Exact Solutions to Electromagnetic Scattering Problems with Multi-Layered Oblate Spheroidal Geometry

Computational electromagnetic methods (CEM) play a critical role in the design and analysis of complex electromagnetic (EM) systems. CEM, such as the finite element method (FEM), finite-difference time-domain (FDTD), and method of moments (MoM), allow researchers and engineers to solve practical EM problems with complex geometries and materials. However, the reliability of these numerical methods requires validation to ensure their accuracy and robustness. One of the primary validation methods involves comparisons with exact solutions derived from canonical EM scattering problems. These analytical solutions serve as benchmarks, providing reference results that numerical solutions must match under the same conditions. The novelty of this work lies in developing a new analytical exact solution for electromagnetic scattering, designed to serve as a critical benchmark for validating numerical approaches in computational electromagnetics. We investigate a semi-oblate spheroidal cavity placed under a double-negative half-space. The cavity has two layers: one is made of double-positive material (DPS), the other is made of double-negative (DNG) metamaterial. The source of the incident field is an electric or magnetic dipole placed in the DNG region (inside or above the cavity). The solution is expressed in terms of series expansions of eigenfunctions, making it a canonical reference for evaluating the accuracy and stability of computational methods. Unlike simpler canonical problems, this configuration includes features such as a cavity and sharp edges, which can be difficult for numerical solvers to handle accurately. We consider exact solutions that can be computed as the sum of series expansions, where the expansion coefficients can be derived analytically and do not require, for example, the approximate solution of a system of equations. After the expansion coefficients are found, we express electric and magnetic fields in all three regions of the considered geometry (two layers of the cavity and the upper half-space). We plot fields with various geometry parameters, investigate some particular cases, and compare results with the full wave numerical model simulation. We also discuss the nuances of series convergence and the possibility of series acceleration.