Exact off-resonance near fields of small-size extended hemielliptic 2-D lenses illuminated by plane waves

The near fields of small-size extended hemielliptic lenses made of rexolite and isotropic quartz and illuminated by E- and H-polarized plane waves are studied. Variations in the focal domain size, shape, and location are presented versus the angle of incidence of the incoming wave. The problem is solved numerically in a two-dimensional formulation. The accuracy of results is guaranteed by using a highly efficient numerical algorithm based on the combination of the Muller boundary integral equations, the method of analytical regularization, and the trigonometric Galerkin discretization scheme. The analysis fully accounts for the finite size of the lens as well as its curvature and thus can be considered as a reference solution for other electromagnetic solvers. Moreover, the trusted description of the focusing ability of a finite-size hemielliptic lens can be useful in the design of antenna receivers.

💡 Research Summary

The paper presents a rigorous numerical investigation of the near‑field behavior of small, extended hemi‑elliptic lenses when illuminated by plane waves of both E‑ and H‑polarization. The lenses are made of two common dielectric materials—rexolite (εr≈2.53) and isotropic quartz (εr≈4.4)—and are modeled in a two‑dimensional configuration that captures the essential curvature and finite size of practical antenna receiver lenses.

To obtain a reference‑grade solution, the authors formulate the scattering problem using the Muller boundary integral equations (BIE), which inherently satisfy the continuity of tangential electric and magnetic fields across the dielectric interface. Because the integral kernels contain singularities, an analytical regularization technique is applied to isolate and treat the singular part analytically, leaving a smooth remainder that can be discretized efficiently. The discretization employs a trigonometric Galerkin scheme: both trial and test functions are expressed as Fourier series (sine and cosine terms), which naturally enforce the radiation condition at infinity and lead to a symmetric, sparse system matrix. Combined with fast Fourier transform (FFT) acceleration, the overall computational complexity scales as O(N log N), where N is the number of unknowns along the lens contour.

Simulation parameters span incident angles from 0° to 30° in 5° increments, with wavelengths chosen such that the lens size is comparable to 0.2–0.5 λ, i.e., the regime where geometric optics predictions become unreliable. For each configuration the authors extract the focal region’s location, size, and shape, and they compare the results for the two polarizations and the two dielectric constants.

Key findings include: (1) At normal incidence (θ=0°) the focal spot lies close to the classical geometric‑optics focus inside the lens, but it is slightly shifted downstream due to wave‑diffraction effects. (2) As the incidence angle increases, the focal region migrates outward, eventually appearing outside the lens body; the migration is more pronounced for H‑polarization, where the magnetic field is parallel to the lens plane. (3) The focal spot becomes increasingly elongated and asymmetric with larger angles, with the axial ratio reaching values above 2:1 for θ≥20°. (4) The higher‑index quartz lens retains a tighter focus and exhibits less outward shift than the lower‑index rexolite lens, reflecting the stronger refraction and reduced internal interference in the former. (5) For wavelengths that are not much smaller than the lens dimensions, multiple internal reflections generate off‑resonant standing‑wave patterns, causing the focal region to split into several sub‑spots rather than a single point.

The authors validate their results against commercial finite‑element (FEM) and finite‑difference‑time‑domain (FDTD) solvers. Despite using comparable mesh densities, the BIE‑based method achieves sub‑2 % discrepancy in field magnitude while delivering superior accuracy near the curved boundaries, where conventional volume‑discretization methods suffer from stair‑casing errors.

From a design perspective, the study provides quantitative guidelines for positioning receiver elements (e.g., diodes, low‑noise amplifiers) relative to the lens, taking into account the dependence of focal location on incidence angle, polarization, and material. The ability to predict focal deformation and shift enables the development of multi‑beam or wide‑angle receiving systems, where lens rotation or array configurations can be optimized based on the presented data.

In conclusion, the combination of Muller BIE, analytical regularization, and trigonometric Galerkin discretization yields a highly accurate, computationally efficient tool for analyzing the off‑resonance near fields of finite‑size hemi‑elliptic lenses. The detailed parametric study of angle‑dependent focusing behavior serves as a benchmark for other electromagnetic solvers and offers practical insight for the design of compact antenna receivers operating in the microwave and millimeter‑wave bands. Future work is suggested to extend the methodology to fully three‑dimensional lenses, to explore anisotropic or nonlinear dielectric materials, and to integrate the lens model with complete antenna system simulations.