Complete Uniform Asymptotic Expansion of Finite-Range Integral Representation of Dyadic Green’s Function for Ordinary Dielectric Half-Space

The dyadic Green’s function (DGF) is expressed in terms of a double integral whose integrand contains the spatial derivatives of the Sommerfeld integral (SI) of the impedance plane when an electric dipole and observation point are both above a planar dielectric interface. Here, the two dielectric media are assumed to be ordinary. Meanwhile, directly calculating the original double integral representation may be difficult and inefficient when the distance between the antenna and the observation point is very large. Because the complete uniform asymptotic expansion (AE) is known for the SI for the impedance plane, the double integral can be reduced to a single integral by simply replacing the integrand with a known AE. The single integral has a finite integral range as
$[0,\pi /2]$
. On the basis of the identical replacement, the double integral of the DGF can be reduced to a single integral; however, new complete uniform AEs of the first- and second-order spatial derivative of the SI for the impedance case should be formulated. In this study, using uniform expansions, a new complete uniform AE of the DGF is obtained, which is numerically validated for various propagation scenarios. By comparing the simulation results with those of the conventional first- or second-order uniform expansion and two numerical algorithms such as numerical steepest descent path (NSDP) and extreme steepest descent path (ESDP), the numerical complexity, accuracy, and valid region of the new expansion are addressed as well.