Negative Refractive Index in Left-Handed Materials

The real part of the refractive index $n\left(\omega \right)$ of a nearly transparent and passive medium is usually taken to have only positive values. Through an analysis of a current source radiating into a 1D “left-handed” material (LHM)—where the permittivity and permeability are simultaneously less than zero—we determine the analytic structure of $n\left(\omega \right)$, demonstrating frequency regions where the sign of $\mathrm{Re}\left[n\right(\omega \left)\right]$ must, in fact, be negative. The regime of negative index, made relevant by a recent demonstration of an effective LHM, leads to unusual electromagnetic wave propagation and merits further exploration.