Bounded phase phenomena in the optically injected laser

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.

Universal generation of devil's staircases near Hopf bifurcations via modulated forcing of nonlinear systems

The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher-order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue. The induced resonances can be precisely controlled via the modulation parameters.

Analysis of laser radiation using the Nonlinear Fourier transform

Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions.

Mathematical approaches for simultaneous characterisation of localized and extended fields in optical signals are not well developed. Here, the authors demonstrate the application of the Nonlinear Fourier transform approach based on the Zakharov-Shabat spectral problem for the analysis of fibre laser radiation.

Cross-predicting the dynamics of an optically injected single-mode semiconductor laser using reservoir computing

In real-world dynamical systems, technical limitations may prevent complete access to their dynamical variables. Such a lack of information may cause significant problems, especially when monitoring or controlling the dynamics of the system is required or when decisions need to be taken based on the dynamical state of the system. Cross-predicting the missing data is, therefore, of considerable interest. Here, we use a machine learning algorithm based on reservoir computing to perform cross-prediction of unknown variables of a chaotic dynamical laser system. In particular, we chose a realistic model of an optically injected single-mode semiconductor laser. While the intensity of the laser can often be acquired easily, measuring the phase of the electric field and the carriers in real time, although possible, requires a more demanding experimental scheme. We demonstrate that the dynamics of two of the three dynamical variables describing the state of the laser can be reconstructed accurately from the knowledge of only one variable, if our algorithm has been trained beforehand with all three variables for a limited period of time. We analyze the accuracy of the method depending on the parameters of the laser system and the reservoir. Finally, we test the robustness of the cross-prediction method when adding noise to the time series. The suggested reservoir computing state observer might be used in many applications, including reconstructing time series, recovering lost time series data and testing data encryption security in cryptography based on chaotic synchronization of lasers.

Excitable interplay between lasing quantum dot states

The optically injected semiconductor laser system has proven to be an excellent source of experimental nonlinear dynamics, particularly regarding the generation of excitable pulses. Typically for low-injection strengths, these pulses are the result of a small above-threshold perturbation of a stable steady state, the underlying physics is well described by the Adler phase equation, and each laser intensity pulse is accompanied by a 2π phase rotation. In this article, we show how, with a dual-state quantum dot laser, a variation of type I excitability is possible that cannot be described by the Adler model. The laser is operated so that emission is from the excited state only. The ground state can be activated and phase locked to the master laser via optical injection while the excited state is completely suppressed. Close to the phase-locking boundary, a region of ground-state emission dropouts correlated to excited-state pulses can be observed. We show that the phase of the ground state undergoes bounded rotations due to interactions with the excited state. We analyze the system both experimentally and numerically and find excellent agreement. Particular attention is devoted to the bifurcation conditions needed for an excitable pulse as well as its time evolution.

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.

Two-color bursting oscillations

Neurons communicate by brief bursts of spikes separated by silent phases and information may be encoded into the burst duration or through the structure of the interspike intervals. Inspired by the importance of bursting activities in neuronal computation, we have investigated the bursting oscillations of an optically injected quantum dot laser. We find experimentally that the laser periodically switches between two distinct operating states with distinct optical frequencies exhibiting either fast oscillatory or nearly steady state evolutions (two-color bursting oscillations). The conditions for their emergence and their control are analyzed by systematic simulations of the laser rate equations. By projecting the bursting solution onto the bifurcation diagram of a fast subsystem, we show how a specific hysteresis phenomenon explains the transitions between active and silent phases. Since size-controlled bursts can contain more information content than single spikes our results open the way to new forms of neuron inspired optical communication.

Frequency-locked chaotic opto-RF oscillator

A driven opto-RF oscillator, consisting of a dual-frequency laser (DFL) submitted to frequency-shifted feedback, is experimentally and numerically studied in a chaotic regime. Precise control of the reinjection strength and detuning permits isolation of a parameter region of bounded-phase chaos, where the opto-RF oscillator is frequency-locked to the master oscillator, in spite of chaotic phase and intensity oscillations. Robust experimental evidence of this synchronization regime is found, and phase noise spectra allow us to compare phase-locking and bounded-phase chaos regimes. In particular, it is found that the long-term phase stability of the master oscillator is well transferred to the opto-RF oscillator, even in the chaotic regime.

Coherence and incoherence in an optical comb

We demonstrate a coexistence of coherent and incoherent modes in the optical comb generated by a passively mode-locked quantum dot laser. This is experimentally achieved by means of optical linewidth, radio frequency spectrum, and optical spectrum measurements and confirmed numerically by a delay-differential equation model showing excellent agreement with the experiment. We interpret the state as a chimera state.

Measuring the universal synchronization properties of driven oscillators across a Hopf instability

When a driven oscillator loses phase-locking to a master oscillator via a Hopf bifurcation, it enters a bounded-phase regime in which its average frequency is still equal to the master frequency, but its phase displays temporal oscillations. Here we characterize these two synchronization regimes in a laser experiment, by measuring the spectrum of the phase fluctuations across the bifurcation. We find experimentally, and confirm numerically, that the low frequency phase noise of the driven oscillator is strongly suppressed in both regimes in the same way. Thus the long-term phase stability of the master oscillator is transferred to the driven one, even in the absence of phase-locking. The numerical study of a generic, minimal model suggests that such behavior is universal for any periodically driven oscillator near a Hopf bifurcation point.