FINITE-DIFFERENCE FREQUENCY-DOMAIN ALGORITHM FOR MODELING ELECTROMAGNETIC SCATTERING FROM GENERAL ANISOTROPIC OBJECTS

Abstract

The finite-difference frequency-domain (FDFD) method is a very simple and powerful approach for rigorous analysis of electromagnetic structures.It may be the simplest of all methods to implement and is excellent for field visualization and for developing new ways to model devices.This paper describes a simple method for incorporating anisotropic materials with arbitrary tensors for both permittivity and permeability into the FDFD method.The algorithm is bench marked by comparing transmission and reflection results for an anisotropic guided-mode resonant filter simulated in HFSS and FDFD.The anisotropic FDFD method is then applied to a lens and cloak designed by transformation optics. INTRODUCTIONThe finite-difference frequency-domain (FDFD) method is a simple and powerful numerical method for solving Maxwell's equations [1][2][3][4].It is rigorous, excellent for visualizing the fields, and able to model structures with complex geometries.It is accurate, stable, and the sources of error are well understood.Lastly, it lends itself very well to parallel processing on graphical processing units (GPUs) [5,6].The method uses central finite-difference approximations to transform Maxwell's equations into a large set of linear algebraic equations that can be written in matrix form as Ax = b.The field stored in the column vector x is calculated by solving x = A -1 b.A complete discussion of the basic FDFD algorithm for ordinary materials can be found in Ref. [7].This paper outlines how to implement an anisotropic FDFD (AFDFD) method that can handle fully anisotropic materials.An anisotropic medium is one where the permeability and/or permittivity depend on the direction of the electromagnetic fields.Anisotropy provides additional design freedom that can be used to produce a wide array of useful phenomena including surface waves [8], slow waves [9], invisibility and cloaking [10], double refraction, polarization control [11], and more.Further, very often devices composed of metamaterials can be modeled more efficiently using effective medium theory."Homogeneous" materials with the effective properties of the metamaterials can be meshed more coarsely compared to the volumetrically complex structures of the metamaterial.In this regard, a method capable of modeling devices with arbitrary dielectric and magnetic anisotropy is very useful.This paper discusses in detail how the three-dimensional (3D) FDFD method described in Ref.[7] can be modified to incorporate anisotropic materials with arbitrary nine-element tensors for both permittivity and permeability.It also discusses some of the subtleties encountered when dealing with tensor quantities like converting them between coordinate systems, rotating them to an arbitrary orientation, combining them with absorbing boundary conditions, and assigning them to points on a grid.Unfortunately, in general anisotropic media Maxwell's equations do not simply for two-dimensional or even one-dimensional analysis due to the complete coupling of the fields.For this reason, it is most straight forward to develop just a 3D FDFD code and use that to model lower dimensional problems.This can be done with virtually no decrease in efficiency if the appropriate precautions are taken.Using