Light propagation in anisotropic and metamaterial media by a Finslerian vector eikonal method

A systematic study of light propagation in anisotropic and metamaterial media is presented by using a Finslerian vector eikonal method, that is, by considering the approximation of very short wavelengths of light in Maxwell’s equations and using particular metric tensors of the general Finsler metric tensor induced by an anisotropic medium. Green’s vector functions, and therefore general solutions, are obtained as a product of a complex scalar function with a local polarization unitary vector. The phase L (r) (wave surface or eikonal function) of such a complex scalar function is calculated using a Fermat integral derived from the Maxwell equations. This integral can be considered as the integral form of the eikonal equation and provides a Finslerian metric tensor which allows, together with energy conservation in anisotropic media, the derivation of a general expression depending on the eikonal function for obtaining the scalar amplitude A(r) of such a complex function. Likewise, the local polarization unitary vector u(r) is calculated by solving an approximated homogeneous linear equations system also depending on the eikonal function. Exact solutions are obtained for uniaxial materials and hyperbolic metamaterials, and approximate solutions are proposed for biaxial materials. Finally, an approximated vector Fresnel–Kirchhoff diffraction formula is presented by using vector Green’s functions in order to study optical propagation. For the sake of completeness, interference patterns produced by anisotropic materials are presented using the results obtained. This systematic study of anisotropic and metamaterial media by a Finslerian vector eikonal method is intended in particular for university physics teachers and as an advanced topic for undergraduate and first year postgraduate students.