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 talk reviews the
basic properties of this spectrum, which typically consists of guided modes,
radiation modes and leaky modes. To motivate the periodic case 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 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 (DtN)
operator on the interfaces of these three domains. While the exterior (DtN) map
can be found analytically, the interior (DtN) map must be computed numerically.
To that end, a symmetric boundary integral formulation is introduced. The
discretized problem is a nonlinear eigenvalue problem, which is solved by
numerical continuation. The talk concludes with numerical results for single-
and double periodic structures.