Bandgaps at high frequency from highly oscillatory metamaterials

We investigate the Bloch spectrum for transverse electric (TE) modes in a 2D high-contrast dielectric metamaterial. Our analysis focuses on the high-frequency regime where the propagating wavelength is commensurate with the microstructural period d, and the permittivity within the inclusions scales as d⁻². Using operator factorization and Sobolev space decomposition, we demonstrate that Bloch eigenvalues ω² can be analytically continued and represented as a power series in d². We establish a rigorous upper bound on the microstructural length scale d, defined by the inclusion geometry, under which this expansion remains valid. Furthermore, we provide a definitive proof for the opening of band gaps in the frequency spectrum for all periods below a critical length scale, i.e., for all d < d̄. The threshold is explicitly characterized as a geometric function determined by the interaction between Dirichlet eigenvalues and the spectrum of the associated Neumann-Poincaré operator.