Bloch Waves in an Arbitrary Two-Dimensional Lattice of Subwavelength Dirichlet Scatterers

TLDR: In this article, a coarse-grained Bloch-wave dispersion problem is solved by a generalized 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 scatterer; there are also empty-lattice waves that are only weakly perturbed.

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 generalized 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. Characterizing the latter is useful in interpreting and potentially designing the dispersion diagrams of such lattices. The method presented, which simplifies and expands on Krynkin and McIver [Waves Random Complex, 19 (2009), pp. 347--365], could be applied in the future to study more sophisticated designs entailing resonant subwavelen...