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.
