  We investigate second harmonic generation (SHG) from hexagonal periodic
arrays of triangular nano-holes of aluminum using a self-consistent methodology
based on the hydrodynamics-Maxwell-Bloch approach. It is shown that angular
polarization patterns of the far-field second harmonic response abide to
three-fold symmetry constraints on tensors. When a molecular layer is added to
the system and its parameters are adjusted to achieve the strong coupling
regime between a localized plasmon mode and molecular excitons, Rabi splitting
is observed from occurrence of both single- and two-photon transition peaks
within the SHG power spectrum. It is argued that the splitting observed for
both transitions results from direct transitions between lower and upper
polaritonic states of the strongly coupled system. This interpretation can be
accounted by a tailored three-level quantum model, with results in agreement
with the unbiased numerical approach. Our results suggest the hybrid states
formed in strongly coupled systems directly contribute to the nonlinear
dynamics. This opens new directions in designing THz sources and nonlinear
frequency converters.
