A Broadband ML-FMA for 3-D Periodic Green’s Function in 2-D Lattice Using Ewald Summation

Abstract

A periodic fast multipole algorithm (P-FMA) is devised for evaluating 3-D periodic Green’s function (PGF) for a 2-D lattice which can be used to solve scattering by a structure with 2-D periodicity. The introduction of periodicity in the Green’s function formulation produces image sources at each lattice site. Like multilevel FMA (ML-FMA), P-FMA takes advantage of the distance between image sources and observation points to factorize the field using multipoles. By substituting known factorizations of the free-space Green’s function into the expression for PGF, one can isolate the summation over the lattice into the translation phase of the FMA. For both plane wave and multipole factorizations, a common term known as lattice constant appears. The lattice constant is an infinite sum over the lattice which does not converge absolutely when expressed as a spatial sum. Using the Ewald summation technique, the lattice constants can be evaluated with exponential convergence and high accuracy. The resulting P-FMA is between $\mathcal {O}(N)$ and $\mathcal {O}(N\log N)$ in memory use and computational complexity, depending on the object size relative to the wavelength.