Upper bounds on photonic bandgaps in two and three dimensions

A 20-year search has been on to find photonic crystals (periodic dielectric structures) with the largest possible full photonic bandgaps. A large, robust bandgap is key to the many applications of these materials, which include near-lossless waveguiding, optical filtering, optical computing, and others. A number of three-dimensional structures with large gaps have been proposed (e.g., a diamond lattice of spheres [1], the ``Woodpile'' structure [2]), and in two dimensions, structural optimizations to find the largest-bandgap structure have been performed, (e.g., [3], [4]). So far, however, there has been no work on finding rigorous limits on how high the bandgap may be. In this talk, I present upper bounds on the bandgaps of two- and three-dimensional photonic crystals.\\ \noindent $[1]$ Phys. Rev. Lett. {\bf 65}, 3152 (1990)\\ $[2]$ J. Mod. Opt. {\bf 41}, 231 (1994)\\ $[3]$ Appl. Phys. B. {\bf 81}, 235 (2005) \\ $[4]$ Phys. Rev. Lett. {\bf 101}, 073902 (2008) \\