Geometric Parametric Instabilities induced by beatings of two modes

The phenomenon of geometric parametric instabilities (GPI) has been initially introduced and studied within the context of multiple modes' beating dynamics in graded-index fibers. The large numbers of participating modes, supported by the specific structure of graded-index fibers, enhance the amplitude of the resulting modulation, and therefore lead to pronounced instabilities. However, this is not a necessary condition for the emergence of GPI. Furthermore, the complex multi-mode nonlinear dynamics hinder theoretical analysis of GPI. In this work, we demonstrate that even two modes beating is sufficient to trigger GPI. As such, the phenomenon of GPI is generic and can be observed in a large class of optical systems. We provide a detailed theoretical analysis of GPI induced by two modes' beatings and derive the generalized phase-matching conditions, which reveal the fundamental physical mechanisms behind the multi-peak structure of such parametric resonances.