Speaker
Fruzsina Agocs, University of Colorado Boulder
Abstract
Periodic surfaces give rise to unique electromagnetic and acoustic phenomena that are exploited for the purposes of guiding or localizing waves. Devices such as metallic and semiconductor diffraction grating spectrometers rely on increased absorption at certain wavelengths; metallic sheets with periodic, sub-wavelength diameter holes admit significantly enhanced transmission at given wavelengths which can be used for manufacturing sub-wavelength features; photonic crystals are increasingly replacing conventional optics for creating parts for quantum computing.
In this talk, I will introduce an efficient boundary integral equation technique for solving acoustic scattering problems near periodic geometries. The problems in focus are described by the two-dimensional Helmholtz equation, defined in the half-plane bounded by an infinite, periodic curve with Neumann boundary conditions and a point source. The Floquet–Bloch transform turns the problem into evaluating a contour integral where the integrand is the solution of “quasiperiodic” boundary value problems. To approximate the integral, one must solve a collection of these problems, which possess a higher degree of symmetry and may be periodized, i.e. solved on a single unit cell of the boundary. Our method is amenable to a large amount of precomputation that can be reused for all of the necessary solves, while being accelerated by the use of low rank linear algebra. I will illustrate the resulting speedup relative to current techniques with numerical experiments and motivate the method’s use for solving three-dimensional problems.