Modes interplay controls the second harmonic generation dynamics of plasmonic nanostructures

A large amount of experimental and theoretical works deals with the second harmonic generation from different plasmonic geometries. Since they often consider relatively long optical pulses, many of these studies are focused on the investigation of a quasi-monochromatic response of the system and can be understood through the excitation of one, possibly two, optical modes. On the other hand, when the excitation pulse duration is short (say, below several tens of fs), the excitation spectrum becomes broader and a very interesting dynamics emerges from the interplay between several optical modes. In this work, the dynamics of modes at the second harmonic frequency for two silver spheres of different diameters and a nanorod is investigated numerically and shown to be quite different. For the pulsed illumination with length close to the modes lifetime, apart from different relative contributions of dipolar and quadrupolar multipoles in the far-field, we have been able to observe and explain non constant phase difference between multipoles, which is not accessible in continuous wave regime. Short pulse durations also allow us to observe only one mode, while another one has already decayed. For the case of the nanorod we also perform an eigenmode analysis, which allows to understand the modes interplay that explains the observed spectra. In the paper, we also show a method allowing a significant reduction of required computational steps to find the response of a plasmonic nanostructure to a pulsed illumination with arbitrary frequency-domain method.