Resolving the wave vector in negative refractive index media

We address the general issue of resolving the wave vector in complex electromagnetic media including negative refractive media. This requires us to make a physical choice of the sign of a square root imposed merely by conditions of causality. By considering the analytic behavior of the wave vector in the complex plane, it is shown that there are a total of eight physically distinct cases in the four quadrants of two Riemann sheets.