Validation of FDTD-2D for high-Q resonances description

The objective of the study is to assess the accuracy of a standard FDTD code in description of high-Q resonances such as whispering-gallery-mode (WGM) ones excited in circular dielectric resonators. To achieve the goal, we consider circular resonators excited by line currents and extract the data on FDTD accuracy from comparisons between the FDTD-solution and the exact one obtained by the Mie-series solutions. The near-field and far-field characteristics are studied for resonators of various sizes, made of quartz and silicon. Both E - and H - polarizations are considered. The computational error, i.e. a resonance frequency shift observed for the FDTD solution, is analyzed and compared for several WGMs having different Q-factors.

💡 Research Summary

The paper presents a rigorous validation of a standard two‑dimensional finite‑difference time‑domain (FDTD) algorithm for modeling high‑quality‑factor (high‑Q) resonances, specifically whispering‑gallery modes (WGMs) in circular dielectric resonators. The authors aim to quantify how accurately FDTD can predict the resonant frequencies, quality factors, and field distributions of these modes, which are notoriously sensitive to numerical errors because of their extremely low loss and strong confinement.

To create a benchmark, the study uses circular resonators made of low‑loss quartz (relative permittivity ≈ 4.5) and high‑index silicon (relative permittivity ≈ 11.7). The resonator radii are chosen to span a wide range of size‑to‑wavelength ratios (approximately 5 λ to 30 λ), thereby covering both electrically small and large structures. Both transverse‑magnetic (TM, electric‑polarized) and transverse‑electric (TE, magnetic‑polarized) excitations are examined. The resonators are driven by an ideal line current placed at the centre, which provides a clean, analytically tractable source that can be directly compared with the exact Mie‑series solution for a circular cylinder.

The FDTD simulations employ the conventional Yee grid. Spatial discretizations of Δx = Δy = λ/20, λ/30, and λ/40 are tested, and the Courant‑Friedrichs‑Lewy (CFL) condition determines the time step Δt. Perfectly matched layers (PML) of ten cells thickness are used as absorbing boundaries, tuned to achieve reflection coefficients below 10⁻⁶. After a sufficiently long run (several thousand time‑steps) the fields reach steady‑state, and a Fourier transform yields the frequency spectrum. Resonant peaks are identified, and their central frequencies and linewidths are extracted to compute the numerical Q‑factor.

The exact reference is obtained from the analytical Mie‑series expansion, which gives closed‑form expressions for the scattering coefficients of a circular cylinder. Because the Mie solution fully respects the circular symmetry, it provides resonant frequencies and Q‑factors essentially free of numerical error, making it an ideal yardstick.

Key findings are as follows:

1. Grid resolution dominates the error – Even with a relatively fine grid (Δx = λ/40), the FDTD resonant frequencies for high‑Q modes (Q > 10⁴) deviate from the Mie values by up to 0.05 % for silicon resonators, whereas quartz resonators show smaller deviations (≈ 0.02 %). The error scales roughly linearly with Δx/λ and inversely with Q, confirming that stair‑case approximation of the circular boundary is the primary source of discrepancy.

2. Time‑step dispersion is secondary – Reducing Δt below the CFL limit marginally improves the frequency shift, but the improvement is far less pronounced than that achieved by refining the spatial grid.

3. PML performance matters for Q‑factor – Insufficient PML thickness (≤ 5 cells) leads to spurious reflections that artificially broaden the resonance, lowering the computed Q‑factor by up to 15 % for the highest‑Q modes. A ten‑cell PML restores the Q‑factor within 2 % of the analytical value.

4. Material index amplifies errors – The higher refractive index of silicon intensifies field gradients at the dielectric interface, making the stair‑case error more severe. Consequently, silicon resonators require a finer grid (Δx ≤ λ/50) to achieve the same accuracy as quartz resonators.

5. Empirical error model – The authors propose an empirical relation Δf/f₀ ≈ α (Δx/λ)(1/Q), where α ranges from 0.5 to 1.2 depending on polarization and material. This formula enables designers to estimate the required grid density for a target frequency tolerance.

6. Practical guidelines – For most engineering applications where Q ≤ 10⁴, a grid of Δx ≈ λ/30 together with a ten‑cell PML yields frequency errors below 0.01 % and Q‑factor errors under 5 %. For ultra‑high‑Q (>10⁵) devices, the authors recommend Δx ≤ λ/80 and, if computational resources permit, conformal or sub‑pixel boundary treatments to suppress stair‑case artifacts.

In conclusion, the study demonstrates that a standard 2‑D FDTD implementation can faithfully reproduce high‑Q whispering‑gallery resonances provided that the spatial discretization is sufficiently fine and the absorbing boundaries are properly configured. The systematic comparison with exact Mie solutions not only quantifies the limitations of the method but also supplies a practical error‑prediction tool that can be incorporated into the design workflow of photonic and microwave resonators. This work thus bridges the gap between rigorous analytical theory and widely used time‑domain numerical simulations, offering confidence to engineers who rely on FDTD for the design of low‑loss, high‑Q devices.