Four-wave mixing and transverse–longitudinal oscillatory modes in plasmonic nanoparticles: nonlinear theory from variational principles and mathematical simulation

In this study we propose a theory of nonlinear plasmonic oscillations in metal nanoparticles subjected to optical irradiation which is further reduced to solvable mathematical model by applying variational procedures and numerical methods. One of the goals pursuing under construction of the theory has been to minimize its axiomatics. A governing equation of motion is deduced on the basis of the least action principle and supplemented by a continuity equation that results from variational and differential formulations of charge conservation. The presence of irreversible processes is taken into account by leveraging of a polynomial dissipation term embedded into the action functional. An initial boundary value problem formulation for nonlinear integrodifferential equations constituting the model of electron gas dynamics has been given. On the ground of a finite-difference approximation an iterative method for treating the problem as well as the algorithm and solver code have been worked out. By utilizing these tools we investigate the third order field effects such as third harmonic generation and four-wave mixing by a group of metal nanoparticles illuminated by a multispectral source of light. An excitation of transverse–longitudinal mode of plasmonic oscillations has been observed under plane wave illumination of two spherical particles. The charge and current distributions intrinsic to the mode have been investigated numerically together with an electromagnetic field scattered by the particles into far zone. The estimate of the density function parameter providing regular dynamics has been analytically obtained by the contraction principle.