Anticipating and identifying the threshold bifurcation of a complex laser with permutation entropy

Data recorded during the laser turn on offers an opportunity to test novel indicators in very noisy data, and evaluate their abilities to characterize and anticipate the bifurcations in complex systems. One of such indicators is permutation entropy [1], a statistical measure that allows to quantify the complexity of a signal, and has become a powerful tool for the analysis of complex signals [2]. Here we show that it is able to anticipate and identify the threshold bifurcation of a complex laser, where thousands of modes compete for gain at the onset of lasing, which are nowadays being developed for complex light structuration and manipulation [3]. In our experimental setup, the cavity roundtrip time is several orders of magnitude longer than the temporal resolution of the detection system, which enables a high statistical sampling of the intensity dynamics per roundtrip. We show that the permutation entropy experiences a clear decrease far below the threshold and reaches a sharp minimum at the threshold bifurcation point, which reveals an abrupt increase of the temporal correlations. The evolution of the entropy is compared with standard quantifiers of approaching bifurcations, the lag-1 autocorrelation and the variance of the signal, which are known to provide only limited characterization of the system transition and may fail to detect bifurcations or critical transitions before they occur [4]. We show that the permutation entropy has a very well-defined and pronounced variation that allows a clearer identification of the threshold. Therefore, our study sets the basis for the use of the permutation entropy as an indicator of the threshold of complex multimode laser systems.¹