Negative refraction of plane electromagnetic waves in non-uniform double-negative media

We propose a new model of a plane-layered medium with a fixed positive refractive index, but variable non-vanishing dielectric permittivity and magnetic permeability. An analytical solution of the problem of plane electromagnetic wave propagation through the medium has been found. It is shown that the negative-refraction effect occurs in this type of medium; in particular, an explicit expression for the field has been first obtained in the region of its formation. This result suggests that the necessary condition for the occurrence of this effect in quite complex media is not the negativity of their refractive indices, but only the requirement of simultaneous negativity of the real parts of dielectric permittivity and magnetic permeability. We also demonstrate that the medium is passive (absorbing) if a plane E-polarized wave propagates in it, and active (amplifying) in the case of H-polarized wave propagation.