1132 OPTICS LETTERS / Vol. 16, No. 15 / August 1, 1991

Intensity statistics and the finesse of electromagnetic radiationin random structures

N. Garcia and A. Z. Genack

Department of Physics, Queens College of the City University of New York, Flushing, New York 11367

Received January 4, 1991

The correlation function of microwave intensity with frequency shift is measured in random mixtures ofTeflon and aluminum spheres at a metallic filling fraction of 0.20. We observe the first three terms in an ex-pansion of the correlation function in a parameter that is the ratio of the spacing to the width of modes of therandom medium. The expansion also applies to optical and electron waves. The expansion parameter isequivalent to the finesse of optical resonators. The intensity distribution at all sample thicknesses is found tobe a stretched exponential.

The graininess of the familiar speckle pattern oflaser light scattered from a sample is a fingerprintof the sample's structure. Deeply modulated inten-sity fluctuations such as those observed in thespeckle pattern also occur inside random media thatare irradiated by a monochromatic wave. Thesefluctuations arise because the field is the sum ofpartial waves that are associated with distinct scat-tering sequences with random amplitudes andphases.' The character of intensity statistics de-pends on the degree of correlation among the partialwaves. In this Letter we report a dramatic modifi-cation of first- and second-order intensity statisticsof microwave radiation in strongly scattering sam-ples due to spectral2 and spatial 3 correlation in themedium. The results are described in terms ofscattering parameters that are well defined for anywave. Thus they are a paradigm for the behavior ofoptical waves or other electromagnetic radiation, ofacoustic waves or other classical waves, and of thequantum-mechanical waves of particles such aselectrons. In the experiments reported here, we ex-ploit the unique properties of microwave radiation toexpose the universal statistical character of wavepropagation. With microwave radiation it is possi-ble to measure intensity fluctuations at a point inan ensemble of random samples with statisticallyequivalent disorder. It is also possible to explorethe strong correlation regime by studying sampleswith locally strong scattering or samples in whichthe wave is confined by reflecting boundaries. Ourexperiments concern the spectral and spatial inten-sity correlation function between points on the out-put plane separated by a distance Ar, C(Av, Ar) =(0I(v,r)5I(v+A1vr + Ar)), where SI is the fractionalfluctuation in intensity from the ensemble averagevalue and with the distribution of intensities for anensemble of sample configurations P(I). The re-sults provide the connection between intensity cor-relation and localization.2 `

We present measurements of propagation of K-band radiation in random mixtures of 3/16-in.-(4.76-mm) diameter aluminum and Teflon spheres

with volume-filling fractions of 0.20 and 0.40, re-spectively. The sample is contained in a 7.3-cm-diameter copper tube that can be rotated about itsaxis to produce new sample configurations. The ra-diation is launched from a horn placed 20 cm infront of the sample to produce nearly plane-wave ra-diation at the sample's input face. The intensity onthe sample's output face is detected by two Schottkydiodes separated by 1.9 cm. The results are basedon the analysis of 2250 pairs of intensity spectracontaining 275 points between 19.5 and 22.5 GHz.The spectra detected by each diode are normalizedto the average of all spectra obtained with thatdiode. Thus the average of the intensity over theensemble of sample configurations studied at posi-tion r and frequency v is unity, (I(vr))=1. Themeasurement of the cumulant intensity autocorrela-tion function C(Av, Ar = 0) for an ensemble of con-figurations for a sample of length 24 cm is shown bythe points in Fig. 1. The measurements of the cumu-lant intensity cross-correlation function C(Av, Ar =1.9 cm) are shown by the points in Fig. 2.

The cumulant intensity correlation function inan ensemble of samples of length L can be expressedas a perturbation expansion in a correlation pa-rameter K,

3' -

C(Av, Ar)= >Ci(Av, Ar)= >AiKi<-F(qL)Hj(Ar).i=1 i=1

(1)

In the limit K << 1, the Ai are constants of orderunity withA, = 1. Fi = 1 for Av = 0, and Hi = 1 atAr = 0. q is the square root with negative imagi-nary part of a2 + i2wrrAv/D, where D is the photondiffusion coefficient, a = (Dra-12 is the absorptioncoefficient for transmission, and T

a is the photon ab-sorption time.

The correlation parameter K is the first-order con-tribution to the intensity correlation between typi-cal points in the output plane owing to long-rangeintensity correlation in the medium. It is theinverse of the effective number of independent

0146-9592/91/151132-03$5.00/0 © 1991 Optical Society of America

August 1, 1991 / Vol. 16, No. 15 / OPTICS LETTERS 1133

1.6

1.4

1.2

-Cl +C2 +c 3

- -C,

-. -C 2

- c.1.0 b_

0.4

0.2\- - * * * - ..

0.0 IC 200 400 1

Frequency Shift (MH:Fig. 1. Measured cumulant intensity afunction with frequency shift for an enseconfigurations. The sample contains aluwat a filling fraction of 0.20 and has a lenfThe contributions of three terms in thiEq. (1) are shown.

0.10

0.08

Ea) 0.06

0.04

00.02

0.00 L)0 500 1000

Frequency Shift (MFFig. 2. Measured cumulant cross-corre.for the same ensemble of samples as in Fitributions of C2 and C3 to C are shown.

parameters (Nind) needed to describemedium, K = l/Nind.2 '3 '6 Nind is the nuiof the electromagnetic field in the meresonant with a monochromatic excitEthe number of modes within of the I8 = (dN/dv) 8v, where dN/dv is the specstates of the sample. 8 may also be exratio of the level width to the level

v1/(dv/dN). This is equivalent to the :tical systems and to the Thouless nuntronic samples. Thus Nind = 8, and K

be associated with the frequency shiftradiation in which the field at the out]sample changes substantially. Wtthe level width quantitatively as t](8v112) of the field correlation functRe[GE(AV)] = Re(E(v, r)E*(v + Av, r)), wcomplex electromagnetic field at a Ioutput face of the sample. Thus(dN/dv) 8v1/2.

300 800

In the lowest order of perturbation theory, thecorrelation between the amplitudes of partial wavesfor different scattering sequences is neglected. Theintensity correlation function is then given bythe field factorization approximation C - C1 =|(E(v, r)E*(v + Av, r + Ar))1 2 = IGE(Azv, Ar)l 2.8,11 Be-cause the extent of field correlation for mono-chromatic radiation decays in a phase coherencelength 1/k, where k is the wave number, the inten-sity correlation length in this approximation is also-1/k. When L > 1, where 1 is the transport meanfree path, F1(qL) is given by the result of the photondiffusion model2' 3

Fl(qL) = Isinh(qa)/sinh(qL)12/[sinh 2(aa)/sinh 2(caL)],(2)

Z) where a is proportional to 1 and includes the effect oftutocorrelation internal reflection at the sample boundaries. In amble of sample weakly scattering sample of polystyrene spheresminum spheres (kl = 25) contained in the sample holder describedYth L = 24 cm.at expansion f above, the intensity autocorrelation function C(AP,

sAr = 0) is given to high accuracy by Eq. (2) forL ' 30 cm. 2

Higher-order terms in C can be seen when there isa high probability that typical partial waves cross

-c 2 +c3 each other in the medium. This can be achieved in-C2 strongly scattering samples or even in weakly scat-

tering samples by using reflecting walls that confine3 C_ the wave in three dimensions. Correlation is ob-

served to increase in the polystyrene sample as thelength L increases. For L > 30 cm, C is not givenby C1 alone. A long tail that falls off as Aiv112 isobserved.2 3 In optical measurements of the corre-lation function of total transmission as a function offrequency shift, which is dominated by C2, van Al-bada et al.12 have found that C decays as AzV-1/2.

C2(Av, Ar) has not been calculated theoretically.The correlation function for intensity in the far-

1500 2000 field speckle pattern has been calculated by Feng1z) et al. as a function of the angle (AO) between thelation function incident and scattered radiation.7 They find thatig. 1. The con- C 2 decays as AO-112 and that C3 is independent of an-

gular separation. These results suggest that C2(Ov)also falls off as Av'-1/2 and that C 3 is independent of

a wave in the spatial separation and of Av. Because H1(Ar) fallsmber of modes rapidly and is small for Ar >> 1/k,12 we expect thatdium that are the contribution from C1 to C(Azv, Ar) is negligible forition. This is Ar = 1.9 cm since k Ar - 10 at 21 GHz. Thereforelevel width 8v, the measurement of the cross-correlation functiontral density of shown in Fig. 2 can be used to find the functionalpressed as the form of F2(qL). C(Av, Ar = 1.9 cm) is well approxi-1 spacing 8 = mated by the sum of a term that falls as (Alv)-112 forfinesse for op- large values of Av that we identify with C2 and alber'° for elec- constant term that we identify with C3.= 8-1. 6v can The contribution of various orders of perturbationof the incident theory to the autocorrelation function C(Av, Ar = 0)put face of the shown in Fig. 1 is found by fitting Eq. (1) to the datae may define using Eq. (2) for F1(qL), the functional form of F2(qL)he half-width obtained from the results of Fig. 2, taking F3(qL) =;ion GE(AV) = 1, and using the value a = 0.17 cm-1 , determinedhere E is the from the measurement of the scale dependence of)oint r at the transmission. Use of D and the coefficients AiKi 1

Nind = K1 as fitting parameters gives D = 4.1 x 109 cm2 /s,

A1 = 1.0, A 2 K1 = 0.37, and A3 K2 = 0.09.

. .

1134 OPTICS LETTERS / Vol. 16, No. 15 / August 1, 1991

-4

-5

-6

Fig. 3intensitionsP(I) -

For negative exponential statistics, var(I) = 1. Thelarger variance in this case, seen from the value ofC(Av = 0, Ar = 0) in Fig. 1, is an indication of the

~ \ P(I) exp _(1)o.60 _ enhanced probability of large fluctuations. Suchlarge fluctuations are expected as the localizationthreshold is approached because 8 is then of orderunity, and transmission can be anomalously largewhen the incident wave is resonant with one of the

- ..:........ sample's eigenmodes.In conclusion, we have observed a constant contri-

bution to the intensity correlation function. Themagnitude of this term is of order K

2, where the cor-

relation parameter K is the inverse of the number ofmodes within the linewidth of the mode. The pres-

I l l l E ence of an analogous term in the degree of correla-O 1 2 3 4 5 6 tion of the current density in random conductors,

I which has not been observed experimentally, givesDistribution of intensities obtained from the rise to universal conductance fluctuations. 7" 3"4

ity spectra used to calculate the correlation func- Such a term may also be expected to occur inin Figs. 1 and 2. The solid curve is a fit of strongly scattering optical systems. We find that asexp - (I)' to the data, which gives * 0.60. the degree of correlation increases, the distribution

The value of K is estimated from K = 8-1 =

[8v112(dN/dv)]-1. The half-width of GE(Av) for thevalues of D, a, and L given above is zv = 12.5 MHz.The density of states is dN/dv = (2k2/wv)V, whereV = (1 - f)AL is the nonmetallic volume of thesample, A is the sample cross section, and v =2.45 x 10l° cm/s is the calculated velocity of the ra-diation in the nonmetallic region of the sample,which is composed of equal volumes of Teflonspheres and air. At v = 21 GHz, dN/dv = 6.1 x10-7 Hz-'. This gives zV112(dN/dv) = 7.6 andK = 0.13. For the polystyrene sample, the value of K

found from measurements of the spatial correlationis within 10% of the value 8-1 obtained from mea-surements of the spectral correlation.3 Use of thisvalue of K gives A2 = 2.8 and A3 = 5.2 for the alu-minum sample. This value of A2 is close to thevalue of 2.6 found in the polystyrene sample, whichsuggests that A2 is nearly constant for K << 1 andthat K can be obtained from measurements ofC2 (Av = 0, Ar = 0).

When K << 1, we expect that the amplitudes ofpartial waves for different scattering sequences areweakly correlated. The assumption that the ampli-tudes for various partial waves arriving at a pointare statistically independent leads to negative expo-nential statistics for the distribution of intensitiesof a single polarization component P(I) = exp(-I).'Indeed, this dependence is observed in the poly-styrene sample for L < 30 cm.2 The distribution ofintensities for the spectra measured here is shownin Fig. 3. P(I) is fit by a stretched exponentialP(I) = exp - (I)It with 1.u = 0.60. Unlike the case ofthe polystyrene spheres, P(I) is fit by a stretched ex-ponential, and not by a negative exponential, evenfor small values of L. The value of C at Ar = Avz =0 is the variance of the intensity ((81)2) = var(I).

of intensity values becomes a stretched exponentialwith enhanced probability of large and small inten-sity values.

We thank E. Kuhner for his technical assistanceand W Polkosnik for assisting with the data analy-sis. This research was supported by the NationalScience Foundation under grant DMR-9001335, bythe Petroleum Research Fund of the AmericanChemical Society under grant 22800-AC7, and bythe Professional Staff Congress-City University ofNew York under award no. 6-69345.

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