materials are opaque for electromagnetic radiation and examples include plasmonic materials such as metals (gold, silver, ...). The foregoing considerations are simplistic for actual materials, which must have complex-valued and . The real parts of both and do not have to be negative for a passive material to display negative refraction. Indeed, a negative refractive index for circularly polarized waves can also arise from chirality. Metamaterials with negative n have numerous interesting properties: Snell's law (n1sinθ1 = n2sinθ2) still describes refraction, but as n2 is negative, incident and refracted rays are on the same side of the surface normal at an interface of positive and negative index materials. Cherenkov radiation points the other way. The time-averaged Poynting vector is antiparallel to phase velocity. However, for waves (energy) to propagate, a –µ must be paired with a –ε in order to satisfy the wave number dependence on the material parameters . Negative index of refraction derives mathematically from the vector triplet E, H and k. For plane waves propagating in electromagnetic metamaterials, the electric field, magnetic field and wave vector follow a left-hand rule, the reverse of the behavior of conventional optical materials. To date, only metamaterials exhibit a negative index of refraction. Single negative Single negative (SNG) metamaterials have either negative relative permittivity (εr) or negative relative permeability (µr), but not both. They act as metamaterials when combined with a different, complementary SNG, jointly acting as a DNG. Epsilon negative media (ENG) display a negative εr while µr is positive. Many plasmas exhibit this characteristic. For example, noble metals such as gold or silver are ENG in the infrared and visible spectrums. Mu-negative media (MNG) display a positive εr and negative µr. Gyrotropic or gyromagnetic materials exhibit this characteristic. A gyrotropic material is one that has been altered by the presence of a quasistatic magnetic field, enabling a magneto-optic effect. A magneto-optic effect is a phenomenon in which an electromagnetic wave propagates through such a medium. In such a material, left- and right-rotating elliptical polarizations can propagate at different speeds. When light is transmitted through a layer of magneto-optic material, the result is called the Faraday effect: the polarization plane can be rotated, forming a Faraday rotator. The results of such a reflection are known as the magneto-optic Kerr effect (not to be confused with the nonlinear Kerr effect). Two gyrotropic materials with reversed rotation directions of the two principal