Statistics of coherent waves inside media with Lévy disorder
SStatistics of coherent waves inside media with L´evy disorder
Luis A. Razo-L´opez , Azriel Z. Genack , , Victor A. Gopar , † Universit`e Cˆote d’Azur, CNRS, Institut de Physique de Nice, Parc Valrose. 06100 Nice, France. Department of Physics, Queens College of the City University of New York, Flushing, NY, 11367 USA. The Graduate Center of the City University of New York, New York, NY, 10016 USA Departamento de F´ısica Te´orica, Facultad de Ciencias, and BIFI,Universidad de Zaragoza, Pedro Cerbuna 12, ES-50009 Zaragoza, Spain.
Structures with heavy-tailed distributions of disorder occur widely in nature. The evolution of suchsystems, as in foraging for food or the occurrence of earthquakes is generally analyzed in terms of anincoherent series of events. But the study of wave propagation or lasing in such systems requires theconsideration of coherent scattering. We consider the distribution of wave energy inside 1D randommedia in which the spacing between scatterers follow a L´evy α -stable distribution characterized bya power-law decay with exponent α . We show that the averages of the intensity and logarithmicintensity are given in terms of the average of the logarithm of transmission and the depth into thesample raised to the power α . Mapping the depth into the sample to the number of scatteringelements yields intensity statistics that are identical to those found for Anderson localization instandard random media. This allows for the separation for the impacts of disorder distribution andwave coherence in random media. I. INTRODUCTION
The average motion of particles in space and timein samples in which at least the first two moments ofthe distribution of spacing are finite follows a diffusionequation in Brownian models. Coherent-transport phe-nomena of classical and quantum waves have been alsostudied within Brownian approaches [1, 2]. Diffusion issuppressed as a result coherent backscattering in whichwaves returning to points in the medium along time-reversed paths interfere constructively. Anderson local-ization occurs as diffusion ceases in sufficiently large sys-tems of dimensions d ≤ (cid:104) T (cid:105) ∼ exp ( − L/(cid:96) ), while (cid:104) ln T (cid:105) = − L/(cid:96) .Indeed, the full statistics of transmission in standardlight-tailed distributions of separation between scatteringelements is determined in accord to the single parameterscaling theory of localization in terms of the dimension-less parameter
L/(cid:96) [1, 7].Most of studies of coherent transport in random mediaconsider standard light-tailed distributions of disorderthat lead to Anderson localization. Such systems includemesoscopic electronic systems in micron scale devices atlow temperatures and classical waves in stationary me-dia. However, heavy tailed distributions are common inbiology and geology [8] and may lead to advantageousmesoscopic devices.Heavy-tailed L´evy α -stable distribution are character-ized by power-law tails. Thus, for a random variable z following a L´evy α -stable distribution ρ ( z ) [9–11]: ρ ( z ) ∼ /z α (1)for z (cid:29) < α <
2. For α <
1, both the first andsecond moments diverge.In L´evy-type disorder, waves can travel long distanceswithout being scattered and thus have a profound impact
FIG. 1. Schematic of a random waveguide with scatterers(slabs) randomly separated according to a L´evy distribution.The scattering processes at the left and right of the observa-tion point x are described by the transfer matrices M l and M r , respectively. on the transport properties. Measurements and analyticcalculations of wave transmission in L´evy α -stable mediagive different scaling than in standard 1D random media[8, 12–26].In this work, we treat the energy inside heavy-tailedL´evy disordered media. The distinctive impacts of L´evy α -stable disorder and Anderson localization upon wavepropagation are manifest. Potential application of novelstates in L´evy disordered media for low threshold lasingare discussed.In particular, we investigate the statistics of waves in-side a 1D L´evy disordered structures via the statistics ofintensity I ( x ) at the observation point x (See Fig. 1).As we show below, for L´evy-type disorder, the averageof the intensity and its logarithm follows a power-law de-pendence with the observation point. We will contrastthese results for disordered systems with standard disor-der, which have been studied experimentally and theoret-ically with random-matrix theory [27–29]. Calculationsof intensity inside L´evy disordered samples using the con-cept of leap-over to compute the density of scatterers [30]gives different results from those presented here in Fig. 4(solid lines). a r X i v : . [ c ond - m a t . d i s - nn ] F e b The intensity inside L´evy disordered samples has beenanalyzed in [30] using the concept of leap-over to cal-culate the density of scatterers, but discrepancies existbetween the calculated average intensity and numericalsimulations.The present manuscript is organized as follows. In Sec-tion II, we present general expressions for the intensityand transmission in a single sample in terms of transfermatrices. We then introduce some known results for thestatistics of the transmission of standard disordered sys-tems that will be contrasted with L´evy disorder in thesubsequent section. In Section III, L´evy type disorderis introduced and compared to transmission in standardsystems. The results for the transmission are useful in thestudy of the statistics of the intensity inside the medium.The averages of the intensity and of the logarithm of in-tensity are given as functions of depth into the sample.Examples of the complete distribution of the logarith-mic intensity are shown in Section III. A summary of theresults and discussion are given in Section IV.
II. PRELIMINARIESA. Transfer matrix: transmission and intensity
Let us assume that we measure the intensity I ( x ) at apoint x . If A and B are the amplitudes of the forwardand backward waves going at this point (see Fig. 1), I ( x ) is given by I ( x ) = | A exp ( ikx ) + B exp ( − ikx ) | , (2)where k is the wavenumber. We now introduce the trans-fer matrices M l and M r associated with the segments ofthe sample at the left and right-hand side of the obser-vation point, respectively: M l ( r ) = (cid:20) γ l ( r ) β l ( r ) β ∗ l ( r ) γ ∗ l ( r ) (cid:21) , (3)where γ l ( r ) and β l ( r ) are complex numbers satisfying | γ l ( r ) | − | β l ( r ) | = 1. The amplitudes A and B can bewritten in terms of the transfer matrices, M l ( r ) and fromEq. (2), the intensity is given by I ( x ) = 1 | γ r | | γ l γ ∗ r + β l β ∗ r | (cid:12)(cid:12)(cid:12)(cid:12) − β ∗ r γ ∗ r exp − ikx (cid:12)(cid:12)(cid:12)(cid:12) = TT r (cid:12)(cid:12)(cid:12)(cid:12) − β ∗ r γ ∗ r exp ( − ikx ) (cid:12)(cid:12)(cid:12)(cid:12) , (4)where T = | γ l γ ∗ r + β l β ∗ r | is the transmission coefficientof the entire sample and T r = 1 / | γ r | is the transmissioncoefficient of the right segment.The transfer matrices M l ( r ) are conveniently writtenin the polar representation as [1] M l ( r ) = (cid:20) (cid:112) λ l ( r ) e iθ l ( r ) (cid:112) λ l ( r ) e i (2 µ l ( r ) − θ l ( r ) ) (cid:112) λ l ( r ) e − i (2 µ l ( r ) − θ l ( r ) ) (cid:112) λ l ( r ) e − iθ l ( r ) (cid:21) , with phases θ l ( r ) , µ l ( r ) ∈ [0 , π ] and λ l ( r ) ≥
0. An advan-tage of using the polar representation is that the radialvariables λ l ( r ) are directly related to the transmissioncoefficients: λ l ( r ) = (cid:0) − T l ( r ) (cid:1) /T l ( r ) . Therefore Eq. (4)can be written as I ( x ) = TT r (cid:12)(cid:12)(cid:12) − (cid:112) − T r exp ( − i ( µ r − θ r + kx )) (cid:12)(cid:12)(cid:12) , (5)while the total transmission T is given by1 T = 1 T r T l (cid:16) T r T l − T r − T l + 2 (cid:112) ( T r − T l −
1) cos 2( µ l − µ r + θ r ) (cid:17) , (6) B. Statistics of the transmission in standarddisordered 1D media
Now that we have obtained analytical expressions forthe intensity and transmission of a single sample in theprevious section, we consider an ensemble of randomsamples. In particular, we assume that the disorderedstructures composed of randomly separated weak scat-terers or slabs. Thus, the intensity I ( x ) is a randomquantity. From Eq. (5), the statistics of I ( x ) depend onthe statistical properties of the transmission and the an-gular variables θ l ( r ) and µ l ( r ) .Before considering the case of disordered samples withL´evy disorder, we introduce the distribution of the trans-mission for standard disorder. The statistics of trans-mission through standard disordered systems with light-tailed distributions have been extensively studied usingrandom matrix theory[1, 2]. The distribution of thetransmission p s ( T ) is given by [31, 32] p s ( T ) = C (cid:104) acosh(1 / √ T ) (cid:105) / T / (1 − T ) / e − s − acosh (1 / √ T ) , (7)where C is a normalization constant, s = L/(cid:96) with L the length of the system and (cid:96) the mean free path. Thecomplete distribution of transmission is determined bythe parameter s , which is proportional to the number ofscatterers n in the sample with proportionality constant a : s = an [33].For later comparisons with systems with L´evy disorder,we point out the asymptotic exponential decay with L ofthe average transmission in standard disordered systems[2]: (cid:104) T (cid:105) ∝ exp ( − L/ (cid:96) ) (8)and the linear behavior of the average of the logarithmictransmission (cid:104)− ln T (cid:105) ∝ L (9)To illustrate some statistical properties of the trans-mission of standard disordered systems, we show inFig. 2(a) the distribution of the logarithmic transmis-sion p s (ln T ), which is obtained from Eq. (7) and thelinear behavior of (cid:104)− ln T (cid:105) with L , given by Eq. (9). Thehistogram and symbols in Fig. 2 are obtained from nu-merical simulations as explained next.The numerical simulations performed in this work arebased on the transfer matrix approach [17, 34]. The nu-merical model consists of layers of thickness 2.5 mm withrefraction index n = 1 .
1, and reflection coefficient 0.007,randomly placed in a background of index of refraction n = 1 with separations following a Gaussian distributionfor standard disorder and a L´evy α -stable distribution forL´evy disorder. We have fixed the frequency at 1 THz inall the calculations. The statistics is collected for 10 realizations of the disorder. III. STATISTICS OF THE INTENSITY INSIDE1D MEDIA WITH L´EVY DISORDER
We utilize the model of L´evy disordered media intro-duced in [15] with the asymptotic decay given in Eq.(1). We briefly summarize the main results of Ref. [15]for the transmission that will be useful for obtaining thestatistical properties of the intensity. We will considerthe case α <
1, where the effects of L´evy disorder ontransport are strong.In L´evy disordered samples of fixed length L , the num-ber of scattering units n is a random variable with strongsample-to-sample fluctuations; thus, it is crucial to knowthe complete distribution of n . The probability densityΠ L ( n ; α ) of these fluctuations is given by [15]Π L ( n ; α ) = 2 α L (2 n ) αα q α,c (cid:16) L/ (2 n ) /α (cid:17) , (10)for 0 < α <
1, in the limit L (cid:29) c /α with c a scalingparameter. The probability density q α,c ( x ) has a power-law tail: q α,c ( x ) ∼ c/x α for x (cid:29) p s ( T ) given in Eq. (7) and the dis-tribution Π L ( n, α ), Eq. (10), we write the distribution ofthe transmission for L´evy disordered systems p α,ξ ( T ) as P α,ξ ( T ) = (cid:90) ∞ p s ( α,ξ,z ) ( T ) q α, ( z )d z, (11)where p s ( α,ξ,z ) ( T ) is given by Eq. (7) with s replaced by s ( α, ξ, z ) = ξ/ (2 z α I α ) and I α is half of the mean value (cid:104) z − α (cid:105) : I α = (1 / (cid:82) z − α q α, dz = cos( πα/ / α ),where Γ denotes the Gamma function [35]. The parame-ter ξ introduced in Eq. (11) is the average of the logarith-mic transmission for a fixed length L : ξ = (cid:104)− ln T (cid:105) L = (cid:82) ∞ an Π L ( n ) dn , which is given by (cid:104) ln T (cid:105) L = − (cid:16) ac I α (cid:17) L α , (12)Since the factor in parentheses in Eq. (12) is a constant,the average of the logarithmic transmission is a power-law function of L , in contrast to the linear dependence FIG. 2. (a) The distribution of the logarithmic transmissionfor standard disorder systems. The distribution is determinedby the parameter s = (cid:104)− ln T (cid:105) = 8. Inset: linear behavior of (cid:104) ln T (cid:105) with the system length. (b) The distribution of thelogarithmic transmission for L´evy disordered systems. Thedistribution is determined by the parameters (cid:104)− ln T (cid:105) = 8and α = 1 /
2. Inset: the power-law behavior of (cid:104) ln T (cid:105) withthe system length, Eq. (12). for standard disorder in Eq. (9). Similarly, a power-lawis found for the average transmission [17] (cid:104) T (cid:105) ∝ L − α incontrast to the exponentially decay for standard disor-dered systems in Eq. (8).In Fig. 2(b), we show an example of the distribution ofthe logarithm of transmission for L´evy disordered struc-tures characterized by α = 1 /
2. The theoretical resultsas given in Eq. (11) are compared to numerical simula-tions show as the histogram. The power-law behaviorof (cid:104) ln T (cid:105) in L´evy structures is shown in the inset in Fig.2(b). Thus, by comparing Fig. 2(a) and 2(b), the stronginfluence of L´evy disorder on the statistical propertiesof the transmission are clearly seen. We also note thatthe only parameters that enter into Eq. (11) are α and ξ = (cid:104) ln T (cid:105) . Thus, the complete statistics of the transmis-sion is determined by these parameters.With the above results for the statistics of transmis-sion, we now study the statistical properties of the inten-sity. As we show next, the presence of L´evy disorder isrevealed in basic statistical quantities such as the ensem-ble averages (cid:104) ln I ( x ) (cid:105) L and (cid:104) I ( x ) (cid:105) L .Let us consider first the average of the logarithmic in-tensity, (cid:104) ln I ( x ) (cid:105) . The calculations are lenghty but a sim-ple analytical result can be provided. This quantity is ofparticular importance since it is directly related to theaverage of the logarithmic transmission, which along α determines all the statistical properties of the transportin L´evy disordered systems.We perform the average over the uniformly distributedphases in Eq. (5) to obtain (cid:104) ln I ( x ) (cid:105) = (cid:104) ln T (cid:105) L −(cid:104) ln T r (cid:105) L − x . Since ln T is an additive quantity, we ob-tain (cid:104) ln T (cid:105) L − x = (cid:104) ln T (cid:105) L − (cid:104) ln T (cid:105) x , and from Eq. (10),we have (cid:104) ln I ( x ) (cid:105) = − (cid:16) ac I α (cid:17) x α = (cid:104) ln T (cid:105) L (cid:16) xL (cid:17) α . (13)Thus, (cid:104) ln I ( x ) (cid:105) ∝ x α has a power-law behavior in L´evydisordered media, in contrast to the linear dependence instandard disordered media [29].We verify numerically the result given in Eq. (13). Theresults (symbols) are shown in Fig. 3(a) together with thetheoretical results (solid lines) from Eq. (13) for α = 1 / (cid:104) ln I ( x ) (cid:105) for standarddisorder is also shown in Fig. 3(a) (green squares).The (cid:104) ln I ( x ) (cid:105) does not fall linearly in L´evy structuresas it does in standard (Fig. 3(a)), however, the role ofcoherent backscattering in inhibiting propagation is un-changed. To gain an insight into this nonlinear behavior,we note that (cid:104) ln I ( x ) (cid:105) is given by the difference betweenaverage of the logarithmic transmission of the completesample ( L ) and the right segment ( L − x ), as we haveshown above. The power-law behavior of (cid:104) ln I ( x ) (cid:105) findsits origin in the power-law of the variation of the num-ber of scatterers up to the depth x . This is illustratedin Fig. 3(b), where the average number of scatterers (cid:104) n (cid:105) = (cid:82) n Π x ( n ) dn = I α x α /c is plotted at the position x/L for L´evy ( α = 1 / , /
4) and standard disorderedstructures. For L´evy disordered samples, the averagenumber of scatterers follows a power law with exponent α . In contrast, in standard disorder, the average num-ber of scatterers is a linear function with the system size.Thus, for both L´evy and standard disorder, (cid:104) ln I (cid:105) is alinear function of (cid:104) n (cid:105) , as shown in Fig. 3(c) and (cid:104) ln I ( x ) (cid:105) is additive, as in standard disordered structures [29, 36].We now study the ensemble average intensity (cid:104) I ( x ) (cid:105) ,after averaging I ( x ), Eq. (4), over the uniformly dis-tributed random phases, we write (cid:104) I ( x ) (cid:105) as [28] (cid:104) I ( x ) (cid:105) = (cid:90) (cid:90) T l (2 − T r ) T l + T r − T l T r P α,ξ l ( T l ) P α,ξ r ( T r ) dT l dT r , (14) FIG. 3. (a) Average of the logarithmic intensity for L´evydisordered systems with α = 1 / x in the case of standard disor-der is also shown (green). In all cases, (cid:104)− ln T (cid:105) L = 5. (b)The average number of slabs (cid:104) n (cid:105) in the L´evy ( α = 1 / , / x . (c) (cid:104) ln I (cid:105) as a function of (cid:104) n (cid:105) for α = 1 / , /
4, andstandard disorder. where the distributions P α,ξ l ( T l ) and P α,ξ r ( T r ) are thedistributions for the left and right segments, respectively,with ξ l = (cid:104) ln T l (cid:105) = ( x/L ) α (cid:104) ln T (cid:105) L . For the right segment, P α,ξ r ( T r ) ξ r = (cid:104) ln T r (cid:105) = (1 − ( x/L ) α ) (cid:104) ln T (cid:105) L , accordingto Eq. (12). We can verify the particular cases at x = 0and x = L : for x = 0, T l = 1 and (cid:104) I (0) (cid:105) = (cid:104) (2 − T r ) (cid:105) =2 − (cid:104) T (cid:105) , while at x = L , T r = 1 and therefore (cid:104) I ( L ) (cid:105) = (cid:104) T r (cid:105) = (cid:104) T (cid:105) .We perform numerical simulations to support Eq. (14),where the double integral is performed numerically. Thedistribution for the left segment P α,ξ l ( T l ) in Eq. (14)is obtained from Eq. (11), while for the right segment, P α,ξ r ( T r ) is obtained by considering the correspondingprobability density of scatterers which is generated nu-merically. The results are shown in Fig. 4(a) for α = 1 / FIG. 4. (a) Ensemble average intensity (cid:104) I (cid:105) for L´evy α = 1 / (cid:104)− ln T (cid:105) = 1. (b) Theaverage (cid:104) I (cid:105) for α = 1 /
2, as in (a), but for waves incident fromthe left and right incidences. The black squares show that thesum of both incidences is constant. in Fig 4(a) to provided a contrast with the power lawdependence found in media with L´evy disorder.The profile of (cid:104) I ( x ) (cid:105) is not symmetric about the center,as it is in standard homogeneously disordered systems.See Fig. 4(a). In L´evy disordered structures, the disorderis inhomogeneous; the density of scatterers is greatestnear the left side of the sample, as can be seen in Fig.3(b), where waves launched causing the intensity to fallmore rapidly there. However, the sum of intensities forwaves incident from the left and right is constant andequal to twice the intensity of the incident beam fromone side, as shown in Fig. 4(b). This can be understoodby noting that the integrand of Eq. (14) for the waveincident from the right, T l and ξ l are replaced by T r and ξ r , respectively, and similarly, T r and ξ r are replaced by T l and ξ l for waves incident from the right. Adding thecontributions from the wave incident from the left andright gives T l (2 − T r ) / ( T l + T r − T l T r ) + T r (2 − T l ) / ( T l + T r − T l T r ) = 2. Since P α,ξ l, ( r ) ( T l, ( r ) ) are normalized,the average of the sum of intensities excited by wavesincident on the left and right is equal to 2. This result isillustrated in Fig. 4 (b) for the case of α = 1 /
2, wherethe black squares are the sum of the averages intensityfor incident waves from the left and right ends of thesamples.
FIG. 5. Distribution of the logarithmic intensity for L´evy andstandard disordered structures. (cid:104)− ln T (cid:105) = 10. The previous discussion is general and gives an averageintensity profile for standard homogeneously disorderedmedia which is symmetric about the center of the sample[36]. This symmetry is summarized by the expression (cid:104) I ( x ) (cid:105) + (cid:104) I ( L − x ) (cid:105) = 2 and reflects the fact that the sumof intensity from the right and left is equal to the localdensity of states (LDOS) relative to LDOS outside themedium, which is unchanged by disorder.The complete distribution of the logarithmic is ob-tained numerically and shown in Fig. 5 for (cid:104) ln T (cid:105) = 10.There is a higher probability of large fluctuations of in-tensity in L´evy disordered samples (blue and red his-tograms) as compared to standard disordered systems(green histogram). IV. SUMMARY AND DISCUSSION
We have studied the wave intensity statistics insiderandom 1D media with disorder described by L´evy typedistributions characterized by an asymptotic power-lawdecay. For both (cid:104) I ( x ) (cid:105) and (cid:104) ln I ( x ) (cid:105) , we find a power-law decay with position. In contrast, (cid:104) I ( x ) (cid:105) falls linearlyin standard disordered systems. The slower decay with x than for the standard disorder indicates that wave lo-calization in space is weaker in L´evy disorder than instandard disorder.The equivalence of the statistics of intensity in α -stableL´evy disordered systems and standard random media atthe corresponding layer number n , suggests opportunitiesfor engineering structures for analyzing and controllingwaves. The forward and background amplitudes withina layer of the medium are constant so that they interfereand create an oscillatory pattern with high peak intensitywithin the layer. When a thick layer is near the spatialand spectral peak of a mode, the lifetime of quasi-normalmode increases and the line narrows as the thickness ofthe layer increases. The material could therefore serveas a filter. If gain is introduced into this system, thecorrespondingly long lifetime of the mode would enhancethe opportunity for emitted photons to stimulate emis-sion before escaping the sample [37]. In addition, thelarge spatial extent of the mode allows the system to beefficiently pumped without saturating the gain medium.The prospects for an α -stable laser will be considered infuture work. ACKNOWLEDGMENTS
We acknowledge discussions with Xujun Ma. Thiswork is supported by the National Science Founda-tion under EAGER Award No. 2022629 and by PSC-CUNY under Award No. 63822-00 51, and by MCIU(Spain) under the Project number PGC2018-094684-B-C22. L.A.R.-L. thanks UCA
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