A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides

Abstract. This works deals with one dimensional infinite perturbation- namely line defects- in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann (DtN) approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. On contrary to existing methods, this one is exact but there is a price to be paid: the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. Key words. Periodic media, line defect, guided waves, spectral analysis, Dirichlet-to-Neumann operator AMS subject classifications. 78A10,78A48,47A70,35P9