New Exact Image Methods for Impedance Boundary Half-Space Green’s Function and Their Fast Multipole Expansion

Two foremost barriers to the development of fast and efficient simulation algorithms for electromagnetic problems in half-space and planar media are efficient Sommerfeld integral evaluation techniques and fast solvers. In this paper, we focus on challenging these two difficulties in the impedance boundary half-space, which is a proper model for common air-lossy media half-space. Based on the Laplace transformation, we first propose an alternative exact image representation for the Sommerfeld integral in half-space Green's function. In addition to its fast and absolute convergence property, this new exact image version does not contain any singularities and interprets the Sommerfeld integral as an integral over real image line. Furthermore, this representation allows a rigorous fast multipole expansion (FME), and thus it can be applied to the development of fast multipole solver with controllable precision. However, this exact image representation is not valid for the all Sommerfeld integrals in half-space Green's function. Therefore, a more generalized single mirror image representation, also exact and computationally efficient, is then presented. The strength of the mirror image is obtained through Weyl's method and steep descent path approach. Subsequently, an approximated FME scheme for this representation is proposed and studied via numerical examples. We find this approximated FME scheme has a higher accuracy than that of previous work. Numerical results show that the proposed work provides exact and efficient evaluations for Sommerfeld integrals in impedance boundary half-space Green's function and valuable insights into the development of fast multipole solver for half-space electromagnetic problems.