Unified matrix representation of Maxwell's and wave equations using generalized differential matrix operators (GDMOs)

Abstract

Summary form only given. Many microwave devices and integrated circuit components, waveguides and absorbers employ anisotropic materials, and, to efficiently formulate boundary value problems associated with these structures, a number of different matrix representations to Maxwell's equations have been proposed in the literature. Most have only dealt with Cartesian coordinates in the spatial and spectral domains, and the matrix operations have been limited solely to Maxwell's curl equations. In this paper, we introduce the concept of generalized differential matrix operators, or GDMOs, in arbitrary orthogonal coordinate systems that are useful for replacing vector differential operations with matrix algebraic manipulation. The use of GDMOs with matrices, whose dimensions normally do not exceed 3/spl times/3, enable one to replace the complicated vector or tensor operations. The GDMOs not only provide a simple and elegant representation of Maxwell's and vector wave equations, but simplify their manipulation in boundary value problems as well. They are especially useful for applications involving anisotropic materials in the problems of guided wave propagation, scattering, and radiation. The use of matrix techniques also has advantage in that it enables us to deal with the three scalar components of the vector equations simultaneously while maintaining the simplicity of mathematical derivation. To illustrate the usefulness of the GDMOs, we derive the Green's functions of a planar arbitrary microwave integrated circuit printed on a substrate, which is both electrically and magnetically anisotropic.