Efficient analysis of periodic dielectric waveguides using Dirichlet-to-Neumann maps

We present a numerical scheme for the analysis of periodic dielectric waveguides using Floquet-Bloch theory. The problem of finding the fundamental propagation modes is reduced to a nonlinear eigenvalue problem involving Dirichlet-to-Neumann maps. This approach leads to much smaller matrix problems than the ones that have appeared previously. By an increase of the discretization fineness, any desired precision of the method can be achieved. We discuss an eigensolver and extend the conventional rule to choose the branches of the transverse wave numbers. This ensures analytic dependence on the Floquet multiplier and convergence of the nonlinear solver. We demonstrate that even for a complicated multilayer waveguide structure the propagation factors can be calculated within seconds to several digits of accuracy.