Mathematical and Numerical Techniques for Open Periodic Waveguides

The propagation of electromagnetic waves in dielectric slab waveguides with periodic corrugations is described by the spectrum of the Helmholtz operator on an infinite strip with quasiperiodic boundary conditions. This chapter reviews the basic properties of this spectrum, which typically consists of guided modes, radiation modes and leaky modes. A great deal of attention will be devoted to planar waveguides which share some of the important features of the periodic case. To compute the eigenmodes and the associated propagation constants numerically, one usually truncates the domain that contains the grating and imposes certain radiation conditions on the artificial boundary. An alternative to this approach is to decompose the infinite strip into a rectangle, which contains the grating, and two semi-infinite domains. The guided and leaky modes can be computed by matching the Dirichlet-to-Neumann operator on the interfaces of these three domains. The discretized eigenvalue problem is nonlinear because of the appearance of the propagation constant in the artificial boundary condition. We will discuss how such problems can be solved by numerical continuation. In this approach, one starts with an approximating planar waveguide and then follows the solutions by a continuous transition to the multilayer periodic structure. The chapter is concluded with a brief description of how the perfectly matched layer can be used to compute the guided modes of a waveguide.