Planewave expansion methods for photonic crystal fibres

Photonic crystal fibres are novel optical devices that can be designed to guide light of a particular frequency. In this paper the performance of planewave expansion methods for computing spectral gaps and trapped eigenmodes in photonic crystal fibres is carefully analysed. The occurrence of discontinuous coefficients in the governing equation means that exponential convergence is impossible due to the limited regularity of the eigenfunctions. We show through a numerical convergence study and rigorous analysis on a simplified problem that the convergence of the planewave expansion method is controlled by the regularity of the eigenfunctions and performs thus no better (but also no worse) than we would expect (non-adaptive) finite element methods to perform, both in terms of error convergence and computational efficiency. We also consider the performance of two variants of the planewave expansion method: (a) coupling the planewave expansion method with a regularisation technique where the discontinuous coefficients in the governing equation are approximated by smooth functions, and (b) approximating the Fourier coefficients of the discontinuous coefficients in the governing equation. There is no evidence that regularisation improves the planewave expansion method, but with the correct choice of parameters both variants can be used efficiently without adding significant errors.