Nonlinear synchronization through vector subharmonic entrainment

Synchronization is ubiquitous across a wide range of fields. Subharmonic entrainment (SHE) is a nonlinear synchronization phenomenon that results in a locking oscillator at a frequency of an external periodic forcing signal with a fraction of the oscillator frequency. Beyond the fundamentals of nonlinear dynamics, SHE has a range of practical applications from stabilizing ultrafast laser pulses to optimizing control in various engineering and natural systems. However, the vectorial nature of SHE remains elusive. Here, we present the results of a theoretical and experimental study of a vector type of subharmonic entrainment (VSHE) using a passively mode-locked fiber laser as a testbed. We unveil the mechanism of vectorial SHE, in which weak external signals can entrain internal laser dynamics through vectorial coupling. Vectorial SHE presents in the form of synchronization between the subharmonic of mode-locking-driven oscillations and continuous wave (CW) signal through an evolving state of polarization. This CW signal, driven by the internal dynamics of the injected signal, causes VSHE with the frequencies ratios of multiples of ten, resulting in a partially mode-locking regime operation. Our findings offer new control techniques over mode-locking and additional dimension such as polarization states.