Computationele Technieken voor Mesoscopische Elektromagnetica

Plasmonics has long relied on classical electromagnetics as its theoretical framework and as justification of a wide variety of phenomena that make such discipline so attractive for applications, such as the squeezing of light at nanoscopic volumes far below the diffraction limit, accompanied with enormous field enhancement. Computational Electromagnetics (CEM), having been used for decades in antenna engineering and microwave frequencies, has reached scientific maturity and has thus been deployed routinely and successfully to the nanoscopic and optical regime that plasmonics requires. However, mounting experimental evidence, enabled by tremendous progress in nanofabrication techniques, has underlined consistently deviations from the predictions of classical electromagnetics (and CEM by extension) when the critical dimensions of the structure probed fall in the deep nanometric regime (roughly below 10 nm). These findings reignited interest in semiclassical modelling, most notably in hydrodynamic models of free electrons such as the Hydrodynamic Drude Model (HDM) and the more advanced Self-Consistent Hydrodynamic Drude Model (SC-HDM). Such approaches aim to bypass the stringent computational limitations that ab initio approaches, such as the Time Dependent Density Functional Theory (TD-DFT), impose. By including in an ad hoc manner phenomena of quantum origin, such as nonlocal electron dynamics, Landau damping, and electron spill-out, semiclassical models have been deployed successfully to explain the aforesaid deviations. The regime they address, where the characteristic dimensions of the objects become comparable to the Fermi wavelength of the electrons they contain, is one where macroscopic averaging and hence the introduction of macroscopic electromagnetics is not clear, while at the same time full microscopic calculations are not--computationally--possible. This intermediate regime is termed mesoscopic electromagnetics. The evolution of such semiclassical models hinges upon the availability of computational recipes, as such models, while set more efficient numerical frameworks than TD-DFT, remain challenging to probe, due to the multiphysics and multiscale nature of the associated simulations; a new branch of CEM arises, namely Computational Mesoscopic Electromagnetics (CMEM). This nascent discipline, which consists of updating, adapting or redesigning from scratch established methods of CEM, is still in its formative years: there are very limited contributions when it comes to specialised software, while numerical methods are still being explored. In this thesis we contribute to CMEM and the discussion around software and numerical methods. In particular, we first present an S matrix-based software, which can deal with nonlocal (within the HDM) multilayered canonical structures and showcase its capabilities in spheres and cylinders. We proceed with a potential-based Boundary Integral-Equation (BIE) formalism, which we argue is particularly appropriate for systems of arbitrary geometry analysed within HDM, and which can be the springboard for efficient numerical methods. Inspired by this work, we proceed to give a derivation of the uniqueness theorem for media treated within HDM and find the material conditions under which uniqueness is guaranteed. Finally, we devise a Volume Integral Equation (VIE) formalism for analysing spherical nanoparticles with both nonlocality and electron spill-out included in their optical response using SC-HDM. We design a Method of Moments (MoM) algorithm using conventional Mie theory as inspiration and manage to circumvent the significant computational requirements associated with such a complicated model.