Abstract:We present an accurate and efficient numerical method, based on integral Nystrom discretizations, for the solution of three dimensional wave propagation problems in piece-wise homogeneous media that have two-dimensional (in-plane) periodicity (e.g. photonic crystal slabs). Our approach uses (1) A fast, high-order algorithm for evaluation of singular integral operators on surfaces in three-dimensional space and (2) A new, representation of the three-dimensional quasi-periodic Green's functions, which, based on use of infinitely-smooth windowing functions, suitable linear combinations (in the spirit of finite differences) of reflected Green's functions, and equivalent-source representations, converges super-algebraically fast throughout the frequency spectrum---even for high-contrast problems and at and around the resonant frequencies known as Wood anomalies. Our fast algorithm for computing periodic Green's functions compare favorably with the classical Ewald's summation method and with other existing methods for fast summations of periodic Green's functions. In addition, we show that the boundary integral equations that incorporate our novel periodic Green's functions are uniquely solvable for all frequencies, including Wood anomalies, in the case of scattering from bi-periodic rough surfaces.
Joint work with O. Bruno (Caltech), S. Shipman (LSU), and S. Venakides (Duke)