Complexity mapping of a photonic integrated circuit laser using a correlation-dimension-based approach

Quantifying complexity from experimental time series generated by nonlinear systems, including laser systems, remains a challenge. Methods that are based on entropy, such as permutation entropy (PE), have proven to be useful tools for the relative measure of time series complexity. However, the numerical value of PE is not readily linked to a specific type of dynamical output. Thus, the quest to calculate quantitatively meaningful fractal dimension values, such as the correlation dimension (CD), from experimental signals, is still important. A protocol for calculating minimum gradient values and their spread, an integral part of CD analysis, is used here. Minimum gradient values with small spread are presented as approximate CD values. Here-in we report mapping these values, derived from analyzing experimental time series, obtained from a 4-section photonic integrated circuit laser (PICL) across a large parameter space. The PICL is an integrated form of a semiconductor laser subject to controllable optical feedback system. The minimum gradient/approximate CD mapping shows it has some qualitatively different map regions in its dynamics as compared to a free-space-based equivalent system. We show that the minimum gradient values give insight into the dynamics even when approximate CD values cannot be determined. The agreement between the qualitative features of permutation entropy mapping and minimum gradient/approximate CD value mapping provides further support for this. Regions of time series with close to periodic and quasi-periodic dynamics are identifiable using minimum gradient value maps.