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.
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 maximum in the transmission coefficient is always related to a QNM, the reverse statement is not necessarily correct. There exists an intermediate state, at which only a part of the QNMs are localized and these QNMs provide a resonant transmission. The rest of the solutions of the eigenvalue problem (denoted as strange quasi-modes) are never found in regular open cavities and resonators, and arise exclusively due to random scatterings. Although these strange QNMs belong to a discrete spectrum, they are not localized and not associated with any anomalies in the transmission. The ratio of the number of the normal QNMs to the total number of QNMs is independent of the type of disorder, and slightly deviates from the constant $sqrt{2/5}$ in rather large ranges of the strength of a single scattering and the length of the random sample.
We explore the optical properties of periodic layered media containing left-handed metamaterials. This study is based on several analogies between the propagation of light in metamaterials and charge transport in graphene. We derive the conditions for these two problems become equivalent, i.e., the equations and the boundary conditions when the corresponding wave functions coincide. We show that the photonic band-gap structure of a periodic system built of alternating left- and right-handed dielectric slabs contains conical singularities similar to the Dirac points in the energy spectrum of charged quasiparticles in graphene. Such singularities in the zone structure of the infinite systems give rise to rather unusual properties of light transport in finite samples. In an insightful numerical experiment (the propagation of a Gaussian beam through a mixed stack of normal and meta-dielectrics) we simultaneously demonstrate four Dirac point-induced anomalies: (i) diffusion-like decay of the intensity at forbidden frequencies, (ii) focusing and defocussing of the beam, (iii) absence of the transverse shift of the beam, and (iv) a spatial analogue of the Zitterbewegung effect. All of these phenomena take place in media with non-zero average refractive index, and can be tuned by changing either the geometrical and electromagnetic parameters of the sample,or the frequency and the polarization of light.
Magnetic barriers in graphene are not easily tunable. However, introducing both electric and magnetic fields, provides tunable and far more controllable electronic states in graphene. Here we study such systems. A one-dimensional channel can be formed in graphene using perpendicular electric and magnetic fields. This channel (quantum wire) supports localized electron-hole states, with parameters that can be controlled by an electric field. Such quantum wire offers peculiar conducting properties, like unidirectional conductivity and robustness to disorder. Two separate quantum wires comprise a waveguide with two types of eigenmodes: one type is similar to traditional waveguides, the other type is formed by coupled surface waves propagating along the boundaries of the waveguide.
We study charge transport in one-dimensional graphene superlattices created by applying layered periodic and disordered potentials. It is shown that the transport and spectral properties of such structures are strongly anisotropic. In the direction perpendicular to the layers, the eigenstates in a disordered sample are delocalized for all energies and provide a minimal non-zero conductivity, which cannot be destroyed by disorder, no matter how strong this is. However, along with extended states, there exist discrete sets of angles and energies with exponentially localized eigenfunctions (disorder-induced resonances). It is shown that, depending on the type of the unperturbed system, the disorder could either suppress or enhance the transmission. Most remarkable properties of the transmission have been found in graphene systems built of alternating p-n and n-p junctions. This transmission has anomalously narrow angular spectrum and, surprisingly, in some range of directions it is practically independent of the amplitude of fluctuations of the potential. Owing to these features, such samples could be used as building blocks in tunable electronic circuits. To better understand the physical implications of the results presented here, most of our results have been contrasted with those for analogous wave systems. Along with similarities, a number of quite surprising differences have been found.
