Out of Plane Propagation of Electromagnetic Waves in a Two-dimensional Periodic Dielectric Medium

By the use of the plane wave method we have calculated the dispersion curves (photonic band structure) of electromagnetic waves propagating in a structure consisting of an infinite array of parallel, infinitely long, dielectric rods of arbitrary cross-section, characterized by a dielectric constant εa, embedded in a medium characterized by a dielectric constant εb, when the intersections of the axes of the rods with a perpendicular plane form a two-dimensional Bravais lattice. In contrast with earlier calculations of the photonic band structures of two-dimensional, periodic, dielectric structures, in the present work the electromagnetic waves are assumed to propagate out of the plane perpendicular to the rods. In numerical calculations we study a triangular lattice of air cylinders in a dielectric medium, which has recently been shown to possess a band gap common to waves of both E and H polarization for propagation in the plane perpendicular to the rods. The shifts of the edges of this absolute band gap as k 3, the component of the wave-vector of the electromagnetic waves parallel to the rods, is increased from zero are studied, as is the closing up of this gap. A new absolute band gap is found to open up below the first band as k 3 is increased. Conclusions concerning the filtering behaviour of two-dimensional, periodic, dielectric structures in the case of out-of-plane propagation of electromagnetic waves through them are drawn.