Excitability in Optical Systems close to Z2 Symmetry: Theory and Experiment

We recently proposed a mechanism for excitability in systems with a weakly broken Z2 symmetry, which includes optical units such as semiconductor ring lasers (SRLs). SRLs are a modern class of semiconductor lasers whose active cavity is characterized by a circular geometry. This enables the laser to support two counterpropagating modes, referred to as the clockwise (CW) and the counterclockwise (CCW) mode. The breaking of the Z2 symmetry is realized by making the linear coupling between the counterpropagating modes asymmetric. This can be achieved by increasing the reflection of, e.g., the CW mode into the CCW mode. This will stabilize lasing in the CCW mode and, when in the excitable regime, the SRL will fire optical pulses in the CW mode as a response to noise perturbations.
In this contribution we experimentally and theoretically characterize these pulses. Our experiments reveal a statistical distribution of the characteristics of the optical pulses that is not observed in regular excitable systems. In particular, an inverse correlation exists between the pulse amplitude and duration. Numerical simulations and an interpretation in an asymptotic phase show that the observed distribution of the pulse amplitude and duration results from the characteristics of the excitability mechanism. Excitable behavior in optical systems has generally been shown to occur close to a fold bifurcation and a homoclinic bifurcation of a stable limit cycle. In these systems, the response to noise is governed by the presence of an accessible saddle point S embedded in the separatrix. Pulses are activated by noise-induced fluctuations that connect the resting state to S, and the large deterministic excursion takes place along the unstable manifold of S. However, in systems with a weakly broken Z2 symmetry, the accessible saddle is not embedded in the separatrix and the unstable manifold of S does not participate in the excitability mechanism. It is the folding of the stable manifold that participates in the pulse generation. The absence of a compressive flow between and perpendicular to this stable manifold explains the observed variety in pulse shapes.
We also consider asymmetric SRLs coupled through a single bus waveguide. This is a first step towards an integrated optical neural network using semiconductor ring lasers as building blocks. We will show that for weak coupling, excitatory excursions still persist due to the similar phase space structure. Moreover, they can excite pulses in each other and can thus function as communicating neurons. This type of neural network can be fully integrated on chip and does not suffer from the drawback of needing extra-cavity measures, such as optical injection or saturable absorbers.