Method for obtaining upper bounds on photonic band gaps

We present a method to calculate upper bounds on the photonic band gaps of two-component photonic crystals. The method involves calculating both upper and lower bounds on the frequency bands for a given structure, and then maximizing over all possible two-component structures. We apply this method to a number of examples, including a one-dimensional photonic crystal (or "Bragg grating") and two-dimensional photonic crystals (in both the TM and TE polarizations) with both four and sixfold rotational symmetries. We compare the bounds to band gaps of numerically optimized structures and find that the bounds are extremely tight. We prove that the bounds are "sharp" in the limit of low dielectric contrast ratio between the two components. This method and the bounds derived here have important implications in the search for optimal photonic band-gap structures.