Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions

Abstract

The analysis of fields in periodic dielectric structures arise in numerous applications of recent interest, ranging from photonic bandgap structures and plasmonically active nanostructures to metamaterials. To achieve an accurate representation of the fields in these structures using numerical methods, dense spatial discretization is required. This, in turn, affects the cost of analysis, particularly for integral-equation-based methods, for which traditional iterative methods require O(N2) operations, N being the number of spatial degrees of freedom. In this paper, we introduce a method for the rapid solution of volumetric electric field integral equations used in the analysis of doubly periodic dielectric structures. The crux of our method is the accelerated Cartesian expansion algorithm, which is used to evaluate the requisite potentials in O(N) cost. Results are provided that corroborate our claims of acceleration without compromising accuracy, as well as the application of our method to a number of compelling photonics applications.

A Discontinuous Galerkin Time Domain Framework for Periodic Structures Subject to Oblique Excitation

“…Computational analysis of fields in increasingly intricate periodic unit cells plays a significant role in their design and optimization. In the frequency domain, Integral Equation (IE) [5], [6], Finite Element (FE) [7], [8], and Discontinuous Galerkin (DG) [9] methods have been successfully applied to a variety of periodic electromagnetic systems. Timedomain (TD) methods for studying periodic systems include FE [10], [11], IE [12], and Finite Difference Time Domain (FDTD) [13], while DG methods remain relatively unexplored.…”

Preconditioning of Periodic Fast Multipole Method for Solving Volume Integral Equations

“…In addition, acceleration of VIEM with FMM has been investigated in [7], [8] and [9]. Attempts to develop fast solvers for periodic structures using the VIEM or VSIE (Volume/Surface integral equation method) are found in [10], where the MultiLevel Green's Function Interpolation Method (MLGFIM) is used for the acceleration, and in [11], where the Accelerated Cartesian Expansion (ACE) method is used. However, applications of the periodic version of the genuine FMM to VIEM have not been investigated so far.…”