Fast Calculation of Nonuniform Plane Waves at Arbitrarily Oriented and Charged Planar Interfaces of Isotropic Lossy Media

A fast method for calculating the reflected and transmitted waves for a given nonuniform plane wave incident on an arbitrarily oriented and charged planar interface between two isotropic and possibly lossy media is proposed based on the decomposition of the complex wave vector and complex wave numbers with respect to the unit normal vector of the interface. According to the complex vector analysis, the exact definition of the complex angles of incidence, reflection and refraction are presented and applied in the complex forms of Snell's law and Fresnel equations to quickly and correctly calculate the complex wave vectors and the complex electric fields of the reflected and refracted waves at a charged interface where the surface charge and current densities are considered. The calculation procedure and two practical examples are also given to demonstrate the validity and powerfulness of the proposed methodology.