Longitudinal Spectral Solutions for the Sommerfeld Half-Space Problem: Presenting New Perspectives for Electromagnetic Field Solutions in an Axially Layered Structure

Abstract

A direct approach based on the reflection and transmission properties of a propagating wave is used to obtain an alternative expression to the Sommerfeld integral for a two-layered, half-space problem. The formulation employs a Fourier longitudinal spectral expansion of the field quantities produced by a Hertzian dipole source, as opposed to the transverse-spectral expansion on which the Sommerfield solution is founded. The nominal reflected and transmitted fields obtained using Fourier longitudinal expansion are not source free; therefore, the solution is incomplete. To make the solution source free, one needs to introduce auxiliary fields that neutralize the spurious source. A branch-cut integral (BCI) on the complex wavenumber plane yields the necessary auxiliary fields. If the BCI is ignored, then the transmitted field becomes nonanalytic and results in a source-conservation violation. The numerical results for the spurious and auxiliary source distributions and the associated reflected and transmitted fields are shown, and the computed fields have excellent agreement with that obtained from Sommerfeld integrals. Finally, the analysis of radially layered structure is discussed with respect to the transverse expansion.