Refraction laws for anisotropic media and their application to left-handed metamaterials

The refraction of phase and power at the interface between free space and anisotropic media is studied. Under a TE incidence (electric field along) and k(y) = 0 incidence (where k(y) is the P component of the wave vector), a closed-form generalization of Snell's laws is proposed. The two relations (one for the phase and one for the power) are expressed in terms of the incident angle, the constitutive parameters of the anisotropic medium (which can take negative values), and the tilting angle of the dispersion relation. In additions both positive and negative dispersions are discussed, making the formulas directly usable for the design of left-handed metamaterials. The validation is done by two methods. First, the electromagnetic fields in layered anisotropic media are computed analytically. The angle of refraction of the wave vector is thus directly obtained, while that of the power is obtained by computing the time-average Poynting power. Second, numerical simulations are performed using HFSS to record the deflection of the power by a prism characterized by positive or negative constitutive parameters. For the specific prism experiment, in which the incident angle is equal to the tilting angle of the dispersion relation, we show that the refraction laws reduce to a simple Snell's law form.