We investigated electrodynamics of structures incorporating composites with negative dielectric permittivity and magnetic permeability, the popular "left-handed metamaterials". We obtained the exact analytical solutions to the Helmholtz equation for gradient-index interfaces between the positive and negative index part in the special case of matched impedances of the two materials. We derived expressions for the field intensity and compared the analytical results with the corresponding results obtained by the numerical simulations using the Finite Element Method (FEM). The model allows for arbitrary spectral dispersion and lossy media.