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A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals

by H. Ammari and E. Orvehed Hiltunen and S. Yu

(Report number 2019-01)

Abstract
In [H. Ammari et al., Honeycomb-lattice Minnaert bubbles. arXiv:1811.03905], the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal comprised of bubbles of arbitrary shape is shown. The aim of this paper is to prove that, near the Dirac points, the Bloch eigenfunctions is the sum of two eigenmodes. Each eigenmode can be decomposed into two components: one which is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is highly oscillating. The slowly oscillating components of the eigenmodes satisfy a Dirac equation. Our results in this paper demonstrate for the first time a near-zero effective refractive index near the Dirac points for the plane-wave envelopes of the Bloch eigenfunctions in a subwavelength metamaterial. They are illustrated by a variety of numerical examples. We also compare and contrast the behaviour of the Bloch eigenfunctions in the honeycomb crystal with that of their counterparts in a bubbly square crystal, near the corner of the Brillouin zone, where the maximum of the first Bloch eigenvalue is attained.

Keywords: Honeycomb lattice, Dirac cone, Dirac equation, bubble, Minnaert resonance, sub-wavelength bandgap, near-zero effective property.

@Techreport{AOY19_805,
  author = {H. Ammari and E. Orvehed Hiltunen and S. Yu},
  title = {A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-01.pdf },
  year = {2019}
}

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