Closed-Form green's functions and their use in the method of moments

Derivation of the spatial-domain, closed-form Green's functions of the vector and scalar potentials are demonstrated for planar media, and their use in conjunction with the method of moments (MoM) is presented. As the first step of the derivation, the Green's functions are obtained analytically in the spectral domain for various sources viz., horizontal and vertical electric and magnetic dipoles embedded in a planar stratified media. The spatial-domain Green's function can be obtained from the Sommerfeld integral which is the Hankel transform of the corresponding Green's function in the spectral domain. The analytical evaluation of this transformation yields the closed-form, spatial-domain Green's functions which can be used in the solution of a mixed-potential integral equation (MPIE) via the MoM. This combination, i.e., the use of the closed-form Green's functions in conjunction with the MoM, results in a significant improvement in the fill-time of MoM matrices. In the conventional application of the spatial-domain MoM, the matrix elements are double integrals and they require the evaluation of the time-consuming Sommerfeld integral for the spatial-domain Green's function. In the approach presented herein, the spatial-domain Green's functions are in closed forms, and the remaining double-integrals in the matrix elements are evaluated analytically. Thus, there are two factors in this approach that contribute to the improvement in the computation time: (i) elimination of the numerical integration to obtain the spatial-domain Green's functions; (ii) circumventing the need to carry out the numerical integration in the calculation of the MoM matrix elements.