Refractory period of an excitable semiconductor laser with optical injection

Characterizing the spike timing of a chaotic laser by using ordinal analysis and machine learning

Semiconductor lasers with optical feedback are well-known nonlinear dynamical systems. Under appropriate feedback conditions, these lasers emit optical pulses that resemble neural spikes. Influenced by feedback delay and various noise sources, including quantum spontaneous emission noise, the dynamics are highly stochastic. A good understanding of the spike timing statistics is needed to develop photonic systems capable of using the fast-spiking laser output for novel applications, such as information processing or random number generation. Here we analyze experimental sequences of inter-spike intervals (ISIs) recorded when a sinusoidal signal was applied to the laser current. Different combinations of the DC value and frequency of the signal applied to the laser lead to ISI sequences with distinct statistical properties. This variability prompts an investigation into the relationship between experimental parameters and ISI sequence statistics, aiming to uncover potential encoding methods for optical spikes, since this can open a new way of encoding and decoding information in sequences of optical spikes. By using ordinal analysis and machine learning, we show that the ISI sequences have statistical ordinal properties that are similar to Flicker noise signals, characterized by a parameter α that varies with the signal that was applied to the laser current when the ISIs were recorded. We also show that for this dataset, the (α, permutation entropy) plane is more informative than the (complexity, permutation entropy) plane because it allows better differentiation of ISI sequences recorded under different experimental conditions, as well as better differentiation of original and surrogate ISI sequences.

Uniform spike trains in optically injected quantum cascade oscillators

It has been experimentally and theoretically analyzed that noise-induced excitability in quantum well and quantum dot semiconductor laser systems usually produces sharp spike patterns of non-uniform amplitude. In this paper, we experimentally record that a quantum cascade oscillator injected externally with a monochromatic laser beam exhibits a series of highly uniform spike trains, which occur in the proximity of the saddle-node bifurcation. Theoretical analysis based on a properly designed single-mode rate equation model endowed with quantum noise reveals that this high uniformity has its primary origin in the ultrashort carrier lifetime of the quantum cascade laser gain medium that is typically close to 1 ps.

Stochastic photonic spiking neuron for Bayesian inference with unsupervised learning

Stochasticity is an inherent feature of biological neural activities. We propose a noise-injection scheme to implement a GHz-rate stochastic photonic spiking neuron (S-PSN). The firing-probability encoding is experimentally demonstrated and exploited for Bayesian inference with unsupervised learning. In a breast diagnosis task, the stochastic photonic spiking neural network (S-PSNN) can not only achieve a classification accuracy of 96.6%, but can also evaluate the diagnosis uncertainty with prediction entropies. As a result, the misdiagnosis rate is reduced by 80% compared to that of a conventional deterministic photonic spiking neural network (D-PSNN) for the same task. The GHz-rate S-PSN endows the neuromorphic photonics with high-speed Bayesian inference for reliable information processing in error-critical scenarios.

Refractory times for excitable dual-state quantum dot laser neurons

Excitable photonic systems show promise for ultrafast analog computation, several orders of magnitude faster than biological neurons. Optically injected quantum dot lasers display several excitable mechanisms with dual-state quantum lasers recently emerging as true all-or-none excitable artificial neurons. For use in applications, deterministic triggering is necessary and this has previously been demonstrated in the literature. In this work we analyze the crucially important refractory time for this dual-state system, which defines the minimum time between distinct pulses in any train.

Inhibitory and excitatory integration with a quantum dot laser neuron

Neuromorphic computing has garnered a lot of attention in recent years. Excitable photonic systems in particular demonstrate great potential for ultrafast, controllable spike processing. Optically injected quantum dot lasers display several distinct excitable regimes. We demonstrate here that optically injected dual-state quantum dot lasers can display the classic leaky integrate-and-fire mechanism where the integration of several sub-threshold perturbations can yield an effective supra-threshold perturbation. Intriguingly, a contrasting integrate-and-inhibit mechanism is demonstrated in this work where the integration of two supra-threshold perturbations yields an effective sub-threshold perturbation similar to the pre-pulse inhibition mechanism of biological neurons. This is the first such mechanism in neuromorphic photonics to the best of our knowledge.

Optical information processing using dual state quantum dot lasers: complexity through simplicity

We review results on the optical injection of dual state InAs quantum dot-based semiconductor lasers. The two states in question are the so-called ground state and first excited state of the laser. This ability to lase from two different energy states is unique amongst semiconductor lasers and in combination with the high, intrinsic relaxation oscillation damping of the material and the novel, inherent cascade like carrier relaxation process, endows optically injected dual state quantum dot lasers with many unique dynamical properties. Particular attention is paid to fast state switching, antiphase excitability, novel information processing techniques and optothermally induced neuronal phenomena. We compare and contrast some of the physical properties of the system with other optically injected two state devices such as vertical cavity surface emitting lasers and ring lasers. Finally, we offer an outlook on the use of quantum dot material in photonic integrated circuits.

Triple point of synchronization, phase singularity, and excitability along the transition between unbounded and bounded phase oscillations in a forced nonlinear oscillator

We report the discovery of a codimension-two phenomenon in the phase diagram of a second-order self-sustained nonlinear oscillator subject to a constant external periodic forcing, around which three regimes associated with the synchronization phenomenon exist, namely phase-locking, frequency-locking without phase-locking, and frequency-unlocking states. The triple point of synchronization arises when a saddle-node homoclinic cycle collides with the zero-amplitude state of the forced oscillator. A line on the phase diagram where limit-cycle solutions contain a phase singularity departs from the triple point, giving rise to a codimension-one transition between the regimes of frequency unlocking and frequency locking without phase locking. At the parameter values where the critical transition occurs, the forced oscillator exhibits a separatrix with a π phase jump, i.e., a particular trajectory in phase space that separates two distinct behaviors of the phase dynamics. Close to the triple point, noise induces excitable pulses where the two variants of type-I excitability, i.e., pulses with and without 2π phase slips, appear stochastically. The impacts of weak noise and some other dynamical aspects associated with the transition induced by the singular phenomenon are also discussed.

Asymmetric excitable phase triggering in an optically injected semiconductor laser

One of the defining characteristics of excitability is the existence of an excitable threshold: the minimum perturbation amplitude necessary to produce an excitable response. We analyze an optically injected dual state quantum dot laser, previously shown to display a dual state stochastic excitable dynamic. We show that deterministic triggering of this dynamic can be achieved via optical phase perturbations. Further, we demonstrate that there are in fact two asymmetric excitable thresholds in this system corresponding to the two possible directions of optical phase perturbations. For fast enough perturbations, an excitable interval arises, and there is a limit to the perturbation amplitude, above which excitations no longer arise, a phenomenon heretofore unobserved in studies of excitability.

The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback

We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop and their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states and homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains.

Hopping and emergent dynamics of optical localized states in a trapping potential

The position and motion of localized states of light in propagative geometries can be controlled via an adequate parameter modulation. Here, we show theoretically and experimentally that this process can be accurately described as the phase locking of oscillators to an external forcing and that non-reciprocal interactions between light bits can drastically modify this picture. Interactions lead to the convective motion of defects and to an unlocking as a collective emerging phenomenon.

Success rate analysis of the response of an excitable laser to periodic perturbations

We use statistical tools to characterize the response of an excitable system to periodic perturbations. The system is an optically injected semiconductor laser under pulsed perturbations of the phase of the injected field. We characterize the laser response by counting the number of pulses emitted by the laser, within a time interval, ΔT, that starts when a perturbation is applied. The success rate, SR(ΔT), is then defined as the number of pulses emitted in the interval ΔT, relative to the number of perturbations. The analysis of the variation of SR with ΔT allows separating a constant lag of technical origin and a frequency-dependent lag of physical and dynamical origin. Once the lag is accounted for, the success rate clearly captures locked and unlocked regimes and the transitions between them. We anticipate that the success rate will be a practical tool for analyzing the output of periodically forced systems, particularly when very regular oscillations need to be generated via small periodic perturbations.

Exploiting a monolithic passively Q-switched Nd:YAG laser to mimic a single neuron cell under periodic stimulation

A monolithic passively Q-switched Nd:YAG laser under periodic pulse pumping is originally exploited to emulate the response of a single neuron cell stimulated by periodic pulse inputs. Experimental results reveal that the output characteristics of the monolithic passively Q-switched laser can analogously manifest not only the firing patterns but also the frequency-locked plateaus of the single neuron cell. Moreover, the sine circle map is innovatively used to generate the output pulse sequences that can exactly correspond to experimental firing patterns. The present exploration indicates that a monolithic passively Q-switched solid-state laser is highly feasible to be developed as a compact artificial neuron cell.

Topological localized states in the time delayed Adler model: Bifurcation analysis and interaction law

The time-delayed Adler equation is the simplest model for an injected semiconductor laser with coherent injection and optical feedback. It is, however, able to reproduce the existence of topological localized structures (LSs) and their rich interactions. In this paper, we perform the first extended bifurcation analysis of this model and we explore the mechanisms by which LSs emerge. We also derive the effective equations governing the motion of distant LSs and we stress how the lack of parity in time-delayed systems leads to exotic, non-reciprocal, interactions between topological localized states.

Two distinct excitable responses for a laser with a saturable absorber

Excitable lasers with saturable absorbers are currently investigated as potential candidates for low level spike processing tasks in integrated optical platforms. Following a small perturbation of a stable equilibrium, a single and intense laser pulse can be generated before returning to rest. Motivated by recent experiments [Selmi et al., Phys. Rev. E 94, 042219 (2016)10.1103/PhysRevE.94.042219], we consider the rate equations for a laser containing a saturable absorber (LSA) and analyze the effects of different initial perturbations. With its three steady states and following Hodgkin classification, the LSA is a Type I excitable system. By contrast to perturbations on the intensity leading to the same intensity pulse, perturbations on the gain generate pulses of different amplitudes. We explain these distinct behaviors by analyzing the slow-fast dynamics of the laser in each case. We first consider a two-variable LSA model for which the conditions of excitability can be explored in the phase plane in a transparent manner. We then concentrate on the full three variable LSA equations and analyze its solutions near a degenerate steady bifurcation point. This analysis generalizes previous results [Dubbeldam et al., Phys. Rev. E 60, 6580 (1999)1063-651X10.1103/PhysRevE.60.6580] for unequal carrier density rates. Last, we discuss a fundamental difference between neuron and laser models.

Generation of extreme pulses on demand in semiconductor lasers with optical injection

The generation of extreme intensity pulses in an optically injected semiconductor laser is studied numerically by using a well-known rate equation model. We show that step-up perturbations of the laser pump current can trigger extreme pulses. We study the perturbation parameters (amplitude, duration) that are more likely to trigger a extreme pulse, and compare the properties of the generated extreme pulses with those spontaneous emitted, which are due to the intrinsic deterministic dynamics of the laser. We study how the phase of the optical field evolves during the pulses and compare both types of pulses (generated by external perturbations and generated by intrinsic nonlinear dynamics). We find that in both cases the phase dynamics is similar with an abrupt rise and fall: as an extreme pulse begins, the phase grows abruptly and reaches a local maximum at the peak of the pulse, then, when the pulse is over, the phase falls down to a value which is similar to the one before the pulse started.

Interactions and collisions of topological solitons in a semiconductor laser with optical injection and feedback

We present experimental and numerical results about dynamical interactions of topological solitons in a semiconductor laser with coherent injection and feedback. We show different kind of interactions such as repulsion, annihilation, or formation of soliton bound states, depending on laser parameters. Collisions between single structures and bound states conserve momentum and charge.