Bloch waves in an arbitrary two-dimensional lattice of subwavelength Dirichlet scatterers

TL;DR

This paper develops a method to analyze wave propagation in two-dimensional lattices of subwavelength Dirichlet scatterers, revealing how these scatterers perturb wave modes and aiding the design of complex periodic structures.

Contribution

It introduces an asymptotic point constraint approach combined with Fourier series to derive dispersion relations for Bloch waves in subwavelength lattices, extending previous methods.

Findings

Derived dispersion relations for Bloch waves in subwavelength lattices.

Identified strongly and weakly perturbed wave modes.

Provided a framework for designing complex lattice structures.

Abstract

We study waves governed by the planar Helmholtz equation, propagating in an infinite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength. Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved by a generalised Fourier series, whose singular asymptotics in the vicinities of scatterers yield the dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed. Characterising the latter is useful in interpreting and potentially designing the dispersion diagrams of such lattices. The method presented, that simplifies and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of the operating wavelength.