Robust high-order numerical scattering from multi-layer dielectric gratings using a new integral representation for quasi-periodic fields

Many numerical problems arising in modern photonic and electromagnetic applications involve the scattering of a plane wave from a piecewise-homogeneous medium, periodic in one direction. Boundary integral equations are an efficient approach to solving such boundary-value problems with high-order convergence. The standard way to periodize is then to replace the free-space Green’s function kernel with its quasi-periodic cousin. However, a major drawback is that the quasi-periodic Green’s function fails to exist for parameter families known as Wood’s anomalies, even though the underlying scattering problem remains well-posed. We propose a new integral representation that relies on the free-space Green’s function alone, adding auxiliary layer potentials on the walls of the unit cell, while enforcing quasi-periodicity with an expanded linear system. The result is a 2nd kind scheme which is immune to Wood’s anomalies, avoids lattice sums, is fastmultipole friendly, and allows arbitrary aspect ratios. Introduction The design and optimization of modern photonic, electromagnetic and acoustic devices relies heavily on numerical modeling. There has been an explosion of recent interest in devices which are periodic in one or more directions, such as thin film solar cells [4], photonic crystals (including slab guides), meta-materials, and specialized dielectric gratings [6]. Computational efficiency and robustness are paramount because predicting real-world device performance often requires thousands of solutions at different incident wavenumbers and angles. In recent work we presented a robust integral equation scheme for the scattering problem of an infinite periodic array of isolated obscacles [5]. However, in most realistic situations, obstacles or inclusions exist near or on a substrate, possibly with multiple homogeneous layers; see Fig. 1a. Here we extend our previous work to cover these more complicated cases. To showcase our approach, we focus on the problem of a periodic single dielectric interface (Fig. 1b). Given an incident wavevector k = (κ, k) := (ω cos θ, ω sin θ), hence an incoming plane wave u = eik·x in Ω ⊂ R2, n3 x z y