3D Diagonalization and Supplementation of Maxwell’s Equations in Fully Bi-anisotropic and Inhomogeneous Media - Part II: Relative Proof of Consistency

Abstract

Consider fully bi-anisotropic and inhomogeneous media supporting the electromagnetic wave propagation. Assume an (x, y, z)–Cartesian coordinate system and a harmonic time-dependence according to exp(–jwt). In the accompanying paper (Part I) it was shown that the Maxwell’s equations can be diagonalized with respect to the z–axis, resulting in the Dc–form. Furthermore, the existence of the associated supplementary matrix equation, the Sc–form, was demonstrated rigorously. In the present paper ‘‘structural,’’ ‘‘differential,’’ and ‘‘material’’ matrices have been introduced to explicate the (Da, Sa)–, (Db, Sb)–, and (Dc, Sc)–forms, relative to the x–, y–, and z–axes, respectively. As the pinnacle of the theory, it has thoroughly been established that the derived combined (Dc, Sc)–forms are sharply equivalent with the joint Maxwell’s and constitutive equations, and thus internally consistent. The presented proof is relative in the sense that its validity hinges on the consistency of Maxwell’s equations and the material realizability conditions.