Photonic crystals are dielectric structures with periodicity on the scale of the wavelength of light. A glance at any optics journal shows their rapidly expanding applications to signal processing, sensing, bandgap and negative-index materials, and the exciting possibility of fast integrated optical computers. Calculating their `band structure' (ie traveling Bloch
waves) is a Laplace eigenvalue problem with (quasi-)periodic boundary conditions, ie a problem on a torus. The wavenumber is usually low, but the problem is now parameter-dependent (the so-called Bloch phases).
We introduce a new approach: imposing the boundary conditions on the unit-cell walls using layer potentials, and a finite number of images, resulting in a second-kind integral equation with smooth data. Unlike standard methods, which construct the quasi-periodic Greens function, our method does not break down at the (spurious) resonances of the empty torus. It couples to existing boundary-integral methods (including high-order quadratures and Fast Multipole acceleration) in a natural way.
It enables piecewise-constant dielectric functions to be analyzed quickly with spectral accuracy, unlike commonly-used finite-difference and plane-wave methods. Given the breadth of the audience, I will review the necessary background ideas.
Joint work with Leslie Greengard (NYU).