Abstract:
We study a plasmonic coupler involving backward (TM_01) and forward (HE_11) modes of dielectric waveguides embedded into infinite metal. The simultaneously achievable contradirectional energy flows and codirectional wavevectors in different channels lead to a spectral gap, despite the absence of periodic structures along the waveguide. We demonstrate that a complete spectral gap can be achieved in a symmetric structure composed of four coupled waveguides.

Abstract:
At the example of two coupled waveguides we construct a periodic second order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary we impose the Neumann boundary condition but cut out a periodic system of small holes, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the holes, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.

Abstract:
We consider the twisted waveguide $\Omega_\theta$, i.e. the domain obtained by the rotation of the bounded cross section $\omega \subset {\mathbb R}^{2}$ of the straight tube $\Omega : = \omega \times {\mathbb R}$ at angle $\theta$ which depends on the variable along the axis of $\Omega$. We study the spectral properties of the Dirichlet Laplacian in $\Omega_\theta$, unitarily equivalent under the diffeomorphism $\Omega_\theta \to \Omega$ to the operator $H_{\theta'}$, self-adjoint in ${\rm L}^2(\Omega)$. We assume that $\theta' = \beta - \epsilon$ where $\beta$ is a $2\pi$-periodic function, and $\epsilon$ decays at infinity. Then in the spectrum $\sigma(H_\beta)$ of the unperturbed operator $H_\beta$ there is a semi-bounded gap $(-\infty, {\mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({\mathcal E}_j^-, {\mathcal E}_j^+)$. Since $\epsilon$ decays at infinity, the essential spectra of $H_\beta$ and $H_{\beta - \epsilon}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{\beta - \epsilon}$ near an arbitrary fixed spectral edge ${\mathcal E}_j^\pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $\sigma_{\rm disc}(H_{\beta-\epsilon})$ in a neighbourhood of ${\mathcal E}_j^\pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $\sigma_{\rm disc}(H_{\beta-\epsilon})$ near ${\mathcal E}_j^\pm$ could be represented as a finite orthogonal sum of operators of the form $-\mu\frac{d^2}{dx^2} - \eta \epsilon$, self-adjoint in ${\rm L}^2({\mathbb R})$; here, $\mu > 0$ is a constant related to the so-called effective mass, while $\eta$ is $2\pi$-periodic function depending on $\beta$ and $\omega$.

Abstract:
The origin of spectral singularities in finite-gap singly periodic PT-symmetric quantum systems is investigated. We show that they emerge from a limit of band-edge states in a doubly periodic finite gap system when the imaginary period tends to infinity. In this limit, the energy gaps are contracted and disappear, every pair of band states of the same periodicity at the edges of a gap coalesces and transforms into a singlet state in the continuum. As a result, these spectral singularities turn out to be analogous to those in the non-periodic systems, where they appear as zero-width resonances. Under the change of topology from a non-compact into a compact one, spectral singularities in the class of periodic systems we study are transformed into exceptional points. The specific degeneration related to the presence of finite number of spectral singularities and exceptional points is shown to be coherently reflected by a hidden, bosonized nonlinear supersymmetry.

Abstract:
Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. In a previous work, two of the authors considered the case of a spectral edge. The main result of this article is finding such asymptotics near a gap edge, for "generic" periodic elliptic operators of second-order with real coefficients in dimension $d \geq 2$, when the gap edge occurs at a symmetry point of the Brillouin zone.

Abstract:
The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N-1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. This work should be viewed as an extension of "Periodic structures on waveguides" by Linton and McIvor.

Abstract:
We analyze nonlinear collective effects near surfaces of semi-infinite periodic systems with multi-gap transmission spectra and introduce a novel concept of multi-gap surface solitons as mutually trapped surface states with the components associated with different spectral gaps. We find numerically discrete surface modes in semi-infinite binary waveguide arrays which can support simultaneously two types of discrete solitons, and analyze different multi-gap states including the soliton-induced waveguides with the guided modes from different gaps and composite vector solitons.

Abstract:
We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, guided mode spectrum can be created by arbitrarily "small" perturbations. Secondly we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum $k_x$ the perturbation generates at most finitely many new eigenvalues inside the gap.

Abstract:
It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap solutions, i. e. standing waves, with the frequency in a spectral gap, that are exponentially localized in spatial variable. The proof is based on the linking theorem in combination with periodic approximations.

Abstract:
A 1D model is developed for defective gap mode (DGM) with two types of boundary conditions: conducting mesh and conducting sleeve. For a periodically modulated system without defect, the normalized width of spectral gaps equals to the modulation factor, which is consistent with previous studies. For a periodic system with local defects introduced by the boundary conditions, it shows that the conducting-mesh-induced DGM is always well confined by spectral gaps while the conducting-sleeve-induced DGM is not. The defect location can be a useful tool to dynamically control the frequency and spatial periodicity of DGM inside spectral gaps. This controllability can be applied to optical microcavities and waveguides in photonic crystals and the interaction between gap eigenmodes and energetic particles in fusion plasmas.