Lattice Sums for the Helmholtz Equation

Abstract

A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d Λ . Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis. LATTICE SUMS FOR THE HELMHOLTZ EQUATION 631to mathematically challenging problems. Attempts to evaluate the sum in (1.1) in closed form have so far been unsuccessful, though some remarkable identities have been derived in the process [23,113].The lattice sum in (1.1) arises from a sum of singular solutions of Laplace's equation. Here we will be concerned with related sums in which the underlying physics is governed by the Helmholtz equationwhich arises naturally in many contexts, particularly when investigating linear wave phenomena at a given frequency. For example, in acoustics u represents fluctuations in pressure, whereas in elasticity it would be some component of the displacement vector, and in electromagnetism a component of the electric or magnetic field. The quantity k (assumed real and positive) is the wavenumber, related to the frequency ω via some appropriate dispersion relation. In diffraction theory we are often faced with trying to solve (1.2) in a region exterior to some scatterer(s) (and maybe inside as well) subject to boundary conditions on the surface of the scatterer(s). The case where the scatterer is periodic represents an important class of such problems; examples include the study of diffraction gratings, scattering by periodic surfaces, and wave propagation through composite materials. Equation (1.2) is also equivalent to the time-independent Schrödinger equation in the presence of a constant potential, in which case k is related to the total energy of the particle under consideration, and as a result the Helmholtz equation finds application in the study of electron scattering in solids. A common simplification used is the so-called muffin-tin approximation (due to Slater [99]), in which the potential is supposed to be spherically symmetric within spheres surrounding each atom and constant in the region inbetween these spheres. This then leads naturally to the need to solve (1.2) on a periodic domain.The study of wave propagation in periodic structures has a long history and leads to a whole range of interesting mathematical problems. The classic text by Brillouin [11] arguably still serves as the best introduction to the subject, though many significant advances have been made since it was written. For example, the mathematical theory of diffraction gratings (based on functional analysis) is developed in books such as [120] and there is an ever-growing literature describing the rapidly developing field of photonic and phononic crystals; an excellent text is [39]. One of the consequences of periodicity is that a problem on...

On acoustic and electric Faraday cages

“…(The estimate (3.5) can be obtained from equation (2.47) in [19] or equation (3.37) in [20].) Using (3.6) for H 0 (ka) and J 0 (ka) ∼ 1 as ka → 0, we obtain…”

Optical properties of two-dimensional magnetoelectric point scattering lattices

“…Decker et al 12 attempted to account for these interactions using numerical summation of retarded electric dipole-dipole interactions on a one-dimensional (1D) chain. However, in this approach, qualitative discrepancies remain compared to full numerical simulations, likely because numerical summation of dipole-dipole interactions in real space is poorly convergent 37,38 because actual lattices in experiments are not 1D and because interactions also involve magnetic dipole-dipole coupling and magnetoelectric coupling. The minimum requirements for a simple dipole lattice model for metamaterials must necessarily include the electrodynamic coupling between electric dipoles, magnetic dipoles, as well as the cross coupling between magnetic and electric dipoles.…”

A new integral representation for quasi-periodic scattering problems in two dimensions

“…In the related case of periodic surface scattering, Zhang and Chandler-Wilde [47] modified the Green's function to that of a half-space, which cures this divergence (this was implemented in [2]); however, this idea fails to help in our case of disconnected obstacles. A final problem is that the quasi-periodic Green's function is often computed using lattice sums [25,26], which is natural when using fast multipole acceleration in large-scale scattering problems [36]. However, since this representation converges in discs (or spheres in 3D), it becomes cumbersome for high aspect ratio geometries.…”