The Sommerfeld half-space problem revisited: from radio frequencies and Zenneck waves to visible light and Fano modes

The classical Sommerfeld half-space problem is revisited, with generalizations to multilayer and plasmonic media and focus on the surface field computation. A new ab initio solution is presented for an arbitrarily oriented Hertzian dipole radiating in the presence of a material half-space with arbitrary horizontal stratification. The solution method combines the vector potential approach and the spectral domain transmission line analog of the medium, which results in the most compact formulation and facilitates the inclusion of any number of layers in the analysis. Following Sommerfeld, the solution is first expressed in terms of the Fourier–Bessel transforms, also known as Sommerfeld integrals. Analytical properties of the integrands in the complex plane are then investigated, including the location of the Sommerfeld pole, which gives rise to the Zenneck wave (ZW) or surface plasmon polariton (SPP), and alternative field representations are developed by a deformation of the integration path and analytic continuation of the integrand functions, using hyperbolic and vertical branch cuts. Closed-form expressions for the asymptotic surface fields are also derived and the rôle of the ZW and SPP is elucidated. Numerical examples are included to illustrate the theory, from radio frequencies to visible light.