Finite-Element Simulations of Light Propagation through Circular Subwavelength Apertures

Light transmission through circular subwavelength apertures in metallic films with surrounding nanostructures is investigated numerically. Numerical results are obtained with a frequency-domain finite-element method. Convergence of the obtained observables to very low levels of numerical error is demonstrated. Very good agreement to experimental results from the literature is reached, and the utility of the method is demonstrated in the investigation of the influence of geometrical parameters on enhanced transmission through the apertures.

💡 Research Summary

The paper presents a comprehensive numerical study of light transmission through circular sub‑wavelength apertures in metallic films surrounded by nanostructured grooves, using a frequency‑domain finite‑element method (FEM). The authors employ the JCMsuite solver, which combines higher‑order vector elements, goal‑oriented error estimation, and adaptive mesh refinement, to solve the time‑harmonic Maxwell equations for rotationally symmetric three‑dimensional geometries. By expanding the incident field in a Fourier series with respect to the azimuthal coordinate, the full 3D problem is reduced to a set of independent 2D problems for each azimuthal mode n, which are solved until the contribution of higher‑order modes becomes negligible.

A convergence study is carried out on a reference configuration (Table 1, parameter set 1). The transmitted far‑field intensity T is computed for increasing numbers of degrees of freedom N. Using second‑order elements and up to ten adaptive refinement steps, the relative error ΔT(N) drops below 1 % when N≈10⁵, corresponding to computation times of less than one minute on a standard workstation (single‑core). This demonstrates that the FEM approach achieves high accuracy with modest computational resources, outperforming previously reported finite‑difference time‑domain (FDTD) simulations that required far larger meshes and still failed to converge.

The method is validated against analytical Mie theory for scattering by a sphere and against a Fourier‑Bessel modal solution for a single circular hole, confirming that the numerical implementation reproduces known solutions. The authors then replicate the experimental setup of Lezec et al. (2004) (Table 1, parameter set 3), performing wavelength scans from 400 nm to 1100 nm and recording transmission at several detection angles (0°–20°). The simulated spectra exhibit a pronounced transmission peak around λ≈660 nm for low observation angles, in excellent agreement with the measured data. Moreover, the angular distribution of the transmitted beam shows a narrow divergence of approximately ±3° (full width at half maximum), reproducing the experimentally observed beaming effect.

A direct comparison with the FDTD results of Baida et al. (Ref. 7) is presented using the same geometric and material parameters (Table 1, parameter set 2). The FEM spectra differ markedly from the FDTD spectra reported in Ref. 7 and align closely with the experimental measurements. The authors attribute the discrepancy in Ref. 7 to insufficient numerical convergence rather than to uncertainties in material dispersion or geometry. A small shift of the transmission peak (≈10 nm) is explained by the difference in hole diameter (d = 330 nm in the simulation versus d = 300 nm in the experiment).

Finally, the influence of geometric parameters on the enhanced transmission is investigated. By varying the groove spacing p from 550 nm to 585 nm in 5 nm increments, the transmission peak shifts by roughly 10 nm, and its amplitude changes slightly. This parametric study demonstrates the sensitivity of the transmission characteristics to nanoscale design variations and provides quantitative guidance for optimizing plasmonic apertures for specific spectral responses.

In conclusion, the paper shows that a rigorously converged FEM solution of Maxwell’s equations can accurately predict the optical response of sub‑wavelength apertures with surrounding nanostructures. The approach delivers high precision with relatively low computational cost, validates experimental observations, and offers a reliable tool for the design of plasmonic devices such as filters, beaming apertures, and metasurfaces.