Unraveling Period-Doubling Bifurcations in Figure-of-9 Lasers via an Energy Transmission Model: From Cascaded Periodic States to Chaos

Period-doubling bifurcation, a crucial route from steady states to chaos in nonlinear dynamical systems, exhibits unique characteristics in laser optics. However, its underlying mechanisms in nonlinear amplifying loop mirror lasers remain incompletely understood. Here, a variational model based on energy transmission and Poincaré map is developed, unraveling that gain-induced nonlinear phase difference between bidirectional pulses trigger bifurcations and validating the existence of 2-based cascaded periodic states and chaos. Experimentally, an erbium-doped passively mode-locked figure-of-9 laser enables experimental observation of a complete cascaded bifurcation route. Furthermore, phenomena new to our knowledge are identified, including the statistical characteristics of chaos and spectral pulsations. These findings not only expand the understanding of nonlinear dynamics, but also offer valuable insights for applications such as metrology, telecommunication, and frequency comb.