Bandgaps at high frequency from highly oscillatory metamaterials
Abstract
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 < d̅. 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.
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BibTeX
@misc{bolanos2026bandgaps,
title = {{Bandgaps at high frequency from highly oscillatory metamaterials}},
author = {Silvia Jiménez Bolaños and Robert P. Lipton and Robert Viator},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04925},
eprint = {2611.04925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04925},
}
RIS
TY - UNPB
TI - Bandgaps at high frequency from highly oscillatory metamaterials
AU - Silvia Jiménez Bolaños
AU - Robert P. Lipton
AU - Robert Viator
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04925
UR - https://arxiv.org/abs/2611.04925
AB - 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 ω^2 can be analytically continued and represented as a power series in d^2. 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 < d̅. 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.
ER -