Dynamic Multi-Mode Mie Model for Gain-Assisted Metal Nano-Spheres

Coupling externally pumped gain materials with plasmonic spherical particles, even in the simplest case of a single spherical nanoparticle in a uniform gain medium, generates an incredibly rich variety of electrodynamic phenomena. The appropriate theoretical description of these systems is dictated by the quantity of the included gain and the size of the nano-particle. On the one hand, when the gain level is below the threshold separating the absorption and the emission regime, a steady-state approach is a rather adequate depiction, yet a time dynamic approach becomes fundamental when this threshold is exceeded. On the other hand, while a quasi-static approximation can be used to model nanoparticles when they are much smaller than the exciting wavelength, a more complete scattering theory is necessary to discuss larger nanoparticles. In this paper, we describe a novel method including a time-dynamical approach to the Mie scattering theory, which is able to account for all the most enticing aspects of the problem without any limitation in the particle's size. Ultimately, although the presented approach does not fully describe the emission regime yet, it does allow us to predict the transient states preceding emission and represents an essential step forward in the direction of a model able to adequately describe the full electromagnetic phenomenology of these systems.