Hidden Modes in Open Disordered Media: Analytical, Numerical, and Experimental Results

We explore numerically, analytically, and experimentally the relationship between quasi-normal modes (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 there exist two types of the eigenstates: ordinary QNMs which are associated with a TR, and hidden QNMs which do not exhibit peaks in transmission or within the sample. The distinctive feature of the hidden modes is that unlike ordinary ones, their lifetimes remain constant in a wide range of the strength of disorder. In this range, the averaged ratio of the number of transmission peaks $N_{\rm res}$ to the number of QNMs $N_{\rm mod}$, $N_{\rm res}/N_{\rm mod}$, is insensitive to the type and degree of disorder and is close to the value $\sqrt{2/5}$, 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 and trapped modes is discussed. When the coupling to the environment is tuned by an external edge reflectors, the superradiace transition is reproduced. Hidden modes have been also found in microwave measurements in quasi-1D open disordered samples. The microwave measurements and modal analysis of transmission in the crossover to localization in quasi-1D systems give a ratio of $N_{\rm res}/N_{\rm mod}$ close to $\sqrt{2/5}$. In diffusive quasi-1D samples, however, $N_{\rm res}/N_{\rm mod}$ falls as the effective number of transmission eigenchannels $M$ increases. Once $N_{\rm mod}$ is divided by $M$, however, the ratio $N_{\rm res}/N_{\rm mod}$ is close to the ratio found in 1D.


I. INTRODUCTION
Two powerful perspectives have helped clarify the nature of wave propagation in open random systems. One of them, relates to the leakage of waves through the boundaries of the system and can be described in terms of quasi-normal modes (QNMs), which are the extension to open structures of the notion of normal modes in closed systems [1][2][3][4][5][6][7][8][9]. The eigenfrequencies of the QNMs are complex, with imaginary parts that are the inverses of the lifetimes of the QNMs. The second perspective is that of transmission through random systems [10][11][12]. For multichannel samples, transmission is most conveniently described in terms of the transmission matrix, t, whose elements are field transmission coefficients [13][14][15]. The transmittance is the sum of eigenvalues of the Hermitian matrix tt † . Some of these eigenvalues are close to unity even in weakly-transmitting samples [13,14,16,17]. Knowledge of the transmission matrix makes it possible to manipulate the incident wavefront to enhance or suppress total transmission through random media [18][19][20][21][22] and to focus transmitted radiation at selected points [23]. The control over transmitted radiation can be exploited to improve images wased out by random scattering and to facilitate the detection and location of objects [23]. The great potential of such algorithms for a host of practical applications has recently attracted attention in both the physics [23] and mathematics communities [1].
In open regular homogeneous systems (e.g. quantum potential wells, optical cavities, or microwave resonators) each peak in transmission, or transmission resonance (TR), is associated with a QNM ( [24] and references therein), so that the resonant frequency is close to the real part of the corresponding eigenvalue. However, despite extensive research and much recent progress the connection between QNMs and TRs in disordered open systems still 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 has a local maximum (peak in transmission), and some of these are close to unity [25][26][27]. In 1D, each peak is associated with an eigenstate which is a solution of the wave equation with outgoing boundary conditions (a pole of the S-matrix).
In this paper we show that the inverse statement, that each QNM manifests itself as a TR, is incorrect. Rather, in 1D disordered systems, there are two types of QNMs: ordinary QNMs, associated with resonant transmission peaks and strange, dark QNMs unrelated to any maxima in the transmission spectrum. The dark modes exist due to random scattering and arise even in the ballistic regime at weak disorder. The imaginary parts of the eigenfrequencies of dark QNMs vary with increasing disorder in an unusual manner. Typically, stronger disorder leads to stronger localization of modes with eigenfrequencies that approach the real axis. However, the imaginary part of a dark mode's eigenfrequency either increases from the onset of disorder or decreases anomalously slowly. Most surprisingly, the average ratio of the number of ordinary modes to the total number of QNMs in a given frequency interval is independent of the type of disorder and remains close to the constant 2/5 over wide ranges of the strength of disorder and of the total length of the system. The value 2/5 follows from the general statistical properties of random trigonometric polynomials [28]. As the scattering strength and the length of the system increase, dark QNMs eventually become ordinary. Thus, in 1D systems, there exists an intermediate domain between the ballistic and strongly localization regimes in which only part of the QNMs are localized and provide resonant transmission.
The situation is different in multi-channel random systems in which a genuine diffusive regime exists. The degree of spectral overlap is expressed in the Thouless number, δ, which gives the ratio of the typical width δν and spacing ∆ν of QNMs, δ = δν/∆ν [8,9,12]. The typical linewidth δν is essentially equal to the field correlation frequency over which there is typically 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 much larger than 1 for diffusive waves so that N res /N mod ∼ 1/δ and may be small. For localized waves, the number of channels that contribute effectively to transmission, M, approached unity and transport becomes effectively one-dimensional [29]. For example, the statistics of transmittance are then in accord with the single parameter scaling hypothesis [30]. We find here 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, MN res /N mod in multichannel systems to the ratio

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. We consider two problems associated with such systems. The first is finding solutions ψ(x, t) of the wave equation satisfying the outgoing boundary conditions, which means that there are no right/left-propagating waves in the left/right lead. A solution ψ m (x, t) of this problem (eigenfunction) is the superposition of two counter-propagating monochromatic waves ψ m (x) (±) e −iωmt . The eigenfunction in the jth layer, ψ m,j e ±ikmx , and the amplitudes a (±) m,j in adjacent layers are connected by a transfer matrix. The wave numbers k m are complex-valued and form The second problem is the transmission of an incident wave through the system. The set of wavenumbers and corresponding fields inside the system for which the transmission coefficient reaches its local maximum are TRs. Evidently these two problems are interrelated.
In particular, the density of QNSs at a frequency ω is proportional to the derivative with respect to frequency of the phase of the complex transmission coefficient [31]. The goal of this paper is to further explore the relation between those two problems, in particular to study the differences between the spectra of TRs and QNMs.
In what follows, the scatterers and the distances between them are characterized by the reflection coefficients r j ≡ r 0 + δr j and thicknesses d j ≡ d 0 + δd j , respectively. The random values δr j and δd j are distributed in certain intervals, and δd j = 0. Here, . . . stands for the value averaged over the sample. The last condition means that the length L of the sample is equal to Nd 0 .
To explicitly introduce a variable strength s of disorder, we replace all reflection coefficients, except for those at the left, r L , and right, r R , edges of the system by sr j , and assume (unless otherwise specified) that the coefficients r j are homogeneously distributed in the interval (−1, 1). This enables to keep track of the evolution of the QNM eigenvalues k When s = 0, (i.e., no disorder) the real and imaginary parts of the QNM eigenvalues where m = 0, 1, 2, . . .
In what follows, instead of the intensity of the mth mode, we consider the quantityĪ m (x) = |ψ m (x)| 2 , which is I m (x) averaged over fast oscillations caused by the interference of the left-and right-propagating waves. Examples of these functions for homogeneous (s = 0) resonators are shown in Fig. 1a,b.
When |r L | = |r R |, the minimum of the intensity is located at the center of the system, and in an asymmetric case shifts to the boundary with higher reflection coefficient. This property will be used when analyzing the behavior of the QNMs as the disorder parameter s grows.
It is easy to show that when s = 0 the wave numbers k (1). Thus, in homogeneous resonators, there is a one-to-one correspondence between QNMs and TRs. The same relation also exists in periodic systems (periodic sets r j and d j ) [32].
The question now is how k (ordinary QNMs). The rest of the points (dark QNMs) either shift down substantially more slowly (#0, 6,9) or move away from the real axis (points 4 and 11). The latter modes are highly unusual because disorder makes them more leaky. This is quite the opposite to the hitherto observed and expected increase of the lifetime of the eigenstates due to multiple scattering.
Note that although the dark modes are not displayed in the amplitude of the transmis- sion coefficient, they manifest themselves in the phase of the transmission coefficient. Our numerical calculations show that in a given frequency interval, the dark modes contribute to the total phase shift of the transmission coefficient exactly in the same way as ordinary QNMs do. The difference between the ordinary and dark QNMs shows up also in the spatial distribution of the quantityĪ m (x) ≡Ī m (j). As an example, the variation ofĪ 5 (j) (ordinary QNM) upon coordinates andĪ 6 (j) (dark mode) is presented in Fig. 3, for different values of s. Note that the difference between the imaginary parts k ′′ of the eigenvectors #5 and #6 increases as s increases (see Fig. 2). Despite the fact that the initial (s = 0) distributions I 5 (j) andĪ 6 (j) are identical (dashed black curve in Fig. 3), even a small amount of disorder (s = 0.05) modifies these distributions in very different ways. The distributionĪ 5 (j)| s=0.05 is similar toĪ 5 (j)| s=0 , but has a much less pronounced minimum. By analogy with a homoge- neous resonator, this can be interpreted as the growth of the effective reflection coefficients r L and r R , that leads to an increase of the eigenmode lifetime. This agrees well with the statement that the eigenmode lifetime increases with disorder. For larger s,Ī 5 (j) tends to manifest the behavior typical of a QNM in the localized regime. In contrast, the intensity evolution of QNM #6 is similar to that of a homogeneous resonator whose right wall becomes more transparent (compare with the bottom panel in Fig. 1). The wave functions of the dark QNMs #4 and #11 also demonstrate the same behavior, but the effective transparency of one of the edges increases much faster when the degree of disorder s grows. It is seen that for ordinary QNMs, k (res) (s) and k ′ (s) practically coincide, whereas there are no resonances associated with strange QNMs (#0,4,6,9,11).
Thus, while each TR has its partner among the QNMs, the reverse is not true: there are dark QNMs that are not associated with any maximum in transmission, as shown in Fig. 5.
Thus, in a given wavenumber interval ∆k, the statistically-averaged number of TRs, N res , is smaller than the statistically-averaged number of QNMs, N mod = ∆kL/π, and does not depend on the degree of disorder. The merging of neighbouring TRs in the propagation through a cavity with attached leads when the coupling to the leads increases has been demonstrated in [33]. Figure 6 illustrates the difference between the regular and dark QNMs. The existence of regular QNMs whose real part of the complex-valued eigenfrequency, Re ω (mod) lies in a given frequency interval, can be determined from the transmittance spectrum T (ω) of 1D disordered samples. The peak in the spectrum, if it exists, corresponds to the frequency whose value ω (res) coincides with Re ω (mod) . Moreover, when disorder is strong enough, the distribution of the wave intensity along the sample reconstructs very closely the shape of the intensity of regular QNM eigenfunctions. In contrast, the dark QNM is invisible in the transmittance spectrum and its intensity distribution is indistinguishable from that at a non-resonant frequency.
The evolution of a dark QNM as the degree of disorder grows is analogous to the evolution of a mode in a regular resonator when one of its edges becomes more transparent. This means that a dark mode may be transformed into an ordinary (i.e., made visible in the transmission) by increasing the reflectivity of the corresponding edge of the sample, as erality is clearly seen in the ratio N res /N mod plotted as a function of the ensemble-averaged transmission coefficient T . At this scaling, all functions N res(s) /N mod presented in Fig. 8 for samples of different lengths merge into a single curve, Fig. 9.

MISSION RESONANCES IN MULTICHANNEL SYSTEMS
It is of interest to explore the ratio of the numbers of local maxima in transmission and QNMs in random multichannel systems and to compare to results for 1D systems. We the sample. In contrast to transmission in 1D samples with a single transmission channel, 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 in quasi-1D samples: the intensity T ba = |t ba | 2 , the total transmission, T a = N chan b=1 T ba , and transmittance, T = N chan a,b=1 |t ba | 2 . The transmittance is analogous to the electronic conductance in units of the quantum of conductance e 2 /h [11,15,34]. The ensemble average value of the transmittance T is equal to the dimensionless conductance, g = T , which characterizes the crossover from diffusive to localized waves. The localization threshold is at g ∼ 1 [10,12].
Significant differences between results in 1D and quasi-1D geometries can be expected since propagation can be diffusive in quasi-1D samples with length greater than the mean free path but smaller than the localization length, ℓ < L < ξ = N chan ℓ, whereas a diffusive regime does not exist in 1D since ξ = ℓ [35]. For diffusive waves, QNMs overlap spectrally and may coalesce into a single peak in the transmittance spectrum. Thus we might expect that the QNMs within a typical linewidth form a single peak in transmission so that the ratio N res /N mod is the ratio of the mode spacing to the mode linewidth. The mode linewidth is related to the correlation frequency in the transmission spectra, but the mode spacing cannot be readily ascertained once modes overlap.
The transmittance can also be expressed as T = N chan n=1 τ n , where the τ n are the eigenvalues of the matrix product tt † [15]. The transmission matrix provides a basis for comparison between results for 1D and quasi-1D, which is often more direct than a comparison based on QNMs, since the statistics of the contribution of different modes to transmission is not well-established, whereas the contribution of different channels is simply the sum of the transmission eigenvalues. In addition, transmission eigenchannels are orthogonal, whereas the waveform in transmission for spectrally-adjacent modes are strongly correlated [8] so that the transmission involves interference between modes.
The transmission eigenvalue may be obtained from the singular value decomposition of the transmission matrix, t = UΛV † [36]. Here, U and V are unitary matrices and Λ is a diagonal matrix 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 are the corresponding outgoing eigenchannels, u n . Only a fraction of the N chan eigenchannels contribute appreciably to the transmission [14]. In diffusive samples, the transmission is dominated by g channels with τ n > 1/e [16,37], while a single eigenchannel dominates transmission for localized samples. The statistics of transmission depend directly on the participation number of transmission eigenhannels, M ≡ ( N chan n=1 τ n ) 2 / N chan n=1 τ 2 n [29]. M is equal to 3g/2 [29] for diffusive waves and approaches unity in the localized limit [29,30].

A. Numerical simulations
To explore this ratio over a broad range of g = T for multichannel disordered waveguides in the crossover from diffusive to localized waves, we carry out numerical simulations for a scalar wave propagating through a two dimensional disordered waveguide with reflecting sides and semi-infinite leads. For diffusive samples in which there is considerable mode overlap since δ = δν/∆ν > 1 [12], (δν and ∆ν are the linewidth and the distance between spectral lines) the density of states (DOS), and from this the number of QNMs within the spectrum, can be obtained from the sum of the derivatives of the composite phase of the transmission eigenchannel [38]. The derivative of the composite phase of the nth eigenchannel is equal to the dwell time of the photon within the sample in the eigenchannel.
The total number of modes N mod in a given frequency interval is then the integral over this interval of the DOS. This has allowed us to determine the ratio N res /N mod in the crossover to localization.
Simulations are carried out by discretizing the wave equation on a square grid and solved via the recursive Green function method [39]. Typical spectra of intensity, total transmission and transmittance are shown in Fig. 10 for a diffusive sample with g = 2.1 and for a localized sample with g = 0.3.
We find that the numbers of peaks in the spectra of intensity, total transmission and transmittance in a single sample are nearly the same for each of the samples shown in Fig. 10. This is seen to be the case over a wide range of T in Fig. 11.
The DOS and so the number of QNMs within the spectrum in the samples of the same size are not affected by the strength of disorder so that the decreasing ratio N res /N mod with increasing T reflects only the decreasing number of peaks in the transmission spectra due 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 might expect the ratio N res /N mod to fall inversely with M, N res /N mod ∼ 1/δ ∼ 1/g ∼ 3/2M. For deeply localized waves, however, this ratio is expected to approach unity as M approaches unity. This suggests that N res /N mod ∼ 1/M. in this limit. A plot of 1/M in Fig. 11 shows 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 the sample changes for fixed length and scattering strength since it is tied to the time of the flight distribution, which is independent of W [40]. Since N res is essentially the width of the spectrum divided by the correlation frequency of the intensity, the number of peaks within the 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. In addition, the propagation in a multichannel disordered sample is essentially 1D, when M is approaching unity [30].
These results suggest that a comparison can be made between propagation in both 1D and multichannel systems via the ratio of the number of peaks in the transmission spectra to the number of modes normalized by M, N res /(N mod /M). This ratio may be expected to be close to unity for L ≫ ξ. We consider the variation with g = T of the ratio MN res /N mod in quasi-1D and compare this with the corresponding ratio in 1D in which M = 1. The values of this ratio in quasi-1D and 1D are close, as seen in Fig. 12. For ballistic waves, each of the N channel transmission eigenvalue is unity so that the transmittance is N channel and all eigenchannels contribute equally to the transmittance so that M = N channel , yielding g/M = 1.

B. Experimental measurements
For quasi-1D samples in the crossover to localization in which spectral overlap is moderate, it is possible to analyze the measured field spectra to obtain the central frequencies calculations [36,41]. We find, however, that the impact of incompleteness upon the statistics of transmittance and transmission eigenvalues is small as long as the number of measured channels is much greater than M, as is the case in these measurements of transmission in localized samples [30]. In this random ensemble, M = 1.23 and therefore the statistics of transmission are not affected by the incompleteness of the measurement [30]. The influence of absorption in these samples is statistically removed by compensating for the enhanced decay of the field due to absorption [42].
The central frequencies and linewidths of the modes are found by simultaneously fitting 45 field spectra. The transmittance as well as the Lorentzian lines for each QNM normalized to unity and the DOS, which is the sum of such Lorentzian lines over all QNMs are shown in Fig. 13 for a single random configuration. The DOS curves for different modes are plotted in different colors so that they can be distinguished more clearly. The DOS is also determined from the sum of the spectral derivatives of the composite phase of each transmission eigenchannel and plotted in Fig. 13. The DOS determined from analyses of the QNMs and of the transmission eigenchannels are seen to be in agreement. The dashed vertical lines in Fig. 13 are drawn from the peaks in the transmittance spectra in (a) to the frequency axis in (b). As found in 1D simulations, each peak in T is close to the frequency of 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 QNMs found from the modal analysis averaged over a random ensemble of 40 configurations is 0.61, with a standard deviation of 0.057. This is indicated by the cross in Fig. 11 and is consistent with values of the ratio found in computer simulations. This ratio is slightly smaller than the ratio of 0.65 for T = 0.37 found in 1D simulations, as seen in Fig. 8. This may be attributed to the value of M of 1.23 in this ensemble being larger than the value of unity in 1D. This reflects the tendency of the ratio to decrease with increasing M as found for diffusive waves.

V. ANALYTICAL CALCULATIONS OF N res /N mod
To calculate the average number of TRs in the limit s ≪ 1, we use the single-scattering approximation and write the total reflection coefficient r(k) of a 1D system as: where x n is the coordinate of the n-th scatterer. The values k max , at which the transmission coefficients, T (k) = 1 − |r(k)| 2 , has a local maxima, are defined as the zeros of the function Assuming first that δd i = 0, we obtain Equation (7) is the trigonometric sum Σ N l=1 a l sin (ν l k) with "frequencies" ν l = 2ld 0 and random coefficients a l . The statistics of the zeroes of random polynomials have been studied in [28], where it is shown that the statistically-averaged number of real roots N root of such sum at a certain interval ∆k is where σ 2 l = Var(a l ) is the variance of the coefficients a L = Σ N −l n=1 r n+l r n l + Σ N n=l r n−l r n l. When the reflection coefficients are uncorrelated, then Var(a l ) = 2(N − l)l 2 σ 4 0 + 2r 2 σ 2 0 , where σ 2 0 = Var(r) andr is the mean value of r i . The sums in Eq. (8) can be calculated using Eq. (9), which yields [46]: From Eqs. (8) and (10) we obtain where L = Nd 0 . Since the number of minima of the reflection coefficient is equal to the number of TRs, N res = N root /2, and the number N mod of QNMs in the same interval ∆k is N res = ∆kL/π, from Eq. (11) it follows that N res /N mod = 2/5.
Although this relation was derived for systems with random reflection coefficients and constant distances between the scatterers, it also holds for samples in which these distances are random (δd i = 0). In this case, the frequencies ν = 2ld d in Eq. (7)

VI. DISCUSSIONS AND CONCLUSIONS
We have found a similarity between (i) the evolution of QNMs in disordered systems when changing of the strength of the random scattering and (ii) the behavior of modes in regular open structures, as the coupling to an environment is altered. The later has been intensively studied in condensed matter, optics, and nuclear, atomic, and microwave physics.
Common to all these studies is the appearance of two time scales when the coupling to the environment via open decay channels increases and QNMs begin to overlap [47][48][49][50][51][52]; for a review, see [53] and references therein. When the coupling to the environment is weak, the lifetimes of all states tend to decrease as the coupling increases. When the coupling reaches a certain critical value, a restructuring of the spectrum of QNMs occurs leading to the segregation of the imaginary parts of the complex eigenvalues and of the decay widths.
Some QNMs become short-lived, while the remaining ones become long-live, suggesting that these are trapped states that are effectively decoupled from the environment and this can be long-lived. This phenomenon is general and, by analogy with quantum optics [54] and atomic physics [55][56][57], is known as the superradiance transition and the newly formed broad resonances called superradiant states. In more complicated structures, such of those consisting of two coupled oscillating subsystems, one with a low and the other with a much higher densities of states, the superradiance transition is closely related to the existence of doorway states [50,51] that strongly couple to short-lived QNMs with external decay channels. The effect of disorder on the superradiance transition has been studied in [47,48].
It was shown that in the regime of Anderson localization, the localization length remains constant as the coupling to the environment changes, however, the critical value of the coupling increases with the strength of disorder.
It is important to stress that in all the situations considered in [3, 47-54, 56, 57], the partitioning of discernible QNMs of an initially strongly isolated system into two groups (long-and short-living states) occurs when the opening of the system to the external world increases and reaches some threshold value. In this paper we show that a similar partitioning is inherent in 1D and quasi-1D open systems of randomly distributed elements. In contrast to the studies mentioned above, we start from a homogeneous open system and demonstrate that two different ranges of the eigenstates' lifetimes arise as soon as random scatterers are introduced.
As the reflectivity of the scattering elements increase, some of the QNMs (ordinary states) gradually decouple from the environment due to Anderson localization, while the rest (dark states) remain coupled or even increase their coupling with the external world. This separation of eigenstates is clearly visible in spectra of the transmittance.
In conclusion, we have studied the relationship between spectra of quasi-normal modes and transmission resonances in open 1D and quasi-1D systems. We start from homogeneous samples, in which each TR is associated with a QNM, and vice versa. As soon as an arbitrarily weak disorder is introduced, this correspondence breaks down: a fraction of the eigenstates become dark, in the sense that the corresponding resonances in transmission disappear. The evolution of the imaginary parts of the eigenfrequencies of the dark QNMs with changing disorder is also rather unusual. Whereas increasing disorder leads to stronger localization of ordinary modes so that their eigenfrequencies approach the real axis, the imaginary parts of the eigenfrequency of dark modes increase (or decrease anomalously slowly) with increasing disorder, and begin to go down only when the disorder becomes strong enough. For weak disorder, the averaged ratio of the number of transmission peaks to the total number of QNMs in a given frequency interval is independent of the type of disorder and deviates only slightly from a constant, 2/5, as the strength of disorder and/or the length of the random sample increase. 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 of disorder grows, ultimately all dark quasimodes become ordinary. This