The entrainment phenomenon, by which an oscillator adjusts its natural rhythm to an external periodic signal, has been observed in many natural systems. Recently, attention has focused on which are the optimal conditions for achieving entrainment. Here we use a semiconductor laser with optical feedback, operating in the low-frequency fluctuations (LFFs) regime, as a testbed for a controlled entrainment experiment. In the LFF regime the laser intensity displays abrupt spikes, which can be entrained to a weak periodic signal that directly modulates the laser pump current. We compare the performance of three modulation waveforms for producing 1:1 locking (one spike is emitted in each modulation cycle), as well as higher order locking regimes. We characterize the parameter regions where high-quality locking occurs, and those where the laser emits spikes which are not entrained to the external signal. The role of the modulation amplitude and frequency, and the role of the dc value of the laser pump current (that controls the natural spike frequency) in the entrainment quality are discussed.

Semiconductor lasers with time-delayed optical feedback display a wide range of dynamical regimes, which have found various practical applications. They also provide excellent testbeds for data analysis tools for characterizing complex signals. Recently, several of us have analyzed experimental intensity time-traces and quantitatively identified the onset of different dynamical regimes, as the laser current increases. Specifically, we identified the onset of low-frequency fluctuations (LFFs), where the laser intensity displays abrupt dropouts, and the onset of coherence collapse (CC), where the intensity fluctuations are highly irregular. Here we map these regimes when both, the laser current and the feedback strength vary. We show that the shape of the distribution of intensity fluctuations (characterized by the standard deviation, the skewness, and the kurtosis) allows to distinguish among noise, LFFs and CC, and to quantitatively determine (in spite of the gradual nature of the transitions) the boundaries of the three regimes. Ordinal analysis of the inter-dropout time intervals consistently identifies the three regimes occurring in the same parameter regions as the analysis of the intensity distribution. Simulations of the well-known time-delayed Lang-Kobayashi model are in good qualitative agreement with the observations.

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