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

Abstract

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.

On the far‐zone electromagnetic field of a vertical Hertzian dipole over an imperfectly conducting half‐space with extensions to plasmonics

“…is a ubiquitous reference integral, which is usually expressed in terms of the complementary error function erfc [Bernard and Ishimaru, 1967;Felsen and Marcuvitz, 1973;Brekhovskikh, 1980]. In the above, as previously stated in Michalski and Mosig [2015b, 2015a, 2016a, we prefer to employ the related and well-researched…”

“…where k 0 is the free-space wave number and A z is the z component of the magnetic vector potential, which may be expressed as [Collin, 2004;Michalski and Mosig, 2016a;Michalski and Nevels, 2017]…”

A Computational Channel Model for Magnetic Induction‐Based Subsurface Applications

“…VMD = vertical magnetic dipole; HMD = horizontal magnetic dipole. Electromagnetic (EM) waves propagation from Hertzian dipoles in the presence of isotropic layered media has been extensively investigated since the pioneering work by Sommerfeld in 1909(Sommerfeld, 1909, 1926 to the present time (Fei et al, 2007;Guzatov et al, 2013;Guzatov & Klimov, 2014;Liao & Sarabandi, 2007;Liu & Li, 2007;Long et al, 2001;Michalski & Mosig, 2016;Wang et al, 2014Wang et al, , 2015. As a result, the EM fields from a vertical magnetic dipole (VMD) or a horizontal magnetic dipole (HMD) in the presence of a two-layered or multilayered region have been presented in analytical closed-form expressions by many researchers, in particular, by Wait (1961Wait ( , 1996Wait ( , 1969Wait ( , 1972Wait ( , 1982.…”

On the Far‐Zone Electromagnetic Field of a Horizontal Electric Dipole Over an Imperfectly Conducting Half‐Space With Extensions to Plasmonics

“…where k 0 is the free-space wave number and A = (̂cos −̂sin ) A x +ẑA z is the magnetic vector potential, with (Michalski & Mosig, 2016a)…”

“…as the "knee point" where the transition from  ( −1 ) to  ( −2 ) begins (Michalski & Mosig, 2016a). In the plasmonic case, when ℑs p > 0, the last term in the asymptotic expansion (A10) contributes…”