Frequency-dependent capacitance matrix formulation for Fabry--Pérot resonances. Part II: One-dimensional periodic media

Abstract

We study Bloch resonances of one-dimensional periodic systems of high-contrast resonators beyond the subwavelength regime. An exact transfer-matrix formulation identifies the limiting wavenumbers with the union of the interior Neumann frequencies and the Dirichlet frequencies of the separating intervals. At each fixed reference wavenumber, we introduce a periodic frequency-dependent capacitance matrix whose positive eigenvalues determine all leading square-root splittings in the contrast parameter. A direct cyclic matching expansion proves the approximation, including multiple eigenvalues and small complex contrasts. A spectral exclusion result outside the limiting set yields a direct bandgap statement. At a common resonance of all intervals, we also derive uniform eigenmode approximations and distinguish the zero-frequency mode from nonzero double Bloch eigenvalues. For positive contrast, exact linear crossings are characterized by the monodromy matrix being proportional to the identity. A complete local classification is obtained for two resonators per cell and, through band folding, for fully symmetric arrays with any number of resonators per cell. We further formulate the weighted Zak phase for isolated bands and its quantization under reflection symmetry. Explicit two-resonator formulas and numerical illustrations complement the analysis.