Numerical simulation of harmonic multiples generation by a group of plasmonic nanoparticles

We propose a theoretical (mathematical) model of nonlinear plasmonic oscillations in metal nanoparticles. The governing equation of motion is deduced from the least action principle and supplemented by a continuity equation resulting from the variational and differential formulations of charge conservation. The presence of irreversible processes is taken into account by the use of a polynomial dissipation term embedded into the action functional. Analysis of the structure of the Euler-Lagrange equation shows that it is possible to derive an expression for the stress tensor of an “electron gas - ionic frame” system. The formulation of an initial boundary value problem for nonlinear integrodifferential equations constituting the model of electron gas dynamics is given. Based on the finite-difference approximation, we developed an iterative method, an algorithm and a solver code for treating the problem. This allowed us to investigate the phenomena of odd (third and fifth) harmonic generation by a group of metal nanoparticles subjected to monochromatic light illumination. It is shown that the presence of higher order terms in the dissipative force expansion is important for providing the stability of the electron gas motion. Estimates for the density function parameter responsible for stable generation were obtained. The estimates were compared versus the analogous ones resulting from the theory of electronic structure of metals as well as from the Drude theory.