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

Figures & data

Figure 1. Hertzian dipole with the current moment above a half-space medium.

Figure 2. Network problem for the determination of the TL Green functions.

Figure 3. Hertzian dipole above a layered half-space and TL network analog.

Figure 4. Schematic diagram of the four-sheeted Riemann surface with fundamental branch cuts associated with and .

Table 1. Taxonomy of surface modes over a homogeneous half-space.

(high-loss)		(low-loss)
ZW	SPP	Brewster mode	Fano mode

Figure 5. Integration paths on the top Riemann sheet. A small loss in the upper half-space is assumed for better drawing clarity.

Figure 6. Illustration of the path deformation on the top Riemann sheet.

Figure 7. Integration path wrapped around vertical branch cuts and poles captured. The shaded areas are swaths of improper Riemann sheets swept in the path deformation.

Figure 8. -plane in the vicinity of the origin, which corresponds to the branch point in the -plane.

Figure 9. Integrand function of a typical Sommerfeld integral in the -plane for a lossy ground case. The Sommerfeld pole, despite its location below the branch cut, strongly affects the integrand behavior along the SIP.

Figure 10. Results for a vertical dipole over dry ground at MHz.

Figure 11. Plot of vs. frequency at a fixed distance from the source for the dry ground case. Nonspectral representation is employed.

Figure 12. Nonspectral representation results for the dry ground case.

Figure 13. Results for a vertical dipole over seawater at MHz.

Figure 14. Nonspectral representation results for a horizontal dipole over seawater at MHz.

Figure 15. Nonspectral representation results for a dipole over ice-covered seawater at MHz.

Figure 16. Nonspectral representation results for the GPR example and a table of the poles included.

Figure 17. Nonspectral representation results for a vertical dipole over aluminum at THz.

Figure 18. Nonspectral representation results for a vertical dipole over silver at nm.

Figure 19. Nonspectral representation results for the near-infrared Cr case.

Figure 20. PF model fit [125] of the tabulated dielectric function of Cr,[128] including the range where .

Figure A1. Riemann sheets associated with in a lossy case.

Michalski KA. On the low-order partial-fraction fitting of dielectric functions at optical wavelengths. IEEE Trans. Antennas Propag. 2013;61:6128–6135.

 Web of Science ®Google Scholar

Lynch DW, Hunter WR. Comments on the optical constants of metals and an introduction to the data for several metals. In: Palik ED, editor. Handbook of optical constants of solids. San Diego (CA): Academic Press; 1998. p. 275–316.

 Google Scholar