Hidden Modes in Open Disordered Media: Analytical, Numerical, and Experimental Results
Yury P. Bliokh, Valentin Freilikher, Z. Shi, A. Z. Genack, Franco Nori
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Hidden Modes in Open Disordered Media: Analytical,Numerical, and Experimental Results.
Yury P. Bliokh,
1, 2
Valentin Freilikher,
3, 2
Z. Shi,
4, 5
A. Z. Genack,
4, 5 and Franco Nori
2, 61
Physics Department, Technion-Israel Institute of Technology, Haifa 32000,Israel Center for Emergent Matter Science (CEMS),RIKEN, Wako-shi, Saitama, 351-0198, Japan Jack and Pearl Resnick Institute of Advanced Technology,Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Department of Physics, Queens College of the CityUniversity of New York, Flushing, NY 11367l, USA The Graduate Center, CUNY, 365 Fifth Avenue, New York, NY 10016, USA Center for Theoretical Physics, Department of Physics, CSCS,University of Michigan, Ann Arbor, Michigan 48109-1040, USA bstract We explore numerically, analytically, and experimentally the relationship between quasi-normalmodes (QNMs) and transmission resonance (TR) peaks in the transmission spectrum of one-dimensional (1D) and quasi-1D open disordered systems. It is shown that for weak disorder thereexist two types of the eigenstates: ordinary QNMs which are associated with a TR, and hiddenQNMs which do not exhibit peaks in transmission or within the sample. The distinctive featureof the hidden modes is that unlike ordinary ones, their lifetimes remain constant in a wide rangeof the strength of disorder. In this range, the averaged ratio of the number of transmission peaks N res to the number of QNMs N mod , N res /N mod , is insensitive to the type and degree of disorderand is close to the value p /
5, which we derive analytically in the weak-scattering approximation.The physical nature of the hidden modes is illustrated in simple examples with a few scatterers.The analogy between ordinary and hidden QNMs and the segregation of superradiant states andtrapped modes is discussed. When the coupling to the environment is tuned by an external edgereflectors, the superradiace transition is reproduced. Hidden modes have been also found in mi-crowave measurements in quasi-1D open disordered samples. The microwave measurements andmodal analysis of transmission in the crossover to localization in quasi-1D systems give a ratio of N res /N mod close to p /
5. In diffusive quasi-1D samples, however, N res /N mod falls as the effectivenumber of transmission eigenchannels M increases. Once N mod is divided by M , however, the ratio N res /N mod is close to the ratio found in 1D. . INTRODUCTION Two powerful perspectives have helped clarify the nature of wave propagation in openrandom systems. One of them, relates to the leakage of waves through the boundaries ofthe system and can be described in terms of quasi-normal modes (QNMs), which are theextension to open structures of the notion of normal modes in closed systems [1–9]. Theeigenfrequencies of the QNMs are complex, with imaginary parts that are the inverses ofthe lifetimes of the QNMs. The second perspective is that of transmission through randomsystems [10–12]. For multichannel samples, transmission is most conveniently described interms of the transmission matrix, t , whose elements are field transmission coefficients [13–15]. The transmittance is the sum of eigenvalues of the Hermitian matrix tt † . Some of theseeigenvalues are close to unity even in weakly-transmitting samples [13, 14, 16, 17]. Knowledgeof the transmission matrix makes it possible to manipulate the incident wavefront to enhanceor suppress total transmission through random media [18–22] and to focus transmittedradiation at selected points [23]. The control over transmitted radiation can be exploited toimprove images washed out by random scattering and to facilitate the detection and locationof objects [23]. The great potential of such algorithms for a host of practical applicationshas recently attracted attention in both the physics [23] and mathematics communities [1]and references therein.In open regular homogeneous systems (e.g. single quantum potential wells, optical cavi-ties, or microwave resonators) each peak in transmission, or transmission resonance (TR), isassociated with a QNM ([24] and references therein), so that the resonant frequency is closeto the real part of the corresponding eigenvalue. However, despite extensive research andmuch recent progress the connection between QNMs and TRs in disordered open systemsstill requires a better physical understanding and mathematical justification,To this end, it is instructive to look for insights in 1D systems. It is well-known [12, 25]that the transmission of a long enough 1D disordered system is typically exponentially small.At the same time, there exists a set of frequencies at which the transmission coefficient hasa local maximum (peak in transmission), and some of these are close to unity [25–27]. In1D, each peak is associated with an eigenstate which is a solution of the wave equation withoutgoing boundary conditions (a pole of the S-matrix).Quite surprisingly, much less are studied the properties of QNMs in 1D systems with3eak disorder where the localization length is smaller than the size of the sample. In thispaper we show for the first time that in completely open one-dimensional disordered sys-tems, two different types of QNMs can exist: ordinary QNMs, associated with resonanttransmission peaks and hidden QNMs unrelated to any maxima in the transmission spec-trum. The hidden modes exist due to random scattering and arise as soon as an arbitrarilysmall disorder is introduced. The imaginary parts of the eigenfrequencies of hidden QNMsvary with increasing disorder in an unusual manner. Typically, stronger disorder leads tostronger localization of modes with eigenfrequencies that approach the real axis. However,the imaginary part of a hidden mode ′ s eigenfrequency, depending on the boundary condi-tions, either is independent of strength of random scattering or even increases from the onsetof disorder. Surprisingly, the average ratio of the number of ordinary modes to the totalnumber of QNMs in a given frequency interval is independent of the type of disorder andremains close to the constant p / p / δ , whichgives the ratio of the typical width δν and spacing ∆ ν of QNMs, δ = δν/ ∆ ν [8, 9, 12]. Thetypical linewidth δν is essentially equal to the field correlation frequency over which there istypically a single peak in the transmission spectrum. The density of peaks is therefore 1 /δν .On the other hand, the inverse level spacing 1 / ∆ ν is equal to the density of states (DOS)of the medium. Thus the ratio N res /N mod can be expected to be close to ∆ ν/δν = 1 /δ for diffusive waves. The localization threshold lies at δ = 1 [9, 10, 12]; δ may be muchlarger than 1 for diffusive waves so that N res /N mod ∼ /δ and may be small. For localizedwaves, the number of channels that contribute effectively to transmission, M , approachedunity and transport becomes effectively one-dimensional [29]. For example, the statisticsof transmittance are then in accord with the single parameter scaling hypothesis [30]. Itis worth noticing that although the statistics of the eigenstates of disordered systems is asubject of intensive investigations for already more that two decades (see, for example [31–37]), the statistics of the transmission resonances (peaks in transmission spectra) is muchless studied. The comparison of these two is a challenging problem for future investigations.4ere we find that a connection can be made between the present 1D calculations of N res /N mod and measurements in multichannel diffusive systems. This is done by comparing ratio of N res to the number of QNMs divided by M , M N res /N mod in multichannel systems to theratio N res /N mod in 1D, where M = 1. II. QUASI-NORMAL MODES OF OPEN SYSTEMS
We first consider a generic 1D system composed of N + 1 scatterers separated by N intervals and attached to two semi-infinite leads. The eigenfunctions ψ m ( x, t ) are solutionsof the wave equation satisfying the outgoing boundary conditions, which means that there areno right/left-propagating waves in the left/right lead. Each eigenfunction is a superpositionof two counter-propagating monochromatic waves ψ m ( x ) ( ± ) e − iω m t . The eigenfunction in the j th layer, ψ ( ± ) m,j ( x ), is equal to a ( ± ) m,j e ± ik m x , and the amplitudes a ( ± ) m,j in adjacent layers areconnected by a transfer matrix. The wave numbers k m are complex-valued and form thediscrete set k (mod) m = k ′ m − ik ′′ m , k ′′ m >
0, so that the frequencies ω (mod) m = ck (mod) m . Theseeigenfunctions are QNMs. Note that all distances hereafter are measured in optical lengths.In what follows, the scatterers and the distances between them are characterized by thereflection coefficients r j ≡ r + δr j and thicknesses d j ≡ d + δd j , respectively. The randomvalues δr j and δd j are distributed in certain intervals, and P δd j = 0. The last conditionmeans that the length L of the sample is equal to N d .To explicitly introduce a variable strength s of disorder, we replace all reflection coeffi-cients by sr j , and assume (unless otherwise specified) that the coefficients r j are homoge-neously distributed in the interval ( − , k (mod) m as the disorder increases. The condition P δd j = 0 ensures that anyrandom realization with the same N contains the same number of QNMs N mod = ∆ kL/π in a given interval ∆ k of the wavenumbers.At the beginning, let us consider the QNMs of a regular resonator of the length L = N d assuming that all reflection coefficients except r = r L and r N +1 = r R are equal to zero. Inthis case the real and imaginary parts of the QNM eigenvalues k (mod) m are k ′ m = 12 L · π + 2 πm, when r L r R > , πm, when r L r R < , (1)5 .0 0.2 0.4 0.6 0.8 1.0-6-4-200.0 0.2 0.4 0.6 0.8 1.0-8-6-4-20 (a)(b) N o r a m a li z ed f i e l d i n t en s i t y , l og sc a l e Normalized axial distance x/L
FIG. 1. Spatial distribution of the intensity I m ( x ) of quasi-normal mode (thin light gray curve) and¯ I m ( x ) (thick red curve) in a regular resonator with (a) symmetric ( | r L | = | r R | ) and (b) asymmetric( | r L | < | r R | ) walls. k ′′ m = − L ln | r L r R | , (2)where m = 0 , , , . . . In what follows, instead of the intensity of the m th mode, I m ( x ) = (cid:12)(cid:12)(cid:12) ψ (+) m ( x ) + ψ ( − ) m ( x ) (cid:12)(cid:12)(cid:12) ,we consider the quantity ¯ I m ( x ) = | ψ (+) m ( x ) | + | ψ ( − ) m ( x ) | , which is I m ( x ) averaged over fastoscillations caused by the interference of the left- and right-propagating waves. Examplesof these functions for resonators are shown in Fig. 1a,b. There ¯ I n ( x ) is distributed alongthe system as ¯ I m ( x ) ∝ cosh[2 k ′′ ( x − x ∗ )], where x ∗ = L [1 − ln( | r R /r L | ) / ln( | r R r L | )] /
2. When | r L | = | r R | , the minimum of the intensity is located at the center of the system, and in anasymmetric case shifts to the boundary with higher reflection coefficient. This property willbe used when analyzing the properties of the QNMs of the disordered system.In a disordered sample, the reflection coefficients r i are random and scaled by the pa-rameter s . The evolution of the eigenvalues k (mod) ( s ) as the parameter s grows shows thatQNMs separate into two essentially different types. There are ordinary QNMs whose life-times, defined by the value of 1 /k ′′ , increase monotonically with s . Simultaneously, there6re “hidden” QNMs (the origin of this term will be explained in the next section), whoselifetimes are substantially smaller than the lifetimes of ordinary QNMs and remain constantwhen s varies over many orders of magnitude. Figures 2 and 3 show trajectories of theQNMs’ eigenvalues k (mod) ( s ) = k ′ ( s ) − ik ′′ ( s ) as the parameter s grows, and dependencies k ′′ ( s ). -2 -2 -2 -2 -2 -2 -2 -2 -2 a R a L c b a k " d k’d FIG. 2. Motion of QNMs’ eigenvalues under disorder strength growth. Eigenvalues move fromabove. Some eigenvalues (examples are marked by “a”, “b”, and“c”) stop their motion and “standstill”.
Our numerical calculations show that when external reflectors are added at the edges ofthe sample, the imaginary parts of some of the hidden modes increase with the strength ofdisorder.The spatial distributions of the intensity ¯ I ( x ) along the system are also different forordinary and hidden QNMs. The evolution of ¯ I ( x ) as the strength of the disorder s growsis shown in Fig. 4.Initially, when s is so small that values of k ′′ are almost equal for both types of modes,the distributions ¯ I ( x ) are practically identical and evolve in the same manner: the minimumis placed near the center of the sample, and slopes (which are k ′′ ) decrease as the disorderstrength s grows (see Fig. 4). These distributions are similar to the distribution of the7 -8 -7 -6 -5 -4 -3 -2 -1 -2 -2 -2 -2 -1 cba k " d Disorder strength s
FIG. 3. Variation of k ′′ ( s ). Some k ′′ (marked by the same letters as in Fig 2) are independent of s in a broad range (several order of magnitude!) of s variation. Note that length of the systemis equal or larger than localization length when s ≥ .
1. Red line shows the dependence k ′′ ( s ),described by Eq. (22) intensity in the regular resonator with a small imbalance between the reflection coefficients r L and r R of the resonator walls.When k ′′ ( s ) of the hidden mode “ a ” reaches its plateau (see Fig. 3), the minimum ofits distribution shifts from the center, as in the resonator with strong imbalance betweenthe reflection coefficients r L and r R . The slope of the distribution of hidden modes remainsconstant ( k ′′ is independent of s on the plateau), whereas the slopes of all ordinary modesare equal and continue to decrease as the parameter s grows. The difference between thedistributions of ordinary and hidden QNMs is that the ordinary modes are concentratednear both edges of the system, while the hidden mode is nestled at one edge.It is important to stress that this separation of the QNMs into two types occurs whenthe disorder strength s is small so that the localization length ℓ loc is large relative to thesystem length L , ℓ loc ≫ L . Thus, this phenomenon is not related to Anderson localization,but, as it will be shown below, manifests itself also when ℓ loc ≪ L .Notwithstanding that at s →
1, the lifetimes of all hidden modes increase, these modes8
50 100 150 200-20-15-10-50 0 50 100 150 200-20-15-10-50 0 50 100 150 200-20-15-10-50
Mode "a L " I n t e n s it y , l og a r it h m i c s ca l e Layer number s -8 -7 -6 -5 -1 Mode "a R " I n t e n s it y , l og a r it h m i c s ca l e Layer number s -8 -7 -6 -5 -1 I n t e n s it y , l og a r it h m i c s ca l e Layer number s -8 -7 -6 -5 -1 Mode "a"
FIG. 4. Intensity distribution ¯ I ( x ) along the system for ordinary QNMs “ a L ” (a) and “ a R ” (b),and hidden QNM “ a ” (c) for different values of the disorder strength s . Notations for the modesare the same as in Figs. 2 and 3. are much more resistant to disorder: they become localized at far stronger disorder thanordinary states. As can be seen, for example, in Fig. 3, at s ∼ . − .
5, when L ∼ − ℓ loc ,the difference between the imaginary parts of ordinary and some of hidden QNMs is aboutof one order of magnitude. 9 II. TRANSMISSION RESONANCES IN 1D SYSTEMS
We now consider the transmission of an incident wave through the system. The wavenum-bers at which the transmission coefficient reaches its local maximum and the correspondingfields inside the system are transmission resonances (TR). The QNMs and TRs are interre-lated. In what follows, we explore the relation between QNMs and TRs, in particular, studythe differences between the spectra of TRs and QNMs.It is easy to show that in a resonator, the wave numbers k (res) m of the transmission res-onances coincide with the real parts k ′ m given by Eq. (1), and there is a one-to-one corre-spondence between QNMs and TRs so that the number of resonances N res is equal to thenumber of QNMs, N mod , in a given frequency interval. The same relation also exists inperiodic systems (periodic sets r j and d j ) [39].In disordered systems, the relation between QNMs and TRs is quite different. While eachTR has its partner among the QNMs, the reverse is not true: there are hidden QNMs thatare not associated with any maximum in transmission as shown in Figs. 5 and 6 T ( k ) k & k’ FIG. 5. Transmission spectrum T ( k ) at s = 0 . L ≃ ℓ loc ). The black solid (dashed) verticallines indicate the k ′ n values of the hidden (ordinary) QNMs. Every maximum in the transmissionspectrum can be associated with ordinary QNM ( Figure 7 illustrates another fundamental difference between the ordinary and hidden10
IG. 6. Variation of the wave numbers k (res) ( s ) (red crosses) and k ′ ( s ) (closed blue circles) withthe strength s of the disorder. QNMs are numbered as in Fig. 5. It is seen that for ordinaryQNMs, k (res) ( s ) and k ′ ( s ) practically coincide, whereas there are no resonances associated withhidden QNMs ( QNMs. The ordinary QNMs whose real parts of the complex-valued eigenfrequency,Re ω (mod) lie in a given frequency interval, can be determined from the transmittancespectrum T ( ω ) of 1D disordered samples, because each peak in the spectrum corresponds toa frequency whose value ω (res) practically coincides with Re ω (mod) . Moreover, when disorderis strong enough, so that L > ℓ loc , the distribution of the transmitted wave intensity alongthe sample reconstructs very closely the shape of the intensity of ordinary QNM eigenfunc-tions. In contrast, a hidden QNM is invisible (this explains the origin of the term ”hidden”)in the transmittance spectrum and its intensity distribution is indistinguishable from thatat a non-resonant frequency.Note that although the hidden modes are not displayed in the amplitude of the trans-mission coefficient, they are manifested in the phase of the transmission coefficient. Thedensity of states at a frequency ω is proportional to the derivative with respect to frequencyof the phase of the complex transmission coefficient [38]. Our numerical calculations show11hat each hidden mode adds π to the total phase shift of the transmission coefficient exactlyin the same way as ordinary QNMs. FIG. 7. Difference between ordinary and hidden QNMs. (a) transmittance spectrum. (b)distribution of the incident wave intensity into the sample as a function of frequency and distance.There are two QNMs with nearby real parts of eigenfrequencies Re ω (mod) , marked by dashed linesin the panels (a) and (b). Distributions of the intensities of eigenfunctions of hidden and ordinaryQNMs along the sample are shown by thick blue lines in panels (c) and (d), correspondingly.Hatched red areas in panels (c) and (d) show intensity distributions of the incident waves whosefrequencies coincide with Re ω (mod) of hidden and ordinary QNMs, corresondingly. The evolution of a hidden QNM as the degree of disorder grows is analogous to theevolution of a mode in a regular resonator when one of its edges becomes less transparent.This means that a hidden mode may be transformed into an ordinary (i.e., made visible inthe transmission) by increasing the reflectivity of the corresponding edge of the sample, asillustrated in Fig. 8.The sample, whose transmission spectrum is shown in Fig. 5, contains three hiddenQNMs ( I ( n ) ( n isthe layer number) for QNMs r R smaller than the left reflection coefficient r L , r R /r L <
1. Theintensity distribution of QNM r R /r L > T ( k ) (a)(b) T ( k ) k & k’ FIG. 8. Hidden modes
Important to stress that the separation of QNMs into two types, ordinary and hidden,occurs already at a very small disorder strength, s →
0, when the localization length islarger than the sample length, ℓ loc ≫ L .The ensemble-averaged of the ratio of the number of transmission resonances, N res , whichis the number of ordinary modes, to the total number of QNMs, N mod , has been calculatednumerically for a variety of randomly layered samples with different types of disorder (ran-dom reflection coefficients of the layers, r j , and/or random thicknesses d j , with rectangularand Gaussian distribution functions) in broad ranges of the disorder strength s and of thelength of the realizations N .Figure 9 shows the average of N res /N mod over 10 random realizations as a function of13he ensemble-averaged transmission coefficient h T i [panel (a)], and as a function of ratioof N to the localization length n loc (measured in numbers of layers), N/n loc [panel (b)] forsamples with N = 50 , , N res ( h T i ) /N mod and N res ( N/n loc ) /N mod for samples of different lengths merge in a single curve. N r e s / N m od T N = 50,100,150,200 (a)(b)
N = 50,100,150,200 N r e s / N m od n loc FIG. 9. Ratio N res /N mod as a function of the ensemble-averaged transmission coefficient h T i [panel(a)], and as a function of ratio of N to the localization length n loc [panel (b)] for systems ofvarious lengths (number of layers N ). The horizontal dashed red line marks p / It is seen in Fig. 9 that the difference between N res and N mod appears when n loc ≫ N ,and the ratio N res /N mod varies weakly even when n loc ≪ N . Moreover, independentlyof the samples parameters, the average ratio N res /N mod tends to the constant p / n loc → ∞ . Thus, the existence of hidden modes and the universality of their relative number14s a general feature of 1D disordered systems not specifically related to localization. IV. MEASUREMENTS OF TRANSMISSION EIGENCHANNELS AND TRANS-MISSION RESONANCES IN MULTICHANNEL SYSTEMS
It is of interest to explore the ratio of the numbers of local maxima in transmission andQNMs in random multichannel systems and to compare to results for 1D systems. Weconsider quasi-one dimensional (quasi-1D) samples with reflecting sides and transverse di-mensions W much smaller than the sample length L and localization length ℓ loc = N chan ℓ , W < ℓ loc , L . Here, N chan is the number of channels or freely-propagating transverse modesin the perfectly conducting leads or empty waveguide leading to the sample and ℓ is thetransport mean free path. The incident channels are thoroughly mixed by scattering withinthe sample. In contrast to transmission in 1D samples with a single transmission chan-nel, transmission through quasi-1D samples is described by the field transmission matrix t with elements t ba between all N chan incident and outgoing channels, a and b , respectively.From the transmission matrix, we may distinguish three types of transmission variables inquasi-1D samples: the intensity T ba = | t ba | , the total transmission, T a = P N chan b =1 T ba , andtransmittance, T = P N chan a,b =1 | t ba | . The transmittance is analogous to the electronic conduc-tance in units of the quantum of conductance e /h [11, 15, 40]. The ensemble average valueof the transmittance T is equal to the dimensionless conductance, g = h T i , which character-izes the crossover from diffusive to localized waves. In diffusive samples, the dimensionlessconductance is equal to the Thouless number, g = δ and the localization threshold is reachedwhen g = δ = 1 [10, 12].Significant differences between results in 1D and quasi-1D geometries can be expectedsince propagation can be diffusive in quasi-1D samples with length greater than the meanfree path but smaller than the localization length, ℓ < L < ℓ loc = N chan ℓ , whereas a diffusiveregime does not exist in 1D since ℓ loc = ℓ [41]. For diffusive waves, QNMs overlap spectrallyand may coalesce into a single peak in the transmittance spectrum. Thus we might expectthat the QNMs within a typical linewidth form a single peak in transmission so that theratio N res /N mod is the ratio of the mode spacing to the mode linewidth. The mode linewidthis related to the correlation frequency in the transmission spectra, but the mode spacingcannot be readily ascertained once modes overlap.15he transmittance can also be expressed as T = P N chan n =1 τ n , where the τ n are the eigenval-ues of the matrix product tt † [15]. The transmission matrix provides a basis for comparisonbetween results for 1D and quasi-1D, which is often more direct than a comparison basedon QNMs, since the statistics of the contribution of different modes to transmission is notwell-established, whereas the contribution of different channels is simply the sum of thetransmission eigenvalues. In addition, transmission eigenchannels are orthogonal, whereasthe waveform in transmission for spectrally-adjacent modes are strongly correlated [8] sothat the transmission involves interference between modes.The transmission eigenvalue may be obtained from the singular value decomposition of thetransmission matrix, t = U Λ V † [42]. Here, U and V are unitary matrices and Λ is a diagonalmatrix with elements √ τ n . The incident fields of the eigenchannels on the incident surface, v n , which are the columns of V , in the singular-value decomposition are orthogonal, as arethe corresponding outgoing eigenchannels, u n . Only a fraction of the N chan eigenchannelscontribute appreciably to the transmission [14]. In diffusive samples, the transmission isdominated by g channels with τ n > /e [16, 43], while a single eigenchannel dominatestransmission for localized samples. The statistics of transmission depend directly on theparticipation number of transmission eigenhannels, M ≡ ( P N chan n =1 τ n ) / P N chan n =1 τ n [29]. M isequal to 3 g/ A. Numerical simulations
To explore the ratio N res /N mod over a broad range of g = h T i for multichannel disorderedwaveguides in the crossover from diffusive to localized waves, we carry out numerical simu-lations for a scalar wave propagating through a two dimensional disordered waveguide withreflecting sides and semi-infinite leads. For diffusive samples in which there is considerablemode overlap since δ = δν/ ∆ ν > δν and ∆ ν are the linewidth and the distancebetween spectral lines) the density of states (DOS), and from this the number of QNMswithin the spectrum, can be obtained from the sum of the derivatives of the compositephase of the transmission eigenchannel [44]. The derivative of the composite phase of the n th eigenchannel is equal to the dwell time of the photon within the sample in the eigen-channel. The total number of modes N mod in a given frequency interval is then the integralover this interval of the DOS. This has allowed us to determine the ratio N res /N mod in the16rossover to localization.Simulations are carried out by discretizing the wave equation ∇ E ( x, y ) + k ǫ ( x, y ) E ( x, y ) = 0 (3)on a square grid and solved via the recursive Green function method [45]. Here, k is thewave vector in the leads. Also, ǫ ( x, y ) = 1 ± δǫ ( x, y ) is the spatially-varying dielectricfunction in the disordered region with δǫ ( x, y ) chosen from a rectangular distribution and ǫ = 1 in the empty leads. Reflections at the sample boundaries are minimal because thesample is index matched to its surroundings. The product of k at 14.7 GHz and the gridspacing is set to unity. In the frequency range studied, the leads attached to the randomwaveguide support N chan = 16 channels which are the propagating waveguide modes. Inour scalar quasi-1D simulations for a sample with a width W , the number of channels atfrequencies above the cutoff frequency is the integer part of 2 W/λ . These channels shouldnot be confused with the QNMs of the random medium which correspond to resonances ofthe medium with Lorentzian lines centered at distinct frequencies. In the simulations, thelength of the sample L is equal to 500 in units of the grid spacing except for one deeplylocalized sample with g = 0 .
12, for which L = 800 and the width of the sample W is 16 π .Typical spectra of intensity, total transmission and transmittance are shown in Fig. 10 fora diffusive sample with g = 2 . g = 0 . h T i in Fig. 11.The DOS and so the number of QNMs within the spectrum in the samples of the samesize are not affected by the strength of disorder so that the decreasing ratio N res /N mod with increasing h T i reflects only the decreasing number of peaks in the transmission spectradue to the broadening of the modes and the consequent increase in their spectral overlap.Since there are typically δ QNMs within the mode linewidth for diffusive waves, we mightexpect the ratio N res /N mod to fall inversely with M , N res /N mod ∼ /δ ∼ /g ∼ / M . Fordeeply localized waves, however, this ratio is expected to approach unity as M approachesunity. This suggests that N res /N mod ∼ /M . in this limit. A plot of 1 /M in Fig. 11shows that towards the diffusive and localized limits 1 /M is close to the ratio N res /N mod .For diffusive waves, the intensity correlation frequency does not change as the width of the17 T b a T a T T b a T a T (b)(c) (d)(e)(f) FIG. 10. Spectra of intensity, total transmission and transmittance for a localized sample drawnfrom a random ensemble with g = 0.3 (a)-(c) and a diffusive sample taken from an ensemble with g = 2.1 (d)-(f). Sharper spectral features are observed and spatial averaging is seen to be lesseffective in smoothing the spectra for localized waves than for diffusive waves. sample changes for fixed length and scattering strength since it is tied to the time of theflight distribution, which is independent of W [46]. Since N res is essentially the width of thespectrum divided by the correlation frequency of the intensity, the number of peaks withinthe intensity spectrum does not change. However, g and the DOS are proportional to N chan ,so that M increases with sample width and N res /N mod is inversely proportional to M . Inaddition, the propagation in a multichannel disordered sample is essentially 1D, when M isapproaching unity [30].These results suggest that a comparison can be made between propagation in both 1Dand multichannel systems via the ratio of the number of peaks in the transmission spectra tothe number of modes normalized by M , N res / ( N mod /M ). This ratio may be expected to beclose to unity for L ≫ ℓ loc . We consider the variation with g = h T i of the ratio M N res /N mod in quasi-1D and compare this with the corresponding ratio in 1D in which M = 1. Thevalues of this ratio in quasi-1D and 1D are close, as seen in Fig. 12.18 N r e s / N m od g =
Measurement T ba T ba T a T a T T FIG. 11. Variation of the N res /N mod for transmission, total transmission and transmittance vs. g = h T i for multichannel random samples in simulations. The ratios obtained from microwavemeasurements of spectra of the three transmission variables in a multichannel localized samplewith g = 0 .
37 are shown as the cross symbols and are in good agreement with the simulations witha similar value of g . The value of 1 /M found in the simulations is shown as overturned triangles. B. Microwave experiment
For quasi-1D samples in the crossover to localization in which spectral overlap is mod-erate, it is possible to analyze the measured field spectra to obtain the central frequenciesof the QNMs and to compare these to peaks in transmission. Spectral measurements of thetransmittance T were made in a copper waveguide of diameter 7.3 cm and of length 40 cmcontaining randomly positioned alumina spheres with index 3.14, over a random ensemblefor which g = 0 .
37. The empty waveguide supports N chan ∼
30 propagation channels in thefrequency range of the experiment: 10.0-10.24 GHz. The transmission matrix is determinedfrom measurements of the field transmission coefficient between points on grids of 49 loca-19
IG. 12. Number of peaks in the transmission spectra per effective transmission eigenchannel, N res / ( N mod /M ), is plotted as a function of T = g/M . The quantity g/M is the effective transmis-sion coefficient per effective transmission eigenvalue of the quasi-1D system. Such normalizationof the conductance g in quasi-1D samples makes possible a comparison with 1D systems. The redline corresponds to a 1D system; the experimental data is shown by the asterisk; the blue dotsshow the results of numerical simulations; and dashed line is drawn at the level p /
5. Beyond thediffusive regime the ratio plotted rises towards unity for ballistic propagation. For ballistic waves,each of the N channel transmission eigenvalue is unity so that the transmittance is N channel and alleigenchannels contribute equally to the transmittance so that M = N channel , yielding g/M = 1. tions for the source antenna and detection antennas on the input and output surfaces of thewaveguide for a single polarization with a grid spacing of 9 mm. Such measurements of thetransmission matrix in real space for a single polarization are incomplete. The distributionof transmission eigenvalues determined from these measurements may differ from theoreticalcalculations [42, 47]. We find, however, that the impact of incompleteness upon the statisticsof transmittance and transmission eigenvalues is small as long as the number of measuredchannels is much greater than M , as is the case in these measurements of transmission inlocalized samples [30]. Here M = 1 .
23 and therefore the statistics of transmission are notaffected by the incompleteness of the measurement [30]. The influence of absorption in thesesamples is statistically removed by compensating for the enhanced decay of the field dueto absorption [48]. Different random sample configurations are obtained by briefly rotating20nd vibrating the sample tube. The probability distribution of the transmittance is in goodagreement with the distribution calculated for this value of g [30, 49–51].We find the central frequencies and linewidths of the QNMs within the frequency rangeof the measurements by carrying out a modal decomposition of the transmitted field. Agiven polarization component of the field can be expressed as a sum of the contributionsfrom each of the QNMs: E ( r , ω ) = Σ m a m ( r ) Γ m / m / i ( ω − ω m ) . (4)Here a m ( r ) are complex-valued amplitudes of QNMs.The central frequencies ω m and linewidths Γ m of the modes are found by simultaneouslyfitting 45 field spectra. The transmittance as well as the Lorentzian lines for each QNMnormalized to unity and the DOS, which is the sum of such Lorentzian lines over all QNMsare shown in Fig. 13 for a single random configuration. The DOS curves for different modesare plotted in different colors so that they can be distinguished more clearly. The DOS isalso determined from the sum of the spectral derivatives of the composite phase of eachtransmission eigenchannel and plotted in Fig. 13. The DOS determined from analyses ofthe QNMs and of the transmission eigenchannels are seen to be in agreement. The dashedvertical lines in Fig. 13 are drawn from the peaks in the transmittance spectra in (a) to thefrequency axis in (b). As found in 1D simulations, each peak in T is close to the frequencyof a QNM, but many QNMs do not correspond to a distinct peak in the transmittance.Frequently, more than one QNM falls within a single peak in T .The ratio of the number of peaks in spectra of transmittance to the number of QNMsaveraged over a random ensemble of 40 configurations is 0.61, with a standard deviation of0.057. This is indicated by the cross in Fig. 11 and is consistent with values of the ratiofound in computer simulations. This value of this ratio is slightly smaller than the value0.65 found in simulations for 1D sample with h T i = 0 .
37 found in 1D simulations, as seen inFig. 9. This may be attributed to the value M = 1 .
23 being larger than the value of unityin 1D. This reflects the tendency of the ratio to decrease with increasing M as found fordiffusive waves.Equation (4) offers an explanation for the fact that the number of transmission resonancescan be smaller than that of all QNMs. If, for example, the transmitted field is a sum oftwo modes, from Eq. (4) it follows that the number of peaks in the transmission spectrum21 IG. 13. (a) Spectrum of transmittance T and the individual modes. The integration of eachLorentzian curve in the lower panel over the frequency yields the density of state of unity. Thereare 22 local maxima in the spectrum of T and the number of modes are 39. (b) Spectrum ofthe density of states. The sum of all the Lorentzian curves above gives the DOS of the sample,which is seen to be in good agreement with the DOS [panel (b)] obtained via the summation ofthe composite phase derivatives of each transmission eigenchannel. is either one or two, depending on the widths of the modes. C. Spatial intensity distribution of QNMs within quasi-1D disordered samples
In order to fully characterize the QNMs and their relationship to peaks in transmittancein quasi-1D samples, it would be desirable to examine the longitudinal profile of QNMswithin the media. Because we do not have access to the interior of the multichannel sample,however, we explore the spatial profile of QNMs using numerical simulations based on therecursive Greens function technique. The Greens function between points on the incidentplane r and within the sample r ′ can be expressed in a manner similar to Eq. (4) as a sum of22ontribution from each of the modes, We find in the simulations that the spatial distributionof the m th mode obtained by decomposing the field into QNMs depends weakly upon theexcitation point r . We therefore average the spatial profile for each QNM over the profilesobtained for all excitation points on the input of the sample.We consider propagation in a sample drawn from an ensemble with a value of g which isbelow unity but still not too small. In this case, QNMs still overlap but it is yet possible toanalyze the field into QNMs. We present in Fig. 14 that a spectrum of transmittance in asample configuration chosen from an ensemble with g = 0 .
26 and h M i = 1 .
16, together withprofiles of a ordinary and a hidden mode within the spectrum. The nature of propagation inthe sample might not differ appreciably from propagation in 1D samples, for which M = 1.We find that the intensity distributions integrated over the transverse direction of the hiddenmode in the transmission spectrum of the quasi-1D samples falls monotonically within thesample, while the ordinary mode associated with peaks in transmission is peaked in themiddle of the sample. V. ANALYTICAL CALCULATIONS OF N res /N mod To calculate the average number of TRs in the limit s ≪
1, we use the single-scatteringapproximation and write the total reflection coefficient r ( k ) of a 1D system as: r ( k ) = Σ Nn =1 r n exp(2 ikx n ) , (5)where x n is the coordinate of the n -th scatterer. The values k max , at which the transmissioncoefficients, T ( k ) = 1 − | r ( k ) | , has a local extrema, are defined as the zeros of the function f ( k ) ≡ d | r ( k ) | /dk = 2Re [ r ( k ) dr ∗ ( k ) /dk ]: f ( k max ) = 4ImΣ Nn =1 Σ Nm =1 r n r m x m e ik max ( x n − x m ) = 0 . (6)Assuming first that δd i = 0, we obtain f ( k ) ∝ Σ Nn =1 Σ Nm =1 r n r m ( m − n ) sin [2 k ( m − n ) d ]= Σ Nl =1 sin (2 kld ) (cid:8) Σ N − ln =1 r n + l r n l +Σ Nn = l r n − l r n l (cid:9) ≡ Σ Nl =1 sin (2 kld ) a l . (7)Equation (7) is the trigonometric sum Σ Nl =1 a l sin ( ν l k ) with “frequencies” ν l = 2 ld andrandom coefficients a l . The statistics of the zeroes of random polynomials have been studied23 IG. 14. Transmittance spectrum and intensity distribution of QNMs in Q1D disordered samples.(a) Spectrum of transmittance T for a localized sample drawn from an ensemble with g = 0 . in [28], where it is shown that the statistically-averaged number of real roots N root of suchsum at a certain interval ∆ k is N root = ∆ kπ s Σ ν l σ l Σ σ l , (8)where σ l = Var( a l ) is the variance of the coefficients a L = Σ N − ln =1 r n + l r n l + Σ Nn = l r n − l r n l . Whenthe reflection coefficients are uncorrelated, thenVar( a l ) = 2( N − l ) l (cid:0) σ + 2¯ r σ (cid:1) , (9)where σ = Var( r ) and ¯ r is the mean value of r i . The sums in Eq. (8) can be calculated24sing Eq. (9), which yields [52]:Σ Nl =1 σ l = 2 (cid:0) σ + 2¯ r σ (cid:1) Σ Nl =1 l ( N − l ) ≃ (cid:0) σ + 2¯ r σ (cid:1) N , Σ Nl =1 ν l σ l = 8 d Σ Nl =1 (cid:0) σ + 2¯ r σ (cid:1) l ( N − l ) ≃ d N (cid:0) σ + 2¯ r σ (cid:1) . (10)From Eqs. (8) and (10) we obtain N root = 2∆ kN d π r
25 = 2 ∆ kLπ r , (11)where L = N d . Since the number of minima of the reflection coefficient is equal to thenumber of TRs, N res = N root /
2, and the number N mod of QNMs in the same interval ∆ k is N mod = ∆ kL/π , from Eq. (11) it follows that N res /N mod = p / . (12)Although this relation was derived for systems with random reflection coefficients andconstant distances between the scatterers, it also holds for samples in which these distancesare random ( δd i = 0). In this case, the frequencies ν = 2 ld d in Eq. (7) should be replacedby ν = 2 | x m − x m ± l | . Since the main contribution to the sums in Eq. (8) is given by theterms with large l ∼ N , the mean value of | x m − x m ± l | can be replaced by ld , in the caseof a homogeneous distribution of the distances d n along the system. This ultimately leadsto the same result Eq. (12). VI. HIDDEN MODES: SIMPLE MODEL
In Sec. II, QNMs were introduced as solutions of the wave equation satisfying the outgoingboundary conditions. Their eigenvalues of QNMs, k (mod) = k ′ − ik ′′ , can be calculated asroots of the equation M = 0, where ˆ M is the transfer matrix, which connects waves’amplitudes at the left and right sides of the whole system. The transfer matrix of thesystem which consists of N + 1 scatterers separated by N intervals has the form:ˆ M = ˆ T N +1 ˆ S N ˆ T N ˆ S N − · · · ˆ S ˆ T ˆ S ˆ T . (13)Here ˆ S i = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e ikd i e − ikd i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (14)25nd ˆ T i is the transfer matrix through the i th scatterer. Assuming that reflection and trans-mission coefficients are real, ˆ T i can be presented asˆ T i = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) /t i − r i /t i − r i /t i /t i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 1 t i (cid:16) ˆ I − r i ˆ σ (cid:17) , (15)where ˆ I is the unit matrix, and ˆ σ is the Pauli matrix.Omitting denominator Q Ni =1 t i , matrix ˆ M can be written as ordered productˆ M = Y (cid:16) ˆ I − sr i ˆ σ (cid:17) ˆ S i , (16)where substitution r i → sr i is used. Eq. (16) allows presenting transfer matrix as a powerseries in s ≪
1: ˆ M = N Y n =1 ˆ S i + N +1 X n =1 s n ˆ A n , (17)where matrix ˆ A n contain various ordered products of matrices ˆ S i and n Pauli matrices.Matrices ˆ S i are diagonal, whereas Pauli matrix is anti-diagonal, so only even combinationsof Pauli matrices contribute to M . Thus, M = e − ikL + [( N +1) / X m =1 s m a m , (18)where L is the sample length, and the coefficients a m contain various combinations of prod-ucts of 2 m reflection coefficients r i with exponential multipliers exp( − ikL + 2 ikd i,j ), where d i,j are the distances between any ordered pairs of scatterers.Neglecting terms with higher than s powers in Eq. (18), the dispersion equation, whichdefines eigenvalues k , can be presented as follows: s X i,j c ij e k ′′ d i,j = − , (19)where the coefficients c ij = r i r j e ik ′ d i,i are formed by various pairs of the scatterers, as it isschematically shown in Fig. 15a. The greater is the distance d i,j between the scatterers, thelarger are exponents e k ′′ d i,j in Eq. (19).The largest exponents are associated with the pairs of scatterers placed near the oppositeends of the sample. When k ′′ d ≪
1, there are many such pairs, located in blue regionsin Fig. 15b, whose associated exponents are of the same order of magnitude, e k ′′ L . Let us26 IG. 15. (a) – coefficients c ij are formed by various pairs of scatterers; (b) – coefficient ˜ c containsall possible pairs of scatterers, linked schematically by red line c is formed by thescatterers from blue and green regions, connected by red lines combine all such pairs in Eq. (19) in one term ˜ c and characterize them by one commonexponent e k ′′ L . The number of scatterers near the sample ends, which form this group, canbe estimated as n eff ≃ ( k ′′ d ) − ≫
1, so that the lengths of blue regions in Fig. 15b are ∼ n eff d ≪ L .The next group, ˜ c , which is associated with the exponent of the order of e k ′′ ( L − n eff d ) ,consists of pairs of scatterers, one from green and another from blue regions in Fig. 15b. Insuch a way, Eq. (19) can be approximately presented as s (cid:16) ˜ c e k ′′ L + ˜ c e k ′′ ( L − n eff d ) + . . . (cid:17) = − , (20)Strictly speaking, the phenomenologically introduced number n eff varies from group to group,but when k ′′ d ≪ n eff is large enough and it is possible to neglect its variation.The coefficients ˜ c n in Eq. (20) are the sums of n eff random vectors in complex plane. Forany given sample the lengths of these vectors are fixed, whereas the phases varies from modeto mode, so that the magnitudes of the coefficients ˜ c n , been averaged over many modes, canbe estimates as h| ˜ c n |i ≃ p h r i n eff ≃ h r i√ n eff . (21)Using Eqs. (20) and (21), one can calculate value of k ′′ , averaged over many modes.When s ≪ h k ′′ ( s ) i is large and the second term in the parentheses in Eq. (20) is smallas compared with the first one ( e − k ′′ n eff d ≪
1) and can be omitted. Then, the average27olution h k ′′ ( s ) i of Eq. (20) is h k ′′ ( s ) i ≃ L ln 1 s ¯ r √ n eff = 12 L (cid:18) ln 1 s ¯ r −
12 ln n eff (cid:19) ≃ L ln 1 s ¯ r . (22)The dependence h k ′′ ( s ) i described by Eq. (22) agrees well with the result of numericalsimulations, presented in Fig. 3 by red line.Expression Eq. (22) describes averaged over many modes dependence k ′′ ( s ), but for anygiven mode this dependence can be different. Indeed, since n eff ≪ N (for example, n eff ≃ s = 10 − in the numerical simulation presented in Figs. 2 and 3)) fluctuation of thevalues of | ˜ c n | for different eigenmodes can be rather large. In particular, | ˜ c | for a certainmode can be much smaller, than | ˜ c | . Presenting Eq. (20) in the form1 + ˜ c ˜ c e − k ′′ n eff d = − e − k ′′ L ˜ c s , (23)it is easy to see that Eq. (23) has solution k ′′ ≃ n eff d ln (cid:12)(cid:12)(cid:12)(cid:12) ˜ c ˜ c (cid:12)(cid:12)(cid:12)(cid:12) , (24)when s exceeds some critical value s , s ≫ s = | ˜ c | − exp (cid:18) − Ln eff d ln (cid:12)(cid:12)(cid:12)(cid:12) ˜ c ˜ c (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (25)Solution Eq. (24) is independent of s and represents the hidden QNM (see Fig. 3).Recall that ˜ c n are formed by different groups of the reflection coefficients. In general,the similar, independent of s , solutions of the dispersion equation appear when magnitudessome first coefficients ˜ c n in Eq. (20) are small as compared with magnitudes of the nextcoefficients.In order to demonstrate that independent on s solutions of the dispersion equation indeedcorrespond to the hidden modes, let us consider the system composed of three scatterersonly. The dispersion equation Eq. (20) for this system is s (cid:2) r r e ik ( d + d ) + r r e ikd + r r e ikd (cid:3) = − . (26)When all r i are of the same order of magnitude, r i ∼ r , and s is so small [ k ′′ ( s ) is so large]that exp[ k ′′ ( s ) d , ] ≫
1, the solution k ′′ of Eq. (26) is k ′′ ≃ d + d ) ln 1 s r . (27)28quations (26) and (27) are particular cases of the general formulas (20) and (22).If, for example, | r | is small as compared with | r , | , but s ≫ s = | r r | − exp h − d + d d ln (cid:12)(cid:12)(cid:12) r r (cid:12)(cid:12)(cid:12)i ,there is another solution of Eq. (26): k ′′ = 12 d ln (cid:12)(cid:12)(cid:12)(cid:12) r r (cid:12)(cid:12)(cid:12)(cid:12) . (28)This solution is independent on s , similarly to the solution Eq. (24).Result of numerical solution of Eq. (26) is shown in Fig. 16. Fig. 17 demonstrates relation -12 -10 -8 -6 -4 -2 00.00.10.20.30.40.50.60.70.8 k " ln(s) FIG. 16. Dependence k ′′ ( s ). Vertical and horizontal red lines mark s crit and k ′′ defined by Eq. (28). between real part of the QNMs’ eigenvalues k ′ ( s ) and position of the peak k (res) ( s ) in thetransmission spectrum. Note that hidden modes are invisible in the transmission spectrumeven when s ≪ s crit . 29
12 -10 -8 -6 -4 -2 010.210.410.610.811.0 k ’ , k (r e s ) ln(s) FIG. 17. k ′ ( s ) (black dots) and k res ( s ) (red dots). When s ≪ s crit , the value of k ′′ is the same forhidden and ordinary modes. Nevertheless, hidden modes are invisible in the transmission spectrum. VII. SUPERRADIANCE AND RESONANCE TRAPPING IN 1D RANDOM SYS-TEMS
The model introduced in the previous Section can be used to study the segregation ofsuperradiant states and trapped modes in regular quantum-mechanical and wave structuresand to illuminate the analogy between this phenomenon and existence of two types of QNMs(hidden and ordinary) in disordered systems considered above. Behavior of modes in regularopen structures as the coupling to an environment is altered, has been intensively studied incondensed matter physics, optics, and nuclear, atomic, and microwave physics. Common toall these studies is the appearance of two time scales when the coupling to the environmentvia open decay channels increases and QNMs begin to overlap [53–58]; for a review, see [59]and references therein. When the coupling to the environment is weak, the lifetimes of allstates tend to decrease as the coupling increases. As the coupling reaches a critical value, arestructuring of the spectrum of QNMs occurs leading to segregation of the imaginary parts30f the complex eigenvalues and of the decay widths. The states separate into short-lived(superradiant) and long-lived (trapped) states. This phenomenon is general and, by analogyto quantum optics [60] and atomic physics [61–63], is known as the superradiance transition.In more complicated structures, such of those consisting of two coupled oscillating subsys-tems, one with a low and the other with a much higher density of states, the superradiancetransition is closely related to the existence of doorway states [56, 57] that strongly coupleto short-lived QNMs with external decay channels.It is important to stress that along with the pronounced similarities between the resonancetrapping in many-particle quantum systems, open microwave cavities, etc., and betweenthe “hidding” of some of quasi-normal modes in disordered samples there are substantialdifferences as well. In particular, resonance trapping happens in regular systems consideredin [55, 59] when the coupling of the large number of QNMs to a much smaller numberof common decay channels increases. Without disorder, the samples that we consider areperfectly coupled to the environment (total transmission at all frequencies). Finite couplingappears due to disorder, as the result of the interference of multiply-scattered random fields,and the role of the coupling parameter is played by the strength of the scattering inside thesystem.To reproduce the superradince phenomena in disordered structures we modify the modelslightly by placing the random sample between two reflectors with reflection coefficients r L and r R , located at distances δ L and δ R from the edge scatterers. For simplicity, we assumethat δ R = δ L = δ . These reflectors can be included in the dispersion equation Eq. (20) asadditional scatterers as follows: s (cid:16) s − r L r R e ik (mod) ( L +2 δ ) + s − ˜ c e k ′′ ( L + δ ) + ˜ c e k ′′ L + ˜ c e k ′′ ( L − n eff d ) + . . . (cid:17) = − . (29)Here ˜ c ∝ r L,R contains the products r L,R r i with corresponding exponential multipliers, thelargest of which, exp [2 k ′′ ( L + δ )], is separated in the explicit form.To approach the conditions at which superradiance and resonance trapping occur, weconsider below (in contrast to the previous sections) the evolution of the eigenvalues of agiven sample with fixed s when r R,L → | r L r R | is large, the first term in the parentheses dominates and thesolution of Eq. (29) is k ′′ = 12( L + 2 δ ) ln (cid:12)(cid:12)(cid:12)(cid:12) r L r R (cid:12)(cid:12)(cid:12)(cid:12) . (30)31f δ = 0, the magnitudes of the exponents in the first three terms are equal. When | r L r R | →
0, the magnitudes of the additional two terms decrease and the solutions of Eq. (29) tendto their solutions in the original sample (without end reflectors), as shown in Fig. 18.
FIG. 18. Two reflectors are placed at the sample ends, r L = r R = r end . Modes marked by letterscorrespond to the same modes in Figs. 2, 3. (a) – Trajectories of eigenvalues as the coupling grows.(b) – k ′′ ( r end ). The life time of the hidden QNMs decreases much faster than the life time of theordinary ones. . δ = 0, the trajectories of the eigenvalues in the complex plane are more complicate.Although most of the eigenvalues finally reach the same positions as in the original sample,there are eigenvalues, for which k ′′ → ∞ as r L,R → δ = 0 and k ′′ → ∞ . In this case Eq. (29) can bewritten as r L r R e ik ( L +2 δ ) + s ( r L r N +1 + r R r ) e ik ( L + δ ) ≃ , (31)where the largest term in ˜ c , which corresponds to the largest distance L + δ between theend reflectors and the sample scatterers, is explicitly presented. Solution of Eq. (31) k ′′ = 12 δ ln s (cid:12)(cid:12)(cid:12)(cid:12) r N +1 r R + r r L (cid:12)(cid:12)(cid:12)(cid:12) (32)tends to infinity, when even one of the reflection coefficients r L,R → kL/π eigenmodesin the given interval ∆ k , whereas the same system surrounded by the reflectors has ∆ k ( L +2 δ ) /π eigenmodes in the same interval. Thus, some of modes should leave this interval ∆ k when the system returns to its original state.The superradiant transition in periodic and disordered quantum system, which consistof a sets of potential wells, was studied in [53] using effective Hamiltonian approach. It wasshown that the transition occurs when the coefficient γ , which characterizes the couplingwith an environment, reaches the value of the coupling Ω between the wells, γ ≃ Ω. Inthe considered above system γ ≃ − | r R,L | and Ω ≃ − s h r i , so that the superradianttransition occurs when | r R,L | ≃ s p h r i . This condition agrees well with presented in Fig. 19bresults.Hidden modes can be associated with superradiant states, while normal modes aretrapped resonances. Thus, p / ≃ .
63 correspond to the fraction of the modes whichare trapped. This result agrees with [55], where this value was estimated as > .
58, and1 − p / ≃ .
37 is the fraction of the modes which are superradiant. Note, that the originaldisordered sample is already coupled to the environment, so that the coupling strength islimited by the intrinsic properties of the sample and cannot exceed this value, even whenthe end reflectors are fully transparent. 33
IG. 19. The same as Fig. 19, but reflectors are placed at a some distance from the sample ends.There is one eigenvalue whose k ′′ grows unlimitedly. VIII. CONCLUSIONS
In conclusion, we have studied the relationship between spectra of quasi-normal modesand transmission resonances in open 1D and quasi-1D systems. We start from homogeneoussamples, in which each TR is associated with a QNM, and vice versa. As soon as anarbitrarily weak disorder is introduced, this correspondence breaks down: a fraction of the34igenstates becomes hidden, in the sense that the corresponding resonances in transmissiondisappear. The evolution of the imaginary parts of the eigenfrequencies of the hidden QNMswith changing disorder is also rather unusual. Whereas increasing disorder leads to strongerlocalization of ordinary modes so that their eigenfrequencies approach the real axis, theimaginary parts of the eigenfrequency of hidden modes changes very slowly (and may evenincrease when external reflectors are added to the edges) with increasing disorder, andbegin to go down only when the disorder becomes strong enough. For weak disorder, theaveraged ratio of the number of transmission peaks to the total number of QNMs in agiven frequency interval is independent of the type of disorder and deviates only slightlyfrom a constant, p /
5, as the strength of disorder and/or the length of the random sampleincrease over a wide range. This constant coincides with the value of the ratio N res /N mod analytically calculated in the weak single-scattering approximation. As the strength s ofdisorder keeps growing, ultimately all hidden quasimodes become ordinary. This means thatin 1D random systems there exists a pre-localized regime, in which only a fraction of theQNMs are long-lived and provide resonant transmission. If the coupling to the environmentis tuned by external edge reflectors, the superradiace transition can be reproduced. Inquasi-1D samples, a genuine diffusive regime exists in which some QNMs coalesce to forma single peak in transmission with width comparable to the typical modal linewidth. Insuch samples, hidden modes have been discovered experimentally and their proportion of allQNMs in the crossover from diffusion to localization was fairly close to the same constant.The number of peaks in spectra of transmission, as well as in total transmission and intransmittance are nearly the same and fall well below the number of QNMs. Though theratio N res /N mod may be small, we find in microwave experiments and numerical simulationsthat once the number of QNMs is divided by the effective number of channels contributingto transmission to give M N p /N m , this function is similar to results in 1D samples. ACKNOWLEDGMENTS
We gratefully acknowledge stimulating discussion with K. Bliokh, S. Rotter, R. Berkovits,J. Page, Hong Chen, and Ping Cheng. We specially thank M. Dennis who drew our attentionto paper [28].This research is partially supported by the: National Science Foundation (DMR-1207446),35IKEN iTHES Project, MURI Center for Dynamic Magneto-Optics, and a Grant-in-Aid forScientific Research (S). [1] F. Cakoni, H. Haddar (Editors), Inverse Problems , 100201 (2013).[2] E.S.C. Ching, P.T. Leung, A. Maassen van den Brink, W.M. Suen, S.S. Tong, and K. Young,Rev. Mod. Phys. , 1545 (1998).[3] N. Hatano and G. Ordonez, J. Math. Phys. , 122106 (2014).[4] C. Sauvan, J.P. Hugonin, I.S. Maksymov, and P. Lalanne, Phys. Rev. Lett. , 237401(2013).[5] C. Vanneste and P. Sebbah, Phys. Rev. A , 041802 (2009).[6] F.A. Pinheiro, M. Rusek, A. Orlowski, and B. van Tiggelen, Phys. Rev. E , 026605 (2004).[7] P. Sebbah, B. Hu, J. Klosner, and A.Z. Genack, Phys. Rev. Lett. , 183902 (2006).[8] J. Wang and A.Z. Genack, Nature, , 345 (2011).[9] D.J. Thouless, Phys. Rep. , 93 (1974).[10] E. Abrahams, P. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. ,673 (1979).[11] E.N. Economou and C.M. Soukoulis, Phys. Rev. Lett. , 618 (1981).[12] D.J. Thouless, Phys. Rev. Lett. , 1167 (1977).[13] P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. , 290-317 (1988).[14] O.N. Dorokhov, Solid State Commun. , 381 (1984).[15] D.S. Fisher and P. A. Lee, Phys. Rev. B , 6851 (1981).[16] Y. Imry, Europhys. Lett. , 249 (1986).[17] J. L. Pichard, N. Zanon, Y. Imry, and A. D. Stone, J. Phys. France , 22 (1990).[18] S.M. Popoff, A. Goetschy, S.F. Liew, A.D. Stone, and H. Cao, Phys. Rev. Lett. , 133903(2014).[19] Z. Shi and A.Z. Genack, Phys. Rev. Lett. , 043901 (2012).[20] M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, Nat. Photon. , 583-587(2012).[21] B. G´erardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, Phys. Rev. Lett. , 173901(2014).
22] M. Kim, Wonjun Choi, C. Yoon, G.H. Kim, and Wonshik Choi, Opt. Lett. , 2994 (2013).[23] A. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nature Phot. , 283 (2012).[24] N. Moiseev, Non-Hermitian Quantum Mechanics , Cambridge Univ. Press (2011).[25] I. Lifshitz, S. Gredeskul, L. Pastur,
Introduction to the Theory of Disordered Systems (Wiley,New York, 1989).[26] K. Bliokh, Y. Bliokh, V. Freilikher, and Franco Nori, “Anderson Localization of Light inLayered Dielectric Structures”, in
Optical properties of photonic structures: interplay of orderand disorder , ed. by. M. Limonov and R. De La Rue, (CRC Press) p.57 - 86 (2012).[27] K. Bliokh, Y. Bliokh, V. Freilikher, S. Savel’ev, and Franko Nori, Rev. Mod. Phys. , 1201(2008).[28] A. Edelman and E. Kostlan, Bull. Am. Math. Soc. , No. 1, 1-37 (1995), p.14.[29] M. Davy, Z. Shi, J. Wang, and A.Z. Genack, Opt. Exp. , 10367 (2013).[30] Z. Shi, J. Wang, and A.Z. Genack, Proc. Natl. Acad. Sci. USA , 2926 (2014).[31] Y.V. Fyodorov and H-J. Sommers, J. Math. Phys. , 1918 (1997).[32] T. Guhr, A. Muller-Groeling, H. A. Weidenmuller, Phys. Rep. , 189 (1998).[33] K. Joshi and A.M. Jayannavar, Solid State Commun. , 363 (1999).[34] S.A. Ramakrishna and N. Kumar, Eur. Phys. J. B , 515 (2001).[35] H. Kunz and B. Shapiro, J. Phys. A , 10155 (2006).[36] H. Kunz and B. Shapiro, Phys. Rev. B , 054203 (2008).[37] E. Gurevich, B. Shapiro, Lithuanian J. Phys. , 115 (2012).[38] A. Avishai, Y. Band, Phys. Rev. B , 2674 (1985).[39] A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, and M. Bertolotti,Phys. Rev. E , 026614 (2003).[40] R. Landauer, Philos. Mag. , 863 (1970).[41] A.A. Abrikosov and I.A. Ryzhkin, Adv. Phys. , 147 (1978); A. A. Abrikosov, Solid StateCommun. , 997 (1981).[42] C.W.J. Beenakker, Rev. Mod. Phys. , 732 (1997).[43] O.N. Dorokhov, Solid State Comun. , 915 (1982).[44] M. Davy, Z. Shi, J. Wang, X. Cheng, and A.Z. Genack, Phys. Rev. Lett. , 033901 (2015).[45] H.U. Baranger, D.P. DiVincenzo, R.A. Jalabert, and A.D. Stone Phys. Rev. B , 10637(1991).
46] A.Z. Genack and J.M. Drake, Europhys. Lett. , 331 (1990).[47] A. Goetschy and A. Stone, Phys. Rev. Lett. , 063901 (2013).[48] R. Weaver, Phys Rev B , 1077 (1993).[49] K.A. Muttalib and P. W¨olfle, Phys. Rev. Lett. , 3013 (1999).[50] L.S. Froufe-P´erez, P. Garc´ıa-Mochales, P. Serena, P. Mello, and J.J. S´aenz, Phys. Rev. Lett. , 246403 (2002).[51] V.A. Gopar, K.A. Muttalib, P. W¨olfle, Phys. Rev. B , 174204 (2002).[52] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products , Seventh Edition,Academic Press, 2007, pp.1,2[53] G.L. Celardo, L. Kaplan, Phys. Rev. B , 155108 (2009).[54] G.L. Celardo, A.M. Smith, S. Sorathia, V.G. Zelevinsky, R.A. Sen’kov, L. Kaplan, Phys. Rev.B , 165437 (2010).[55] E. Persson, I. Rotter, H.-J. St¨ockmann, M. Barth, Phys. Rev. Lett. , 2478 (2000).[56] C. Sanchez-Perez, K. Volke-Sepulveda, J. Flores, Progress In Electromagnetics Research Sym-posium Proceedings, p. 209 (2011).[57] A. Morales, A. Diaz-de-Anda, J. Flores, L. Gutierrez, R. Mendez-Sanchez, G. Monsivais, P.Mora, EPL, , 204101 (2008)[59] N. Auerbach, V. Zelevinsky, Rep. Prog. Phys. , 106301 (2011).[60] R. Dicke, Phys. Rev.
99 (1954)[61] E. Akkermans, A. Gero, R. Kaiser, Phys. Rev. Lett. , 103602 (2008).[62] A. Gero, E. Akkermans, Phys. Rev. A , 023839 (2013).[63] L. Bellando, A. Gero, E. Akkermans, R. Kaiser, Phys. Rev. A , 063822 (2015)., 063822 (2015).