Closed-form expressions for the far-field due to a VMD placed on the surface of a planar two-layered conducting earth are obtained fast and accurate by using the Shoeib’s technique. This technique hinges on the use of some integral identities and the identification of the stationary phase point of the integrand. The location of the stationary point is determined easily, and the slowly part in SIs has no singularities at the stationary point. The results using this method agree with the conventional stationary phase method or saddle-point method, additionally it is fast and more accurate than the previous methods, and it can be used without difficulty to calculate the far-fields for comparable troubles.

**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 Fast and Accurate Far-Field Expressions for VMD on Two-Layered Earth

**Research Summary**:

- The study aims to determine an exact solution for the
**far-region scattering electromagnetic fields**of a Vertical Magnetic Dipole (VMD) placed on a**planar two-layered earth structure**. - The solution is derived without any approximation, resulting in fast and more accurate closed-form expressions for the far-field due to the VMD in the sea (planar layered media).
- The method developed in this research allows for reducing the integral representations for the field components generated by the dipole source to a well-known standard integral, making assessment easier.
- Numerical simulations validate the advanced method’s validity, which remains applicable irrespective of the operating frequency and saves time compared to other techniques [1].

**Methodology**:

- The derivation consists of three steps:
**Reduction of Field Integrals**: The field integrals are reduced to combinations of recognized Sommerfeld integrals (SIs) expressed as an infinite series.**Complex Plane Transformation**: In the complex plane, the reflection coefficient is replaced using an exponential function that approximates the summation of algebraic series.**Shoeib Technique**: The integrand is divided into two factors using the Shoeib technique, which helps calculate the stationary phase point. Closed-form expressions for the far-field are then derived.- The resulting expressions provide a fast and accurate solution for the surface fields scattering of the dipole antenna on the multi-layered earth.

**Applications**:

- These results can be used to evaluate numerical solutions of more complicated modeling algorithms.
- The findings are valuable for applications such as
**remote sensing**and**radar signal processing**.

**References**

- 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, Weilin and McLeod, Euan. “Fast and accurate electromagnetic field calculation for substrate-supported metasurfaces using the discrete dipole approximation” Nanophotonics, vol. 12, no. 22, 2023, pp. 4157-4173. https://doi.org/10.1515/nanoph-2023-0423