Numerical Investigation of Photonic Crystal Fibers by Spectral and Multipole Methods

Abstract

In this paper, we investigate the propagation of electromagnetic waves along microstructured optical fibres (MOF) with a crystal cladding of finite extension in the transverse plane. In these so-called Photonic Crystal Fibres (PCF), many interesting and unusual phenomena may be observed. For instance, it appears that, for some range of frequencies, a defect in the crystal cladding gives rise to localized waves in the low index core (i.e. in the defect). It is worth noting that the methods reducing the system of equations to a single longitudinal (electric or magnetic) component cannot tackle these genuine vectorial propagation problems. We first achieve this study with a variational numerical method i.e. a Finite Element Formulation with the electric field as the variable, where edge elements are used for discretizing the transverse field and nodal elements for the longitudinal field [4]. Finally, we compare our finite element method — which leads to a generalized eigenvalue problem- with another method co-developed by some of us [18, 11], which is based on the determination of poles of the scattering operator (multipole expansion method).