Dirichlet-to-Neumann Transparent Boundary Conditions for Photonic Crystal Waveguides

In this paper, we present a complete algorithm for the exact computation of the guided mode band structure in photonic crystal (PhC) waveguides. In contrast to the supercell method, the used approach does not introduce any modeling error and is hence independent of the confinement of the modes. The approach is based on Dirichlet-to-Neumann transparent boundary conditions that yield a nonlinear eigenvalue problem. For the solution of this nonlinear eigenvalue problem, we present a direct technique using Chebyshev interpolation that requires a bandgap calculation of the PhC in advance. For this bandgap calculation, we introduce as a very efficient tool a Taylor expansion of the PhC band structure. We show that our algorithm-like the supercell method-converges exponentially, however, its computational costs-in comparison with the supercell method-only increase moderately since the size of the matrix to be inverted remains constant.