An hybrid method for the calculation of the band structure of 2D photonic crystals

In recent years the interaction of electromagnetic waves with periodic structures has received much attention, and several applications in the area of antennas have been suggested. In this paper we present a new hybrid integral equation-plane wave expansion method, for the calculation of the band structure of a dielectric photonic crystal. We derive an enlarged problem whose band structure contains that of the photonic crystal, but whose eigenfunctions are much smoother. This causes their plane wave expansion to converge much faster than that of the original problem. Here we limit our analysis to 2-dimensional structures with cylindrical inclusions (the axis of the cylinders is oriented in the z-direction). In this case, TM-z and TE-z polarizations decouple. We present the formalism corresponding to the TE-z polarization.