The mathematics of photonic crystals
Mathematical modeling in optical science
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
SIAM Journal on Numerical Analysis
Convergence of the Supercell Method for Defect Modes Calculations in Photonic Crystals
SIAM Journal on Numerical Analysis
Discontinuous Galerkin Approximation of the Maxwell Eigenproblem
SIAM Journal on Numerical Analysis
Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes
Journal of Computational and Applied Mathematics
Photonic Crystals: Molding the Flow of Light
Photonic Crystals: Molding the Flow of Light
SIAM Journal on Numerical Analysis
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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.