Lossy Wave Propagation through a Graded Interface to a Negative Index Material – Case of Constant Impedance

We investigate the transmission and reflection properties at the interfaces between the lossy positive-index materials (PIM) and negative-index materials (NIM) with graded permittivity and permeability. We present an exact analytical solution to Helmholtz’ equation for the oblique incidence on profile with the graded real parts of permittivity and permeability profile changing according to a hyperbolic tangent function. Thereafter we restrict our considerations to a special case of the normal incidence and present an exact analytical solution to Helmholtz' equation for a lossy case The simple analytical solutions and graphical results for the field intensity along the graded structure are presented. The model allows for arbitrary temporal dispersion.