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A New Framework Based on Geometric Algebra for the Analysis of Materials and Metamaterials with Electric and Magnetic Anisotropy

Matos, S.A. ; Canto, João R. ; Paiva, C. R. ; Barbosa, A.

A New Framework Based on Geometric Algebra for the Analysis of Materials and Metamaterials with Electric and Magnetic Anisotropy, Proc IEEE AP-S/URSI International Symp., San Diego, United States, Vol. -, pp. - - -, July, 2008.

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Abstract
Anisotropy is an old topic in electromagnetism as a wide range of materials with electric or magnetic anisotropy is available in nature. However, only recently did materials with both electric and magnetic anisotropy become available through the development of
metamaterials technology. These media have singular properties that can be used for new exciting applications such as invisibility cloaking devices. Nevertheless, in this new area, a general theory that can fully describe the physical nature of this anisotropy is still
lacking. In fact, few studies address this type of material in a general way. Plain tensors and dyadics give a purely mathematical solution for this problem,thereby obnubilating its physical interpretation. By using the new mathematical language of geometric algebra, a new framework that enables a fresh physical insight into this problem will be presented herein. This new formalism can shed new light into the classical problem of plane wave propagation in a biaxial crystal. In this communication we study general anisotropy, i.e., media with both electric and magnetic anisotropy. The only restriction, apart from reciprocity, is that we consider that the eingenvectors of the permeability and permittivity functions should be parallel. Through this new perspective on anisotropy we revisit the old concepts of biaxial and uniaxial media by establishing broader definitions for these concepts. Namely, by defining a new constitutive function ζ , that we call the anisotropic function, we are able to develop a global classification in terms of just three new types of media: (i)pseudo-isotropic; (ii) uniaxial; (iii) biaxial. We stress, however, that this classification (uniaxial/biaxial) should not be confused with the usual meaning given to these terms (i.e., when only electric or magnetic anisotropy is involved). Pseudo-isotropic media, on
the other hand, corresponds to a new classification where only one eigenwave exists as in the isotropic case (singly refracting medium) – although the eigenwave depends on direction. Our
analysis presents a new global scheme for the general problem at hand. Moreover, in the design of new electromagnetic devices that use anisotropic metamaterials such a general framework is indeed needed. This is particularly relevant when designing metamaterials used in anisotropic cloaks for invisibility devices.