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

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.

On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions

“…ε . Taking into account ( 5), ( 7), ( 8), and when j = k, using the integration by parts one can represent the boundary integrals in (12) via the integrals over F (j) ε , and hence the representation (12) takes the form…”

“…Analysis of waves in a plane with semi-infinite arrays of isotropic scatterers is discussed in [11]. Waves governed by the Helmholtz equation in a doubly periodic array with an elementary cell containing several scatterers were analysed in [12], based on the approach of [10], [11] and asymptotics representing singular perturbation leading-order approximations, similar to simplified cases of [2], [6] and [7]. Dispersion of waves analysed in [12] shows dynamic anisotropy linked to the scatterers within the elementary cell.…”

Plasmonic resonances of slender nanometallic rings

“…Nonetheless, it can be shown that (3.4) still holds at intermediate radial distances from the centerline, i.e., b r a, if only the cross-sectional radius bf (φ) is replaced by the so-called 'conformal radius', say bf * (φ), of the cross-sectional geometry at the azimuthal angle φ. Working in the corresponding crosssectional plane, the conformal radius bf * (φ) can be extracted from a conformal mapping from the exterior of a circle of that radius to the domain exterior to the true cross section (see, e.g., [33,Chapter 5] and [59]). In particular, for elliptical cross sections with semi-diameters bσ 1 (φ) and bσ 2 (φ), one finds bf * (φ) = (bσ 1 + bσ 2 (φ))/2.…”

Hybrid topological guiding mechanisms for photonic crystal fibers

“…Equations (2) and (4) are, for metallic scatterers and with r constant, the Helmholtz equation and we can draw upon efficient, rapid, semi-analytical methods created for small inclusions as described in [58]. These are derived in [25] (based upon an extension of [69]) utilizing matched asymptotic expansions, each circular scatterer -of radius ηa -is approximated by a monopole and dipole source term whose coefficients, a mono and b di are a priori unknown constants. These unknown constants can trivially absorb factors of a or r , the analysis from [25] remains largely unchanged, that is for a single scatterer…”