A Complete Uniform Asymptotic Expansion of the Sommerfeld Integral of an Impedance Plane for Imperfectly Homogeneous Half-Space

The Sommerfeld integral for an impedance plane has played an important role in many electromagnetic applications. A direct numerical computation may not be efficient and cannot provide a robust and accurate result especially for highly conductive media and/or antennas located near the surface. Two complete asymptotic series are formulated for the integral, which is a power series of the reciprocal of the distance between the source and observation point. The asymptotic series are nonuniform, so the application of the series is limited and is not accurate for the aforementioned cases. Hence, in this article, a new uniform asymptotic expansion of the integral is derived, which is represented in terms of repeated integrals of the complementary error function. The representation is very similar to the conventional uniform expansion of the integral. However, the formulated expansion is a complete power series of the reciprocal of the distance, while the conventional series contains only the first-order term. For a far-field region, the proposed expansion is mathematically shown to reduce the known complete nonuniform expansion for the first few terms of the expansion. For some scenarios that include near-earth propagation and/or highly conductive media, the proposed formulation is numerically verified and its mathematical properties are examined.