Semi-Vectorial Response of Three-Dimensional Reflecting Structures via Iterative Solution of the Helmholtz Equation

Waveguiding structures that generate reflections are currently modeled by a number of different methods, most of which are of very limited generality. Of these methods, only the finite-difference time-domain (FDTD) method1 has been shown to be useful over a wide class of structures. Unfortunately, this technique is very costly in terms of computer resources, thus limiting its utility. In order to address this need, we recently reported a new modeling technique2 that involves the iterative solution of the scalar Helmholtz Equation using the ADI (alternating direction implicit) method. That technique was limited by two major restrictions: (1) Proper convergence of the iteration required an appropriate absorbing layer around the problem region periphery, and (2) The method as stated was only valid for 2D problems, although it was pointed out that an extension to three dimensions seemed feasible. In this paper we remove both restrictions and model three-dimensional structures by iterative solution of the scalar Helmholtz Equation with normalized coordinates.