Anisotropy done right: a geometric algebra approach

S. A. Matos, C. R. Paiva and A. M. Barbosa

Instituto de Telecomunicações and Department of Electrical and Computer Engineering, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Corresponding author: sergio.matos@lx.it.pt

Received: 10 April 2009
Accepted: 17 June 2009

Abstract

For simple electric (magnetic) anisotropy a single function – one that maps a given direction of space to a specific value of permittivity (permeability) – is able to describe the electromagnetic behavior of the medium. Accordingly, the well-known classification of non-magnetic anisotropic crystals, as either uniaxial or biaxial, depends only on the characteristics of the permittivity function. However, when studying metamaterials, we frequently deal with general anisotropy characterized by two linear constitutive operators: the permittivity and permeability functions. Using the mathematical language of Clifford (geometric) algebra, we show – for general (reciprocal) anisotropy – that the direct interpretation of those two constitutive operators cannot provide an accurate description of the medium anymore. Namely, a new operator – one that depends on both those two constitutive operators – is needed, thereby leading to a new classification scheme. Therefore, although the uniaxial/biaxial characterization is still possible, the corresponding physical meaning is completely restated. Furthermore, a new concept – the pseudo-isotropic medium – emerges as a natural consequence of the new classification scheme.