  Photonic band gap (PBG) materials are attractive for cavity QED experiments
because they provide extremely small mode volumes and are monolithic,
integratable structures. As such, PBG cavities are a promising alternative to
Fabry-Perot resonators. However, the cavity requirements imposed by QED
experiments, such as the need for high Q (low cavity damping) and small mode
volumes, present significant design challenges for photonic band gap materials.
Here, we pose the PBG design problem as a mathematical inversion and provide an
analytical solution for a two-dimensional crystal. We then address a planar (2D
crystal with finite thickness) structure using numerical techniques.
