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Isotropic-Uniaxial Interfaces for Anisotropic Media with Biaxial Electric and Magnetic Functions: A New Approach with Geometric Algebra

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

Isotropic-Uniaxial Interfaces for Anisotropic Media with Biaxial Electric and Magnetic Functions: A New Approach with Geometric Algebra, Proc IEEE AP-S/URSI International Symp., San Diego, United States, Vol. 1-9, pp. 1380 - 1383, July, 2008.

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Abstract
The study of the electromagnetic properties of anisotropic media has always been a keystone of electromagnetism. This concept has been widely used in optics and photonics and, more recently, due to the advent of metamaterials, several new exciting possibilities have risen such as the design of cloaking devices. A coordinate-free approach reveals itself essential when addressing anisotropy. The classical adopted tool for this is the dyadic (or tensor) calculus. More recently some attempts at using differential forms have been also presented. Both have revealed themselves to be quite cumbersome as problems escalate and tend to overshadow a direct geometric interpretation. Here we intend to show that such insight can be obtained when addressing general anisotropy through the geometric framework given by Clifford’s algebras. In fact, recently this approach has been adopted to address biaxial nonmagnetic crystals, thereby unveiling new results that were hidden under the dryness of the dyadic formalism. Further work applying this geometric framework to reciprocal anisotropic media, where both electric and magnetic anisotropy were considered, led to new results – as explained in another communication to this Conference. In that communication we have shown that general anisotropic media, with both electric and magnetic functions, must be classified according to the eigenvalues of an anisotropic function defined taking into account both the permeability and the permitivity. The only restriction to the previous analysis is that the axes of the permitivity and permeability functions should be aligned. This clearly diverges from the classical criteria of classifying a nonmagnetic crystal according to the eigenvalues of the permitivity (likewise, a magnetic crystal can be classified according to the permeabillity) – a criterion that loses its meaning here. Furthermore, none of the eigenwaves of the (general) uniaxial medium is ordinary and neither is the eigenwave of the pseudo-isotropic medium. These newly found results are of major importance when addressing the problem of interfaces between isotropic and general anisotropic media. This paper will then focus on the study of isotropic – uniaxial and isotropic – pseudoisotropic interfaces. Closed form expressions for the transmitted wavenumbers and for their normal components as well are then presented as functions of the incidence angle. Furthermore, the conditions for total reflection are also found. A complete analysis of these interfaces, including the reflection and transmission coefficients, will be presented elsewhere.