A Dirichlet-to-Neumann Approach for The Exact Computation of Guided Modes in Photonic Crystal Waveguides

This work deals with one dimensional in nite 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 (the Supercell method for instance), 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 xed point nature.