Refraction at the Interface of a Lossy Metamaterial

The refraction phenomenon at the interface of an ordinary material and a lossy metamaterial has been investigated. For oblique incidence on the lossy metamaterial, the planes of constant amplitude of the refracted wave are parallel to the interface and the plane of constant phases make a real angle with the interface (real refraction angle). The real refraction angle and hence, the real refraction index corresponds to the real refraction angle which satisfies the real version of Snell's law are negative in two different regimes. In one regime, the metamaterial is double-negative, while in the other one it is single-negative. Moreover, we show that the plane wave solution for the refracted wave is causal in both double-negative and single-negative regimes