Electromagnetic plane waves in anisotropic media: an approach using bivectors

Abstract

The propagation of time-harmonic electromagnetic plane waves in non-absorbing, non-optically active, electrically and magnetically anisotropic media is considered. Both homogeneous and inhomogeneous plane waves are considered. All such solutions to Maxwell equations are obtained for crystals with arbitrary uniform magnetic anisotropy to the electrical anisotropy introduces qualitative changes. For example, for homogeneous linearly polarized waves in magnetically isotropic media the electric displacement vector D and the magnetic induction vector B are always orthogonal, whereas for magnetically anisotropic media these vectors are generally along the common conjugate radii of pairs of ellipses and are only orthogonal in special cases. Also in magnetically isotropic media a homogeneous wave with D and B both circularly polarized may propagate along an optic axis. However, for magnetically and electrically anisotropic media there is in general no homogeneous wave for which D and B are both circularly polarized. For inhomogeneous waves there are similar qualitative changes for magnetically anisotropic media. The description of an inhomogeneous plane wave involves two complex vectors, or bivectors: the amplitude and slowness bivectors. By a systematic use of the properties of bivectors and their associated directional ellipses, many of the results obtained are given a direct geometrical interpretation.