Abstract:
The Laplace operator in infinite quantum waveguides (e.g., a bent strip or a twisted tube) often has a point-like eigenvalue below the essential spectrum that corresponds to a trapped eigenmode of finite L2 norm. We revisit this statement for resonators with long but finite branches that we call "finite waveguides". Although now there is no essential spectrum and all eigenfunctions have finite L2 norm, the trapping can be understood as an exponential decay of the eigenfunction inside the branches. We describe a general variational formalism for detecting trapped modes in such resonators. For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode. Varying the branch lengths may switch certain eigenmodes from non-trapped to trapped states. These concepts are illustrated for several typical waveguides (L-shape, bent strip, crossing of two stripes, etc.). We conclude that the well-established theory of trapping in infinite waveguides may be incomplete and require further development for being applied to microscopic quantum devices.

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 suggest the numerical approach to detect eigenfrequencies of trapped modes in waveguides or guided waves in diffraction gratings. At the same time, the approach works perfectly for computation of systems with finitely many scattering channels. The most attractive example concerns the possibility of control on electron transport in nano-dimensions system consisting of a resonator and finitely many adjoined channels due to external variable electric field.

Abstract:
We investigate the characteristics of guided wave modes in planar coupled waveguides. In particular, we calculate the dispersion relations for TM modes in which one or both of the guiding layers consists of negative index media (NIM)-where the permittivity and permeability are both negative. We find that the Poynting vector within the NIM waveguide axis can change sign and magnitude, a feature that is reflected in the dispersion curves.

Abstract:
It has been shown that a small discontinuity such as an enlargement or a hole on circular waveguides can produce trapped electromagnetic modes with frequencies slightly below the waveguide cutoff. The trapped modes due to multiple discontinuities can lead to high narrow-band contributions to the beam-chamber coupling impedance, especially when the wall conductivity is high enough. To make more reliable estimates of these contributions for real machines, an analytical theory of the trapped modes is developed in this paper for a general case of the vacuum chamber with an arbitrary single-connected cross section. The resonant frequencies and coupling impedances due to trapped modes are calculated, and simple explicit expressions are given for circular and rectangular cross sections. The estimates for the LHC are presented.

Abstract:
We assert that the physics underlying the extraordinary light transmission (reflection) in nanostructured materials can be understood from rather general principles based on the formal scattering theory developed in quantum mechanics. The Maxwell equations in passive (dispersive and absorptive) linear media are written in the form of the Schr\"{o}dinger equation to which the quantum mechanical resonant scattering theory (the Lippmann-Schwinger formalism) is applied. It is demonstrated that the existence of long-lived quasistationary eigenstates of the effective Hamiltonian for the Maxwell theory naturally explains the extraordinary transmission properties observed in various nanostructured materials. Such states correspond to quasistationary electromagnetic modes trapped in the scattering structure. Our general approach is also illustrated with an example of the zero-order transmission of the TE-polarized light through a metal-dielectric grating structure. Here a direct on-the-grid solution of the time-dependent Maxwell equations demonstrates the significance of resonances (or trapped modes) for extraordinary light transmissio

Abstract:
The spectrum of electromagnetic waves propagating in a strongly coupled magnetized fully ionized hydrogen plasma is found. The ion motion and damping being neglected, the influence of the Coulomb coupling on the electromagnetic spectrum is analyzed.

Abstract:
We study long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We decompose the wave field in propagating and evanescent transverse electric and magnetic modes with random amplitudes that encode the cumulative scattering effects. They satisfy a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion approximation theorem, under the assumption that the fluctuations decorrelate rapidly in the range direction. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering.

Abstract:
The recurrence dispersion equation of coupled single-mode waveguides is modified by eliminating redundant singularities from the dispersion function. A recurrence zero-bracketing (RZB) technique is proposed in which the zeros of the dispersion function at one recurrence step bracket those of the next recurrence step. Numerical examples verify the utility of the RZB technique in computing the roots of the dispersion equation of the TE and TM modes of both uniform and non-uniform arrays.

Abstract:
We study an analogue of the equations describing TE and TM modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schr\"odinger equation. The effective potential energies involved in the Schr\"odinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permeability is constant, it means that the refractive index is constant, we find that the TE and TM modes are described by the supersymmtric quantum mechanics.