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Anisotropic and bianisotropic media in electromagnetics with geometric algebra

Matos, S.A. ; Paiva, C. R.

Anisotropic and bianisotropic media in electromagnetics with geometric algebra, Proc International Conf. on Clifford Algebras and their Applications - ICCA9, Weimar, Germany, Vol. -, pp. - - -, July, 2011.

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Abstract
In this work we present a new approach to the theory of electromagnetic wave propagation in unbounded complex media and metamaterials using Clifford geometric algebras in both Euclidean and Minkowskian spaces. It is a common belief that geometric algebra is particularly useful for studying relativistic electromagnetics or relativistic quantum mechanics. Nevertheless, the new algebraic techniques brought up to linear and multilinear functions, also make geometric algebra a valuable tool to understand (and work on) complex media and metamaterials in Euclidean 3D space. Indeed, we show that the usual 3D dyadic approach can be reinterpreted on a new coordinate-free setting, leading to new insights. Actually, the geometric insight that this new approach provides has led to a new classification scheme for media with both electric and magnetic anisotropy. Herein a generalization of this classification scheme will be presented for bianisotropic media. However, we cannot overlook the fact that spacetime algebra – the geometric algebra of Minkowskian space – provides an ideal setting to display the real power of the graded structure of multivectors in geometric algebra. Therefore, a geometric analysis of moving media – namely, in the special case of moving bi-isotropic media – is also developed. The manifestly covariant form of the spacetime constitutive relation allows, in a single stroke, to address the same medium either seen from stationary or from moving inertial frames. Furthermore, spacetime algebra is also useful to investigate the fundamental properties of a given medium – even in a 3D setting. In fact, the concept of perfect electromagnetic conductor (PEMC) constitutes a paradigmatic example in which spacetime algebra clarifies the physical meaning – demystifying incorrect interpretations and overcoming the unnecessary mathematical complexity inherent in the dyadic algebra associated with differential forms.