Complex Dispersion of a Dielectric-Coated Cylindrical Conductor: A Spectral Study of the Sommerfeld–Goubau Line

The complex dispersion and modal sensitivity of an axisymmetric transverse magnetic surface wave supported by a dielectric-coated perfectly conducting cylinder are investigated. Starting from Maxwell’s equations, the boundary-value problem is reduced to a nonlinear complex dispersion equation for the longitudinal propagation constant β. A numerical framework combining zero-level localization of the real and imaginary parts of the dispersion function, nonlinear root refinement, numerical clustering, and adaptive continuation in the complex coating permittivity is used to identify and track a selected spectral branch. One- and two-parameter computations characterize the mapping ε↦β(ε) over prescribed subsets of the complex-permittivity plane. At fixed ℑε, increasing ℜε increases ℜβ and decreases ℑβ, whereas at fixed ℜε, increasing ℑε increases both components of β in the investigated parameter range. The rectangular-grid, concentric-circle, and radial-beam experiments show that the spectral response is smooth on the considered parameter sets but non-affine, coupled, and direction-dependent. The corresponding longitudinal electric field is reconstructed, normalized, and phase-aligned along the tracked branch. Difference fields, radial localization measures, a global modal distance, and a normalized correlation coefficient show that the same qualitative radial TM mode is retained throughout the sampled parameter domain, while its propagation constant and spatial localization vary continuously with the complex coating permittivity.