Revisiting the "Quasi-Periodic" Route to Chaos

International audienceIt is well known that external-cavity lasers (ECLs) subjected to optical feedback undergo bifurcations which lead to chaos. The most commonly cited route is a quasi-periodic route. We show here that the previously claimed quasi-periodic route is not the typical route given by the Lang-Kobayashi equations. Further, we demonstrate the mechanisms which lie behind this route to chaos. Laser diodes subjected to optical feedback from a distant reflector have been studied intensely for decades as an experimental testbed for nonlinear dynamics [1], as well as, their promise in high-speed, high energy-efficiency applications [2]. The dynamics behind such systems can be characterized by three variables: photon number, carrier number, and optical phase. Previously, most experiments have focused on the optical intensity and carrier number [3-4]. One example of all three being measured is for a fixed optical feedback [5]. Here we measure all three variables simultaneously and vary the optical feedback to see how the dynamics unfold [4-7]. We uncover that the previously identified quasi-periodic route is unlike the LK equations and present an interpretation of the route to chaos that differs from the previous one [3]. The dynamics observed in these systems are manifested from the interplay of two main frequencies, the relaxation oscillation and external cavity frequencies. The relaxation oscillation frequency results from the exchange between photon and carrier populations in the cavity. The external cavity frequency is given by the inverse of the delay time between the laser field and the field re-injected by the external reflective surface. The external-cavity frequency gives the location in optical frequency of attractors known as external cavity modes (ECMs). The separation between each ECM is given by the external cavity frequency and ECMs are labeled with integer numbers. Positive (negative) integers correspond to a higher (lower) frequency from the free running optical frequency of the laser which is labeled ECM 0. These steady state solutions form an ellipse in the optical phase and carrier population plane, which is a projection of phase space. In our experiments for low levels of feedback, the optical intensity, carrier number, and optical phase are characterized by stable operation around a single attractor ECM 0. As feedback is increased, an instability occurs in which the laser operates stably around ECM +1 and the laser’s optical frequency moves between ECM +1 and negative ECMs as seen in Fig. 1. As the laser’s optical frequency is moving toward negative ECMs, the photon and carrier populations undergo periodic oscillations near the relaxation oscillation frequency. We observe that the laser switches periodically between stable emission around ECM +1 and movement toward negative ECMs. For larger feedback, the laser moves around ECM +2 and the laser is characterized in the photon and carrier populations by limit cycles near the relaxation oscillation frequency, while the optical phase is stably around ECM 2