Lossy transmision with arbitrary gradient permittivity and permeability and constant impedance throughout the structure

We present a remarkably simple exact analytical solution for the electromagnetic field distribution across an infinite metamaterial composite with an arbitrary graded variation of complex effective permittivity and permeability for the case of constant impedance across the structure. Arbitrary temporal dispersion and losses are allowed and the model is generally applicable to different inhomogeneous and anisotropic media simultaneously containing positive and negative refractive index constituents, as long as the effective medium approximation remains valid. The analytical solution is validated by a dispersive numerical model of lossy metamaterials that uses a transmission line matrix method based on Z-transforms, where a close agreement between the analytic and numerical results is obtained.