Efficient formulation of closed-form Greens functions for general electric and magnetic sources in multilayered media

A complete set of closed-form multilayered media Greenʹs functions with general electric and magnetic sources is presented. The Greenʹs functions are written in the mixed potential integral equation formulation which is consistent with the MichalskiZheng (1990) C-formulation. In addition, the differentiations of the curl operator are taken in the spectral domain. This leads to calculation of the first- and second-order Sommerfeld integrals and differentiation with respect to z. Traditionally only the zeroth-order Sommerfeld integrals are expressed in the closed form by the discrete complex image method with Sommerfeld identity. Here, we present a generalized Sommerfeld identity by which also the higher order Sommerfeld integrals can be expressed in closed form. The closed-form expressions are derived so that all required differentiations are obtained in closed form and numerical differentiation can be avoided. In addition, the number of required closed-form Greenʹs functions is minimized. In the source layer, only three and in other layers four spectral domain functions have to be considered in writing the mixed potential Greenʹs functions with general electric and magnetic source in closed form. In the source layer the derived closed-form Greenʹs expressions are valid for all field and source points and in the other layers they are valid for a fixed z coordinate of a field point. The power of the formulation becomes evident when both the electric and magnetic sources are present, e.g., in the electromagnetic scattering by dielectric buried objects.