Disorder-induced mutation of quasi-normal modes in 1D open systems
aa r X i v : . [ c ond - m a t . d i s - nn ] J u l Disorder-induced mutation of quasi-normal modes in 1D open systems
Yury Bliokh,
1, 2
Valentin Freilikher,
3, 2 and Franco Nori
2, 4 Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel, CEMS, RIKEN, Wako-shi, Saitama 351-0198, Japan Department of Physics, Jack and Pearl Resnick Institute, Bar-Ilan University, Israel Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
We study the relation between quasi-normal modes (QNMs) and transmission resonances (TRs)in one-dimensional (1D) disordered systems. We show for the first time that while each maximumin the transmission coefficient is always related to a QNM, the reverse statement is not necessarilycorrect. There exists an intermediate state, where only part of the QNMs are localized and theseQNMs provide a resonant transmission. The rest of the solutions of the eigenvalue problem (denotedas strange quasi-modes) are never found in regular open cavities and resonators, and arise exclusivelydue to random scatterings. Although these strange QNMs belong to a discrete spectrum, they arenot localized and not associated with any anomalies in the transmission. The ratio of the number ofthe normal QNMs to the total number of QNMs is independent of the type of disorder, and deviatesonly slightly from the constant p / Wave processes in open systems can be described interms of quasi-normal modes (QNMs), which are a gen-eralization of the notion of normal modes for closed sys-tems, to open structures, [1–9]. The corresponding eigen-frequencies are complex, so that the imaginary partscharacterize the lifetime of the quasi-normal states. Re-garding the transmission of radiation through randommedia, it is more appropriate to use an alternative ap-proach based on transmission resonances (TR): openchannels, through which the radiation transmits withhigh efficiency [3, 10–22].Recently, physicists came to realize that focusing radi-ation into such channels could not only enhance the totalintensity transmitted through strongly-scattering media,but also: significantly improve images blurred by ran-dom scattering, facilitate the detection and location ofobjects, provide optical tomography at very large depths,etc.[13, 17, 18, 20, 23]. To efficiently excite transmissionresonances, it is preferable to treat them as superposi-tions of QNMs, with which the incident signal can becoupled by a properly-shaped wavefront [13, 14]. Thegreat potential of such algorithms for a host of prac-tical applications is obvious. This is why the relationbetween transmission resonances and QNMs have re-cently attracted particular attention of both the physical[12, 24–28] and mathematical communities [1].It is now universally accepted that in open systems(e.g., quantum potential wells, optical cavities, or mi-crowave resonators) each maximum in the transmissioncoefficient (i.e., transmission resonance) is associatedwith a QNM, so that the resonant frequency is close tothe real part of the corresponding eigenvalue. QNMsand TRs are often considered identical. For example,the solutions of the eigenvalue problem (with no incom-ing waves), which in physics are unambiguously calledQNMs, in the mathematical community dealing withthe scattering inverse problem, are termed transmission eigenvalues [1]. However, the connection between QNMsand TRs is not that simple and, despite extensive re-search and much recent progress, still needs a betterphysical understanding and mathematical justification,at least for disordered systems.To this end, it is instructive to look for insights the 1Dlimit because its spectral and transport properties arebetter understood. It is well-known [29] that the trans-mission of a long enough 1D disordered system is typi-cally (for most of the frequencies) exponentially small. Atthe same time, there exists a set of frequencies where thetransmission coefficient has local maxima (resonances intransmission), some of them close to one [30]. Each reso-nance is a transmission channel and is always associatedwith a QNM determined in a standard way as a solutionwith outgoing boundary conditions. The reverse state-ment, that each QNM manifests itself as a transmissionresonance, although never has been questioned, is usuallytaken as obvious and self-evident, perhaps because it isalways the case in all regular (homogeneous or periodic)quantum-mechanical and optical open structures.Here we show, both numerically and analytically, thatin 1D disordered systems there exist two types of QNMs:ordinary QNMs, that provide resonance transmissionpeaks, and “strange” QNMs unrelated to any anoma-lies in the transmission spectrum. These strange modesexist exclusively due to random scatterings and arise al-ready in the ballistic regime with weak disorder. Al-though they belong to the discrete spectrum, their eigen-functions are not localized. The imaginary parts of thestrange QNMs eigenfrequencies vary with increasing dis-order in a highly unusual manner. Indeed, typically, thestronger the disorder is, the more confined the systembecomes, which implies that the eigenfrequencies shouldapproach the real axis. However, the imaginary part of astrange mode’s eigenfrequency either increases from theonset of disorder, or goes down anomalously slowly. Mostsurprisingly, up to rather strong disorder, the average ra-tio of the density (in the frequency domain) of strangemodes to the total density of QNMs, being independentof the type of disorder, remains close to the constant p / p / N + 1scatterers separated by N intervals and attached to twosemi-infinite leads. Two problems are associated withsuch systems. The first one is finding solutions ψ ( x, t )of the wave equation satisfying the outgoing boundaryconditions, which means that there are no right/left-propagating waves in the left/right lead. The eigenfunc-tion solution ψ n ( x, t ) of this problem is the superposi-tion of two counter-propagating monochromatic waves ψ n ( x ) ( ± ) e − iω n t . In any j th layer ψ ( ± ) n ( x ) = ψ ( ± ) n,j ( x ) = a ( ± ) n,j e ± ik n x . The amplitudes a ( ± ) n,j in adjacent layers areconnected by a transfer matrix. The wave numbers k n are complex-valued and form the discrete set (poles ofthe scattering matrix) k (mod) n = k ′ n − ik ′′ n , k ′′ > , andfrequencies ω (mod) n = ck n . The corresponding eigenfunc-tions are the so-called QNMs. Note that all distanceshereafter are measured in optical lengths. The secondproblem is the transmission of an incident wave throughthe system. The set of wave numbers and correspondingfields inside the system for which the transmission coef-ficient reaches its local maximum are the so-called TRs.Evidently these two problems are interrelated. In partic-ular, the density of QNSs at a frequency ω is proportionalto the derivative with respect to frequency of the phaseof the complex transmission coefficient [32, 33].The goal of this paper is to establish the relation be-tween the spectra and wave functions of QNMs and TRs.In what follows, the scatterers and the distances be-tween them are characterized by the reflection coefficients r i ≡ r + δr i and lengths d i ≡ d + δd i , respectively. Therandom values δr i and δd i are distributed in certain in-tervals, and h δr i i = 0 and h δd i i = 0. Here, h . . . i standsfor the value averaged over the sample. The last condi-tion means that the total length L of the system is equalto N d and therefore any random realization with thesame N contains the same number of QNMs.To explicitly introduce the tunable strength s of disor-der, we replace all reflection coefficients, except for thoseat the left, r L , and right, r R , edges of the system by sr i ,and assume (unless otherwise specified) that the coeffi-cients r i are homogeneously distributed in the interval ( − , . This notation enables keeping track of the evolu-tion of the QNM eigenvalues k (mod) n and of the resonantwave vectors k (res) when the disorder increases from zero( s = 0) while the reflection coefficients r L and r R at thesemitransparent boundaries remain constant.When s = 0, (i.e., no disorder) the real and imaginaryparts of the QNM eigenvalues k (mod) n are k ′ n = 12 L · (cid:26) π + 2 πn, when r L r R > , πn, when r L r R < , (1) k ′′ n = − L ln | r L r R | . (2)The wave intensity, defined as I n,j = | ψ (+) n,j | + | ψ ( − ) n,j | isdistributed along the system as I n ( x j ) ∝ cosh[2 k ′′ ( x j − 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 atthe centre of the system, and in an asymmetric case shiftsto the boundary with a higher reflection coefficient. Thisproperty will be used when analyzing the behavior of theQNMs when the disorder parameter s grows.It is easy to show that when s = 0 the wave num-bers k (res) n of the transmission resonances coincide withthe real parts k ′ n given by Eq. (1). Thus, in the homo-geneous resonator, there is a one-to-one correspondencebetween QNMs and TRs. The same correlation existsalso in periodic systems (periodic sets r i and d i ) [34].The question now is whether this relationship survivesin the disordered system, when s = 0. There is strongevidence [3, 11–13, 15] that for every resonance there isa corresponding QNM. However, as we show below, thereverse statement is not valid: there are certain QNMswhich cannot be associated with any resonance.Figure 1 shows the evolution of the eigennumbers k mod n in the complex plane ( k ′ , k ′′ ) as s grows. Initially, when s = 0, all eigenumbers are equidistantly located on theline k ′′ = const, in agreement with Eqs. (1, 2). As soon asdisorder arises ( s = 0) and increases, the eigenvalues sep-arate into two essentially different types. Indeed, with s increasing, the points k ′′ i decrease) with approximatelythe same “velocity” (ordinary QNMs). The rest of thepoints (strange QNMs) either shift down substantiallymore slowly ( I ,j and I ,j of QNMs s . Note that the difference be- lg k d k d FIG. 1. (color online) Motion of the QNMs eigenvalues, k (mod) n = k ′ n − ik ′′ n , as the degree s of disorder grows. Redasterisks mark the initial positions with no disorder ( s = 0).Red open circles and blue solid circles show the positions ofthe QNMs eigenvalues at s = 0 . s = 0 .
2, respectively.Note that ordinary QNMs shift down when increasing disor-der while some strange QNMs (e.g. s=0.3 s=0.05 l n (I) Layer number s=0
FIG. 2. (color online) Spatial distribution of the intensity I ( j ) of QNMs s . The dashed blackcurve corresponds to a homogeneous resonator ( s = 0, r L = − r R = 0 . s =0 . tween the imaginary parts k ′′ of the eigenvectors 5 and 6increases as s increases (see Fig. 1). Despite the fact thatthe initial ( s = 0) distributions are identical, even smalldisorder ( s = 0 .
05) deforms the distributions I ,j and I ,j in very different ways. The distribution I ,j | s =0 . issimilar to I ,j | s =0 , but has a much less pronounced min-imum. By analogy with a homogeneous resonator, thiscan be interpreted as the growth of the effective reflec-tion coefficients r L and r R , which agrees well with thestatement that the wave lifetime increases when disorderbecomes stronger. For larger s , I ,j tends to manifestthe behaviour typical for QNM in the localized regime.In contrast, the intensity evolution of QNM s grows.We also consider the propagation of a monochromaticwave through the same system. When s = 0, the num-ber of resonances N res is equal to the number of QNMs, N mod , and all k (res) n coincide with the real parts k ′ n ofQNMs. When disorder is introduced, s = 0, each k (res) n remains close to the k ′ n of the corresponding ordinaryQNM: k (res) n ( s ) ≃ k ′ n ( s ). The spatial intensity distribu-tions of QNM s = 0 are equal to the real parts of the eigenvaluesof the strange QNMs, disappear when the mean value ofthe reflection coefficients s h r i i becomes of the order of r L,R . Figure 3 demonstrates this behavior. Here, therole of the reflection coefficients r R,L ( r R = r L = 0 . s = 0 . k ’ d & k ( r e s ) d Disorder strength, s FIG. 3. (color online) Dependencies k res ( s ) (thick blue solidcircles) and k ′ ( s ) (thin open red circles). QNMs are numberedas in Fig. 2. It is seen that for ordinary QNMs, k res ( s ) and k ′ ( s ) practically coincide, whereas there are no resonancesassociated with strange QNMs ( Thus, any TR has its partner among QNMs, but thereverse is not true: there are strange QNMs that are notassociated with any maxima in the transmission, as it isshown in Fig. 4, and therefore do not have co-partnersbetween resonances. In other words, in a given wavenumber interval ∆ k , the statistically-averaged numberof TRs, N res , is smaller than the statistically-averagednumber of QNMs, N mod = ∆ kL/π , and does not dependon the degree of disorder. This fact was noticed in thenumerical calculations in [28].Surprisingly, when s →
0, the ratio N res /N mod is a universal constant p /
5, independent of the type of dis-order, and remains practically independent on the degreeof disorder and the length L of the system in a rather T ( k ) kd FIG. 4. (color online) Transmission spectrum T ( k ) at s =0 .
15. The black dashed (red solid) vertical lines indicate the k ′ n values of ordinary (strange) QNMs. broad range of these parameters.Figure 5 shows the ratio N res /N mod as a function of s ,statistically averaged over 10 random realizations andnormalized by p /
5, for various lengths L in the case r L,R = 0. It is important to note that the localiza- res mod N o r m a li ze d nu m b e r o f r e s on a n s e s Disorder strengh s FIG. 5. (color online) Normalized ratio p / N res /N mod ver-sus the degree of disorder s for systems of various lengths L = Nd ( N is the number of layers). tion length (measured in numbers of layers) N loc ∝ s − ,and this is less than 20 for s = 0 .
3. This means that N res /N mod ≃ p / N res and N mod appears when s is very small so that N loc ≫ N ,and remains practically unchanged even when s is ratherlarge so that N loc ≪ N . This means that the origin ofthis phenomenon is not specifically related to localizationand can be studied when s is arbitrarily small.To calculate the average number of TRs in the limit s ≪
1, we use the single-scattering approximation andwrite the total reflection coefficient r ( k ) of the whole sys-tem as: r ( k ) = Σ Nn =1 r n e ikx n , (3) where x n is the coordinate of the n -th scatterer. Thevalues k max , where the transmission coefficient, T ( k ) =1 − | r ( k ) | , has local maxima, are defined as the zeros ofthe 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 . (4)Assuming first that δd i = 0, then f ( k ) becomes 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 . (5)Eq. (5) is the trigonometric sum Σ Nl =1 a l sin ( ν l k ) with“frequencies” ν l = 2 ld and random coefficients a l . Thestatistics of the zeroes of random polynomials have beenstudied in [31], where it is shown that the statistically-averaged number of real roots N root of the sum of thistype at a certain interval ∆ k is N root = ∆ kπ s Σ ν l σ l Σ σ l , (6)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 . When N ≫ a l ) ≃ N − l ) l σ , (7)where σ = Var( r ). The sums in Eq. (6) can be calcu-lated using Eq. (7), which yields [35]:Σ Nl =1 σ l = 2 σ Σ Nl =1 l ( N − l ) ≃ σ N , Σ Nl =1 ν l σ l = 8 d Σ Nl =1 σ l ( N − l ) ≃ d N σ . (8)From Eqs. (6) and (8) we obtain N root = 2∆ kN d π r
25 = 2 ∆ kLπ r , (9)where L = N d . Since the number of minima of thereflection coefficient is equal to the number of TRs, N res = N root /
2, and the number N mod of QNMs in thesame interval ∆ k is N res = ∆ kL/π , from Eq. 9 it followsthat N res /N mod = p / with-out assuming any periodicity of the scatterers. To calcu-late this ratio for more general situations, when the dis-tances between the scatterers are also random ( δd i = 0) , the frequencies ν = 2 ld d in Eq. (5) should be replaced by ν = 2 | x m − x m ± l | . Since the main contribution to thesums in Eq. (6) is given by the terms with large l ∼ N ,the mean value of | x m − x m ± l | can be replaced by ld , inthe case of a homogeneous distribution of the distances d n along the system. This ultimately leads to the sameresult Eq. (10).In summary, it is well known that there is a one-to-one correspondence between the QNMs of a regular opensystem (wave resonator or quantum cavity) and its trans-mission resonances: each QNM is unambiguously associ-ated with a TR, and vice versa. In this paper, we showfor the first time that in 1D random structures, this reci-procity is broken: any weak disorder mutates part of theeigenstates so that the corresponding resonances in thetransmission disappear and the density of TR becomessmaller than the total density of states. Although thestrange modes belong to a discrete spectrum, the spa-tial structure of the eigenfunctions differs drastically fromthat of the ordinary states and show no sign of localiza-tion. It is significant that while the strange modes do notshow up in the amplitude of the transmission coefficient,in the phase of the transmitted field they manifest them-selves in just the same way as the ordinary modes do.Indeed, the numerical calculations show that the deriva-tive of this phase with respect to the frequency gives thetotal density of QNMs, which includes both the normalordinary and strange ones. When the disorder is weak(but strong enough to localize the ordinary modes), theratio of the number of TRs to the total number of QNMsin a frequency interval ∆ ω → ∞ is independent of thetype of disorder and anomalously weakly deviates from auniversal constant, p /