Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy

Abstract

A semiconductor laser with delayed optical feedback is an experimental implementation of a nominally infinite dimensional dynamical system. As such, time series analysis of the output power from this laser system is an excellent test of complexity analysis tools, as applied to experimental data. Additionally, the systematic characterization of the range and variation in complexity that can be obtained in the output power from the system, which is available to be used in applications like secure communication, is of interest. Output power time series from a semiconductor laser system, as a function of the optical feedback level and the laser injection current, have been analyzed for complexity using permutation entropy. High resolution maps of permutation entropy as a function of optical feedback level and injection current have been achieved for the first time. This confirms prior research that identifies a coherence collapse region which is found to be uninterrupted with respect to any embedded islands with different dynamics. The results also show new observations of low optical feedback dynamics which occur in a region below that for coherence collapse. The map of the complexity shows a strong dependence on the delay time used in the permutation entropy calculation. Short delay times, which sample information at the complete measurement bandwidth, produce maps with drastically different systematic variation in complexity throughout the coherence collapse region, compared to maps generated with a delay time that matches the optical feedback delay. Evaluating the complexity with a permutation entropy delay equal to the external cavity delay produces results consistent with the notion of weak/strong chaos, as well as categorizing the dynamics as being of high complexity where the external cavity delay time is harder to identify. These are both desirable features for secure communication applications. The results also show permutation entropy as a function of delay time can be used to detect key frequencies driving the dynamics, including any that may exist due to, or arise from, technicalities of device fabrication and/or noise. A more complete insight into complexity as measured by permutation entropy is gained by considering multiple delay times.

Physics and applications of laser diode chaos

“…Several means for concealling the time-delay value are available : setting the time-delay close to the time-period of the relaxation oscillations 98 , using more than one time-delayed feedback 99 , using stochastically varying or modulated time-delay values in the feedback loop 94 , or exploiting the polarization properties of VCSELs 99 . Optimization of chaos communication parameters also benefits from recent investigations of complexity measures from experimental time-series [100][101][102] . Besides conventional algorithms that compute Lyapunov exponents, recent approaches using permutation entropy have shown a greater robustness of the complexity analysis against noise as inevitably present in experimental chaotic timeseries 103 .…”

“…Besides conventional algorithms that compute Lyapunov exponents, recent approaches using permutation entropy have shown a greater robustness of the complexity analysis against noise as inevitably present in experimental chaotic timeseries 103 . Detailed mappings of the laser diode dynamics with optical feedback based on permutation entropy have identified those parameter regions where high complexity (large value of permutation entropy) and weak signature of laser parameters in the chaotic dynamics (in particular time-delay concealment) can be achieved simultaneously 102 .…”

Effects of periodic forcing on the temporally correlated spikes of a semiconductor laser with feedback

“…This methodology is capable of identifying the presence of serial correlations in datasets, and by analyzing the statistics of the different symbols (ordinal patterns, OPs), it can capture subtle variations in temporally correlated data [35][36][37][38]. We unveiled serial correlations between consecutive spikes both, in the unforced laser (without modulation) and in the forced laser (with current modulation).…”

Identifying the Most Relevant Lag with Runs

“…Particularly, in [5,6], the use of permutation entropy, evaluated at several delay times, is theoretically motived and then applied to identify from a time series the characteristic lag of the generating system. Interesting physical applications of this approach are [7,8]. In order to complete the paper, the new proposal technique is compared with the widely applied autocorrelation function and with recent techniques based on permutation entropy.…”