Quantitative identification of dynamical transitions in a semiconductor laser with optical feedback

Identifying transitions to complex dynamical regimes is a fundamental open problem with many practical applications. Semi- conductor lasers with optical feedback are excellent testbeds for studying such transitions, as they can generate a rich variety of output signals. Here we apply three analysis tools to quantify various aspects of the dynamical transitions that occur as the laser pump current increases. These tools allow to quantitatively detect the onset of two different regimes, low-frequency fluctuations and coherence collapse, and can be used for identifying the operating conditions that result in specific dynamical properties of the laser output. These tools can also be valuable for analyzing regime transitions in other complex systems.


Video
In the video included as supplementary material we observe, as the pump current is ramped, the gradual transitions studied in the main text: from noise intensity fluctuations, to a regime where occasionally rare intensity dropouts occur, which then become more regular and frequent in the low frequency fluctuations region, and finally, with further increase of the laser current, the regular and well-defined intensity dropouts transform into fast and highly irregular intensity fluctuations. As discussed in the main text, in the LFF regime, due to intensity dropouts, the probabil- in the CC regime (I/I th = 1.2), the intensity pdf has a well defined cutoff. While at low current the pdf is Gaussian, in the CC regime the pdf is not Gaussian. We note that for I/I th = 1.02 the pdf displays a nontrivial structure which is due to the step-like recovery that occurs after a dropout, as shown in Fig. 3. These observations are in agreement and consistent with previous findings [1,2,3,4,5].

Second set of experimental observations
Here we present experiments performed with a different laser and feedback conditions compared to those in the main text, and we find qualitatively very similar results. The laser is a 685 nm HL6750MG semiconductor laser (Opnext HL6750MG) with solitary threshold current of I th = 28.29 mA. The feedback-induced threshold reduction and the feedback delay time are 15.42% and 5.3 ns respectively. Figure 4 displays the standard deviation, σ, of the intensity time-series vs. the laser pump current, for a sampling frequency of 5 GSa/s of the oscilloscope, and a very good agreement is seen with Fig. 3 in the main text. Figure 5(a) displays the number of events vs. the detection threshold, and here again a qualitative good agreement is found with Fig. 4(a) of the main text. It is worthwhile to note that the plateau also exists with a different detection method, as shows Fig. 5(c): instead of normalizing the time series to standard deviation equal to one, we normalize such that the maximum and minimum are equal to one and zero respectively. We note that these two methods differ in the sense that with the second method any threshold value within (0,1) will detect a certain number of events, while with the first method (used in the main text), the interval of detection thresholds depends on the pump current [as shown in Fig. 5(b) and Fig.  4(b) of the main text]. Nevertheless, with this alternative normalization one can also observe the existence of the plateau. Figure 6 displays the six OP probabilities vs. the pump current. We note a variation very similar to that shown in Fig. 5(a) in the main text.

MODEL
In order to further demonstrate the robustness of the experimental findings presented in the main text, we performed simulations of the Lang and Kobayashi (LK) rate equations [6] for the slowly varying complex electric field E and the carrier density N . The model equations are: where α is the linewidth enhancement factor, τ p and τ N are the photon and carrier lifetimes respectively, G = N/(1 + ε |E| 2 ) is the optical gain (with ε a saturation coefficient), µ is the pump current parameter (which is equal to the experimental control parameter -the normalized pump current-only at the solitary threshold [12], where both are equal to 1), η is the feedback coupling coefficient, τ is the feedback delay time, ω 0 is the solitary laser frequency, ω 0 τ is the feedback phase, β sp is the noise strength, representing spontaneous emission, and ξ is a Gaussian distribution with zero mean and unit variance. The model equations were simulated with typical parameters as in [11] [τ p = 0.00167 ns, τ N = 1 ns, α = 4.0, ε = 0.01, η = 10 ns −1 , and τ = 5 ns, β sp = 5 × 10 −5 ns −1 ].

Numerical results
In the framework of the LK model, it has been shown that the LFF intensity dropouts can be either transient or sustained [7,8], with the probability of observing sustained LFFs or stable emission depending on the relative widths of the win-dows where these regimes occur. For typical parameters, however, the LFF are a transient dynamics with a duration that increases with the pump current parameter [9,10]. Typical intensity time-series are shown in Fig. 7.
To compare with experimental observations we need to generate a sufficiently large number of dropouts, therefore, for each value of the pump current parameter, 20 trajectories of 50 µs were generated from random initial conditions.
In Fig. 8 we show that, taken together, the results of the analysis of the simulated data are in very good qualitative agreement with the experimental observations: the variation of the standard deviation, Fig. 8(a), the variation of the number of threshold-crossing, Fig. 8(b), and the variation of the OP probabilities, Fig. 8(c), with the pump current parameter are very similar to those encountered in the experimental data. The comparison between the shape of the experimental and simulated σ curve, shown in Fig. 9, allowed us to determine the five values of the pump current parameter that correspond to the experimental pump currents analyzed in the main text. For those values, as shown in Fig. 8(b), the variation of the number of events is very similar to that seen in the experiments.
However, it is worthwhile to note that the agreement is only qualitative: we note that the simulated dropouts are less depth than the experimental ones [in Fig. 8(b) the lowest detection threshold is -4σ]. A second discrepancy is seen in Fig. 9, where the experimental and simulated σ curves agree qualitatively well only if the horizontal axes are shifted (i.e., µ = 1 is shifted with respect to I/I th =1) and the vertical axes are re-scaled. The origin of these discrepancies could be the fact that in the simulations the LFFs are transient; also, the simple filtering used (a moving average in a time-window of 5 ns) might play a role. We remark that our goal here is only to demonstrate the robustness of our findings though a comparison with model simulations.
To conclude this comparison, in Fig. 10 we present the equivalent of Fig. 1, computed from the simulated time-series. Here again we observe a good qualitative agreement model simulations -experimental observations.    Figure 10: As Fig. 1 but computed from simulated data.