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Quasi-elastic light scattering for intermittent dynamics

Pierre-Anthony Lemieux and Douglas J. Durian

Dynamic light-scattering techniques provide noninvasive probes of diverse media such as colloidalsuspensions, granular materials, and foams. Traditional analysis relies on the Gaussian properties ofthe scattering process found in most experimental situations and uses second-order intensity-correlationfunctions. This approach fails in the presence of, among other things, the collective intermittent dy-namics found in systems such as granular materials. By extending the existing formalism and intro-ducing higher-order intensity-correlation functions, we show how to detect and quantify the intrinsicdynamics and switching statistics of intermittent processes. We then explore two systems: ~1! anauger-driven granular column for which the granular dynamics are controlled and the formalism is testedand ~2! a granular heap whose dynamics are a priori unknown but may, now, be characterized. © 2001Optical Society of America

OCIS codes: 300.6480, 120.2920, 290.4210, 170.5280.

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1. Introduction

Dynamic light-scattering techniques1–4 are widelyused to noninvasively probe a medium’s dynamicsover a wide range of length and time scales: Ång-stroms to micrometers and nanoseconds to minutes,respectively. In a typical experiment a sample isilluminated by coherent light of a single frequency,say, v0. Because of quasi-elastic scattering off the

oving scattering sites, the power spectrum uE~v!u2 ofhe scattered field is broadened by an amount dv.s such, the statistical fluctuations of the scatteredeld E~t! provide information on the average dynam-

ics within the medium. Because in most cases dv ,,v0 and E~t! oscillates at approximately v0 ' 1015

radys, one instead measures the scattered-light in-tensity I~t! 5 E~t!E*~t!, which fluctuates more slowlyat approximately dv. The objective of any dynamiclight-scattering formalism is hence to relate the mea-sured fluctuations in I~t! to the dynamics of the scat-tering sites.

The traditional approach reflects the ordinary dy-namics found in most experimental situations.When large numbers of independent scattering sitesare present in the scattering process the superposi-

The authors are with the Department of Physics and Astronomy,University of California, Los Angeles, California 90095-1547.D. J. Durian’s e-mail address is durian@physics.ucla.edu.

Received 7 December 2000; revised manuscript received 9 May2001.

0003-6935y01y243984-11$15.00y0© 2001 Optical Society of America

3984 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

tion principle and the central-limit theorem guaran-tee that E~t! is a Gaussian variable with zero mean.The fluctuations of E~t! are therefore uniquely de-scribed by the spectrum uE~v!u2, a second-order quan-tity that itself is strictly a function of the averagedynamics of the scattering sites. Furthermore, thesecond-order intensity-correlation function g~2!~t! 5I~0!I~t!&y^I&2 can, in turn, be unambiguously related

to uE~v!u2 through the Siegert relation

g~2!~t! 5 1 1u^E~0! E*~t&u2

u^EE*&u2

; 1 1 ug~t!u2, (1)

where the electric field autocorrelation function g~t!is simply the Fourier transform of uE~v!u2. In gen-eral, g~t! is a complex function that decays from 1 to0 on the time scale 1ydv.

The traditional approach is unfortunately limitedy the requirement that the scattering process beaussian. This condition can break down not onlyecause of common experimental caveats5 but also,

and more importantly, because of the very nature ofthe dynamics of the scattering sites one set out tostudy. For instance, if the dynamics within the scat-tering volume are intermittent, i.e., oscillating be-tween two or more states, E~t! becomes stronglycorrelated in time and loses in the process its Gauss-ian properties. Now intermittent dynamics are ofinterest because they occur in a large number of sys-tems, particularly those susceptible to jamming,6,7

such as granular flows,8,9 colloidal glasses,10 and

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foams. The dynamics of these systems can alter-nate between fluid states in which the scattering sitesundergo rapid motion to jammed states in which thescattering sites are essentially at rest. There is,therefore, a need for a formalism that would permitthe application of light-scattering techniques to suchsystems. By taking advantage of higher-order in-tensity correlations, we developed a means of char-acterizing intermittent dynamics in systems that areon the verge of jamming. In Ref. 9, we describe theapplication of this technique to the study of ava-lanches down a granular heap. In the present pa-per, we discuss this technique in detail.

In intermittent systems the fluctuations of thescattered field E~t!, or, equivalently, its spectrumuE~v!u2, are a combination of both the intrinsic motionof the scattering sites and the statistics associatedwith the intermittent switching. If, for instance, oneconsiders an ON–OFF intermittent process in whichthe scattering-site dynamics alternate between astatic state ~the 0 state! and a fluid state ~the 1 state!,we can show that

g~2!~t! 5 1 1 ~1 2 f0!ug~t!u2 1 f0 P0~t!. (2)

The measured second-order intensity-correlationfunction will thus decay on multiple time scales.The fastest decay scale is given, just as above, by g~t!,which depends solely on the dynamics of the scatter-ing sites during the 1 state. The remaining decayscales are contained within P0~t! and depend on thedetails of the switching statistics. Finally, f0 is sim-ply the fraction of time spent in the 0 state.

Although it contains significant information aboutthe intermittent process, g~2! alone cannot unambig-uously characterize the dynamics of the scatteringsites. In other words, there is no equivalence be-tween the presence of multiple time scales in g~2! andhe existence of an intermittent-scattering process.ust as a non-Gaussian probability distribution is notully characterized by its second moment, the non-aussian fluctuations of E~t! are not fully character-

ized by second-order measures such as g~t!, or,equivalently, g~2!. It is therefore natural to consider

igher-order intensity-correlation functions. Wehow that the fourth-order function

gT~4!~t! 5 ^I~0!I~T!I~t!I~t 1 T!&y^I&4 (3)

is particularly well suited to the study ofintermittent-scattering processes and can be mea-sured experimentally. The delay T is held fixedwithin a range from 50 ns to 50 ms by a custom-builtdigital delay line, whereas t is swept over a range ofvalues by a commercial digital correlator.5 Usingagain an ON–OFF intermittent process as an examplemakes it easy to show why gT

~4!~t! contains a strongsignal arising from the intermittency. If T is chosento fall between the characteristic time scales of the 0state and the 1 state, I~t!I~t 1 T! averages to ^I2& if ts restricted to within a 0 state or to ^I&2 if t is within1 state. As gT

~4!~t! is essentially the autocorrelation

f the function I~t!I~t 1 T!, it contains information onthe intermittency that is not available to g~2! alone.

The idea of using higher-order statistical measuresto quantify complex phenomena can be found inmany areas of science. In the space domain third-order correlation functions are used, for instance, toremove speckle noise from images12,13; in the timedomain they are used to capture laser-lightstatistics.14–16 In the frequency domain bispectrumand trispectrum methods are commonly used in sig-nal analysis.17 A hybrid technique, the second spec-trum, which involves both time and frequencyvariables, has been used to characterize resistancefluctuations.18,19 These techniques require the post-processing of an accumulated signal, which limitsboth the dynamic range and the signal-to-noise ratio.Also, little or no formalism exists for the analysis ofthese higher-order statistical measurements. Incontrast, gT

~4! can be measured in real time by use ofoff-the-shelf instrumentation and straightforwardelectronics.5 Furthermore, we propose here a com-plete framework for the analysis of gT

~4! in the contextof intermittent dynamics.

This paper is organized as follows: To begin, werecall results from Ref. 5 in which higher-orderintensity-correlation functions are derived for ordi-nary dynamics, i.e., the scattering process is Gauss-ian. This derivation provides us with a startingpoint for deriving equivalent relations in the presenceof ON–OFF intermittent dynamics. The intermittentstatistics of such dynamics are characterized byswitching probabilities such as P0~t!, which, in turn,can systematically be expressed in terms of the un-derlying probability distributions for the durations of0 states and 1 states. This approach allows us topresent three concrete examples of switching pro-cesses that emphasize a few key features. We alsogive a short overview of how the formalism could beapplied to other types of intermittent dynamics. Fi-nally, we present applications of this formalism totwo experimental systems. The first is a systemthat was specifically designed to exhibit ON–OFF in-ermittent dynamics, namely, a granular columnriven by an auger that successively rotates at con-tant speed and remains at rest. We show how g~4!

clearly contains information that is not available ing~2! alone. The second is a granular heap to whichgrains are added at a constant rate9 and for whichknowledge of both g~2! and g~4! is imperative to prop-erly extract dynamic information.

2. Ordinary Dynamics

As a baseline for comparison, we begin by recallingthe intensity-correlation results found in Ref. 5 forthe most usual experimental situation in which thetotal detected electric field has stationary Gaussianstatistics. The general nth-order normalized inten-sity correlation

g~n!~t1, t2, . . . , tn21! 5 ^I~0!I~t1!I~t2! . . . I~tn21!&y^I&n (4)

20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 3985

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is a function of n 2 1 delay times. Because I~t! 5E~t!E*~t!, g~n! is more fundamentally a 2nth-orderelectric field correlation. And because the field hasGaussian statistics, as we assume in this sectiononly, g~n! can be factored as the sum of all possibleproducts of n field autocorrelations.17 The simplestexample is for g~2!~t!, which in unnormalized form isE~0!E*~0!E~t!E*~t!&5^E~0!E*~0!&^E~t!E*~t!&1^E~0!

E~t!&^E*~0!E*~t!& 1 ^E~0!E*~t!&^E*~0!E~t!&. Note thatthe first term on the right-hand side is the squareof the average intensity, the second is zero, and thethird is the magnitude squared of the field autocorre-lation; this is the usual Siegert relation noted in theintroduction. Repeating this procedure for third- andfourth-order intensity correlations and additionally ac-counting for speckle-averaging effects in the detectionprocess gives the following generalized Siegert rela-tions5:

g~2!~t1! 5 1 1 bug01u2, (5)

g~3!~t1, t2! 5 1 1 b~ug01u2 1 ug12u2 1 ug20u2!

1 2b2 Re~g01g12g20!, (6)

g~4!~t1, t2, t3! 5 1 1 b~ug01u2 1 ug02u2 1 ug03u2 1 ug12u2

1 ug13u2 1 ug23u2! 1 b2~ug01u2ug23u2

1 ug02u2ug13u2 1 ug03u2ug12u2!

1 2b2 Re~g01g12g20 1 g01g13g30

1 g02g23g30 1 g12g23g31!

1 2b3 Re~g01g12g23g30 1 g02g23g31g10

1 g02g21g13g30!. (7)

In Eqs. ~5!–~7!, b is roughly the reciprocal of the num-er speckles collected, and gij is the normalized elec-

tric field autocorrelation evaluated at delay time ~tj 2ti!. If the power spectrum is symmetrical, as is theusual case, then g~t! 5 ug~t!uexp~iv0t!, and only themagnitude ug~t!u enters into expressions ~5!–~7!; hencehigher orders can easily be predicted from the mea-surement of g~2!~t! alone. As was demonstrated inRef. 5, this method makes it possible to verify that thedetected field is Gaussian, hence to rule out artifactsthat are due to, e.g., number fluctuations, laser drift,stray light, or partial coherence of the incident light.For asymmetric power spectra there is a phase factorthat could be extracted by the analysis of differentorders. Note that the well-known intensity-moments result

^In&y^I&n 5 1~1 1 b!~1 1 2b!. . .@1 1 ~n 2 1!b# (8)

an be verified by one’s taking the limit of gij 3 1 inthe generalized Siegert relations.

One of the most widely investigated non-Gaussianprocesses is number fluctuations20 in which the totalnumber of scattering sites within the scattering vol-ume is too small for the central-limit theorem to beinvoked. Such number fluctuations can be detectedby a violation of the Siegert relations or, more simply,by a violation of the intensity-moments result @Eq.

986 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

~8!#. In a nearly transparent single-scattering sam-ple, for example, the intensity distribution will bebroader than expected, and the moments will belarger than in Eq. ~8!, if large dust particles occasion-ally sediment through the scattering volume. Forthe case of ON–OFF intermittency, we demonstratethat, in contrast, the Siegert relations are violatedbut the intensity moments are preserved. The Sieg-ert relations thus provide a much more stringent testthan do the intensity moments for Gaussian statisticsof the electric field.

3. Intermittent Dynamics

A. ON–OFF Intermittency

If the electric field does not have Gaussian statisticshigher-order intensity correlations cannot be pre-dicted from the intensity autocorrelation by use of thegeneralized Siegert relations @Eqs. ~5!–~7!#. In other

ords, extra information is contained in the higher-rder correlations that could, in principle, be ex-racted from an analysis of the experimental data.f course, all non-Gaussian processes are different,nd each requires its own analysis. Explicit analy-es for laser drift and a static ~heterodyne! component

are given in Ref. 5. In this subsection, which is themain part of the paper, we propose a theoreticalmodel for the non-Gaussian example of intermittentswitching between a dynamic and a static state.The results are intended for use in the analysis ofintensity-correlation data from experiments of thegeneral type depicted in Figs. 1~a! and 1~b!. In botha granular medium is driven in such a way that thegrains in the scattering volume alternate intermit-tently between a flowing state and a static state. InFig. 1~a! the switching between these states is set as

esired by the intermittent rotation of an auger thats plunged into a tube of sand; this example is a toyontrol system that can be used to test our model. Inig. 1~b! the switching is set entirely by the dynamicsf the sand itself; the only element that we control ishe overall flow rate, and we wish to characterize theesultant intermittent switching by measurement ofhe intensity correlations.

To see why the field is non-Gaussian for this type ofntermittency, consider the intensity trace shown inhe inset of Fig. 1~a! in which the intensity fluctuatesuring flow but has a different constant value forach static state. During flow, all the grains withinhe scattering volume are moving, and their instan-aneous velocities have a large, random, uncorrelatedomponent. Just as for the Brownian motion of mol-cules in a flowing liquid, coarse-grain spatial or timeveraging gives smooth hydrodynamic behavior.he uncorrelated motion and the large number ofrains ensure that the central-limit theorem applies;herefore the intensity fluctuations during flow statesave Gaussian statistics. During static states, inontrast, all grains within the scattering volume aretatic, hence perfectly correlated; therefore the inten-ity is constant and, just like a plane wave, does notave Gaussian statistics. We stress that it is the

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presence of correlations during static states and not asmall number of scattering sites that invalidates thecentral-limit theorem. Nevertheless, the distribu-tion of intensities for a large number of static statesmust be the same as that during the flow states;therefore the intensity moments must obey Eq. ~8!even though the Siegert relations are violated.

To analyze experimental g~n! data for systems ex-ibiting ON–OFF intermittency requires that correla-

tions be computed from a model of intensityfluctuations of the kind shown in the inset of Fig. 1~a!.

he first ingredient is the intensity versus the time1~t! for a hypothetical system in which the flowingtate persists indefinitely. The corresponding elec-ric field has Gaussian statistics, in accord with theiegert relations and the intensity moments of Eqs.

5!–~8!; hence it is described completely in terms of itsormalized autocorrelation g~t!. The decay of ug~t!u

s set by the dynamics of the scattering sites, as insual applications of photon-correlation spectroscopy

n either single- or multiple-scattering limits. Theecond ingredient is a function x~t! that equals 1hen the system is flowing and 0 when the system is

tatic. It is also useful to introduce x#~t! [ 1 2 x~t!and x#9~t!, which equals 0 during flow states and arandom value with the same moments as ^In&y^I&n

during OFF states. With these ingredients the inten-sity fluctuations can be written as

I~t! 5 I1~t! x~t! 1 ^I&x# 9~t!. (9)

Fig. 1. Experimental apparatus for the two sample systems con-sidered: ~a! A toy system composed of a granular column drivenn a predictable fashion by an auger. It exhibits, by design, ideal

ON–OFF intermittent dynamics. ~b! A granular-heap systemriven by the addition of grains at its apex at a controlled flow rate. It exhibits intrinsically intermittent dynamics for Q smaller

han a certain threshold.

his model makes a few assumptions worth noting.irst, it requires negligible transients in thecattering-site dynamics when the flow is switchedither on or off. Second, it requires that the flowtates last longer than the decay time of g~t! so thatuccessive static states have uncorrelated intensities.hird, it requires that the static states be truly staticcross the entire scattering volume with no slow driftf the intensity. This model can, however, be ap-lied equally well to single- or multiple-scatteringystems.It is now straightforward to compute intensity cor-

elations simply by one’s writing the intensity at eachelay time according to Eq. ~9! and expanding. Thisrocess gives a sum of terms that are products of I1~t!

correlations and x~t! or x#9~t! correlations. Consider-able simplification can be achieved by use of the gen-eralized Siegert relations for the I1~t! correlationsand by the definition of y~t! 5 x~t! 1 x#9~t!. The re-sults, although not yet ready for data analysis, ex-hibit a nice general pattern when compared with theusual Siegert relations:

g~2!~t1! 5 ^y0 y1& 1 bug01u2^x0 x1&, (10)

g~3!~t1, t2! 5 ^y0 y1 y2& 1 b~ug01u2^x0 x1 y2&

1 ug12u2^y0 x1 x2& 1 ug20u2^x0 y1 x2&!

1 2b2 Re~g01g12g20!^x0 x1 x2&, (11)

g~4!~t1, t2, t3! 5 ^y0 y1 y2 y3& 1 b~ug01u2^x0 x1 y2 y3&

1 ug02u2^x0 y1 x2 y3& 1 ug03u2^x0 y1 y2 x3&!

1 b~ug12u2^y0 x1 x2 y3& 1 ug13u2^y0 x1 y2 x3&

1 ug23u2^y0 y1 x2 x3&! 1 b2~ug01u2ug23u2

1 ug02u2ug13u2 1 ug03u2ug12u2!^x0 x1 x2 x3&

1 2b2 Re~g01g12g20^x0 x1 x2 y3&

1 g01g13g30^x0 x1 y2 x3&

1 g02g23g30^x0 y1 x2 x3&

1 g12g23g31^y0 x1 x2 x3&!

1 2b3 Re~g01g12g23g30 1 g02g23g31g10

1 g02g21g13g30!^x0 x1 x2 x3&. (12)

As above, the subscripts denote delay times in thecorrelation functions.

To make the connection with the experiment, it isnecessary to evaluate the correlations of the switch-ing functions x~t! and y~t! that appear in Eqs. ~10!–~12!.

his evaluation can be done in terms of time-ependent probabilities of the system’s switching be-ween the flowing ~1 state! and the static ~0 state!tates. Most important is the probability P0~t! that

the system will be in the same static state after a timeinterval t. As t increases from 0 to infinity, thisfunction decays monotonically from 1 to zero. Nextare the interrelated probabilities Pij~t! that the sys-em, if it is initially in state i, will be in state j after

time interval t, irrespective of the number of

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switches experienced. These functions decay fromeither 1 for i 5 j or 0 for i Þ j to the fraction of timefj that the system spends in state j. Finally, there isthe probability P00

S ~t! that the system, if initially in astatic state, will be in a different static state after aime interval t because of some even, nonzero numberf switches. These switching probabilities are notarticular to ON–OFF intermittency processes but can

be used to characterize any type of two-state switch-ing process. In Subsection 3.C, we both show how toderive switching probabilities for different processesand present concrete examples.

In accord with the above definitions, there are sev-eral simple identities that are useful for simplifyingthe x~t! and the y~t! correlations. Because the sys-tem can be in only the 0 state or the 1 state, we have1 5 f0 1 f1, 1 5 P11~t! 1 P10~t!, 1 5 P00~t! 1 P01~t!,and P00~t! 5 P0~t! 1 P00

S ~t! plus a variety of relationssuch as P11~t! 5 P11~t9!P11~t 2 t9! 1 P10~t!P01~t 2 t9!.Less obviously, by time-reversal symmetry, we alsohave f0P01~t! 5 f1P10~t!. It is thus possible to usejust one of the Pij~t! functions to deduce the otherthree.

We now compute the x and the y correlations thatppear in Eqs. ~10!–~12!. This computation is doney straightforward enumeration of the possible casesor whether the system is flowing or is in either theame or a different static state at the specified delayimes. For correlations of x~t! alone this process is

simple because x~t! equals 1 ~0! when flowing ~static!,and therefore the only contribution to the ensembleaverage comes from the case in which the system isflowing at each delay time:

^x0 x1 x2. . .& 5 f1 P11~t01! P11~t12!. . . . (13)

s above, the time intervals are denoted by tij 5 tj 2ti. Intuitively, f1 is the probability that the systemwill be flowing at any t0, P11~t01! is the probabilitythat the system will be flowing again at t1, P11~t12! ishe probability that the system will be flowing againt t2, and so on. Here, and also in what follows, wessumed that the delay times are ordered from small-st to largest, t0 , t1 , t2 , . . . . Also, because the

dynamics are stationary and the correlations aretherefore not a function of t0, we can set t0 5 0.

The next simplest correlations are ones in whichy~t! is not evaluated at successive delay times. Thenonly the first moment of y, which equals 1 whetherflowing or static, enters the problem. Therefore thecorrelations can be limited to ones involving x~t!alone, as given by Eq. ~13!. For example, ^x0y1x2&and ^x0y1x2y3& both equal ^x0x2& 5 f0P11~t02!. Cor-relations involving y~t! at successive delay times aremore complicated because higher moments of y are

^I&m^I&nf1 P11~t! 1 ^I&m^In&f1 P10~t! 1 ^Im&^I

988 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

involved, in accord with the probability that the sys-tem will be in the same static state across the rele-vant time interval.

An instructive example of correlations involving y~t!alone is that of the ~m 1 n!th-order intensity correla-tions of the specific form ^I~0!I~T!I~2T! . . . I@~m 2 1!T#3 I~t!I~t 1 T!I~t 1 2T! . . . I@t 1 ~n 2 1!T#&. In thelimit of t . mT and g~T! 5 0, according to the patternin Eqs. ~10!–~12!, this term equals ^I&m1n times the~m 1 n!th-order y correlation with the same list ofdelay times. Straightforward enumeration of the fivepossible cases gives

Intuitively, the first term is the value times the prob-ability of flowing at 0 times the probability of flowingagain at t; the second term is the value times theprobability of flowing at 0 times the probability ofbeing static at t; the third term is the value times theprobability of being static at 0 times the probability offlowing at t; the fourth term is the value times theprobability of being static at 0 times the probability ofbeing in a different static state at t; the fifth term isthe value times the probability of being static at 0times the probability of being in the same static stateat t. This list exhausts the possibilities. To sim-plify further, one can evaluate the intensity momentsaccording to Eq. ~8!, and the elementary identities forthe switching functions can be invoked. For exam-ple, the two-time intensity autocorrelation, m 5 n 51, in the limit of g~t! 5 0, is

^I~0!I~t!&

^I&2 3 ^y~0! y~t!&

5 f1 P11~t! 1 f1 P10~t! 1 f0 P01~t!

1 f0 P00S ~t! 1 ~1 1 b! f0 P0~t!

5 1 1 bf0P0~t!. (15)

Similarly, the four-time intensity correlation, m 5n 5 2, in the limit of g~T! 5 0 and t . 2T, is

^I~0!I~T!I~t!I~t 1 T!&

^I&4 3 ^y~0! y~T! y~t! y~t 1 T!&

5 1 1 bf0@2 1 bP00~t! 1 ~4 1 10b 1 6b2! P0~t!#.

(16)

hese two special examples are, in fact, the mostmportant results for extracting switching functionsrom experimental data, as is shown in Section 4.

Note that more general correlations involving y~t!an also be evaluated by exhaustive enumeration ofhe possibilities. For example, ^y0y1y2& has contri-

butions from five cases when flowing at t0 and eightcases when static at t0; ^y0y1y2y3& has contributions

P01~t! 1 ^Im&^In&f0 P00S ~t! 1 ^Im1n&f0 P0~t!. (14)

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from 13 cases when flowing at t0 and 21 cases whenstatic at t0. It is similar too for the x~t! and the y~t!cross correlations. For example, ^x0x1y2y3& has con-tributions from only cases in which the system isflowing both at t0 and at t1; it differs from ^x0x1& onlyin that it also includes the possibility that the systemwill be in the same static state at t2 and t3, giving^x0x1y2y3& 5 f1P11~t01!@1 1 bP10~t21!P0~t32!#.

The general results for intensity correlations incases of ON–OFF intermittency can now be found by theinsertion of the x and the y correlation results into Eqs.10!–~12!. For the intensity autocorrelation the x andhe y correlations explicitly computed above yield

g~2!~t! 5 1 1 bf0 P0~t! 1 bf1 P11~t!ug~t!u2. (17)

s an important check, this correlation decays from 1 1at t 5 0, the expected normalized second intensityoment, to 1 at t 3 `. Note, however, that any role

layed by the variation of P11~t! is inconsistent with theassumption that flow states last long enough to com-pletely decorrelate the light. In other words, Eq. ~17! iscorrect only if P11~t! decays so slowly that it does notnoticeably differ from P11~0! 5 1 during the range ofelay times over which ug~t!u is nonzero. Thus to avoidny such trouble, it is best to set P11~t! 5 1 in Eq. ~17!.imilarly, enforcing the assumption that ug~t!u decaysuch more rapidly than the switching functions gives

he following final intensity-correlation results:

g~2!~t1! 5 1 1 bf0 P0~t01! 1 bf1ug01u2, (18)

g~3!~t1, t2! 5 1 1 bf0@P0~t01! 1 P0~t12! 1 ~1

1 2b! P0~t02!# 1 bf1~ug01u2 1 ug12u2

1 ug20u2! 1 2b2f1 Re~g01g12g20!, (19)

g~4!~t1, t2, t3! 5 1 1 bf0@P0~t01! 1 P0~t12! 1 P0~t23!

1 ~1 1 2b! P0~t02! 1 ~1 1 2b! P0~t13!#

1 bf0@bP0~t01! P00~t12! P0~t23! 1 ~1

1 6b 1 6b2! P0~t03!# 1 bf1$ug01u2@1

1 bP10~t12! P0~t23!# 1 ug02u2 1 ug03u2

1 ug12u2 1 ug13u2 1 ug23u2@1

1 bP0~t01! P10~t12!#% 1 b2f1~ug01u2ug23u2

1 ug02u2ug13u2 1 ug03u2ug12u2!

1 2b2f1 Re~g01g12g20 1 g01g13g30

1 g02g23g30 1 g12g23g31!

1 2b3f1 Re~g01g12g23g30 1 g02g23g31g10

1 g02g21g13g30!. (20)

he 1 and the f0 terms come from the ^y0y1 . . . & correla-ions, whereas the f1 terms come from g~t! and correla-ions of x~t! with itself and with y~t!. Two importanthecks are that the correct intensity moments are recov-red when all delays are taken to zero and that the Sieg-rt relations for Gaussian statistics are recovered whenhe intermittency is turned off, f0 5 0 and f1 5 1.

In our experiments, we measure a particular slice

of the full fourth-order intensity correlation, gT ~t! 5^I~0!I~T!I~t!I~t 1 T!&y^I&4. The delay times in Eq. ~20!

ust be ordered, so our slice is given as g~4!~T, t, t 1 T!or T , t and as g~4!~t, T, T 1 t! for t , T. To takedvantage of the necessary separation of time scales inhe decays of g~t! and the switching functions, it is best tohoose an intermediate value of T so that g~T! 5 g~T 1 t!

5 0 and P~t 6 T! 5 P~t! for all five switching functions.Then the fourth-order correlation simplifies to one ex-pression that holds for all t:

gT~4!~t! 5 1 1 bf0@2 1 bP00~t! 1 ~4 1 10b

1 6b2! P0~t!# 1 bf1$ug~t!u2@2 1 bug~t!u2#

1 ug~t 2 T!u2@1 1 bug~t 2 T!u2#%. (21)

Equation ~21! is the central prediction that will beused for the analysis of g~4! data, whereas Eq. ~18! isthe central prediction that will be used for the anal-ysis of g~2! data.

B. Other Types of Intermittency

It is instructive to note briefly that there are other typesof intermittency that can be treated by suitable modifi-cation of the approach given in Subsection 3.A. Oneexample is intermittent switching between dynamic anddark states, for example, by the chopping of the incidentlight beam. Then the detected intensity is I~t! 5

1~t!x~t!, where, as above, I1~t! is the intensity in the dy-amic state and x~t! equals 1 ~0! in the dynamic ~dark!tate. The average detected intensity is then ^I& 5 ^I1&f1,

reduced by the fraction of time that the system is dy-namic. If the electric field underlying I1~t! has Gaussianstatistics the intensity autocorrelation is

^I~0!I~t!&y^I&2 5 ^I1~0!I1~t!&^x~0! x~t!&y^I&2

5 @1 1 bug~t!u2#P11~t!yf1. (22)

imilarly, the nth-order intensity correlation equals f12n

times the nth-order Siegert relation times Eq. ~13!.Another example is intermittent switching be-

ween two dynamic states with the same averagentensity. Then the detected intensity is Id~t! 5

I~t!x~t! 1 I#~t!x#~t! with ^Id& 5 ^I& 5 ^I#&. If the field hasGaussian statistics in both the 1 and the 0 states,with respective correlations of g~t! and g# ~t!, the in-tensity correlations are simply computed as

g~2!~t! 5 1 1 bug~t!u2f1 P11~t! 1 bug# ~t!u2f0 P00~t!,

(23)

g~3!~t1, t2! 5 1 1 b@ug01u2f1 P11~t01! 1 ug12u2f1 P11~t12!

1 ug20u2f1 P11~t20! 1 ug# 01u2f0 P00~t01!

1 ug# 12u2f0 P00~t12! 1 ug# 20u2f0 P00~t20!#

1 2b2Re~g01g12g20! f1 P11~t01! P11~t12!

1 2b2Re~g# 01g# 12g# 20! f0 P00~t01! P00~t12!.(24)

The computations are similar for higher orders.Note that ON–OFF intermittency is not obtained by

20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 3989

Rtd

P

~2n!

s

t

t

b

ati

2Wswiii

i

3

one’s taking two-state intermittency to the limit ofg# ~t!3 1; for static states the field is a plane wave anddoes not have Gaussian statistics.

C. Switching Processes

Before we jump into the experimental data analysis itis useful to gain some intuition on the switching-probability functions P0 and Pij. Indeed, theintensity-correlation functions developed in Subsec-tions 3.A and 3.B make no assumptions about thenature of the underlying switching process. Wetherefore present in this subsection some generalproperties of P0 and Pij as well as concrete examples.

1. Switching-Probability FunctionsAlthough the switching-probability functions P0 andPij are the natural quantities to extract from actualintensity-correlation data, switching processes areperhaps more intuitively described in terms of thedistributions for the durations of the flowing and thestatic states, p0~t! and p1~t!, respectively. For in-stance, random-telegraph switching is characterizedby p0~t! 5 exp~2tyt0!yt0 and p1~t! 5 exp~2tyt1!yt1.To facilitate the analysis of the experimental data, wetherefore need a systematic way to express both P0~t!and P00 in terms of p0~t! and p1~t!. The remainingPij are, as was seen above, easily related to P00.

We start with P0~t!, the probability that one startsin a 0 state and remains in the same 0 state for a timet. Letting t0 be the duration of a given 0 state and ube the time at which we enter the same 0 state showsthat the probability that such an event will occur isp0~t0!yt0, where t0 5 *0

` p0~t!tdt is the average dura-tion of a 0 state. Summing over all the possiblevalues of t0 and u, we obtain

P0~t! 5 *0

` dut0 *

u1t

`

p0~t0!dt0 (25)

5 *t

`

p0~t0!~t0 2 t!dt0. (26)

elations ~25! and ~26! have two interesting proper-ies: We obtain t0 from the first derivative dP0~t!ytut50 5 1yt0, and we recover p0~t! from the second

derivative d2P0ydt2 5 t0p0~t!.We now tackle the more arduous task of computing

00, the probability that one starts in a 0 state andlands within any 0 state after a time t. DefiningP00

~2n! to be the probability that, after starting in a 0state, one lands in a 0 state after switching states 2ntimes, we can write

P00~t! 5 P0~t! 1 (n51

`

P00~2n!~t!. (27)

Note that P0~t! is maximum at t 5 0 and the remain-ing terms peak about ~t0 1 t1!n. Therefore P00~t!can easily be approximated up to any arbitrary timeby the calculation of successive terms.

This process leads us to the general problem of

990 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

computing P00 ~t!. In that task it is instructive tofirst consider P00

~2!~t! as a proxy for the remainingterms. Counting all the possible configurations inwhich one starts at time u in a 0 state of duration t0,witches to a 1 state of duration t1, and finally lands

after a total time t in a 0 state of duration t2, we findthat

P00~2!~t! 5 *

0

` dut0 *

u

u1t

p0~t0!dt0 *0

u1t2t0

p1~t1!dt1

3 *u1t2t02t12t2

`

p0~t2!dt2. (28)

We transform this convoluted integral problem intoan algebraic problem by taking its Laplace trans-form:

P̃00~2!~s! 5 *

0

`

P00~2!~t!exp~2st!dt

5 p̃1~s!@1 2 p̃0~s!#2yt0 s2. (29)

In fact, if we follow the same basic idea this result@Eq. ~29!# can be generalized to all P̃00

~2n!~s!. We findhat

P̃00~2n.0!~s! 5 p̃1~s!np̃0~s!n21@1 2 p̃0~s!#2yt0 s2. (30)

Provided that these transforms can be inverted ana-lytically or numerically, one can compute anyP00

~2n.0!~t! with the sole knowledge of the Laplaceransforms of p0 and p1, a relatively easy problem.

Moreover, the Laplace transform of P00~t! @Eq. ~27!#ecomes a geometric sum and converges to

P̃00~s! 5 P̃0~s! 1 (n51

`

P̃00~2n!~s!

51

s2t0Hst0 2 1 1 p̃0~s! 1 p̃1~s!

@1 2 p̃0~s!#2

1 2 p̃0~s!p̃1~s!J ,

(31)

as long as P̃00~2n.0!~s! , 1.

In summary, it is now possible to both predict andnalyze intermittent light-scattering data in terms ofhe underlying distributions for the durations of thentermittent states.

. Examplese now give concrete examples for three types of

witching, all of which are important for comparisonith the experimental data. The simplest example

s random-telegraph switching in which the probabil-ty of switching from a 0 state to a 1 state after annfinitesimal time interval is dtyt0 and the probabil-

ity of switching from a 1 state to a 0 state after annfinitesimal time interval is dtyt1. The times ti are

the respective average lifetimes of the respectivestates, so f0 5 t0y~t0 1 t1! and f1 5 t1y~t0 1 t1!. It is

w

t

ps

w

t

w

i

oad

d

straightforward to compute the switching and thedistribution functions

P0~t! 5 exp~2tyt0!,

P00~t! 5 f0 1 f1 exp~2tytA!,

p0~t! 5 exp~2tyt0!yt0,

p1~t! 5 exp~2tyt1!yt1, (32)

here 1ytA 5 1yt0 1 1yt1 represents an averageswitching rate and the other Pij~t! can be deducedeasily from P00~t!.

The second example is periodic switching in whichhe lifetime of each static state ~0 state! is precisely t0

and the lifetime of each flowing state ~1 state! isrecisely t1. It is straightforward to compute thewitching probabilities

P0~t! 5 H1 2 tyt0 if t # t0

0 if t $ t0,

P00~t! 5 H1 2 t9yt0 if t9 , min~t0, t1!0 if t0 , t9 , t1

1 2 t1yt0 if t1 , t9 , t0

,

p0~t! 5 d~t 2 t0!,

p1~t! 5 d~t 2 t1!, (33)

here t9 5 minut 2 n~t0 1 t1!u is the amount of timebetween t and the nearest integer multiple of theperiod.

The third example is intermediate betweenrandom-telegraph and periodic switching. In aquasi-periodic-switching process, p0 and p1 arepeaked distributions with finite width, unlike thedelta-function distributions of periodic switching,and with zero probability for short events, unlike theexponential distributions of random-telegraphswitching. The family of normalized distributionfunctions, tp@~p 1 1!yti#

p11 exp@2~p 1 1!tyti#yp!, sat-isfies such conditions if p . 0 and ti is the averagestate duration. Taking p 5 3, for instance, we findhat

here P0~t! was computed by use of Eq. ~25!. It isjust as straightforward to compute P00~t! by use ofEq. ~31!, but this yields a cumbersome expressionthat we do not write out here.

The various probability functions for the three ex-amples are compared in Fig. 2. One should imme-diately note that the initial slope of P0~t! is identicaln all cases and equal to 1yt0, a general consequence

of Eq. ~25!. Furthermore, P0~t! decays linearly at

P0~t! 5@8~tyt0!

3 1 12~tyt0!2 1

p0~t! 5@t3y~t0y4!4#exp@2ty~t0y

6

first in both the periodic and the quasi-periodic cases.Because the curvature of P0~t! is given by p0~t!, an-ther general result from Eq. ~25!, the linear decay istelltale sign that there are no 0 states of short

uration and that the underlying p0~t! distributionis peaked. Last but not least is the presence ofoscillations in P00 for both the periodic and the quasi-periodic cases; these are absent in the random-

telegraph case. The period of the oscillations isgiven by the intermittency period t0 1 t1, whereastheir strength is given by the amount of correlation inthe location and the duration of successive 0 states.

4. Intermittent Granular Motion

As a test of the validity of the ON–OFF intermittencyformalism developed here and as a proxy for the anal-ysis of systems that exhibit intermittent dynamics,

t0!2 1 3~tyt0!#exp@2ty~t0y4!#

3,

p1~t! 5@t3y~t1y4!4#exp@2ty~t1y4!#

6, (34)

Fig. 2. Intermittency statistics: The switching-probability func-tions P00 and P0 are shown along with the underlying state-

uration probability function p0 for a variety of switching processes@Eqs. ~32!–~34!#. For the sake of simplicity, we chose identicalfunctional forms for both p1 and p0, all with t0 5 2 s and t1 5 3 s.

9~ty

4!#,

20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 3991

tw2

~nwp

3

we consider the two granular systems shown in Fig.1. The first, a granular column stirred by an auger,is one that is engineered to display the characteristicsof an ideal periodic intermittent system. The sec-ond, a granular heap driven by the addition of mate-rial at its apex, is one in which the intermittencystatistics are unknown a priori and need to be deter-mined.

A. Ideal System: Auger Stirring

The first system is a 1-in. inner-diameter verticalpolycarbonate column filled with 10-mm-diameterglass beads. The grains are intermittently excitedby a rotating auger that is connected to a steppermotor. The auger is successively rotated at constantspeed for t1s and stopped for t0s. To speed up thegrain–grain dynamics and limit transient effects, wemaximize the angular speed and acceleration of theauger within the torque limitations of the steppermotor.

This system is an idealized periodic ON–OFF inter-mittent system with well-separated time scales.When the auger rotates, the grains are agitated, andthe speckle pattern fluctuates rapidly, reflecting therapid grain–grain dynamics during this fluid state.When the auger comes to rest the speckle patternfreezes in a random configuration for the duration ofthe static state. We can therefore directly verify theintensity-correlation predictions of Eqs. ~18! and ~21!,along with the periodic-switching predictions of Eqs.~33!. We are, in particular, looking for periodic os-cillations in the baseline of gT

~4! that arise from thepresence of P00 and are a telltale sign of periodicswitching. To that effect, we set the nominal valuesof t1 and t0 to 1, 2, or 3 s, keeping the intermittencyperiod of t0 1 t1 5 4s. Accounting for the finitedeceleration time of the auger ~100 ms! and the set-ling time of the system ~also of the order of 100 ms!,e see that ~t0, t1! take actual values of ~0.8, 3.2!, ~1.8,.2!, and ~2.8, 1.2! s.We illuminate the column with an Ar1 laser with a

wavelength of l 5 514 nm over a 1-mm region. Theintensity fluctuations in the speckle pattern createdby backscattered photons are captured by a single-mode optical fiber coupled to a photon-counting as-sembly. Following Ref. 5, we simultaneouslymeasure both the second- and the fourth-orderintensity-correlation functions g~2! and gT

~4!, respec-tively. In addition, we also capture gT

~4! with a linearcorrelator, as the periodic intermittency features arebest observed on a linear scale. The resulting g~2!

and gT~4! are shown in Fig. 1~b! for both continuous and

intermittent rotation. We choose a value of T 5 6.5ms, somewhere between the time scale of the decaycaused by the grain–grain dynamics and that result-ing from the intermittency. As was pointed outabove, this value not only makes the analysis easierbut also maximizes the intermittency signal, as gT

~4!

will contain terms correlated on both time scales.In the continuous case, g~2! exhibits a single decay

that is due to the grains of sand being agitated by theauger. The intermittent case is obviously more in-

992 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

teresting. During the fluid state, the beads are ag-itated, and, accordingly, g~2! exhibits a decay at earlytimes that is similar to that observed for continuousrotation. During the static state, the grains are atrest, the scattered intensity is constant in time, andg~2! exhibits a late decay on the time scale of thestatic-state duration. As expected, we note the pres-ence of a periodic structure in the baseline of gT

~4!,which is best seen in Fig. 3. The periodic oscillationsare not experimental noise because they are absent ing~2!. In fact, over the course of an experiment theybecome sharper as the baseline of g~2! stabilizes.Note that this is a clear sign that gT

~4! contains infor-mation not present in g~2! alone and that the scatter-ing process is not Gaussian.

To verify the combined equations ~18!, ~20!, and33!, we need a few ingredients. The first is theormalized electric field autocorrelation function g~t!,hose Fourier transform gives the broadening of theower spectrum caused by the relative motion of the

Fig. 3. Second- and fourth-order intensity-correlation functionsfor the auger system shown in Fig. 1~a!. The first decay of g~2! isdue to the rapid motion of the grains of sand while the auger isturning, whereas the second decay of g~2! is a sign of the intermit-tent nature of the dynamics. The periodic structure present inthe baseline of g~4! and absent from g~2! is a telltale sign of periodicintermittency. To verify the combined equations ~18!, ~20!, and~33!, we generate predictions ~shown as the solid curves! for theremaining correlation functions by using solely g~2! from the con-tinuous case and the known values of t0 and t1.

ittf

fIdhbt

vwa

~2! ~4!

Hm

tf

c

w

scattering sites while the auger is rotating. Al-though g~t! is unknown a priori, we assume that it isdentical for both continuous and intermittent rota-ion. Because the scattering process is Gaussian inhe continuous regime, g~t! can readily be extractedrom g~2! by use of the usual Siegert relation @Eq. ~5!#.

The next ingredient is b, which corresponds roughlyto the inverse of the number of speckles viewed by thedetector and is solely a function of the scatteringgeometry. The value of b can therefore be extractedonce from the continuous g~2!. The final ingredientsare the intermittency parameters, namely, t0 and t1,and the resultant predictions are shown as solidcurves in Fig. 3.

The agreement between predictions and experi-mental data is good both in magnitude and in shape.There are, however, some discrepancies worth men-tioning. The early decay of g~2! is slightly slower inthe intermittent-rotation case, as can be seen fromthe slight discrepancy between the predicted and themeasured g~2! at approximately 1023 s. We also notethat there are some fluctuations in the height of thepeaks and that the predictions are systematicallylower than the actual data. These differences arevery likely signatures of grain-scale transients thatoccur during the acceleration–deceleration of the au-ger.

B. Intermittent Heap Flow

We now consider a granular heap driven by the con-tinuous addition of grains at its apex. This examplewas discussed in detail in Ref. 9, so here we presentonly a short overview of the results that are relevantto intermittency statistics. The experimental setup,shown in Fig. 1~b!, consists of parallel static-dissipating plastic walls, 30 cm 3 30 cm, that form a9.5-mm-wide vertical channel that is closed at thebottom and on one side and filled with spherical glassbeads with a diameter of 0.33 6 0.03 mm. The rateQ at which grains are poured onto the heap is con-trolled by a set of valves. As Q decreases, the sur-ace flow evolves from continuous to intermittent.n the continuous regime the flowing layer thicknessand the heap angle u are constant over the entireeap. In the intermittent regime, u oscillates locallyetween some critical angle and the angle of repose ofhe heap, and d is approximately 5 mm whenever

there is flow, independently of the flow rate. In thisdiscussion, we, of course, focus solely on the intermit-tent regime.

The light-scattering measurements proceed in es-sentially the same manner as for the auger experi-ment. We illuminate the top surface of the flow overa circular region ~8 mm diameter! that is located 18cm from the closed end. The fluctuations of thebackscattered light intensity effectively capture thegrain dynamics averaged over a depth of approxi-mately 2 mm ~6 grains!, twice the photon mean freepath. We simultaneously measure g~2! and gT

~4! for aariety of flow rates in the intermittent regime inhich the grain motion within the scattering volumelternates between uniform jostling and complete

rest. Both g and gT , shown in Fig. 2 of Ref. 9,exhibit at short times a partial decay that is followedby a plateau and a final decay at long times. It istherefore natural to suspect that the scattering pro-cess can be described by use of the same ON–OFF in-termittency model used in the auger systemdescribed in Subsection 4.A. In other words, bothg~2! and gT

~4! should be described by Eqs. ~18! and ~21!.owever, unlike the previous example, the switchingodel here is a priori unknown, i.e., both P0 and P00

need to be determined. And because gT~4! does not

exhibit any baseline oscillations, we can immediatelyrule out periodic switching. Determining the natureof the switching process is crucial here, as it dictatesthe temporal distribution of the rearrangementevents in the heap, i.e., avalanches, regarding whichmuch controversy has occurred ~see Ref. 8, for in-stance!.

Before attempting to extract any switching func-tions from the light-scattering data, we must firstensure that the system can indeed be adequatelydescribed by an ON– OFF intermittency model. Thisprocess requires knowledge of both g~2! and g~4! andcan be accomplished without knowledge of eitherthe intermittency statistics or the grain dynamics.Indeed, expressions ~15! and ~16! show that, be-ween the two characteristic decays, the correlationorms a plateau at values of 1 1 bf0 for g~2! and 1 1

b~6 1 11b 1 6b2! f0 for gT~4!. Knowing that b ' 1,

we extract f0 from both sets of correlation functionsand show that the two sets agree well for all flowrates, i.e., for values of f0 from 0.01 to 0.85. Aneven more stringent test is extracting the functionalform of the switching function P0~t! from both gT

~4!

and g~2!, neglecting the small contribution in Eq.~21! from P00~t!. Again, the results from the twosets of correlation functions agree for all flow rates.Unfortunately, the very fact that we ignored con-tributions from P00~t! in gT

~4! means that, aside fromt1, only static-state-related statistics can be ex-tracted.

We show in Fig. 4 the values of P0~t! that wereextracted independently from the g~2! and the gT

~4!

data at several flow rates. The results are scaled byt0, extracted from the slope at t 5 0, as dictated by Eq.~25!. This scaling makes comparison easier becauseas the flow rate decreases so does the time betweenrearrangement events ~1 states!. Although somescatter is present in the data, it does not follow anysystematic trend as a function of flow rate and islikely due to statistics: The scatter is more pro-nounced at large t, where a smaller sample of 1 statescontributes to P0~t!.

We are now in position to draw two importantonclusions: First, as all P0~t! collapse to the same

functional form, the distribution of time betweenrearrangement events remains unchanged over anorder of magnitude of flow rate. Second, concen-trating on the shape of P0~t!, we note that the firsthalf of the decay is linear, whereas the second halfdefinitely shows some curvature. Using Eq. ~25!,

e can readily conclude that there are no static

20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 3993

mn

ioadptdjcdwc

ttbmp

TND

s

n

3

states of short duration, unlike, say, in the case ofrandom-telegraph switching, whose correspondingP0~t! is also shown in Fig. 4. It is no accident thatearlier we considered the example of quasi-periodicswitching in the form of Eqs. ~34!. Indeed, the corre-sponding P0~t!, shown in Fig. 4, matches the experi-

ental data perfectly, with no fitting parameterseeded. Although this particular form for P0~t! is

phenomenological, we can confidently conclude thatthe time between avalanches definitely peaks aboutsome average t0, similar to the behavior reported inRef. 21, although with better statistics.

5. Conclusions

In summary, we have developed a framework fordetecting and analyzing photon-correlation datafrom systems that exhibit temporal intermittency.The central principle is that such dynamics destroythe Gaussian nature of the electric field so that itbecomes necessary to measure and to model higher-order temporal intensity correlations. This papercomplements our first report on the subject ~Ref. 5!,n which we applied our new technique to the studyf avalanche statistics and grain dynamics insidevalanches of a granular heap, by providing a fullescription of the theoretical details and by com-aring predictions with data for a system with con-rolled periodic intermittency. The unusualynamics associated with materials on the verge ofamming are at the very forefront of currentondensed-matter research. This subject includesriven athermal systems like sand and foam, asell as dense thermal systems like molecular and

olloidal glasses. These systems are all amenable

Fig. 4. Intermittency statistics for the granular-heap flow shownin Fig. 1~b!. P0~t! is the probability of being in the same statictate after a time interval t, and t0 is the average time between

successive avalanches. Data are shown for flow rates rangingfrom 0.03 ~lighter-shade symbols! to 0.3 ~darker-shade symbols!gys. Over this range t0 varies from 6 to 96 s. Note that there is

o systematic dependence between the shape of P0~t! and the flowrate. The scatter in the data results from the relatively smallnumber of statistical samples at large t and, accordingly, vanishesas t3 0. For comparison, we plot P0~t! for the quasi-periodic @Eq.~34!# and the random-telegraph switching processes @Eqs. ~32!# assolid and dashed curves, respectively.

994 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

o study with photon-correlation spectroscopies ofhe type described here. Such experiments coulde the key to a fuller characterization and, ulti-ately, a better fundamental understanding of

hoton-correlation phenomena.

We thank J. Rudnick for many useful discussions.his study was supported in part by NASA grantAG3-1419 and National Science Foundation grantMR-0070329.

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