Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers

Abstract

We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.

Localising elastic edge waves via the topological rainbow effect

“…A fascinating chiral beaming phenomenon has been observed for classical and electronic valley-Hall systems [12,49]. By exciting a valley-Hall crystal near the periphery of the topological band gap (Fig.…”

“…6(b). The excitation of the π/3 separated ± pseudospin states stems from the gradient direction of the triangular isofrequency contours [12,49].…”

Hybrid topological guiding mechanisms for photonic crystal fibers

“…We proceed by numerically solving the spectral problem ( 2)-( 4) using the finite element package Comsol Multiphysics [28] to compute the eigenfrequencies Ω k and associated Bloch eigensolutions H k l when k spans the Brillouin zone (BZ) in the reciprocal space. These finite element numerics are complemented by a semi-analytic method based around monopolar and dipolar scattering by small metallic cylinders, the details of which are in [29] and covered briefly in Appendix A. These eigenfrequencies form a discrete spectrum ω j , j = 1, 2 • • • , and noting that Ω ≥ 0 in (2) it is easily seen [4] that these are bounded below with…”

“…We opt to use an alternative semi-analytical approach that is less computationally expensive than directly using finite element methods and which is relevant to open systems. This method is explained in detail in [29] and is briefly summarised in Appendix A. The usefulness of this semi-analytical scheme will become apparent in Sec.…”

“…It is well-known, [4], that this metallic outer boundary allows us to estimate the real part of the eigenfrequencies of an open system, provided the outer radial boundary is sufficiently large; we systematically checked that numerical values of eigenfrequencies of modes of interest are robust against the diameter of PCFs. To further confirm that these modes persist in the case of open PCFs we also utilise a semi-analytic method, designed for the rapid simulation of small metallic cylinders that asymptotically represents them as monopole and dipole scatterers as detailed in [29] and these simulations have no outer metallic boundary.…”