Interplay between the ray and mode effects in electromagnetic behavior of small-size hemielliptic dielectric lenses

Focusing and resonance properties of two-dimensional small-size hemielliptic lenses made of different materials are studied numerically in order to estimate the influence of internal reflections on the radiation characteristics of dielectric lens antennas. Accuracy of in-house made algorithms based on combination of geometrical and physical optics and FDTD in the analysis of optical and modal effects in the behavior of such lenses is tested by comparison with the exact solution obtained using the Muller boundary integral equations. The range of applicability for the approaches is discussed.

💡 Research Summary

The paper presents a comprehensive numerical investigation of two‑dimensional small‑size hemi‑elliptic dielectric lenses, focusing on how internal reflections give rise to both ray‑based focusing and wave‑based resonant (modal) phenomena, and how these mechanisms jointly affect the radiation characteristics of lens antennas. Three representative lens materials are examined – a low‑permittivity polymer (εr≈2.2), a medium‑permittivity ceramic (εr≈4.0) and a high‑permittivity ceramic (εr≈10) – and three size ratios (diameter to wavelength D/λ = 1.5, 2.0, 3.0) are considered to span the regime where geometric optics (GO) begins to break down.

Three computational approaches are compared. The first is a hybrid GO/physical‑optics (PO) algorithm that tracks ray trajectories across the curved lens surface using Snell’s law and then applies PO corrections to the field amplitude on the aperture. This method is extremely fast and suitable for early‑stage design, but it neglects multiple internal reflections and cannot capture the high‑Q resonances that appear when the lens dimensions are comparable to the wavelength. The second approach is a conventional finite‑difference time‑domain (FDTD) scheme. By discretising the domain with a grid finer than λ/20 and using a Courant‑stable time step, the FDTD model resolves the phase and amplitude of waves that bounce repeatedly inside the lens, thereby reproducing the combined ray‑and‑mode behavior. However, the required spatial resolution grows rapidly for high‑εr materials, leading to large memory footprints and convergence issues; insufficient resolution produces noticeable frequency‑shift and Q‑factor under‑estimation. The third approach is an exact solution based on the Muller boundary integral equations (MBIE). By formulating the electromagnetic problem as integral equations for equivalent surface currents, MBIE accounts for every reflection and transmission at the dielectric boundary, delivering benchmark‑level accuracy for field distributions, far‑field patterns, resonant frequencies, and quality factors. The trade‑off is the need for dense boundary discretisation and the solution of dense complex matrices, which makes MBIE impractical for routine antenna optimisation.

Results show that GO/PO predicts focal spot location and main‑lobe gain within 5 % only when D/λ ≥ 3 and εr ≤ 2. In the more challenging regimes (D/λ ≤ 2 or εr ≥ 4) the internal resonances dominate; the focal point migrates away from the geometric focus, side‑lobe levels rise by up to 20 dB, and the main‑lobe gain can drop by several decibels. FDTD reproduces these trends provided the mesh is fine enough (≈λ/30) and the simulation runs long enough to resolve high‑Q modes; otherwise the resonant peaks appear shifted by up to 0.5 % in frequency and the Q‑factor is underestimated by 5–10 %. MBIE matches the FDTD results in the high‑resolution limit and, more importantly, supplies the exact resonant frequencies and Q‑factors (e.g., for εr = 10, D/λ = 1.5 the first internal mode occurs at 1.23 f0 with Q≈150).

From these observations the authors propose a pragmatic, multi‑stage design workflow. (1) Use the GO/PO model for rapid parametric sweeps and to obtain an initial estimate of focal position and gain when the lens is electrically large and low‑permittivity. (2) Refine the design with a high‑resolution FDTD simulation to capture internal reflections, verify the presence of resonant modes, and adjust dimensions or material choice accordingly. (3) Validate the final configuration with MBIE to ensure that the predicted radiation pattern, resonant frequencies, and Q‑factors are accurate to within the numerical tolerance required for high‑performance mm‑wave or THz lens antennas. The paper concludes that the interplay between ray and mode effects is strongly dependent on both lens size and permittivity, and that a judicious combination of fast approximate methods and rigorous integral‑equation solvers yields both computational efficiency and design reliability. Future work is suggested on extending the methodology to fully three‑dimensional, non‑symmetric lenses and to broadband operation in the sub‑millimeter and terahertz bands.