## Imaging and Computing Seminar

### Alex Barnett

**Title:**

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

**Abstract:**

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).