An attempt to derive field expressions that are not subject to restrictive assumptions has been made by Chew, where an analytical procedure has been presented that allows to derived simple formula for SIs. Long et al., advanced the Chew’s technique and derived the far-field approximation of the SIs rapidly depending on the stationary phase-point. Bishay et al. and Shoeib used the technique (2001) with numerous approximation and transformations, to get closed-form expressions for the far-field due to a VMD in a rough/planar layered media. Recently, Shoeib used the technique (2001) without any approximation and she gets fast and more accurate closed-form expressions for the far-field due to a VMD in the sea (planar layered media).

Lately, Wang H. and Ren Y. study the EMW propagation produced by horizontal magnetic dipole (HMD) and VMD located both in seawater, based on vector potential. Their studies are useful in EM communications near the sea. Also, Hu et al. introduced a quasi-static approximation method to solve the SIs challenge within the near-field region, employing the horizontal electric dipole (HED) as a model.

**Author(s) Details:**

**Sh. Shoeib
**Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt and El-Gazeera Higher Institute for Engineering and Technology, Egypt.

**Recent global research developments in ****Exact Far-Region EM Scattering by a Dipole on Multi-Layered Earth **

- This paper, published in
*Physical Review B*, discusses the role of multipole contributions in the scattering spectra of arbitrarily shaped magnetic particles. - The method for calculating exact multipole moments, including induced electric polarization and magnetization, is suggested and applied for dipole and quadrupole terms.
- The derived expressions for dipole and quadrupole moments can be easily implemented in numerical solvers.
- The study expands the capabilities for modeling metamaterial properties in spectral ranges where natural materials have a relative permeability different from unity.

- In this research, a fast and accurate solution is derived for EM scattering from a dipole antenna placed on a planar multi-layered earth.
- The derivation involves three steps:
- Reducing field integrals to combinations of recognized Sommerfeld integrals (SIs).
- Expressing the solution as an infinite series.
- This approach is applicable irrespective of the operating frequency and saves computation time compared to other techniques.

**“Exact Solution of Electromagnetic Scattering From a Dipole Antenna Inside a Two-Layered Cavity”[3]** :

- Although not directly related to multi-layered earth, this study provides an exact analytical solution for electric and magnetic dipoles radiating inside a two-layered cavity involving anti-isorefractive materials.
- It serves as an additional benchmark for validating computational software.

**“Electromagnetic Scattering”** :

- This resource discusses scattering processes, where particles continuously remove energy from incident electromagnetic waves and re-radiate it into the total solid angle centered at the particle.
- The study focuses on the far-field solution for a single homogeneous sphere.

** ****References**

- Evlyukhin, A. B., & Tuz, V. R. (2023). Electromagnetic scattering by arbitrary-shaped magnetic particles and multipole decomposition: Analytical and numerical approaches. Physical Review B, 107(15), 155425.
- Shoeib, H. S. (2021). Fast and accurate solution for the surface fields scattering of a dipole antenna placed on planar multi-layered earth. Radio Science, 56(7), 1-7.
- Liu, Y., Negishi, T., & Erricolo, D. (2021). Exact solution of electromagnetic scattering from a dipole antenna located inside a multilayer metamaterial oblate spheroidal cavity. alR, 1(2lž1šĄicŽN1), 2lž1šĄicŽ.