Geometric properties of optimal photonic crystals

Photonic crystals can be designed to control and confine light. Since the introduction of the concept by Yablonovitch and John two decades ago, there has been a quest for the optimal structure, i.e., the periodic arrangement of dielectric and air that maximizes the photonic band gap. Based on numerical optimization studies, we have discovered some surprisingly simple geometric properties of optimal planar band gap structures. We conjecture that optimal structures for gaps between bands n and n+1 correspond to n elliptic rods with centers defined by the generators of an optimal centroidal Voronoi tessellation (transverse magnetic polarization) and to the walls of this tessellation (transverse electric polarization).