Eigenvalue branches of the perturbed Maxwell operator M+λD in a gap of σ(M)

Abstract

The propagation of guided waves in photonic crystal fibers (PCFs) is studied. A PCF can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction. This problem can be treated as an eigenvalue problem for a family of noncompact self-adjoint operators. Under the assumption that the background spectrum has a gap, we prove that a line defect can create an eigenvalue of any given fixed value in the gap, provided that the defect is strong enough. Based on a decoupling of regions in R2 by means of Dirichlet and Neumann boundaries, then using the trace ideal estimates, we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches. In particular, we show that if the defect is weak enough, no eigenvalues can be created inside the gap.