Dispersive and effective properties of two-dimensional periodic media

We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of circular cylinders. Propagation of waves is described by the Helmholtz equation with the continuity conditions for the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the cell size is small compared with the wavelength, but large compared with the radius a of the inclusions. Explicit formulae are obtained for asymptotic expansion of the solution of the problem in terms of the dimensionless magnitude q of the wavevector and radius a. This leads to explicit formulae for the effective dielectric tensor and the dispersion relation with the rigorously justified error of order O((q ² + a ²)^5/2).