Mathematical analysis of the guided waves in photonic crystal fibers

The propagation of guided waves in photonic crystal fibers (PCFs) is studied. The structure of a PCF can be regarded as a perfect two-dimensional photonic crystal with a line defect along the invariant direction. This problem can be treated as an eigenvalue problem for a family of noncompact self-adjoint operators. We prove that line defects do not change the essential spectrum of the associated “background” medium. This result plays a key role for studying the influence of line defects on the “background” spectrum. A modified Combes-Thomas estimate is also formulated.