A new framework based on geometric algebra for the analysis of materials and metamaterials with electric and magnetic anisotropy

Few studies address media with both electric and magnetic anisotropy. However, the advent of metamaterials has prompted a fresh look into the old problem of anisotropic media in electromagnetics. In this communication we have presented a new general approach to solve this problem by using the grammar of geometric algebra. In fact, with this novel approach, we have shown how geometric algebra can provide a mathematical framework for general anisotropy that is far better than plain tensors and dyadics: through the direct manipulation of coordinate-free objects such as vectors, bivectors and trivectors, geometric algebra reveals itself as the most natural setting to study anisotropy by providing a deeper physical and geometrical insight while avoiding cumbersome calculations. We have shown that general anisotropy can be understood through a new function zeta that plays a central role: when zeta is uniaxial (biaxial) the medium is uniaxial (biaxial). Furthermore, a generalization of uniaxial and biaxial media has revealed new features when compared with media with only electric or magnetic anisotropy (but not with both). We have also defined a new state of anisotropy that we have called pseudo-isotropy. This state, although stemming from both electrical and magnetic anisotropy, corresponds to an isotropic zeta. In summary: we have shown that general anisotropy does not depend on the mathematical form that each constitutive function (either epsiv or mu) separately assumes but rather on a new single constitutive function: zeta = epsiv^-1(mu).