Permutation entropy-based characterization of speckle patterns generated by semiconductor laser light

Semiconductor lasers with optical feedback are stochastic nonlinear systems that can display complex dynamics and abrupt changes when their operation conditions change. Even very small changes can lead to large variations in the spatial and spectral properties of the laser emission. This makes a semiconductor laser with feedback an ideal system for conducting controlled experiments to test data analysis tools to detect and characterize transitions. Here, we identify feedback-induced transitions by analyzing speckle patterns that are generated after the laser light propagates in an optical fiber. Speckle patterns result from the interference of multiple modes, and their statistical properties are understood, but a direct mathematical model does not exist. Here we show the versatility of the correlation length and the permutation entropy as measures for characterizing speckle patterns. Combining entropy and correlation analysis with speckle contrast analysis, we uncover changes that occur when the laser current increases from below to well above the threshold, which unveils the effects of optical feedback on the coherence of the laser emission.