Fast and high accuracy FDTD using numerical Green's functions based on a nonstandard finite difference model for photonics design

Abstract

The use of Green’s functions to solve inhomogeneous differential equations, such as the Maxwell’s equations with source currents is well known. Unfortunately, it is usually difficult – if not impossible – to find a Green’s function which satisfies the boundary conditions. The finite difference time domain (FDTD) algorithm is derived from a finite difference equation (FDE) of Maxwell’s equations. FDTD, which automatically takes boundary conditions into account, is often used to solve the FDE, but its computational cost increases much faster than the accuracy as the grid spacing (<i>h</i>) decreases; moreover small <i>h</i> must be used to capture fine features of structures such as subwavelength gratings. A discrete Green’s function (DGF), computed using FDTD, can be used to overcome some of the shortcomings of FDTD alone. A DGF computed using FDTD automatically includes the boundary conditions of the problem. In this paper we use a DGF based on what is called a nonstandard finite difference model of Maxwell’s equations to compute scattering off subwavelength structures. We verify the accuracy of our method by comparing our calculations with analytic solutions given by Mie theory.