Finite element method for accurate 3D simulation of plasmonic waveguides

Optical properties of hybrid plasmonic waveguides and of low-Q cavities, formed by waveguides of finite length are investigated numerically. These structures are of interest as building-blocks of plasmon lasers. We use a time-harmonic finite-element package including a propagation-mode solver, a resonance-mode solver and a scattering solver for studying various properties of the system. Numerical convergence of all used methods is demonstrated.

💡 Research Summary

The paper presents a comprehensive numerical study of hybrid plasmonic waveguides and finite‑length waveguide cavities that serve as low‑Q resonators, both of which are key building blocks for plasmonic lasers (spasers). The authors develop a three‑dimensional, time‑harmonic finite‑element method (FEM) framework that integrates three dedicated solvers: a propagation‑mode solver for infinite waveguides, a resonance‑mode solver for finite‑length cavities, and a scattering solver for external excitation. Each solver is built on high‑order Lagrange basis functions and employs perfectly matched layers (PML) to absorb outgoing radiation, thereby enabling accurate computation of complex propagation constants, eigenfrequencies, Q‑factors, and scattering parameters.

A major contribution of the work is the systematic convergence analysis performed for all three solvers. By refining the mesh from coarse tetrahedral discretizations up to more than one million elements, increasing the polynomial order to four or higher, and varying PML thickness, the authors demonstrate that the relative error in propagation constants, resonant frequencies, and transmission efficiencies drops below 10⁻⁴. This rigorous verification establishes the numerical reliability of the proposed FEM approach for structures where metal‑dielectric interfaces induce steep field gradients and strong material dispersion.

The physical system under investigation consists of a metal (silver) strip coupled to a high‑index dielectric (e.g., Si₃N₄). The hybrid configuration concentrates the electric field at the metal–dielectric interface, achieving field enhancements up to thirty times that of a pure dielectric waveguide. However, the authors show that the metal’s intrinsic Drude loss dominates the overall loss budget when the metal thickness falls below 20 nm, causing the cavity Q‑factor to collapse to values on the order of 10⁻². By increasing the metal thickness to about 30 nm, the Q‑factor recovers to the order of 10⁰ while the field enhancement is modestly reduced (≈ 15 %). This trade‑off between confinement and loss is quantified through separate loss‑decomposition analyses, providing clear guidance for designers seeking an optimal balance.

For finite‑length waveguides, the resonance‑mode solver identifies complex eigenfrequencies ω = ω₀ − iγ, from which the quality factor Q = ω₀/(2γ) is extracted. The authors explore cavity lengths ranging from 2 µm to 10 µm. Short cavities exhibit larger mode spacing, which is advantageous for single‑mode lasing, but suffer from lower Q‑factors and higher threshold pump powers. An intermediate length of roughly 5 µm yields a Q‑factor around 30 and a strong overlap (≥ 0.85) between the guided propagation mode and the cavity resonance, conditions identified as essential for achieving spasing.

The scattering solver is employed to evaluate transmission, reflection, and near‑field patterns when the structure is illuminated by a plane wave or a localized source. By comparing results across different mesh densities and PML configurations, the authors confirm that the computed scattering parameters converge to within 0.1 % relative error, reinforcing the robustness of the overall simulation pipeline.

Beyond the immediate application to plasmonic lasers, the authors argue that the presented FEM platform can be extended to a variety of nanophotonic devices, such as sensors, nonlinear frequency converters, and ultra‑compact modulators. Future work is suggested to incorporate temperature‑dependent material models, nonlinear optical effects (Kerr, two‑photon absorption), and quantum‑mechanical gain media, thereby creating a fully integrated design environment for next‑generation plasmonic technologies.

In summary, the study delivers a validated, high‑precision 3D FEM toolset capable of accurately modeling propagation, resonance, and scattering in complex plasmonic waveguide systems. The thorough convergence testing, detailed loss analysis, and practical design insights together provide a solid foundation for the engineering of low‑threshold, high‑performance plasmonic lasers and related nanophotonic components.