Waveguiding in planar photonic crystals

Abstract—We analyze, in three dimensions, the dispersion prop-erties of dielectric slabs perforated with two-dimensional photonic crystals (PCs) of square symmetry. The band diagrams are calcu-lated for all-vectors in the first Brillouin zone, and not only along the characteristic high-symmetry directions. We have analyzed the equal-frequency contours of the first two bands, and we found that the square lattice planar photonic crystal is a good candidate for the self-collimation of light beams. We map out the group veloc-ities for the second band of a square lattice planar PC and show that the group velocity is the highest in the region of maximum self-collimation. Such a self-collimated beam is predicted to show beating patterns due to the specific shape of the equal-frequency contours. A geometrical transformation maps the region of the first and second photonic bands where self-collimation takes place one onto the other and gives additional insights on the structural simi-larities of self-collimation in those two bands. Index Terms—Autocollimation, finite difference time domain (FDTD), planar photonic crystals, self-collimation. I