Practical Implementation of Accurate Finite-Element Calculations for Electromagnetic Scattering by Nanoparticles

The finite-element method (FEM) is increasingly used as a numerical tool to support experimental and theoretical studies of the optical properties of nanoparticles, in contexts such as surface-enhanced spectroscopy, molecular plasmonics, metamaterials, and optical trapping. Here, we investigate the validity of such calculations, focusing in particular on numerically challenging cases involving strong optical (plasmon) resonances and solutions with large electric field gradients and intensities. These are exemplified by elongated metallic nanoparticles and two closely spaced metallic spheres (dimer), where highly localized regions of intense electric field enhancements occur at the tip or in the gap, so-called electromagnetic hot-spots. We assess the accuracy of the FEM solutions by comparing the result to exact analytic solutions based on the T-matrix method for an elongated particle and generalized Mie theory for a dimer. Particular attention is given to the electromagnetic properties that have seldom been studied in this context, notably near-field properties such as surface-field enhancement factors and far-field radiation profiles. We also demonstrate explicitly how the accuracy of the FEM predictions can be inferred from the solution of two problems with different mesh and bounding box parameters. Such a numerical check is crucial in practice as no exact solutions are in general available to compare with. While we chose the commercial software COMSOL to illustrate our results, the methods and conclusions are equally applicable to other FEM implementations. We provide for convenience full details of how to set up these calculations in COMSOL, which we hope will allow readers to easily reproduce them and seamlessly adapt them to their modeling needs. We expect this work will cement the FEM as a reliable method for routine calculation of electromagnetic scattering by nanoparticles.