Exact analytical approach to TM-wave oblique incidence on impedance-matched graded interfaces between RHM and LHM media

We investigate TM-wave propagation over an impedance-matched graded interface between a right-handed material (RHM) and a left-handed material (LHM) with graded permittivity and permeability. We assume a lossy case with permittivity and permeability profile changing according to a hyperbolic tangent function along the direction perpendicular to the boundary plane between the two materials. A remarkably simple exact analytical solution to Helmholtz equation is obtained. The exact analytical expression for the field intensity along the graded LHM-RHM structure confirms the expected properties of LHM media. Furthermore, in the special case of normal incidence, we reconfirm that there is an excellent agreement between the present analytical results and numerical simulations.