General analytical treatment of the graded interfaces with negative-index nanocomposites

We investigate the electrodynamics of structures incorporating nanocomposites with negative dielectric permittivity and magnetic permeability, the popular “left-handed metamaterials”. We obtain analytical solutions to the Helmholtz equation for gradient-index interfaces between the positive and negative index part. We derive expressions for the field intensity as well as for the transmission- and reflection-coefficients, assuming that the graded region across the interface is smaller than the operating wavelength. We then compare the analytical results with the numerical results obtained by the Transfer Matrix Method (TMM). The model allows for arbitrary frequency dependencies and lossy media.