Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers

Abstract

We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green's function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green's function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 10⁵ unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation.

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

“…3.4] and references within, and [17, thm. 6], by using equation (13) from that paper. We do not know of a reference stating this for the Laplace case, but note that the proof is identical to that in [4, thm.…”

“…We do not know of a reference stating this for the Laplace case, but note that the proof is identical to that in [4, thm. 6], by using equation (13) from that paper. The restriction R p < is crucial for high accuracy, since when R p > , although convergence occurs, k .M / k 2 blows up, causing accuracy loss due to catastrophic cancellation [4, thm.…”

“…Thus by the continuity of the single-layer potential [45, thm. 6.18], the interior limit of v 1 It would also be possible to write a purely continuous version by replacing the proxy circle by a single-layer potential; however, sometimes an intrinsically discrete basis f j g is useful [5,13].…”

“…However, with the exception of layered media that vary in only one dimension, finding this tensor requires the numerical solution of "cell problems" [62,66], namely BVPs in which the solution is periodic up to some additive constant expressing the macroscopic driving. In this work we focus on the first two (nonoscillatory) PDEs above, noting that the methods that we present also apply with minor changes to the oscillatory Helmholtz and Maxwell cases, at least up to moderate frequencies [5,13,56]. In this work we focus on the first two (nonoscillatory) PDEs above, noting that the methods that we present also apply with minor changes to the oscillatory Helmholtz and Maxwell cases, at least up to moderate frequencies [5,13,56].…”

“…Application areas span all of the major elliptic PDEs, including the Laplace equation (thermal/electrical conductivity, electrostatics and magnetostatics of composites [12,27,35,38]); the Stokes equations (porous flow in periodic solids [18,26,50,75], sedimentation [1], mobility [69], transport by cilia carpets [16], vesicle dynamics in microfluidic flows [58]); elastostatics (microstructured periodic or random composites [29,36,61,64]); and the Helmholtz and Maxwell equations (phononic and photonic crystals, bandgap materials [42,65]). In this work we focus on the first two (nonoscillatory) PDEs above, noting that the methods that we present also apply with minor changes to the oscillatory Helmholtz and Maxwell cases, at least up to moderate frequencies [5,13,56].…”