Convergence Analysis of Planewave Expansion for Band Gap Computations in Photonic Crystal Fibres

In this paper we consider the numerical computation of band gaps in photonic crystal fibres. We approximate the solution to a variational eigenvalue problem using the planewave expansion method (spectral Galerkin method). As well as presenting implementation and error analysis results we consider solving a modified problem where the piecewise constant coefficient function is replaced with a smooth function. The error analysis for the smooth problem is also presented and we answer the question: Is smoothing worth it? Introduction Photonic Crystal Fibres (PCFs) [1] are optical fibres that have a core surrounded by cladding. The structure of a PCF is described by its refractive index n. n is constant along the length of the fibre (z-axis) and we write n = n(x, y). The function n(x, y) is a piecewise constant function representing the refractive index of two materials. Light will be described by its frequency ω and its propagation constant in the z-direction β. For certain designs of cladding and fixed ω, light propagation in the cladding may be forbidden for some values of β. These values of β are called ‘band gaps’. Our task is to approximate the band gaps of a particular PCF and ω by approximating the solution to Maxwell’s equations. We will restrict ourselves to the case where n = n(x). Physically, this is the case of a radially symmetric PCF or a planar PCF. Using time harmonic, source free Maxwell’s equations for a non-magnetic material the problem decouples into two eigenvalue problems dhx dx2 + γ(x)hx = β hx (TE) dhy dx2 + γ(x)hy − dη(x) dx dhy dx = βhy. (TM)