Tensorial mathematic formulation for the generalization of effective dielectric concept in electromagnetism: superlattices of any symmetry with one and two directions of periodicity
Résumé
I introduce a new theoretical formulation based on a composition method and a statistical discretization approach by matrical block so as to shape analytical expressions of effective dielectric tensors of superlattices (SLs) in electromagnetism by considering anisotropic properties and boundary conditions. Such a general framework describing the effective dielectric tensors of idealized free-standing SLs possessing respectively one and two directions of periodicity (1D-SLs and 2D-SLs), composed of thin alternating layers and bars of any symmetry is derived, as a function of the dielectric tensors of each of the N and (N £M) constituents. For the case of 1D-SLs of any symmetry, it is possible to unify and to infer from such global formulation the particular relationships on effective dielectric tensors known as the 'Vegard rules' in solid state physics. Moreover, it is worth noting that in the simpli¯ed case of a 2D-SL made of only two different isotropic materials showing of the same periodicity in both directions, our general matrix formulation, due to the alternative composition laws, leads to the well-established results called, respectively, 'Wiener's and Lichtenecker's bounds' regarding the dielectric constants. This new formalism refashions the concept of bounds of effective dielectric tensors and the notion of form birefringence applied to 1D- and 2D-SLs, with respectively relevant N multilayers and (N £ M) rectangular anisotropic columns for arbitrary symmetries. The results are applied to SLs with layers of all classes of all symmetries (triclinic, monoclinic, orthorhombic, hexagonal, tetragonal, and cubic).