Complete Nonuniform Asymptotic Expansion of Sommerfeld Integral for Dielectric Half-Space

Generally, the Sommerfeld integral (SI) for a dielectric half-space is approximated based on the steepest descent method. A higher order approximation of the integral requires higher order derivatives of the integrand in the steepest descent path in the complex domain, whose explicit formulation may be cumbersome. Recently, a new finite-range integral representation of the SI has been proposed, whose integrand is written in terms of the SI for an impedance half-plane. Based on the known complete nonuniform asymptotic expansion of the SI for an impedance half-plane, the complete nonuniform asymptotic expansion can be analytically formulated for the dielectric half-space. Moreover, the completeness is mathematically proven such that the expansion satisfies the recurrence relation for the Wilcox expansion. The accuracy of the proposed expansion is numerically examined for several propagation scenarios. Furthermore, a numerical procedure is proposed to calculate the higher order terms of the expansion by partially using the impedance approximation for a large relative permittivity of the half-space.

Automated summary

A new finite-range integral representation is proposed for the approximation of the Sommerfeld integral for a dielectric half-space. The complete nonuniform asymptotic expansion of the integral is analytically formulated based on the known expansion for an impedance half-plane. The accuracy of the expansion is numerically examined, and a numerical procedure is proposed for calculating the higher order terms.