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Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A 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 for any fixed quasi-momentum. 

"This paper generalizes the results obtained by the authors in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021) concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021), that significantly narrowed the applicability of the homogenization results obtained is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global L∞-bound for the gradient of the solution of the scalar equation −divax∕$$\epsilon$$∇$$\phi$$\epsilon$$(x)=f(x){\$$}{\$$}- {\backslash}operatorname {\{}{\{}{\backslash}mathrm {\{}div{\}}{\}}{\}} {\backslash}left [ {\backslash}mathbf {\{}a{\}} {\backslash}left ( x/{\backslash}varepsilon {\backslash}right ){\backslash}nabla {\backslash}varphi ^{\{}{\backslash}varepsilon {\}}(x) {\backslash}right ] = f(x){\$$}{\$$}, uniform with respect to microstructure scale parameter $$\epsilon$${\thinspace}≪{\thinspace}1 in a small interval (0, $$\epsilon$$0), where the coefficient a is only piecewise H{\"o}lder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021)."