Laws of geometrical optics for isotropic left-handed metamaterials

Annotation: Peculiarities of the phenomenon of negative refraction of forward and backward plane electromagnetic waves are considered within the framework of geometric optics in the presence of left-handed isotropic metamaterials with two negative permeabilities. Various scalar forms of the Snell law, recorded without using the term «negative index of refraction», are compared. A formula is given for the generalized law of ray refraction using a wave type identifier. The formulas for the angle of refraction, depending on the angle of ray deviation, and for the position of an imaginary source in the case of negative refraction were derived. In accordance with the Hamilton and MacCullagh rule, possible ray trajectories for one interface are considered. Using the Fermat's least time principle, generalized bipolar differential relations are obtained for the non-planar boundary, which differ for real and imaginary images. These relationships are amenable to integration and can be taken to calculate refractive and reflective surfaces with focusing properties applied to microwave antennas using metamaterials.

Keywords: electromagnetic metamaterials, forward and backward plane waves, geometrical optics, ray of light, snell’s law, refractive index, real and imaginary images