Static and propagating exponential solutions of Maxwell’s equations for crystals

Abstract

Solutions of Maxwell’s equations are considered for anisotropic media for which the electric permittivity κ and magnetic permeability µ are assumed to be arbitrary real positive definite symmetric second order tensors. The propagation of time-harmonic electromagnetic inhomogeneous plane waves or ‘propagating exponential solutions’ in such media has been presented previously. These solutions were systematically obtained by prescribing an ellipse - the directional ellipse associated with a bivector (complex vector) C - and finding the corresponding slowness bivectors. Here, it is shown that for some prescribed directional ellipses, not only propagating exponential solutions (PES), but also static exponential solutions (SES), may be obtained. There are a variety of possibilities. For example, for one choice of directional ellipse it is found that two SES and one PES are possible, whereas, for some other choices, only one SES or only one PES is possible. By a systematic use of bivectors and their associated ellipses, all the possible SES and PES are classified. To complete the classification it is necessary to examine special elliptical sections of the ellipsoids associated with the tensors κ , µ , κ -1 , µ -1 . In particular, sections by planes orthogonal to special directions called ‘generalized optic axes’ and ‘generalized ray axes’ play a major role. These axes reduce to the standard optic axes and ray axes in the special case of magnetically isotropic media.