A sum rule and physical bound for electromagnetic band gap structures

We give a physical bound on the product of bandwidth and attenuation per unit length for general electromagnetic band gap structures. The analysis is based on the Kramers-Kronig relations for the Bloch refractive index for a fixed direction of propagation through the crystal. The bound is given in terms of the low- and high-frequency asymptotes of the refractive index, which can be given simple estimates in terms of the volume fractions of the component materials and their static permittivities.