Regularity for Maxwell eigenproblems in photonic crystal fibre modelling

Abstract

The convergence behaviour and the design of numerical methods for modelling the flow of light in photonic crystal fibres depend critically on an understanding of the regularity of solutions to time-harmonic Maxwell equations in a three-dimensional, periodic, translationally invariant, heterogeneous medium. In this paper we determine the strength of the dominant singularities that occur at the interface between materials. By modifying earlier regularity theory for polygonal interfaces we find that on each subdomain, where the material in the fibre is constant, the regularity of in-plane components of the magnetic field are $H^{2-η}$ for all $η> 0$. This estimate is sharp in the sense that these components do not belong to $H^2$, in general. However, global regularity is restricted by the presence of an interface between these subdomains and the interface conditions imply only $H^{3/2-η}$ regularity across the interface. The results are useful to anyone applying a numerical method such as a finite element method or a planewave expansion method to model photonic crystal fibres or similar materials.