1506 Volume 54, Number 10, 2000 APPLIED SPECTROSCOPY0003-7028 / 00 / 5410-1506$2.00 / 0q 2000 Society for Applied Spectroscopy

Characterization of Random Media byLow-Coherence Interferometry

S. R. THURBER, A. M. BRODSKY, and L. W. BURGESS*CPAC, University of Washington, Seattle, Washington 98195-1700

We describe the use of low-coherence interferometry (LCI) for thestructural characterization of nonuniform media with mesoscopicheterogeneities. The high sensitivity of LCI to the phase propertiesof scattered light makes it a suitable technique for the direct deter-mination of size, concentration, and uniformity of heterogeneities inmulti-phase systems. To demonstrate this, we examine using LCI tostudy a range of polystyrene microsphere suspensions with particlesizes ranging from 41 to 818 nm and concentrations from 0.25% to10% by weight. The particle size and concentration information wasextracted from the amplitude of the signal by using the theory oflight scattering in nonuniform media. We have shown that the � uc-tuations in the unaveraged LCI signal may be utilized to extractadditional information about system uniformity.

Index Headings: Low-coherence interferometry; LCI; Interferome-try; Particle size.

INTRODUCTION

Low-coherence interferometry (LCI) is a broadbandinterferometric technique, which was originally devel-oped in the mid-1980s to inspect miniaturized opticalcomponents, such as laser diode structures, for use in thelightwave communications industry.1 It has since becomeof interest for applications in the � eld of medicine, as apart of a new imaging technique, called optical coherencetomography.2,3 Recently, LCI research has expanded tothe analysis of the effects that individual system charac-teristics have on the signal, and how they can be extractedfrom it, or deconvoluted, to produce meaningful infor-mation about a sample.4,5

In LCI, one utilizes a broadband near-infrared (NIR)light source and a � ber-optic Michelson interferometer(in which the � xed mirror is replaced by the sample) tomonitor the position and magnitude of the interferencefringes created by the interaction of light returning froma sample and a moving mirror. LCI measures only a smallportion of the backscattered light that remains coherentwith the light in the reference channel of the instrument.The size, concentration, spatial distribution, and relativemotion of the scatterers, as well as the dielectric contrastsbetween the suspended and continuous phases of the me-dia, affect the coherence properties of light returning afterbeing scattered multiple times. Because LCI is extremelysensitive to these small phase changes in the backscat-tered light, induced by the structural characteristics of thesample, we investigated LCI as a tool for the rapid anddirect morphological characterization of nonuniform ma-trices.

Information about the phase properties of scatteredlight and, correspondingly, the morphology of the studied

Received 10 April 2000; accepted 7 June 2000.* Author to whom correspondence should be sent.

sample is contained, albeit in a convoluted way, in theLCI scattering signal. The amplitude of the signal, its rateof decay, and the � uctuation patterns in the unaveragedsignal all contain information about the morphology ofthe system. Understanding LCI scattering signal has im-plications for the advancement of optical coherence to-mography to diagnose changes in highly scattering tis-sues, as well as for the analysis and control of manyindustrial processes involving heterogenous materials.

Our research shows that LCI experiments can, in fact,provide substantial information about structural nonuni-formity of heterogeneous media. In our previous workwe demonstrated that both the degree of particle agglom-eration and the diffusion of nonuniformities affect thesignal.4 In this article, we examine the LCI signal as wevary the mean size, concentration, and spatial distributionof nonuniformities. To accomplish this, we conducted theLCI measurements on a concentration series of nominallymonodispersed suspensions of mesoscopic, spherical,polystyrene particles, freely diffusing in water. We thencompare our experimental results with independent the-oretical estimations.

Even though virtually all the features of the LCI signalunveil characteristics of the studied nonuniform media,in this article we will focus on the information stored inthe signal amplitude and the � uctuations in the decaypro� le of the signal. Speci� cally, we demonstrate that themean particle size can be extracted from the signal am-plitude if the density of the suspension is known. It isimportant to note that LCI can be utilized in highly con-centrated systems of particles without a need for dilution.We also demonstrate that the � uctuation patterns in theLCI signal contain information about the uniformity ofthe samples as well. The nonuniformity characteristicsare estimated from the observed deviation of these � uc-tuations from Gaussian distribution. The � uctuation anal-ysis is especially interesting since it has the potential tobe employed to characterize particle size and density si-multaneously. In addition, there is promise of elucidatingparticle spatial distribution (i.e., clustering) and size dis-tribution in polydispersed mixtures even in the systemswhose particle (nonuniformity) dimensions are smallerthan the wavelength. Clustering can manifest itself spe-ci� cally in the formation of wave-packet traps, whichmay cause quasi-localization of wave packets if theircharacteristic dimensions are on the order of the coher-ence length of the incident light.

This paper is organized as follows. The experimentaldata are presented below in the next section. In the fol-lowing section we compare the morphological informa-tion extracted from the signal amplitude and � uctuationswith simple theoretical estimations. The � nal section is

APPLIED SPECTROSCOPY 1507

FIG. 1. Block diagram of the LCI experimental device.

devoted to conclusions and descriptions of possible ap-plications of LCI measurements to the characterization ofnonuniform media.

EXPERIMENTAL

Instrumentation used in this work is similar to that de-scribed in our earlier work.4 The LCI device we use is aHewlett-Packard 8504A optical low-coherence re� ectom-eter, which is a � ber-optic Michelson interferometer witha broadband, low-coherence-length source (see Fig. 1).This re� ectometer utilizes 9 m m single-mode optical � -bers optimized for 1300 nm. The low-coherence sourceis an edge-emitting, light-emitting diode with a centralwavelength of 1300 nm and a bandwidth of 60 nm (fullwidth at half-maximum). A wavelength-independent, 3dB, single-mode, � ber-optic coupler also functions as abeamsplitter. The single-mode test � ber acts as bothtransmitter and detector for the signal from the sampleunder test (SUT). Thus, the SUT replaces a � xed mirrorin the Michelson interferometer, and the detection ge-ometry is exclusively 180 8 . The detection scheme is aheterodyne system with a polarization diversity receiver.

The re� ected beams from the mirror and the SUT addtogether at the coupler and are detected at the receiver asa photocurrent. The photocurrent for a single test re� ec-tion, JD, can be presented as a sum

JD 5 Rd (Jc 1 JInt( )̀) (1)

where Jc is a constant sum of the intensities of the re-� ected light from the two arms of the interferometer;JInt( )̀ is the � uctuating term representing the sum of theinterference signal in the interferometer and the � uctua-tions in the measuring device at the mirror position .̀The values of the photocurrents are proportional to theconstant responsivity of the detector, Rd. For pathlengthdifferences within the coherence length of the source (i.e.,10 m m for our instrument), the interference term producesa beat frequency equal to the Doppler frequency shiftcaused by the moving mirror. In our device the Dopplershift is equal to 27 kHz. The raw beat signal is prepro-cessed with an envelope detector to give the normalizedpositive envelope of the last term in (Eq. 1) as the output

signal, JSUT( )̀. The technique permits measurement of theintensity of the backscattered light, which remains co-herent with the source, as a function of .̀ This capabilityallows both the measurement of the relative position ofsharp re� ecting interfaces and the measurement of thelow-intensity portion of the photon � ux resulting frombackscattering by nonuniformities in highly scatteringsamples.

In our LCI experiments, we used two sets of particlesize standards as the random media to be tested. We pur-chased the � rst set from Duke Scienti� c Corporation.6 Itconsisted of eight nominally monodispersed solutions ofuniform polystyrene microspheres in deionized water.Each solution was 1% by weight, made with a differentparticle size standard. The standard particle diameters, ascalibrated with NIST (National Institute of Standards andTechnology) traceable technology, were 41 6 1.8, 50 62.0, 60 6 2.5, 73 6 2.6, 83 6 2.7, 96 6 3.1, 126 6 3,and 204 6 6 nm. We created a concentration series ofthese standards by serially diluting them with deionizedwater to 0.5% and 0.25% by weight.

The second set of particles, purchased from SeradynInc., consisted of six carboxyl-modi� ed, uniform, colloi-dally stable, monodispersed polystyrene micro-spherestandard solutions.7 The nominal particle diameters ofthese standards were 199, 223, 308, 401 6 3.4, 543 65.7, and 818 6 7.7 nm. The measurement standard de-viation was not available for the � rst three samples. EachSeradyn solution was originally 10% by weight. Theywere serially diluted to 5%, 2.5%, 1.25%, 1%, 0.5%, and0.25%. We sonicated each of the Duke and Seradyn so-lutions for 30 s immediately prior to LCI measurementof that particular solution, in order to break up possibleparticle agglomerates.

For each of the solutions, we collected 50 re� ectometerscans using a � ber-optic probe, polished to a 15 8 anglein order to eliminate the collection of light that was back-re� ected from the � ber end face.

A typical LCI signal created by a scattering matrix ispresented in Fig. 2. The x-axis represents the distancedifference between the mirror position and the sample, ,̀with respect to the instrument coupler. The y-axis rep-resents the intensity of the envelope around the interfer-

1508 Volume 54, Number 10, 2000

FIG. 2. LCI signal for a highly scattering system: A single scan (thehighly � uctuating line) and an average of 50 individual scans (bold line)for a 2.5% (by weight) solution of 308 nm in diameter polystyrenemicrospheres in deionized water. The x-axis represents the distance dif-ference between the mirror position and the sample, ,̀ with respect tothe instrument coupler. The y-axis represents the intensity of the en-velope around the interferogram on a logarithmic scale. The intensityof the high-frequency � uctuations at least partly depends on the Brown-ian motion of the particles and the nonuniformities. The dotted linecorresponds to the exponential decay law (Beer’s law). The decay isdue to incoherent scattering within the matrix.

FIG. 3. The left part of the � gure represents the average of 50 re� ectometer scans for 1% (by weight) suspensions of polystyrene microspheres indeionized water, as a function of particle diameters. The signal traces correspond to the following particle diameters: 41, 50, 60, 73, 83, 96, 126,and 204 nm. The right portion of the � gure represents the averaged re� ectometer signals for 0.25%, 0.5%, 1%, 1.25%, 2.5%, 5%, and 10%concentrations (by weight) of 308 nm diameter polystyrene microspheres in deionized water.

ogram on a logarithmic scale. The intensity of the high-frequency � uctuations at least partly depends on theBrownian motion of the particles and the nonuniformi-ties. As a result of this motion, the � uctuations in theLCI signal can be averaged out in liquid samples, butthey remain strong even after averaging in solid samples.The dotted line in Fig. 2 corresponds to the exponential

decay law (Beer’s law). The decay is due to incoherentscattering within the matrix. In our experiments the signaldecay is caused solely by the loss of coherence duringconsecutive scattering events, since the samples do notabsorb in the wavelength region spanned by the lightsource. The deviation from linearity may be attributed tostrong localization of wave packets. This localization,which occurs within clusters of particles or nonunifor-mities, preserves the phase properties of the wave packetsbut causes their delayed return to the receiver of the in-strument.

Figure 3 shows examples of the interferogram enve-lopes for different particle sizes and concentrations. Theprovided data show that the shape of the interferogramenvelopes, the rate of decay, and their amplitudes varywith the size and concentration of particles (nonunifor-mities) randomly suspended in studied samples.

DISCUSSION

Information Contained in the Amplitude of the LCISignal. The full theoretical description of an LCI inter-ferogram of nonuniform media is a rather complex prob-lem. It represents a key issue in the currently very active� eld of research, aimed at developing a theory of wavepropagation in nonuniform media.8 In this section we willfocus only on the information stored in the signal ampli-tude.

We get a relatively simple analytical description forLCI data with the help of the following approximation:

2ln ù n 1 N A (0) (2)Om a a2 p a

where n is the overall optical refractive index of the me-dium with suspended particles (nonuniformities); nm isthe refractive index of the suspending medium; l is thewavelength in the medium; N a is the mean concentrationof particles of type a ; and A a (0) is the forward-scatteringamplitude of light by those particles.9 This approximation

APPLIED SPECTROSCOPY 1509

holds in the case of not-very-high volume concentrationsof particles and not-very-large optical contrasts betweenthe particles and the suspending medium.

The backscattering measured by LCI results from therelatively long-range (comparable with the wavelength)spatial � uctuations of particle concentration, d N a , aroundthe mean value ^ N a & . The angular brackets, used through-out this text, denote averaging over particle space distri-bution. The LCI signal, JSUT ( )̀, measures the intensity ofcoherently backscattered light as a function of the dis-tance ` proportional to the dwelling time of wave packetsin the sample. In a general case, JSUT( )̀ is a function of� uctuations in the refractive index of the medium, d n(x):

JSUT ( )̀ 5 F { d n(x); `} (3)

where, according to Eq. 2,2l

d n(x ) 5 d N (x)A (0) (4)O a a2 p a

In the last equality it is assumed that the spatial averaginghas been made over the distances on the order of thewave-packet length. If the condition

^ ( d n(x)) 2 & , ^ n & 2 (5)

is ful� lled, we can present expression 3 in the form of aseries:

J ( )̀ ù C 1 C ( ^ n & )̀ d N A (0)OSUT 0 1 a a7 ) ) 8a

2

1 C ( ^ n & )̀ d N A (0)O2 a a7 ) ) 8a

3

1 C ( ^ n & )̀ d N A (0) 1 · · · (6)O3 a a7 ) ) 8a

By retaining only the � rst nonvanishing term in the series(Eq. 6), which describes � uctuations, we get the follow-ing approximation for JSUT( )̀:

2

J ( )̀ ù C ( ^ n & )̀ d N A (0)OSUT 2 a a7 ) ) 8a

2d N a25 C ( ^ n & )̀ z ^ N & A (0) z (7)O2 a a 7 1 2 8^ N &a a

where coef� cient C2( ^ n & )̀ depends on the optical dwelldistance of a wave packet, which is a function of thegeometric dwell distance ` and the mean dielectric prop-erties of the sample ^ n & . In Eqs. 6 and 7 we have takeninto account that C0 5 0 and that the term proportionalto C1 in Eq. 6 disappears since, by de� nition,

^ d N a & 5 0 (8)

Note that Eq. 7 holds only if the inequality (Eq. 5) isful� lled. This inequality can be ful� lled even in the caseof close-packing, when the defects in the structure playthe role of nonuniformities. The averaging is applied inEq. 7 because it is only the signal averaged over manyoptical paths that is observed in LCI.

In the case of noninteracting, freely diffusing particlesor nonuniformities, whose mean dimension (i.e., radiusof gyration) R̄ a is less than the wavelength, we can use

the following expressions for the amplitude A a (0) and the� uctuation intensity D a :

3¯ ¯R Ra aA (0) ù · c 1 ic · ;a 1 22 [ ]l l

2 3¯d N 1 4 Ra aD [ ; 5 p (9)a 7 1 2 8N N 3 ra a a

where c1 and c2 are real constants, and r a is the volumeconcentration of particles a . The constants c1 and c2 canbe calculated within the framework of the Rayleigh–Gansapproximation.9

The � rst of the expressions in Eq. 9 is a reasonableapproximation even when the volume fraction of particlesapproaches close-packing, as long as their dimensions areless than or on the order of the wavelength. Its structurefollows from the very general requirement of time in-variance, according to which the real and imaginary partsof A a (0) have to be proportional to the even and oddpowers of (1/ l ), respectively. It follows from the theoryof light scattering by particles of simple shapes that thereal part of A a (0) initially has to increase to its maximumvalue, as the ratio (R̄ a / l ) increases. After R̄ a reaches somecritical value Rcr , which is for spherical particles de� nedby Mie theory as

cR ù (10)cr

2 v Ï Î ( Ï Î 2 Ï Î )m p m

the real part of A a (0) has to start declining steeply withincreasing R̄ a . The equality R̄ a ù Rcr is also the conditiona particle radius has to ful� ll in order for the Mie reso-nances to be formed.

LCI signals for different samples could be analyzed atany position .̀ However, the comparison is the most con-venient at ` 5 0 (i.e., where the center of the incidentwave packet reaches the sample surface and the coherentbackscattering enhancement is maximal). At this positionthe coef� cient C2 in Eq. 6 does not depend on meandielectric properties and is the same for different particlecharacteristics.

Table I gives the values of the signal JSUT(0) at ` 5 0,corresponding to all original and diluted Duke and Ser-adyn solutions. The signal intensity changes monotoni-cally with particle size and concentration. A noticeablediscrepancy from the trend is observed in the cases whenrelatively small micro-spheres (e.g., less than 50 nm indiameter) were highly diluted and when very large par-ticles (e.g., 818 nm in diameter) were at high concentra-tions. For both cases, there is a reasonable explanation.At high dilutions, deviation from the assumed random,Gaussian distribution of scatterers is expected, due to par-ticle agglomeration and attraction to vessel walls. Forlarge particles approaching the dimension of the wave-length (R̄ a . Rcr), the Rayleigh–Gans scattering approx-imation is no longer valid.

According to Eqs. 7 and 9, the value of the JSUT(0)created by suspensions of small particles has to be propor-tional to r 0R̄ a

3

2

3¯J (0) ; z A (0) ^ N & z D ; r R (11)O OSUT a a a 0 a7 8a a

1510 Volume 54, Number 10, 2000

TABLE I. The maximum intensity of the signal as a function ofparticle size and concentration for Duke and Seradyn polystyrenemicrospheres in deionized water.

Particleconcentration

(% w/w)

Maximum intensity of the re� ectometer signal

Duke samples—particle diameter (nm)

41 50 60 73 83 96 126 204

10 2 7 310.50.25

1.40.381.5

0.781.20.69

1.00.640.50

2.21.00.91

3.11.50.84

4.71.91.1

155.22.7

382116

Seradyn samples—particle diameter (nm)

199 223 308 401 543 818

10 2 7 31052.51.2510.50.25

1701601004646239.3

25020014065452617

9405003001301006428

12006703001701308834

14005602801801107734

6003202401101003623

FIG. 4. The maximum signal intensities as a function of particle concentration (by weight) in deionized water and cubed radius of the particles.The particle volume fractions are ( ) 0.0025; (V) 0.0050; (v) 0.010; ( ) 0.025; ( ) 0.050; and ( ) 0.10.

In Fig. 4, we present the dependence of JSUT (0) at ` 5 0on r 0R̄ a

3 for different particle concentrations and radii onthe logarithmic scale. According to Fig. 4, the relation inEq. 11 is ful� lled as long as R̄ a is smaller than the criticalvalue Rcr . The deviation from linearity in Fig. 4 occurswhen R̄ a reaches Rcr . The dependence of this critical valueon density r 0 is presented in Fig. 5. The found universalvalue of Rcr ù 1·10 2 5 cm � ts the estimation in Eq. 10.

The analysis of the information contained in the meanLCI signal at ` . 0 is substantially more complex thanthe analysis of the signal at ` 5 0. The observed meanstructure of the interferogram can be approximated by thesmooth matching of two exponential regions:

ì 2`

exp 2 for ` , `01 2ï1̀íJ ( )̀ ; (12)SUT

`exp 2 for ` . `ï 01 2`2î

where 1̀ and 2̀ are the wave-packet dwell distance pa-rameters, and 0̀ is the position of the matching point.This position changes monotonically on the sample prop-erties (see Fig. 2).

Information Contained in the Signal Fluctuations.To analyze the � uctuation patterns of LCI signals, weconsidered the change of the signal intensity, log JSUT,after small changes of wave-packet dwell distance, d :̀

d log J ( )̀ 5 log J (` 1 d )̀ 2 log J ( )̀ (13)SUT SUT SUT

This change in the signal corresponds to the relativelysmall change in the number of particles encountered bythe light on the d ` interval. The smallest interval ofwave-packet dwell distance measurable in our instrument,d ,̀ is limited by the step size of the mirror in our Mi-chelson interferometer. For these experiments d ` is equalto 5 m m. The -̀dependence of the signal intensity is con-

APPLIED SPECTROSCOPY 1511

FIG. 5. The critical particle radius as a function of particle volume fraction.

FIG. 6. Normalized signal change caused by a 5 m m increase in wave-packet dwell distance at every 5 m m increment on the trajectory. Datashown are for 2.5% by weight suspension of 308 nm in diameter poly-styrene microspheres in deionized water.

venient for the analysis since d log JSUT( )̀ is approxi-mately proportional to

dJ ( )̀SUT

d log J ( )̀ d`SUTd log J ( )̀ ù d ` 5 d `SUT d` J ( )̀SUT

d J ( )̀ ^ d J ( )̀ &SUT SUTù 1 1 O (14)1 2[ ]^ J ( )̀ & ^ J ( )̀ &SUT SUT

where d JSUT( )̀ is the � uctuating jump of JSUT( )̀ on theinterval [ ,̀ ` 1 d ]̀. Correspondingly, this analysis allowsus to evaluate a normalized signal jump on a small in-terval d .̀ Fluctuations in d log JSUT( )̀ provide more di-rect information about the system than the � uctuations inthe overall signal JSUT( )̀, because the overall signal, for` . d ,̀ is dominated by long-range material character-istics. The dependence of d log JSUT( )̀ on ,̀ averagedover 50 individual scans of 2.5%/308 nm polystyreneparticles in deionized water, is shown in Fig. 6.

The � rst step of the statistical analysis of � uctuationsis the calculation of the squared deviations of d logJSUT( )̀ from its mean value. The mean value is calculatedfrom 50 individual measurements taken in every suspen-sion. The mean value is then subtracted from each indi-vidual measurement, and the difference is squared. Theresults of the calculations are presented in Fig. 7. Foreasier viewing of the trends, the Savitsky–Golay 51-point� ltering technique was applied to the data corresponding

to different concentrations of 308 nm microspheres.10 Thesmoothed data are plotted in Fig. 8.

The interaction of light with the � uctuating regions ofspeci� c optical properties gives rise to the changes in thesignal. In the case of freely diffusing, monodispersed,

1512 Volume 54, Number 10, 2000

FIG. 7. Normalized signal change squared, for a 5 m m increase inwave-packet dwell distance at every 5 m m increment on the trajectory.Data shown are for 10% and 0.5% by weight suspensions of 308 nmin diameter polystyrene microspheres in deionized water.

FIG. 8. Savitsky–Golay 51-point smoothed data: normalized signalchange squared, for a 5 m m increase in wave-packet dwell distance atevery 5 m m increment on the trajectory. Data shown are for 10%, 2.5%,and 0.5% by weight suspensions of 308 nm in diameter polystyrenemicrospheres in deionized water.

TABLE II. The measure of system nonuniformity for Gaussiandistribution of particles. In the case of Gaussian distribution ^ f 4 & /3 ^ f2)2 5 1, and ^ f3 & / ^ f2 & 3/2 5 0, where angle brackets denote averaging.In this case, f 5 d log JSUT( )̀ 2 ^ d log JSUT( )̀ & .

Particleconcentration

(% w/w)

Particle diameter (nm)

199 223 308 401 543 818

^ f 4 & /3 ^ f 2 & 2

1052.51.2510.50.25

1.101.101.091.101.101.091.04

1.081.041.131.121.111.081.06

1.081.081.081.101.111.101.09

1.021.081.081.081.081.101.10

1.061.041.081.071.071.091.09

1.031.041.051.031.071.071.06

^ f 2 & / ^ f 2 & 3/2

1052.51.2510.50.25

0.250.250.300.400.320.350.25

0.170.170.290.320.260.310.26

0.090.100.110.180.200.280.31

0.070.070.090.110.120.220.31

0.090.070.080.090.100.170.22

0.070.070.070.060.090.180.13

noninteracting small particles, the squared deviations ofd log JSUT( )̀ from its mean value then must be propor-tional to the number density of particles (or nonunifor-mities), N, and their concentration, r :

1 12^ ^ [ d log J ( )̀ 2 ^ d log J ( )̀ & ] & & ; D 5 ; (15)SUT SUT ^ N & r

where double angle brackets denote the averaging of all50 scans for each sample, followed by additional aver-aging over .̀ The corresponding dependencies for a rangeof particle sizes are presented in Fig. 9. The data in the� gure show monotonic changes in the squared deviationsof d log JSUT( )̀ from its mean value, as a function ofparticle concentration.

The additional information about particle density � uc-tuations can be inferred from the relations of the meanvalues of the even and odd powers of � uctuations. Spe-ci� cally, in the case of random Gaussian particle spacedistribution, when the � rst ones determine the higher sta-tistical momentums, the following relations must hold:

3^ ^ [ d log J ( )̀ 2 ^ d log J ( )̀ & ] & &SUT SUT5 0 (16)

2 3/2( ^ ^ [ d log J ( )̀ 2 ^ d log J ( )̀ & ] & & )SUT SUT

and

4^ ^ [ d log J ( )̀ 2 ^ d log J ( )̀ & ] & &SUT SUT5 1 (17)

2 23 · ( ^ ^ [ d log J ( )̀ 2 ^ d log J ( )̀ & ] & & )SUT SUT

The values obtained by using these two expressionsfor a range of particle sizes and concentrations are sum-marized in Table II. The values for Eq. 16 do approach0, and those for Eq. 17 approach 1, as expected for ran-dom Gaussian systems. The deviation from Eq. 16 showsa clear trend of increasing values when it is reasonableto expect less randomness in the system. However, ad-ditional analysis with more experimental data is neces-sary to be conducted in order to con� rm that the valueswithin Table II are statistically different from each other.

By autocorrelation of signal � uctuations of a speci� csample, the correlation length can be determined. Cor-

relation will exist between re� ections from particles ornonuniformities whose position is well de� ned with re-spect to each other. Figure 10 depicts the dependence ofthe correlation:

Cor(` 1 k · d ;̀ )̀

5 ^ ^ [ d log J (` 1 k · d )̀ 2 ^ d log J (` 1 k · d )̀ & ]SUT SUT

3 [ d log J ( )̀ 2 ^ d log J ( )̀ & ] & &SUT SUT

k 5 1, 2, 3, . . . (18)

on the distance k· d ` between measuring points. The datain the � gure represent the average of all 50 measurementsfor each sample.

Figure 10 shows that the correlation rapidly decreasesto zero on dwell distances on the order of 10 m m, indi-cating that the correlation length must also be 10 m m orless. This � nding is in agreement with the correlationlength value L calculated from the diffusion theory, ac-cording to which

APPLIED SPECTROSCOPY 1513

FIG. 9. The averaged normalized square of the signal change, as a function of particle concentration. Data shown are for 10%, 2.5%, 1.25%, 1%,and 0.25% by weight suspensions of 308 nm in diameter polystyrene microspheres in deionized water.

FIG. 11. Normalized signal change squared, for suspensions of micro-spheres that gave similar averaged values in Fig. 9. A Savitsky–Golay51-point smoothing � lter has been applied.

FIG. 10. The mean correlation intensity, Cor(` 1 d ;̀ )̀, of the nor-malized re� ectometer signal as a function of small changes in wave-packet dwell distance d .̀

k TBL ; t! h R

where kB is a Boltzmann constant, T is temperature, h isthe dynamic viscosity of the suspending medium, and tis the time duration of a single LCI measurement. The� uctuations lasting shorter than t are averaged duringmeasurements and cannot contribute to an observed sig-nal.

The � uctuations in the LCI signal contain an abun-dance of information about the sample nonuniformity.However, the analysis described above does not extractall that information. For example, it is evident that com-

binations of many different particle sizes and concentra-tions yield the same values for the average squared de-viations of d log JSUT ( )̀ from its mean value (shown onthe y-axis in Fig. 9). We compared the unaveraged datafor sample pairs that exhibited similar y-axis values afteraveraging over .̀ Some of those pairs are presented inFigs. 11 and 12. Even though their average values arethe same, the unaveraged data take different shapes, re-sulting in slight differences in the ‘‘center of mass’’ ofthe plot. In this case the center of mass is de� ned as thepoint on the x-axis for which the area underneath thecurve on the left is equal to the area underneath the curveon the right. For example, in Fig. 11 the center of mass

1514 Volume 54, Number 10, 2000

FIG. 12. Normalized signal change squared, for suspensions of micro-spheres that gave similar averaged values in Fig. 9. A Savitsky–Golay51-point smoothing � lter has been applied.

for 1%/543 nm suspension is at 794 m m, while that of0.5%/818 nm suspension is at 740 m m. The difference of54 m m is signi� cant. In Fig. 12 the difference is small-er—about 10 m m—but may still be useful in distinguish-ing samples. These � uctuation data, in conjunction withthe initial peak-intensity data presented in Fig. 4, indicatethe possibility of deconvoluting particle size and concen-tration information stored in the re� ectometer signal.

CONCLUSION

In this work we have used simple theoretical formulasto obtain quantitative information about these samplecharacteristics from both the signal amplitude and the sig-nal � uctuations. We have demonstrated that the analysisof LCI experimental data provides a simple method fordirect determination of mean particle radius in mesos-copic � nite density suspensions, if the density of the sus-pension is known. Currently, the extraction of size andconcentration of particles requires the help of � nite di-lution. In future work, we hope to utilize the informationcontained in the signal � uctuations to simultaneously de-termine these parameters. Since the characteristic lengthsof the nonuniformities in systems described here weresubstantially less than the wavelength of the incident

light, we can conclude that LCI can be used for studyingnanoparticle suspensions with NIR and visible light. Fur-ther understanding and additional extraction of informa-tion stored in the LCI signal awaits the advancement ofthe general theory of light propagation in nonuniformmedia.

Compared to existing imaging methods of randommultiscattering objects, the described technique repre-sents advancement. Other methods generally require thatthe scattering centers have dimensions comparable withthe wavelength and be only weakly scattering (e.g., al-lowing use of the � rst-order Born approximation) evenin the case of low concentration of scatterers.11 This ad-vancement is important for both scienti� c and practicalapplications. In particular, our results show that coherentnonionizing optical scattering may serve as an effectivesubstitute for X-ray based techniques in such applicationsas medical imaging.12 In addition, LCI may be used forcharacterization of dynamic processes in nanosystems, in-cluding particle distribution in � ows, particle agglomer-ation, polymerization, and crystallization.

ACKNOW LEDGMENTS

We would like to thank the CPAC sponsors and the Boeing Companyfor their support.

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