Convergence analysis of planewave expansion for band gap computations in photonic crystal fibres

Richard Norton

Oct. 16, 2007, 3:30 p.m. T 1010

Photonic Crystal Fibres (PCFs) are silicon or glass fibres designed with a special 2D structure that allows them to guide certain frequencies of light. For a fixed frequency we solve Maxwell's equations for the z-component of the wave vector to determine the modes of light permissible in the fibre. We transform the problem onto a bounded domain with periodic boundary conditions using the Floquet transform and then apply the planewave expansion method (spectral Galerkin method) to the resulting variational eigenvalue problem. A piece-wise constant coefficient function reduces the regularity of the eigenfunctions and the method does not exhibit exponential convergence. As well as presenting the error analysis we have considered solving a modified problem where the piece-wise constant coefficient function is replaced with a smooth coefficient function. The error analysis for the smooth problem is also presented and we answer the question: Is smoothing worth it? Finally, our theoretical results and the answer to this question are supported with 1D and 2D numerical computations.