Farey tree and devil’s staircase of frequency-locked breathers in ultrafast lasers

Abstract

Nonlinear systems with two competing frequencies show locking or resonances. In lasers, the two interacting frequencies can be the cavity repetition rate and a frequency externally applied to the system. Conversely, the excitation of breather oscillations in lasers naturally triggers a second characteristic frequency in the system, therefore showing competition between the cavity repetition rate and the breathing frequency. Yet, the link between breathing solitons and frequency locking is missing. Here we demonstrate frequency locking at Farey fractions of a breather laser. The winding numbers exhibit the hierarchy of the Farey tree and the structure of a devil’s staircase. Numerical simulations of a discrete laser model confirm the experimental findings. The breather laser may therefore serve as a simple test bed to explore ubiquitous synchronization dynamics of nonlinear systems. The locked breathing frequencies feature a high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications.

Resonant Kushi-comb-like multi-frequency radiation of oscillating two-color soliton molecules

“…Recently it was demonstrated that the oscillation period of breathing solitons in a mode-locked fiber laser, i.e. a dissipative system, depends on the pump power with their relationship following that of a devil's staircase [49]. This relationship is clearly different from that discussed in section 3.1 for our conservative system, which could lead to expect a very different dynamics of two-color soliton molecules in dissipative and conservative systems.…”

Real‐Time Investigation of Ultrafast Dynamics through Time‐Stretched Dispersive Fourier Transform in Mode‐Locked Fiber Lasers

“…A recent demonstration of a fiber laser working in the breathing soliton regime shows frequency locking at Farey fractions. [ 128 ] This frequency‐locked breather states are robust toward pump power and polarization variations, and breathing frequency with a narrow linewidth and a high signal‐to‐noise ratio. A fractal dimension of the winding numbers of breathers approaches that of devil's staircase, denoting the universal nature of a laser.…”

Emergent Phenomena of Vector Solitons Induced by the Linear Coupling

“…The parameter region of the breathing VS is narrow between the stationary and chaotic VSs, and the enlargement version of the parameter space is present in Figure S3, Supporting Information, which agrees qualitatively with the scalar breather in lasers. [67][68][69][70][71][72] In the vicinity of the maximum linear coupling, attributed to the intense energy exchange between two modes, the VS presents the period bifurcation manifesting as a fast switching of pulse states in a few cavity roundtrips (RTs). Each state controlled by the linear coupling has unique characteristics in certain aspects, such as temporal morphology, spectral characteristics, and evolutionary dynamics.…”