On band gaps in photonic crystal fibers

We consider the Maxwell's system for a periodic array of dielectric `fibers' embedded into a `matrix', with respective electric permittivities \(\epsilon_0\) and \(\epsilon_1\), which serves as a model for cladding in photonic crystal fibers (PCF). The interest is in describing admissible and forbidden (gap) pairs \((\omega,k)\) of frequencies \(\omega\) and propagation constants \(k\) along the fibers, for a Bloch wave solution on the cross-section. We show that, for "pre-critical" values of \(k(\omega)\) i.e. those just below \(\omega (\min\{\epsilon_0,\epsilon_1\}\mu)^{1/2}\) (where \(\mu\) is the magnetic permeability assumed constant for simplicity), the coupling specific to the Maxwell's systems leads to a particular partially degenerating PDE system for the axial components of the electromagnetic field. Its asymptotic analysis allows to derive the limit spectral problem where the fields are constrained in one of the phases by Cauchy-Riemann type relations. We prove related spectral convergence. We finally give some examples, in particular of small size "arrow" fibers (\(\epsilon_0>\epsilon_1\)) where the existence of the gaps near appropriate "micro-resonances" is demonstrated by a further asymptotic analysis.