Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals

TL;DR

This paper models closely packed photonic and phononic crystals as asymptotic networks of resonators, providing analytical dispersion relations that match simulations and revealing novel topological features like Dirac points.

Contribution

It introduces a novel asymptotic network model for subwavelength metamaterials formed by closely packed crystals, enabling explicit derivation of their dispersion properties.

Findings

Discrete models accurately predict continuous crystal dispersion curves.

Hexagonal lattice behaves like a honeycomb network with Dirac points.

Symmetry-breaking defects can control degeneracies in the lattice.

Abstract

We demonstrate that photonic and phononic crystals consisting of closely spaced inclusions constitute a versatile class of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystal form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous photonic (phononic) crystals are in fact asymptotically equivalent, at low frequencies, to discrete capacitor-inductor (mass-spring) networks whose lumped parameters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch, or branches when the discrete analogue is polyatomic, is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power 1/4; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals. The insight gained from the network approach is used to show that, surprisingly, the continuum created by a closely packed hexagonal lattice of cylinders is represented by a discrete honeycomb lattice. The analogy is utilised to show that the hexagonal continuum lattice has a Dirac-point degeneracy that is lifted in a controlled manner by specifying the area of a symmetry-breaking defect.