Dynamics and Synchronization of Nonlinear Oscillators with Time Delay: A Study with Fiber Lasers

Abstract

The effect of time delay on nonlinear oscillators is an important problem in the study of dynamical systems. Erbium-doped fiber ring lasers have an internal time scale set by the length of the laser's electromagnetic cavity. Long cavities allow thousands of modes to experience gain making it very difficult to model the lasers. We examine the effect of adding external time delays through feedback and coupling.

In the first experiment an external time delay is added to a laser by adding a feedback loop to the cavity. These delay times are varied over four orders of magnitude by changing the length of fiber in the feedback loop. The laser intensity dynamics are examined using time series, power spectra, time delay embeddings, and spatiotemporal representations. We apply Karhunen-Loeve (KL) decomposition on the spatiotemporal representations and use the Shannon entropy as calculated from the KL eigenvalue spectra as a measure of the complexity of the dynamics. For long delays we find that the complexity increases as expected, but also that the fluctuation size increases.

In the second experiment two lasers are mutually coupled together with a coupling time delay that is varied over four orders of magnitude. The analysis is repeated and we find the surprising result that the dynamical complexity decreases for short coupling delays as compared to the uncoupled lasers. Measurements of the optical spectra indicate a narrowing of the spectra indicating that the simplification in dynamics could be due to the reduction in the number of electromagnetic modes experiencing gain. The fluctuation size increases for all delay times and is largest when the internal and external time delays match.

Lag-synchrony is also observed for the mutually coupled lasers. Recent modeling using Ikeda ring oscillators showed that stable isochronal synchrony could be achieved if a third drive laser was unidirectionally coupled with enough strength. We experimentally find that increasing the coupling strength of a third drive laser added to the mutually coupled lasers above quenches the lag-synchrony. The two response lasers become more synchronized to the drive than to each other, however the levels of isochronal synchrony are low.