Trapped modes for periodic structures in waveguides

TL;DR

This paper proves the existence of multiple trapped modes in waveguides with periodic obstacles under Neumann or Dirichlet boundary conditions, extending previous work on waveguide structures with periodic features.

Contribution

It establishes the existence of at least N or N-1 trapped modes in waveguides with periodic obstacles, generalizing prior results to more complex obstacle configurations.

Findings

At least N trapped modes under Neumann conditions.

At least N-1 trapped modes under Dirichlet conditions.

Extension of previous work on periodic waveguide structures.

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.