Imaging and Computing Seminar

Alex Barnett

Robust and efficient computation of two-dimensional photonic crystal band structure using second-kind integral equations

Photonic crystals are dielectric structures with periodicity on the scale of the wavelength of light. They have a rapidly growing range of applications to signal processing, sensing, negative-index materials, and the exciting possibility of integrated optical computing. Calculating their `band structure' (propagating Bloch waves) is an elliptic PDE eigenvalue problem with (quasi-)periodic boundary conditions on the unit cell, i.e. eigenmodes on a torus. Since the material is piecewise homogeneous, boundary integral equations (BIE) are natural for high-accuracy solution.

In such geometries BIEs are usually periodized by replacement of the free space Greens function kernel by its quasi-periodic cousin. We show why this approach fails near the (spurious) resonances of the empty torus. We introduce a new approach which cures this problem: imposing the boundary conditions on the unit-cell walls using layer potentials, and a finite number of neighboring images, resulting in a second-kind integral equation with smooth data. This couples to existing BIE tools (including high-order quadratures and Fast Multipole acceleration) in a natural way, allowing accuracies near machine precision. We also discuss inclusions which intersect the unit cell walls, and how we use a small number of evaluations to interpolate over the Brillouin zone to spectral accuracy. Joint work with Leslie Greengard (NYU).