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

A complete set of closed-form multilayered media Green´s functions with general electric and magnetic sources is presented. The Green´s functions are written in the mixed potential integral equation formulation which is consistent with the Michalski-Zheng (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 Green´s 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 Green´s functions with general electric and magnetic source in closed form. In the source layer the derived closed-form Green´s 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.