  In this work, we utilize thin dielectric meta-atoms placed on a silver
substrate to efficiently enhance and manipulate the third harmonic generation.
We theoretically and experimentally reveal that when the structural symmetry of
the meta-atom is incompatible with the lattice symmetry of an array, some
generalized nonlinear geometric phases appear, which offers new possibilities
for harmonic generation control beyond the accessible symmetries governed by
the selection rule. The underlying mechanism is attributed to the modified
rotation of the effective principal axis of a dense meta-atom array, where the
strong coupling among the units gives rise to a generalized linear geometric
phase modulation on the pump light. Therefore, nonlinear geometric phases
carried by the third-harmonic emissions are the natural result of the
wave-mixing process among the modes excited at the fundamental frequency. This
mechanism further points out a new strategy to predict the nonlinear geometric
phases delivered by the nanostructures according to their linear responses. Our
design is simple and efficient, and offers alternatives for the nonlinear
meta-devices that are capable of flexible photon generation and manipulation.
