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 to the number of QNMs , , is insensitive to the type and degree of disorder and is close to the value , 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 close to . In diffusive quasi-1D samples, however, falls as the effective number of transmission eigenchannels increases. Once is divided by , however, the ratio is close to the ratio found in 1D.
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 (); 5 (); new J (); newM (); newR (); newS (); newT (); 1974c (). 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 1977a (); 1979a (); 10a (). For multichannel samples, transmission is most conveniently described in terms of the transmission matrix, , whose elements are field transmission coefficients 1981c (); 1984a (); 1988c (). The transmittance is the sum of eigenvalues of the Hermitian matrix . Some of these eigenvalues are close to unity even in weakly-transmitting samples 1984a (); 1988c (); Imry (); 1990g (). Knowledge of the transmission matrix makes it possible to manipulate the incident wavefront to enhance or suppress total transmission through random media newA1 (); 8a (); 2012g (); 12 (); 22a () and to focus transmitted radiation at selected points new F (). The control over transmitted radiation can be exploited to improve images washed out by random scattering and to facilitate the detection and location of objects new F (). The great potential of such algorithms for a host of practical applications has recently attracted attention in both the physics new F () and mathematics communities 1 () and references therein.
In open regular homogeneous systems (e.g. single quantum potential wells, optical cavities, or microwave resonators) each peak in transmission, or transmission resonance (TR), is associated with a QNM (Moiseev () 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 1977a (); Lifshitz () 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 Lifshitz (); we2 (); we1 (). 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).
Quite surprisingly, much less are studied the properties of QNMs in 1D systems with weak disorder where the localization length is smaller than the size of the sample. In this paper we show for the first time that in completely open one-dimensional disordered systems, two different types of QNMs can exist: ordinary QNMs, associated with resonant transmission peaks and hidden QNMs unrelated to any maxima in the transmission spectrum. The hidden modes exist due to random scattering and arise as soon as an arbitrarily small disorder is introduced. The imaginary parts of the eigenfrequencies of hidden 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 hidden modes eigenfrequency, depending on the boundary conditions, either is independent of strength of random scattering or even increases from the onset of disorder. 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 over wide ranges of the strength of disorder and of the total length of the system. The value follows from the general statistical properties of random trigonometric polynomials Edelman (). As the scattering strength and/or the length of the system increase, hidden QNMs eventually become ordinary.
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, , which gives the ratio of the typical width and spacing of QNMs, 5 (); 1974c (); 1977a (). The typical linewidth 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 . On the other hand, the inverse level spacing is equal to the density of states (DOS) of the medium. Thus the ratio can be expected to be close to for diffusive waves. The localization threshold lies at 1974c (); 1977a (); 1979a (); may be much larger than 1 for diffusive waves so that and may be small. For localized waves, the number of channels that contribute effectively to transmission, , approached unity and transport becomes effectively one-dimensional M (). For example, the statistics of transmittance are then in accord with the single parameter scaling hypothesis Genack PNAS (). It is worth noticing that although the statistics of the eigenstates of disordered systems is a subject of intensive investigations for already more that two decades (see, for example RefA (); RefB (); RefC (); RefD (); RefE (); RefF (); RefG ()), the statistics of the transmission resonances (peaks in transmission spectra) is much less studied. The comparison of these two is a challenging problem for future investigations. Here we find that a connection can be made between the present 1D calculations of and measurements in multichannel diffusive systems. This is done by comparing ratio of to the number of QNMs divided by , in multichannel systems to the ratio in 1D, where .
Ii Quasi-normal modes of open systems
We first consider a generic 1D system composed of scatterers separated by intervals and attached to two semi-infinite leads. The eigenfunctions are solutions of the wave equation satisfying the outgoing boundary conditions, which means that there are no right/left-propagating waves in the left/right lead. Each eigenfunction is a superposition of two counter-propagating monochromatic waves . The eigenfunction in the th layer, , is equal to , and the amplitudes in adjacent layers are connected by a transfer matrix. The wave numbers are complex-valued and form the discrete set , , so that the frequencies These eigenfunctions 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 the reflection coefficients and thicknesses , respectively. The random values and are distributed in certain intervals, and . The last condition means that the length of the sample is equal to .
To explicitly introduce a variable strength of disorder, we replace all reflection coefficients by , and assume (unless otherwise specified) that the coefficients are homogeneously distributed in the interval . This enables to keep track of the evolution of the QNM eigenvalues as the disorder increases. The condition ensures that any random realization with the same contains the same number of QNMs in a given interval of the wavenumbers.
At the beginning, let us consider the QNMs of a regular resonator of the length assuming that all reflection coefficients except and are equal to zero. In this case the real and imaginary parts of the QNM eigenvalues are
In what follows, instead of the intensity of the th mode, , we consider the quantity , which is averaged over fast oscillations caused by the interference of the left- and right-propagating waves. Examples of these functions for resonators are shown in Fig. 1a,b.
There is distributed along the system as , where . When , 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 properties of the QNMs of the disordered system.
In a disordered sample, the reflection coefficients are random and scaled by the parameter . The evolution of the eigenvalues as the parameter grows shows that QNMs separate into two essentially different types. There are ordinary QNMs whose lifetimes, defined by the value of , increase monotonically with . Simultaneously, there are “hidden” QNMs (the origin of this term will be explained in the next section), whose lifetimes are substantially smaller than the lifetimes of ordinary QNMs and remain constant when varies over many orders of magnitude. Figures 2 and 3 show trajectories of the QNMs’ eigenvalues as the parameter grows, and dependencies .
Our numerical calculations show that when external reflectors are added at the edges of the sample, the imaginary parts of some of the hidden modes increase with the strength of disorder.
The spatial distributions of the intensity along the system are also different for ordinary and hidden QNMs. The evolution of as the strength of the disorder grows is shown in Fig. 4.
Initially, when is so small that values of are almost equal for both types of modes, the distributions are practically identical and evolve in the same manner: the minimum is placed near the center of the sample, and slopes (which are ) decrease as the disorder strength grows (see Fig. 4). These distributions are similar to the distribution of the intensity in the regular resonator with a small imbalance between the reflection coefficients and of the resonator walls.
When of the hidden mode “” reaches its plateau (see Fig. 3), the minimum of its distribution shifts from the center, as in the resonator with strong imbalance between the reflection coefficients and . The slope of the distribution of hidden modes remains constant ( is independent of on the plateau), whereas the slopes of all ordinary modes are equal and continue to decrease as the parameter grows. The difference between the distributions of ordinary and hidden QNMs is that the ordinary modes are concentrated near 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 when the disorder strength is small so that the localization length is large relative to the system length , . Thus, this phenomenon is not related to Anderson localization, but, as it will be shown below, manifests itself also when .
Notwithstanding that at , the lifetimes of all hidden modes increase, these modes are much more resistant to disorder: they become localized at far stronger disorder than ordinary states. As can be seen, for example, in Fig. 3, at , when , the difference between the imaginary parts of ordinary and some of hidden QNMs is about of one order of magnitude.
Iii Transmission resonances in 1D systems
We now consider the transmission of an incident wave through the system. The wavenumbers at which the transmission coefficient reaches its local maximum and the corresponding fields inside the system are transmission resonances (TR). The QNMs and TRs are interrelated. In what follows, we explore the relation between QNMs and TRs, in particular, study the differences between the spectra of TRs and QNMs.
It is easy to show that in a resonator, the wave numbers of the transmission resonances coincide with the real parts given by Eq. (1), and there is a one-to-one correspondence between QNMs and TRs so that the number of resonances is equal to the number of QNMs, , in a given frequency interval. The same relation also exists in periodic systems (periodic sets and ) Settimi ().
In disordered systems, the relation between QNMs and TRs is quite different. While each TR has its partner among the QNMs, the reverse is not true: there are hidden QNMs that are not associated with any maximum in transmission as shown in Figs. 5 and 6
Figure 7 illustrates another fundamental difference between the ordinary and hidden QNMs. The ordinary QNMs whose real parts of the complex-valued eigenfrequency, lie in a given frequency interval, can be determined from the transmittance spectrum of 1D disordered samples, because each peak in the spectrum corresponds to a frequency whose value practically coincides with . Moreover, when disorder is strong enough, so that , the distribution of the transmitted wave intensity along the sample reconstructs very closely the shape of the intensity of ordinary QNM eigenfunctions. 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 that at a non-resonant frequency.
Note that although the hidden modes are not displayed in the amplitude of the transmission coefficient, they are manifested in the phase of the transmission coefficient. The density of states at a frequency is proportional to the derivative with respect to frequency of the phase of the complex transmission coefficient avish (). Our numerical calculations show that each hidden mode adds to the total phase shift of the transmission coefficient exactly in the same way as ordinary QNMs.
The evolution of a hidden 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 less transparent. This means that a hidden 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 illustrated in Fig. 8.
The sample, whose transmission spectrum is shown in Fig. 5, contains three hidden QNMs (# 1,5, and 6) in the given spectral range. Distributions of the intensity ( is the layer number) for QNMs #1 and 6 are similar to the distributions in resonators with right reflection coefficient smaller than the left reflection coefficient , . The intensity distribution of QNM #5 is characterized by the opposite inequality . When the value of the right-end reflection coefficient is increased, new resonances appear in transmission for the initially hidden modes #1 and 6, while mode #5 remains hidden (Fig. 8a). In contrast, increasing the left-end reflection coefficient transforms QNM #5 into a ordinary mode, whereas QNMs #1 and 6 remain hidden in the transmission spectrum (Fig. 8b).
Important to stress that the separation of QNMs into two types, ordinary and hidden, occurs already at a very small disorder strength, , when the localization length is larger than the sample length, .
The ensemble-averaged of the ratio of the number of transmission resonances, , which is the number of ordinary modes, to the total number of QNMs, , has been calculated numerically for a variety of randomly layered samples with different types of disorder (random reflection coefficients of the layers, , and/or random thicknesses , with rectangular and Gaussian distribution functions) in broad ranges of the disorder strength and of the length of the realizations .
Figure 9 shows the average of over random realizations as a function of the ensemble-averaged transmission coefficient [panel (a)], and as a function of ratio of to the localization length (measured in numbers of layers), [panel (b)] for samples with , and layers. At this scaling, all functions and for samples of different lengths merge in a single curve.
It is seen in Fig. 9 that the difference between and appears when , and the ratio varies weakly even when . Moreover, independently of the samples parameters, the average ratio tends to the constant when . Thus, the existence of hidden modes and the universality of their relative number is a general feature of 1D disordered systems not specifically related to localization.
Iv Measurements of transmission eigenchannels and transmission 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 consider quasi-one dimensional (quasi-1D) samples with reflecting sides and transverse dimensions much smaller than the sample length and localization length , . Here, is the number of channels or freely-propagating transverse modes in the perfectly conducting leads or empty waveguide leading to the sample and is the transport mean free path. The incident channels are thoroughly mixed by scattering within 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 with elements between all incident and outgoing channels, and , respectively. From the transmission matrix, we may distinguish three types of transmission variables in quasi-1D samples: the intensity , the total transmission, , and transmittance, . The transmittance is analogous to the electronic conductance in units of the quantum of conductance Landauer (); 10a (); 1981c (). The ensemble average value of the transmittance is equal to the dimensionless conductance, , which characterizes the crossover from diffusive to localized waves. In diffusive samples, the dimensionless conductance is equal to the Thouless number, and the localization threshold is reached when 1977a (); 1979a ().
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, , whereas a diffusive regime does not exist in 1D since Abr (). 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 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 , where the are the eigenvalues of the matrix product 1981c (). 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 5 () so that the transmission involves interference between modes.
The transmission eigenvalue may be obtained from the singular value decomposition of the transmission matrix, Been (). Here, and are unitary matrices and is a diagonal matrix with elements . The incident fields of the eigenchannels on the incident surface, , which are the columns of , in the singular-value decomposition are orthogonal, as are the corresponding outgoing eigenchannels, . Only a fraction of the eigenchannels contribute appreciably to the transmission 1984a (). In diffusive samples, the transmission is dominated by channels with 3 (); Imry (), while a single eigenchannel dominates transmission for localized samples. The statistics of transmission depend directly on the participation number of transmission eigenhannels, M (). is equal to M () for diffusive waves and approaches unity in the localized limit M (); Genack PNAS ().
iv.1 Numerical simulations
To explore the ratio over a broad range of 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 1977a (), ( and 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 2015a (). The derivative of the composite phase of the th eigenchannel is equal to the dwell time of the photon within the sample in the eigenchannel. The total number of modes in a given frequency interval is then the integral over this interval of the DOS. This has allowed us to determine the ratio 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 stone91 (). Here, is the wave vector in the leads. Also, is the spatially-varying dielectric function in the disordered region with chosen from a rectangular distribution and in the empty leads. Reflections at the sample boundaries are minimal because the sample is index matched to its surroundings. The product of at 14.7 GHz and the grid spacing is set to unity. In the frequency range studied, the leads attached to the random waveguide support channels which are the propagating waveguide modes. In our scalar quasi-1D simulations for a sample with a width , the number of channels at frequencies above the cutoff frequency is the integer part of . These channels should not be confused with the QNMs of the random medium which correspond to resonances of the medium with Lorentzian lines centered at distinct frequencies. In the simulations, the length of the sample is equal to 500 in units of the grid spacing except for one deeply localized sample with , for which = 800 and the width of the sample is . Typical spectra of intensity, total transmission and transmittance are shown in Fig. 10 for a diffusive sample with and for a localized sample with .
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 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 with increasing 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 QNMs within the mode linewidth for diffusive waves, we might expect the ratio to fall inversely with , . For deeply localized waves, however, this ratio is expected to approach unity as approaches unity. This suggests that . in this limit. A plot of in Fig. 11 shows that towards the diffusive and localized limits is close to the ratio . 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 Drake (). Since 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, and the DOS are proportional to , so that increases with sample width and is inversely proportional to . In addition, the propagation in a multichannel disordered sample is essentially 1D, when is approaching unity Genack PNAS ().
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 , . This ratio may be expected to be close to unity for . We consider the variation with of the ratio in quasi-1D and compare this with the corresponding ratio in 1D in which . The values of this ratio in quasi-1D and 1D are close, as seen in Fig. 12.
iv.2 Microwave experiment
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 of the QNMs and to compare these to peaks in transmission. Spectral measurements of the transmittance were made in a copper waveguide of diameter 7.3 cm and of length 40 cm containing randomly positioned alumina spheres with index 3.14, over a random ensemble for which . The empty waveguide supports propagation channels in the frequency range of the experiment: 10.0-10.24 GHz. The transmission matrix is determined from measurements of the field transmission coefficient between points on grids of 49 locations for the source antenna and detection antennas on the input and output surfaces of the waveguide for a single polarization with a grid spacing of 9 mm. Such measurements of the transmission matrix in real space for a single polarization are incomplete. The distribution of transmission eigenvalues determined from these measurements may differ from theoretical calculations Been (); 2013j (). 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 , as is the case in these measurements of transmission in localized samples Genack PNAS (). Here and therefore the statistics of transmission are not affected by the incompleteness of the measurement Genack PNAS (). The influence of absorption in these samples is statistically removed by compensating for the enhanced decay of the field due to absorption 1993a (). Different random sample configurations are obtained by briefly rotating and vibrating the sample tube. The probability distribution of the transmittance is in good agreement with the distribution calculated for this value of Mutt (); Gopar (); Froufe (); Genack PNAS ().
We find the central frequencies and linewidths of the QNMs within the frequency range of the measurements by carrying out a modal decomposition of the transmitted field. A given polarization component of the field can be expressed as a sum of the contributions from each of the QNMs:
Here are complex-valued amplitudes of QNMs.
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 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 .
The ratio of the number of peaks in spectra of transmittance to the number of QNMs 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 value of this ratio is slightly smaller than the value 0.65 found in simulations for 1D sample with found in 1D simulations, as seen in Fig. 9. This may be attributed to the value being larger than the value of unity in 1D. This reflects the tendency of the ratio to decrease with increasing as found for diffusive waves.
Equation (4) offers an explanation for the fact that the number of transmission resonances can be smaller than that of all QNMs. If, for example, the transmitted field is a sum of two modes, from Eq. (4) it follows that the number of peaks in the transmission spectrum is either one or two, depending on the widths of the modes.
iv.3 Spatial intensity distribution of QNMs within quasi-1D disordered samples
In order to fully characterize the QNMs and their relationship to peaks in transmittance in quasi-1D samples, it would be desirable to examine the longitudinal profile of QNMs within 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 the recursive Greenâs function technique. The Greenâs function between points on the incident plane and within the sample can be expressed in a manner similar to Eq. (4) as a sum of contribution from each of the modes, We find in the simulations that the spatial distribution of the th mode obtained by decomposing the field into QNMs depends weakly upon the excitation point . We therefore average the spatial profile for each QNM over the profiles obtained for all excitation points on the input of the sample.
We consider propagation in a sample drawn from an ensemble with a value of which is below unity but still not too small. In this case, QNMs still overlap but it is yet possible to analyze the field into QNMs. We present in Fig. 14 that a spectrum of transmittance in a sample configuration chosen from an ensemble with and , together with profiles of a ordinary and a hidden mode within the spectrum. The nature of propagation in the sample might not differ appreciably from propagation in 1D samples, for which . We find that the intensity distributions integrated over the transverse direction of the hidden mode in the transmission spectrum of the quasi-1D samples falls monotonically within the sample, while the ordinary mode associated with peaks in transmission is peaked in the middle of the sample.
V Analytical calculations of
To calculate the average number of TRs in the limit , we use the single-scattering approximation and write the total reflection coefficient of a 1D system as:
where is the coordinate of the -th scatterer. The values , at which the transmission coefficients, , has a local extrema, are defined as the zeros of the function :
Assuming first that , we obtain
Equation (7) is the trigonometric sum with “frequencies” and random coefficients . The statistics of the zeroes of random polynomials have been studied in Edelman (), where it is shown that the statistically-averaged number of real roots of such sum at a certain interval is
where is the variance of the coefficients . When the reflection coefficients are uncorrelated, then
where . Since the number of minima of the reflection coefficient is equal to the number of TRs, , and the number of QNMs in the same interval is , from Eq. (11) it follows that
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 (). In this case, the frequencies in Eq. (7) should be replaced by . Since the main contribution to the sums in Eq. (8) is given by the terms with large , the mean value of can be replaced by , in the case of a homogeneous distribution of the distances along the system. This ultimately leads to the same result Eq. (12).
Vi Hidden modes: simple model
In Sec. II, QNMs were introduced as solutions of the wave equation satisfying the outgoing boundary conditions. Their eigenvalues of QNMs, , can be calculated as roots of the equation , where is the transfer matrix, which connects waves’ amplitudes at the left and right sides of the whole system. The transfer matrix of the system which consists of scatterers separated by intervals has the form:
and is the transfer matrix through the th scatterer. Assuming that reflection and transmission coefficients are real, can be presented as
where is the unit matrix, and is the Pauli matrix.
Omitting denominator , matrix can be written as ordered product
where substitution is used. Eq. (16) allows presenting transfer matrix as a power series in :
where matrix contain various ordered products of matrices and Pauli matrices. Matrices are diagonal, whereas Pauli matrix is anti-diagonal, so only even combinations of Pauli matrices contribute to . Thus,
where is the sample length, and the coefficients contain various combinations of products of reflection coefficients with exponential multipliers , where are the distances between any ordered pairs of scatterers.
Neglecting terms with higher than powers in Eq. (18), the dispersion equation, which defines eigenvalues , can be presented as follows:
where the coefficients are formed by various pairs of the scatterers, as it is schematically shown in Fig. 15a.
The greater is the distance between the scatterers, the larger are exponents in Eq. (19).
The largest exponents are associated with the pairs of scatterers placed near the opposite ends of the sample. When , there are many such pairs, located in blue regions in Fig. 15b, whose associated exponents are of the same order of magnitude, . Let us combine all such pairs in Eq. (19) in one term and characterize them by one common exponent . The number of scatterers near the sample ends, which form this group, can be estimated as , so that the lengths of blue regions in Fig. 15b are .
The next group, , which is associated with the exponent of the order of , consists of pairs of scatterers, one from green and another from blue regions in Fig. 15b. In such a way, Eq. (19) can be approximately presented as
Strictly speaking, the phenomenologically introduced number varies from group to group, but when , is large enough and it is possible to neglect its variation.
The coefficients in Eq. (20) are the sums of random vectors in complex plane. For any given sample the lengths of these vectors are fixed, whereas the phases varies from mode to mode, so that the magnitudes of the coefficients , been averaged over many modes, can be estimates as
Using Eqs. (20) and (21), one can calculate value of , averaged over many modes. When , is large and the second term in the parentheses in Eq. (20) is small as compared with the first one () and can be omitted. Then, the average solution of Eq. (20) is
Expression Eq. (22) describes averaged over many modes dependence , but for any given mode this dependence can be different. Indeed, since (for example, for in the numerical simulation presented in Figs. 2 and 3)) fluctuation of the values of for different eigenmodes can be rather large. In particular, for a certain mode can be much smaller, than . Presenting Eq. (20) in the form
it is easy to see that Eq. (23) has solution
when exceeds some critical value ,
Recall that are formed by different groups of the reflection coefficients. In general, the similar, independent of , solutions of the dispersion equation appear when magnitudes some first coefficients in Eq. (20) are small as compared with magnitudes of the next coefficients.
In order to demonstrate that independent on solutions of the dispersion equation indeed correspond to the hidden modes, let us consider the system composed of three scatterers only. The dispersion equation Eq. (20) for this system is
When all are of the same order of magnitude, , and is so small [ is so large] that , the solution of Eq. (26) is
If, for example, is small as compared with , but , there is another solution of Eq. (26):
This solution is independent on , similarly to the solution Eq. (24).
Fig. 17 demonstrates relation between real part of the QNMs’ eigenvalues and position of the peak in the transmission spectrum.
Note that hidden modes are invisible in the transmission spectrum even when .
Vii Superradiance and resonance trapping in 1D random systems
The model introduced in the previous Section can be used to study the segregation of superradiant states and trapped modes in regular quantum-mechanical and wave structures and 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 regular open structures as the coupling to an environment is altered, has been intensively studied in condensed matter physics, 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 new A (); new B (); new C (); new D (); new E (); new F (); for a review, see new G () and references therein. When the coupling to the environment is weak, the lifetimes of all states tend to decrease as the coupling increases. As the coupling reaches a critical value, a restructuring of the spectrum of QNMs occurs leading to segregation of the imaginary parts of 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 analogy to quantum optics new H () and atomic physics new J (); new i (); new i (), is known as the superradiance transition. In more complicated structures, such of those consisting of two coupled oscillating subsystems, one with a low and the other with a much higher density of states, the superradiance transition is closely related to the existence of doorway states new D (); new E () that strongly couple to short-lived QNMs with external decay channels.
It is important to stress that along with the pronounced similarities between the resonance trapping in many-particle quantum systems, open microwave cavities, etc., and between the “hidding” of some of quasi-normal modes in disordered samples there are substantial differences as well. In particular, resonance trapping happens in regular systems considered in new C (); new G () when the coupling of the large number of QNMs to a much smaller number of common decay channels increases. Without disorder, the samples that we consider are perfectly coupled to the environment (total transmission at all frequencies). Finite coupling appears 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 the system.
To reproduce the superradince phenomena in disordered structures we modify the model slightly by placing the random sample between two reflectors with reflection coefficients and , located at distances and from the edge scatterers. For simplicity, we assume that . These reflectors can be included in the dispersion equation Eq. (20) as additional scatterers as follows:
Here contains the products with corresponding exponential multipliers, the largest of which, , is separated in the explicit form.
To approach the conditions at which superradiance and resonance trapping occur, we consider below (in contrast to the previous sections) the evolution of the eigenvalues of a given sample with fixed when .
When the product is large, the first term in the parentheses dominates and the solution of Eq. (29) is
If , the magnitudes of the exponents in the first three terms are equal. When , the magnitudes of the additional two terms decrease and the solutions of Eq. (29) tend to their solutions in the original sample (without end reflectors), as shown in Fig. 18.
When , 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 as (see Fig. 19). Indeed, the first two terms in Eq. (29) always dominate when and . In this case Eq. (29) can be written as
where the largest term in , which corresponds to the largest distance between the end reflectors and the sample scatterers, is explicitly presented. Solution of Eq. (31)
tends to infinity, when even one of the reflection coefficients .
The reason why these solutions “run away” when the coupling to the environment is maximal, is very simple. The original system without end reflectors has eigenmodes in the given interval , whereas the same system surrounded by the reflectors has eigenmodes in the same interval. Thus, some of modes should leave this interval when the system returns to its original state.
The superradiant transition in periodic and disordered quantum system, which consist of a sets of potential wells, was studied in new A () using effective Hamiltonian approach. It was shown that the transition occurs when the coefficient , which characterizes the coupling with an environment, reaches the value of the coupling between the wells, . In the considered above system and , so that the superradiant transition occurs when . This condition agrees well with presented in Fig. 19b results.
Hidden modes can be associated with superradiant states, while normal modes are trapped resonances. Thus, correspond to the fraction of the modes which are trapped. This result agrees with new C (), where this value was estimated as , and is the fraction of the modes which are superradiant. Note, that the original disordered sample is already coupled to the environment, so that the coupling strength is limited by the intrinsic properties of the sample and cannot exceed this value, even when the end reflectors are fully transparent.
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 becomes hidden, in the sense that the corresponding resonances in transmission disappear. The evolution of the imaginary parts of the eigenfrequencies of the hidden 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 hidden modes changes very slowly (and may even increase when external reflectors are added to the edges) 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, , as the strength of disorder and/or the length of the random sample increase over a wide range. This constant coincides with the value of the ratio analytically calculated in the weak single-scattering approximation. As the strength of disorder keeps growing, ultimately all hidden quasimodes become ordinary. This means that in 1D random systems there exists a pre-localized regime, in which only a fraction of the QNMs are long-lived and provide resonant transmission. If the coupling to the environment is tuned by external edge reflectors, the superradiace transition can be reproduced. In quasi-1D samples, a genuine diffusive regime exists in which some QNMs coalesce to form a single peak in transmission with width comparable to the typical modal linewidth. In such samples, hidden modes have been discovered experimentally and their proportion of all QNMs 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 in transmittance are nearly the same and fall well below the number of QNMs. Though the ratio may be small, we find in microwave experiments and numerical simulations that once the number of QNMs is divided by the effective number of channels contributing to transmission to give , this function is similar to results in 1D samples.
Acknowledgements.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 attention to paper Edelman (). This research is partially supported by the: National Science Foundation (DMR-1207446), RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics, and a Grant-in-Aid for Scientific Research (S).
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