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

TL;DR

This paper explores how breather oscillations in lasers can lock to specific frequencies, forming a Farey tree and devil's staircase structure, with potential applications in dense radio-frequency combs.

Contribution

It demonstrates the existence of frequency locking in breather lasers and links it to Farey fractions and devil's staircase structures, supported by experiments and simulations.

Findings

Frequency locking occurs at Farey fractions in breather lasers.

The hierarchy of locking frequencies forms a devil's staircase.

Locked frequencies produce high signal-to-noise ratio signals.

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 show 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 model system to explore universal synchronization dynamics of nonlinear systems. The locked breathing frequencies feature high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications.