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As sources of short, high-amplitude light pulses, self-pulsing lasers are key elements in many applications, including telecommunications and optical processing of information. We consider here a semiconductor micropillar laser subject to delayed optical feedback. Without feedback, the laser is excitable, and, as such, displays an all-or-none response to external perturbations. Recent experiments demonstrated that, in the presence of feedback, a single external optical perturbation can trigger a train of optical pulses, whose repetition rate is determined by the delay time. These pulse trains can be controlled reliably through external optical perturbations. In particular, several pulse trains can be switched on and sustained simultaneously in the external cavity with different interpulse timing; moreover, they can be switched off or retimed, depending on the timing of the external perturbation. Such pulse trains are also referred to as localised structures or temporal dissipative solitons in the literature.
We focus on the theoretical investigation of such pulsing dynamics. It has been verified experimentally and theoretically that, as long as the pulses are short compared to the delay time, the phase of the electric field is not relevant. Therefore, the Yamada model with feedback - a system of three delay differential equations for the gain G, absorption Q and intensity I - is a suitable mathematical model. We show that its temporal integration produces a wealth of pulsing dynamics in very good agreement with the experiment. The model allows us to explain the control and interaction of pulses by the interplay of the dynamics of the gain G and that of the net gain G-Q-1.
We perform a bifurcation analysis of the Yamada model to unveil its complex dynamics. In particular, we show that several periodic solutions coexist and are stable. Each stable periodic solution corresponds to a pulsing regime with a given repetition rate, close to a submultiple of the delay time. These correspond to equidistant pulses in the external cavity. Importantly, no stable periodic solution with non-equidistant pulses are found. Although coexisting pulse trains may seem independent from each other on the timescale of the experiment, we demonstrate that they rather correspond to extremely long transients toward one of the available stable periodic solutions. Hence, the different pulses in the external cavity become equidistant in the long term. The rate of convergence toward the stable regime depends on the number of pulses in the external cavity, and can be determined theoretically. The maximum number of pulse trains that can be sustained simultaneously corresponds to the number of coexisting stable pulsing periodic solutions, and it strongly depends on the delay time and strength of the feedback.
By providing a better understanding of pulsing dynamics in excitable lasers with feedback, these results constitute a step toward an all-optical control of pulse train duration, which may have applications in photonics. Because the mechanism for self-pulsations described here is typical and relies only on excitability and delayed feedback, our results might be of interest beyond the specific device considered here, for example for cell dynamics.
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