The negative index of refraction demystified

We study electromagnetic wave propagation in media in which the effective relative permittivity and the effective relative permeability are allowed to take any value in the upper half of the complex plane. A general condition is derived for the phase velocity to be oppositely directed to the power flow. That extends the recently studied case of propagation in media for which the relative permittivity and relative permeability are both simultaneously negative, to include dissipation as well. An illustrative case study demonstrates that in general the spectrum divides into five distinct regions.