On the spectrum of a waveguide with periodic cracks

The spectral problem on a periodic domain with cracks is studied. An asymptotic form of dispersion relations is calculated under assumption that the opening of the cracks is small

Homogenization of the planar waveguide with frequently alternating boundary conditions

We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under the certain condition the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

We consider a magnetic Schroedinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum

Bound states in coupled guides. II. Three dimensions.

We compute bound-state energies in two three-dimensional coupled waveguides, each obtained from the two-dimensional configuration considered in part I by ro- tating the geometry about a different axis. The first geometry consists of two concentric circular cylindrical waveguides coupled by a finite length gap along the axis of the inner cylinder and the second is a pair of planar layers coupled laterally by a circular hole. We have also extended the theory for this latter case to include the possibility of multiple circular windows. Both problems are formulated using a mode-matching technique, and in the cylindrical guide case the same residue calcu- lus theory as used in I is employed to find the bound-state energies. For the coupled planar layers we proceed differently, computing the zeros of a matrix derived from the matching analysis directly

Bound states in coupled guides. I. Two dimensions.

Bound states that can occur in coupled quantum wires are investigated. We consider a two-dimensional configuration in which two parallel waveguides (of dif- ferent widths) are coupled laterally through a finite length window and construct modes which exist local to the window connecting the two guides. We study both modes above and below the first cut-off for energy propagation down the coupled guide. The main tool used in the analysis is the so-called residue calculus technique in which complex variable theory is used to solve a system of equations which is derived from a mode-matching approach. For bound states below the first cut-off a single existence condition is derived, but for modes above this cut-off (but below the second cut-off), two conditions must be satisfied simultaneously. A number of results have been presented which show how the bound-state energies vary with the other parameters in the problem

Trapped modes in the presence of thin obstacles

In this thesis we use various techniques to investigate the occurrence of trapped modes in the presence of thin obstacles. Physically trapped modes are oscillations of finite energy in a fluid which is unbounded in at least one direction. These oscillations mainly occur locally to some structure and decay to zero at large distances away from it. Trapped modes are important as they have been found to exist in a wide range of physical situations. We consider a number of problems in two and three dimensions including waveguides containing bodies and arrays of identical structures. A modified residue calculus technique, a variational technique and a method based on the truncation of matched eigenfunction expansions are used to solve the problems, with numerous results presented

Wannier-Function based Scattering-Matrix Formalism for Photonic Crystal Circuitry

In this dissertation, I develop and investigate in detail a scattering-matrix approach for the efficient treatment of Photonic Crystal circuits and demonstrate its applicability to large-scale 2D circuits. The approach relies on the efficient calculation of the scattering matrices of basic functional elements using an expansion of fields into photonic Wannier functions, which represent a basis ideally suited to describe localized fields within defect structures in Photonic Crystals