIZ(N+k+1) for-N/2 k<0 (19)Z(k + 1) for 0 < k < N2,

where Z(J) is the Jth element of the input array.In order to simulate Gaussian-Lorentzian radiation,

the elements of the input array were filled with zero-mean Gaussian random numbers with variance

E[(ReZk)2 ] = E[(ImZk) 2] = a2k = AI[1 + (Bk) 2 ], (20)

where A and B are constants. With the substitutionsgiven in Eqs. (3) and (18) it can be seen to correspondto a Lorentzian spectrum with half-power radianbandwidth 21 = 4ir/BNAt, when the rate of the Ny-quist samples is 1/At. After evaluation of the FFT, theresults were tested by direct evaluation of the autocor-relation of the sequence of the real parts and of theimaginary parts of the output array, and of the cross-correlation between the two. For a Lorentzian spec-trum we expect the two autocorrelations to be equal andto fall off exponentially, while the cross-correlationshould be virtually zero, since the spectrum is sym-metric [see Eq. (10)].

The simulation and test program was run 100 timeswith the parameters N = 212 = 4096 and r = (/1oo).(1/At). Since the truncation frequency is given by(N/2)Aw = ir(1/At), we see that errors due to spectraltruncation will be negligible. Furthermore the differ-ence between consecutive samples will be very small, sothat it will be possible to produce the simulated wave-form by putting out the samples through a digital-to-analog converter at a regular rate, without any specialfiltering. This corresponds to trading off computer

execution time for simplicity of implementation, whichcan be advantageous, since the simulation program doesnot have to be run in real time but can be run in ad-vance, its output being stored on a tape or similar me-dium for use later in real time. The value of Awo =27r/NAt = (7r/2048)(1/At) is considerably less than thevalue of F, so that the shape of the Lorentzian spectrumcan be accurately reproduced with discrete terms.

The average results after 100 executions of the pro-gram are presented in Figs. 1 and 2. Figure 1 shows theautocorrelation coefficient for the simulated x (t) andy(t) as a function of sample separation. The 100 dis-crete points have been connected by straight-line seg-ments. The two curves can be seen to be very close toone another and virtually straight lines in a log-linearplot, corresponding to the expected exponential falloff.Figure 2 shows the cross-correlation coefficient as afunction of sample separation. Its absolute value canbe seen to be always much less than unity, indicatingthat the two random processes are virtually uncorre-lated.

V. Poisson Statistics Using a Wideband Source

It is well known that, for an arbitrary light source, thenumber of photons observed during a time interval offixed duration is a random variable whose distributionis, in general, different from Poisson. This distributioneffectively reduces to Poisson under a broad range ofconditions, however, so that the observation of photo-counting distributions different from Poisson usuallyrequires carefully designed experiments." 2 These ex-periments often make use of an artificial light sourcespecifically designed to exhibit such non-Poisson be-havior, such as those discussed in the introduction.3-9

It has been shown by Mandel'5 that the photo-counting distribution for a chaotic source reduces toPoisson when the degeneracy parameter of the radiationis much less than unity. Furthermore, Troup andLyons16 have demonstrated that under certain condi-tions the photocounting distribution obtained by

*0.1

zw

0LL

a:0U)

U)

n

0

-0.1

-2n

-100 0SAMPLE SEPARATION

+100 +200

Fig. 2. Cross-correlation coefficient between the in-phase andquadrature components of computer-simulated Gaussian-Lorentzianradiation as a function of Nyquist sample separation. It can be seenthat the absolute value of the coefficient is always considerably lessthan unity, indicating that the two components are virtually

uncorrelated.

15 February 1980 / Vol. 19, No. 4 / APPLIED OPTICS 551

oo

I I I I

modulating a narrowband amplitude-stabilized sourcecan also be obtained by modulating a widebandsource.

We shall show that the similarity of the photo-counting distributions obtained from a wideband sourceand from an amplitude-stabilized source is a conse-quence of a more fundamental similarity in the statisticsof the underlying random point processes corre-sponding to photon registrations from the two types ofsource. We can usefully exploit this similarity by usinga wideband source in place of an amplitude-stabilizedsource, not only in experiments where the photo-counting distributions are important, but also in ex-periments where the details of the underlying pointprocess are relevant.

The degeneracy parameter of an electromagnetic fieldis defined as the average number of photons per co-herence volume. 3 In the case of a (1-D) doubly sto-chastic Poisson point process, the degeneracy parameteris defined as the average number of events per coher-ence time. When a quantum photodetector is exposedto optical radiation, quantum absorptions occur, ingeneral, as the events of a doubly stochastic Poissonprocess whose coherence time corresponds to the inverseof the bandwidth of the radiation.3 Because the co-herence time provides a measure of the memory of theprocess, there will be correlation between events oc-curring closer to one another than the coherence time.Events occurring further apart will be less correlated,and events occurring several coherence times apart willbe virtually uncorrelated.17 As a consequence, if theaverage time interval between photons is much largerthan the coherence time, it is seldom that photons occurclosely enough to one another to be significantly cor-related. Under these conditions, the doubly stochasticPoisson process will be very similar to a simple Poissonprocess with constant rate; indeed, in the limit of van-ishing degeneracy parameter they will be identical.

The light from a typical light-emitting diode (LED),for example, has a coherence time of the order of 10-14sec. Thus, if a photomultiplier exposed to light froman LED generates, on the average, 109 photon regis-trations (pulses)/sec, the average interval betweenconsecutive pulses will be 10-9 sec. This is much largerthan the coherence time. Under these conditions, formost practical purposes, the photomultiplier pulses areindistinguishable from a Poisson point process whoserate is proportional to the average intensity of the light.Since photons from a narrowband amplitude-stabilizedsource (ideal single-mode laser) also form a Poissonpoint process whose rate is proportional to the intensityof the light, the laser can be replaced by an LED (or, ingeneral, by a wideband source). This is true for appli-cations where the statistical behavior is of interest andwhere the spectral characteristics are unimportant, suchas the simulation method discussed in the previoussections. It is sufficient to modulate the average in-tensity of the wideband source (e.g., by varying thecurrent through the LED).

We note, furthermore, that for a typical multimodeHe-Ne laser, intensity fluctuations over a time scale of

the order of 1 nsec arise from mode beating. It is thuspossible to generate a Poisson point process using sucha source if the light is sufficiently attenuated so that theaverage interval between photons is much larger than1 nsec. If these conditions are adhered to, a multimodelaser can be used in place of a single-mode laser togenerate pseudothermal light through scattering asmentioned in the Introduction. An LED would not besuited to this kind of application, however, because itswider bandwidth would destroy the interference effectsthat are the basis for these methods.

We performed an experiment to demonstrate thevalidity of the discussion presented above. The lightsource was an inexpensive general-purpose gallium-arsenide-phosphide LED whose emission band wascentered about 660 nm. The intensity of the source wasmaintained constant by means of a feedback systememploying a Centronic OSD-50-0 silicon photodetectorand an RCA CA3140 operational amplifier. The ra-diation was attenuated by a filter with neutral density5 and detected by an RCA type-8575 photomultipliertube. The output pulses from the anode of the photo-multiplier tube were converted to square TTL pulsesand fed into a PDP 11/03 computer. The computer wascapable of recording the time of arrival of the pulseswith a resolution of 1 ,gsec.

The times of arrival, as recorded by the computer,were processed to test whether their statistical behaviorwas in accord with a Poisson point process with constantrate. Figures 3 and 4 show the results of the two sta-tistical tests performed. The experimental histogramof the distribution of the time interval between con-secutive pulses is presented in Fig. 3. For a Poissonpoint process with constant rate, this distributionshould be exponential; the solid straight line in Fig. 3corresponds to the fitted exponential distribution forthe observed mean (2.25 msec). The experimentalhistogram is clearly in excellent agreement with thefitted distribution.' 8 In addition, for a Poisson pointprocess with constant rate, the time interval betweentwo consecutive pulses is a random variable that is in-dependent of the time interval between any other setof consecutive pulses (this is a renewal process). Ac-cordingly, the second test consisted of evaluating thecorrelation coefficient between interpulse intervalsseparated by a fixed number of intervening intervals.The results are summarized in Fig. 4. The correlationcoefficient can take on values in the range from -1 to+1. It is clear that, as expected, the experimental val-ues observed are virtually indistinguishable from zero,as none is larger than 0.01 in absolute value. In Fig. 4,the abscissa represents the separation between the in-tervals: the first data point corresponds to consecutiveintervals, whereas the last data point corresponds tointervals separated by eighteen intervening intervals.The data in both figures were obtained from the ob-servation of 32,769 photomultiplier pulses.

We also performed the same experiment with sun-light obtaining similar results.

552 APPLIED OPTICS / Vol. 19, No. 4 / 15 February 1980

4 8 12 16 20

INTERVAL SEPARATION

INTERPULSE INTERVAL, msec

Fig. 3. Statistics of photons from a light-emitting diode (LED).Experimental histogram and theoretical distribution of the time in-terval between consecutive detected photons. The solid straight linerepresents the theoretical (exponential) probability density functionfor a Poisson point process with constant rate. The mean time in-

terval is 2.25 msec.

VI. Conclusion

The technique presented here can be used on a smalllaboratory computer, together with a laser and a lasermodulator, to simulate a superposition of coherent andchaotic radiation with arbitrary spectral shape. Fur-thermore, as indicated in Sec. V, the laser and lasermodulator can usually be replaced by a wideband sourcesuch as a light-emitting diode, thus making the wholesystem extremely simple to set up and operate.

The same technique can be used in general for thecomputer generation of a Gaussian random process witha given spectrum, and in particular, it can be used tosimulate lognormal intensity statistics, such as thoseencountered in the propagation of laser light throughthe turbulent atmosphere.

The most useful application of this technique isprobably in conjunction with another program thatsimulates a Poisson point process with a given rate, togenerate computer-simulated photocounting statisticsunder arbitrary conditions. This will be very useful forthe study of systems that are not amenable to analyticsolutions, such as photocounting with different typesof dead time in conjunction with chaotic light of a givenspectrum.'9

This work was supported by the Joint ServicesElectronics Program (U.S. Army, U.S. Navy, and U.S.Air Force) under contract DAAG29-79-C-0079 and bythe National Science Foundation.

Fig. 4. Statistics of photons from a light-emitting diode (LED).Correlation between interphoton intervals for different intervalseparations. The first data point corresponds to consecutive inter-vals, whereas the last data point corresponds to intervals separatedby eighteen intervening intervals. Observe that no correlation point

exhibits an absolute value larger than 0.01.

References1. R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).2. R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 (1956).3. B. Saleh, Photoelectron Statistics (Springer, Heidelberg, 1978),

pp. 160 ff.4. H. Z. Cummins, N. Knable, and Y. Yeh, Phys. Rev. Lett. 12,150

(1964).5. H. Z. Cummins, in Photon Counting and Light Beating Spec-

troscopy, H. Z. Cummins and E. R. Pike, Eds., (Plenum, NewYork, 1974).

6. W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).7. F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).8. M. C. Teich, R. J. Keyes, and R. H. Kingston, Appl. Phys. Lett.

9, 357 (1966).9. B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Prop-

erties of Scattered Light (Academic, New York, 1975), pp.159-212.

10. N. F. Ruggieri, D. 0. Cummings, and G. Lachs, J. Appl. Phys. 43,1118 (1972).

11. G. Lachs and R. K. Koval, J. Appl. Phys. 43, 2918 (1972).12. J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).13. B. Gold and C. M. Rader, Digital Processing of Signals

(McGraw-Hill, New York, 1969), Chap. 6.14. M. Schwartz and L. Shaw, Discrete Spectral Analysis, Detection

and Estimation (McGraw-Hill, New York, 1975), p. 33ff.15. L. Mandel, Proc. Phys. Soc. 74, 233 (1959).16. G. J. Troup and J. Lyons, Phys. Lett. 29A, 705 (1969).17. B. L. Morgan and L. Mandel, Phys. Rev. Lett. 16, 1012 (1966).18. The first column of the histogram is seen to be slightly depressed

below the fitted curve. This results from computer dead time;i.e., the computer software cannot resolve pulses occurring closerthan -20,vsec. The observed reduction of the column height isfound to be in accord with the calculated effect of dead time.

19. G. Vannucci and M. C. Teich, Opt. Commun. 25, 267 (1978).

15 February 1980 / Vol. 19, No. 4 / APPLIED OPTICS 553

0 .1 , I I I

* 0 *

I-zw-

LL.LL00z0

-Jwrr

0U-

0.1