Analysis of planewave methods for photonic crystal fibres

Photonic crystal fibres are novel optical devices that can be designed to guide light of a particular frequency. This requires two phenomena to occur. First, the frequency of light must be in a gap of the spectrum of the fibre’s cladding (usually a periodic arrangement of air holes), so that the cladding acts as a barrier to that frequency of light. Second, the perturbation or defect in the middle of the fibre must allow a localised or trapped mode to exist in a spectral gap of the cladding so that light may propagate inside the defect. In this paper the performance of the planewave expansion method for computing such spectral gaps and trapped eigenmodes in photonic crystal fibres is carefully analysed. The analysis is complicated by the occurrence of discontinuous coefficients in the governing equation which means that exponential convergence is impossible due to limited regularity of the eigenfunctions. However, we show through rigorous analysis on simplified model problems and through numerical experiments on the un-simplified problem, that the planewave expansion method is at least as good as standard finite element schemes on uniform meshes in both error convergence and computational efficiency. More importantly, we also consider the performance of two commonly used 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, however, we determine the optimal choices for the required parameters so that both of these variant methods do not add significant errors.