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

## TL;DR

This paper develops a high-frequency homogenization method to analyze Bloch eigenfunctions near Dirac points in bubbly honeycomb crystals, revealing a near-zero effective refractive index and detailed eigenmode decomposition.

## Contribution

It introduces a novel homogenization approach that decomposes eigenmodes into slowly varying and oscillatory parts, demonstrating a near-zero refractive index near Dirac points in sub-wavelength metamaterials.

## Key findings

- Eigenfunctions decompose into two components: slowly varying and highly oscillating.
- Slowly varying components satisfy Dirac equations.
- Near-zero effective refractive index is achieved at Dirac points.

## 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 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 system of Dirac equations. Our results in this paper proves 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 sub-wavelength 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.