Characterizing the optical chaos in a special type of small networks of semiconductor lasers using permutation entropy

A critical issue in the usage of permutation entropy (PE) for the complexity measure is the selection of the embedding time delay. It is known that PE evaluated with multiple delays can help us gain additional insight into the unpredictability degree at different timescales. In this paper, we investigate the unpredictability degree of the optical chaos in a special type of small networks of semiconductor lasers (SLs) using permutation entropy. These laser systems, such as SLs subject to delayed self-feedback, mutually delay-coupled SLs, and a ring configuration consisting of three unidirectionally delay-coupled SLs, exhibit similar correlation properties characterized by autocorrelation function. The results show that, for a given embedding time delay, the curves for the PE follow the same trends upon monotonically increasing the coupling rate, regardless of how complex the system structure actually is. Furthermore, the results corroborate the effectiveness of PE as an unpredictability measure for small networks consisting of SLs with a single coupling delay, which can be extracted from the intensity autocorrelation function. Finally, we show an example for generating unpredictability-enhanced optical chaos based on small networks of SLs.