A Recursive-Doubling Dirichlet-to-Neumann-Map Method for Periodic Waveguides

For optical-waveguiding structures composed of uniform segments that are invariant in the longitudinal direction, the Dirichlet-to-Neumann (DtN)-map method is highly competitive. Instead of computing the eigenmodes, it calculates the DtN operators of the uniform segments using an efficient Chebyshev collocation scheme. In this paper, a new formulation of the DtN-map method is developed for periodic piecewise uniform waveguiding structures. The method involves a recursive-doubling process so that the overall required computation time is proportional to log₂ N, where N is the number of periods. We illustrate our method by analyzing the scattering of surface plasmon polaritons due to Bragg gratings on metal surfaces and thin metal films.