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