  Analytic representation formulas and power series are developed describing
the band structure inside periodic photonic and acoustic crystals made from
high contrast inclusions. Central to this approach is the identification and
utilization of a resonance spectrum for quasi-periodic source free modes. These
modes are used to represent solution operators associated with electromagnetic
and acoustic waves inside periodic high contrast media. Convergent power series
for the Bloch wave spectrum is recovered from the representation formulas.
Explicit conditions on the contrast are found that provide lower bounds on the
convergence radius. These conditions are sufficient for the separation of
spectral branches of the dispersion relation.
