Soizic Terrien

Bifurcation Analysis of the Yamada Model for a Pulsing Semiconductor Laser with Saturable Absorber and Delayed Optical Feedback

Semiconductor lasers exhibit a wealth of dynamics, from emission of a constant beam of light to periodic oscillations and excitability. Self-pulsing regimes, where the laser periodically releases a short pulse of light, are particularly interesting for many applications, from material science to telecommunications. Self-pulsing regimes need to produce pulses very regularly and, as such, they are also known to be particularly sensitive to perturbations, such as noise or light injection. We investigate the effect of delayed optical feedback on the dynamics of a self-pulsing semiconductor laser with saturable absorber (SLSA). More precisely, we consider the Yamada model with delay---a system of three delay-differential equations (DDEs) for two slow and one fast variables---which has been shown to reproduce accurately self-pulsing features as observed in SLSA experimentally. This model is also of broader interest because it is quite closely related to mathematical models of other self-pulsing systems, such as...

Quasiperiodic Dynamics in a Micropillar Laser with Saturable Absorber and Delayed Optical Feedback

By considering a simple delay differential equation model, we show that modulated pulse trains measured in a micropillar semiconductor laser with saturable absorber and optical feedback can be interpreted as quasiperiodic oscillations on a torus.

Dynamics of a micropillar laser with saturable absorber and delayed optical feedback (Conference Presentation)

We consider a VCSEL micropillar laser with an internal absorber section, which has been shown to show excitable behaviour before its threshold. The addition of optical feedback offers the possibility to generate a pulse train of low jitter with a repetition time that is controlled by the length of the feedback loop. We perform a bifurcation analysis of the governing Yamada equations for the intensity, gain and absorption and with delayed feedback. This reveals an increasing variety of new dynamics and a considerable degree of multistability when the delay time is increased. More concretely, we present a bifurcation/stability diagram in the plane of feedback delay and feedback strength --- the two parameters of the external feedback loop. The organising feature is a winding curve of Hopf bifurcations, which develops an increasing amount of self-intersections as the delay of the feedback is increased. Along certain parts of the Hopf curve stable periodic solutions are created, which are physically self-pulsations with an increasing number of pulses in the external cavity. Several of these self-pulsations may coexist stably, and their basins of attractions are intermingled in a complicated way. As a result, the micropillar laser is very sensitive to small perturbations and noise while in such a multistable configuration. We also find multifrequency dynamics, where the amplitude of the self-pulsations is strongly modulated.

Pulse train control in an excitable micropillar laser with delayed optical feedback (Conference Presentation)

Recent experiments with an excitable VCSEL micropillar laser with delayed optical feedback demonstrated that the system is able to sustain trains of optical pulses. The laser has two layers of gain and one layer of absorption in the VCSEL cavity, and it is an excitable single longitudinal and transverse mode laser. With optical feedback, a past pulse can trigger a new pulse, creating a pulse train with repetition rate given by the delay time. It is possible to trigger and retime pulses by appropriate external perturbations, in the form of appropriately timed short optical pulses. In particular, several pulse trains can be triggered independently by optical perturbations, and sustained simultaneously in the external cavity, with different timing in between pulses. Such dynamics are also called localised structures, and are investigated here theoretically. It has been verified experimentally and theoretically that the phase of the electric field is not relevant. The Yamada model – a well-established system of ordinary differential equations for intensity, gain and absorption – is thus a suitable model. As we show, the Yamada model with delayed intensity feedback describes the pulsing micropillar laser system in good agreement with the experiment. A bifurcation analysis of this model shows that several pulsing periodic solution with different repetition rates coexist and are stable. Although coexisting pulse trains can seem independent on the timescale of the experiment, we show that they correspond here to extremely long transient dynamics toward one of the stable periodic solutions, with equidistant pulses.

Spiking and pulse train dynamics in a neuromimetic micropillar laser (Conference Presentation)

Processing of information with optical spikes could present an alternative path with a reduced energy consumption. It could also be well suited in the framework of novel brain-inspired computation paradigms. We investigate the spiking and pulse train dynamics in a micropillar laser with integrated saturable absorber. The optically-pumped microcavity laser is based on a specifically optimized design. The solitary laser can emit sub-nanosecond Q-switched pulses above laser threshold. Below threshold, the laser is in the so-called excitable regime, a generic all-or-none kind of response also found in biological neurons. We demonstrate several neuromimetic properties of the micropillar laser including the relative and absolute refractory periods and the temporal summation. The latter gives rise to sensitive and fast coincidence detectors of optical signals. In the configuration with delayed optical feedback, the system is shown experimentally and theoretically to sustain controllable trains of dissipative temporal solitons controlled by adequate optical perturbations. We show that the pulse train can be started or resynchronized (retiming) with a single perturbation and that the system can store a large variety of temporal pulse patterns. We discuss the role of pump noise that may terminate a pulse train. We demonstrate a strong asymmetry in the effect of noise on the switch on and off processes, as well as a peculiar role played by noise timing. Besides its interest as a compact source of controllable pulses, this system can be arranged if needed in arrays leading to interesting prospects for artificial optical neural networks.

Stability and long-term behaviour of pulse trains in an excitable microlaser with delayed optical feedback (Conference Presentation)

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.

Bifurcation analysis of nonlinear coupled photonic nanocavities

Recently, various research groups have studied spontaneous symmetry breaking and bistable behaviour in both active and passively driven coupled photonic nanocavities[1]. This work is of interest both in itself and because it naturally extends to coupled nonlinear quantum systems described by the Bose-Hubbard model [2]. Here we present a theoretical study of the dynamics in such a system by considering the bifurcation behaviour as the different parameters are varied. In normalised units the model can be written as: dA/dt = i(δ τ + |A|²)A-A+(iκ + γ)τB+f (1) dB/dt = i(δ τ + |B|²)B-B+(iκ + γ)τA+f (2) where A and B are the slowly varying normalised amplitudes of the fields in each nano-cavity. Further, δ is the detuning from the cavity resonance, τ is the photon lifetime, κ and γ describe the linear coupling between the cavities (if γ = 0 and κ < 0 the model is identical to the Bose-Hubbard model [2] in the semiclassical regime), while f represents a coherent driving term. In deriving the above equations it is assumed that the cavities are identical, and so a focus is the switching between the symmetrical state [A(t) = B(t)] and asymmetrical states, which must come in pairs. Fig. 1(a) shows a typical boundary curve dividing the (δ, f) plane into the regions where the symmetric and asymmetric solutions exist. Further analysis shows that more complicated dynamics arise from the asymmetric state. Fig. 1(b) shows the field intensity for the symmetric and asymmetric steady-states as a function of f, where stable solutions are represented by solid and unstable ones by dashed curves. Hopf bifurcations, marked by H, cause the asymmetric solution to change stability. A typical bifurcating periodic solution is shown in Fig. 1(c) for f = 6.

Noise-induced death of temporal external cavity solitons

As compact sources of light pulses, Q-switched semiconductor lasers are suitable for many applications, including, in particular, telecommunications. We investigate here the dynamics of a micropillar semiconductor laser with an internal saturable absorber subject to delayed optical feedback [1]. Both theoretically and experimentally, this device has been shown to produce pulse-like periodic solutions, also referred to as temporal external cavity solitons [1, 2].

Multistable dynamics in a micropillar laser with saturable absorber and delayed optical feedback

Multistability between several pulsing solutions in a model of laser with saturable absorber and optical feedback is studied. High sensitivity to small perturbations is discussed and emerges as an interpretation of experimentally-observed dynamics.