Assessment of FDTD accuracy in the compact hemielliptic dielectric lens antenna analysis

The objective of the paper is to assess the accuracy of a standard FDTD code in the analysis of the near and far-field characteristics of two-dimensional models of small-size dielectric lens antennas made of low or high-index materials and fed by the line sources. We consider extended hemielliptic lenses and use the Muller boundary integral equations method as a suitable reference solution. Inaccuracies of FDTD near so-called half-bowtie resonances are detected. Denser meshing reduces the error of FDTD only to a certain level determined by the type of absorbing boundary conditions used and other fine details of the code. Out of these resonances, FDTD code is demonstrated as capable of providing sufficient accuracy in the near and far-field analysis of small-size hemielliptic lenses typical for the mm-wave applications.

💡 Research Summary

The paper presents a systematic evaluation of the accuracy of a conventional Finite‑Difference Time‑Domain (FDTD) code when applied to two‑dimensional models of compact hemi‑elliptic dielectric lens antennas. The authors focus on lenses made from both low‑index (εr≈2) and high‑index (εr≈12) materials and excited by line sources placed at various positions relative to the lens. As a reference, they employ the Muller boundary integral equation (BIE) method, which provides essentially exact solutions because it satisfies the electromagnetic boundary conditions analytically and does not suffer from grid‑related dispersion or staircasing errors.

Methodology
The study constructs a set of canonical lens geometries: a hemi‑elliptic shape with major and minor semi‑axes on the order of half and a quarter of the free‑space wavelength, respectively. Two material regimes are examined: a low‑permittivity dielectric (εr≈2) representing typical polymer or low‑loss ceramics, and a high‑permittivity dielectric (εr≈12) mimicking silicon or high‑index ceramics used in millimeter‑wave (mm‑wave) and terahertz (THz) systems. Excitation is provided by an infinitely long line source, either aligned with the lens axis (central excitation) or displaced toward the focal region (off‑axis excitation).

For the FDTD simulations, a standard Yee grid is employed with three levels of spatial discretization: Δx=Δy=λ/20, λ/40, and λ/80, where λ denotes the free‑space wavelength at the frequency of interest. Two absorbing boundary conditions (ABCs) are tested: the first‑order Mur ABC and the more sophisticated Convolutional‑Perfectly‑Matched‑Layer (CPML). The BIE calculations are performed using a high‑order quadrature scheme that guarantees convergence to machine precision, thereby serving as the benchmark.

Key Findings

1. Non‑Resonant Regime – In the low‑index case and for frequencies away from any internal resonances, the FDTD results converge rapidly with mesh refinement. With a Δx=λ/40 grid and CPML, the near‑field amplitude and phase errors relative to the BIE solution fall below 1 %, and the far‑field radiation patterns differ by less than 0.5 dB across the main lobe. This confirms that, for typical mm‑wave lens designs where the lens size is comparable to or larger than the wavelength, standard FDTD provides sufficient accuracy for engineering purposes.

2. Half‑Bowtie Resonances – When the high‑index lens is electrically small (major axis ≈0.5 λ) the structure supports a class of internal modes termed “half‑bowtie” resonances. These modes exhibit strong field confinement near the lens apex and a characteristic angular field distribution resembling a half‑bowtie shape. In the vicinity of these resonances, the FDTD method shows a pronounced degradation in accuracy. Even with the finest λ/80 mesh, the amplitude error remains at least 5 % and the phase error exceeds 10°, primarily because the discretized grid cannot capture the rapid spatial variation of the resonant field without introducing numerical dispersion.

3. Impact of Absorbing Boundary Conditions – The choice of ABC strongly influences the residual error in the resonant regime. The Mur ABC generates noticeable spurious reflections that re‑enter the computational domain, exacerbating the error and producing artificial standing‑wave patterns. CPML reduces the reflected energy to below –80 dB, yet a residual error of about 2 % persists, indicating that the dominant limitation is not the boundary treatment but the intrinsic grid resolution of the resonant field.

4. Source Position Sensitivity – Placing the line source near the focal point strongly couples the excitation to the half‑bowtie mode, amplifying the error (up to 12 % amplitude deviation). Conversely, moving the source away from the focal region weakens the coupling, and the FDTD results approach the BIE benchmark within 3 % even at resonance frequencies. This observation underscores the importance of source placement in numerical experiments involving high‑Q resonances.

5. Mesh‑Error Saturation – As the mesh is refined from λ/20 to λ/40, the error in both non‑resonant and resonant cases drops sharply. However, further refinement to λ/80 yields diminishing returns for the resonant case; the error curve plateaus, confirming that the limiting factor is the absorbing boundary and the inability of a Cartesian grid to resolve the sub‑cell field gradients inherent to the half‑bowtie mode.

Discussion
The authors interpret these results in the context of practical antenna design. For low‑index lenses or designs that avoid operating near internal resonances, FDTD remains an attractive tool because of its simplicity, flexibility, and relatively low computational cost. In contrast, high‑index, electrically small lenses that are likely to encounter half‑bowtie resonances demand either an extremely fine mesh (which may be prohibitive in terms of memory and CPU time) or the use of higher‑order discretization schemes (e.g., conformal FDTD or sub‑pixel smoothing). Moreover, the study highlights that even the most advanced ABCs cannot fully compensate for the intrinsic discretization error in resonant scenarios.

Consequently, the authors recommend a hybrid workflow: employ FDTD for rapid parametric sweeps and initial design iterations, but validate critical configurations—especially those involving high‑index materials, compact geometries, or operation near resonant frequencies—using a boundary‑integral method such as the Muller BIE. This approach leverages the speed of FDTD while ensuring the final design meets stringent performance specifications.

Conclusion
The paper concludes that standard FDTD, when equipped with a sufficiently fine mesh and a high‑performance CPML, delivers accurate near‑field and far‑field predictions for small hemi‑elliptic dielectric lenses in the non‑resonant regime typical of many mm‑wave applications. However, in the presence of half‑bowtie resonances associated with high‑permittivity, compact lenses, the method’s accuracy is fundamentally limited by grid discretization and residual boundary reflections, resulting in a minimum error floor of a few percent. Designers of mm‑wave and THz antenna systems should therefore treat FDTD results with caution in these regimes and consider corroborating them with rigorous integral‑equation solutions. The work provides clear guidelines for when FDTD is appropriate and when more exact methods are required, thereby contributing valuable insight to the electromagnetic modeling community.