The devil's staircase in a figure-of-nine laser

Breathers in ultrafast lasers serve as a promising platform for investigating fascinating fractal dynamics—the devil's staircase. This phenomenon has been exclusively investigated in lasers with ring configurations. It is natural to ask whether other cavity configurations can also present such fractal dynamics. The figure-of-nine laser cavity is widely employed in engineering by virtue of its stability. Here we show numerically that devil's staircase can also be observed in a figure-of-nine laser, thereby validating the universality of the fractal phenomena in ultrafast laser systems. While devil's staircase was previously obtained by tuning the pump power in a ring laser, the figure-of-nine laser provides a new control parameter to observe the fractal phenomenon—the phase bias. A discrete model based on the generalized nonlinear Schrödinger equation is employed in this work.