Exact analytical solution for oblique incidence on a graded index interface between a right-handed and a left-handed material

We have determined the scattering parameters of optical structures incorporating both left-handed metamaterials (LHM) and conventional ("right-handed") materials (RHM) for the case when the refractive index at the LHM-RHM interface is graded, for oblique incidence at an arbitrary angle and for arbitrary spectral dispersion. We derived an accurate analytical solution to Helmholtz' equation for the case of the refractive index gradient varying as a hyperbolic tangent where the steepness of the index transition may be arbitrary, even of the order of vacuum wavelength. We determined the expressions for the field intensity along the LHM-RHM structure with refractive index gradient. We show that there is an excellent agreement between our analytical results and the accurate numerical simulations done by finite element method