Optoelectronic chaos

Neuromorphic regenerative memory optoelectronic oscillator

Neuromorphic spiking information processing based on neuron-like excitable effect has achieved rapid development in recent years due to its advantages such as ultra-high operation speed, programming-free implementation and low power consumption. However, the current physical platforms lack building blocks like compilers, logic gates, and more importantly, data memory. These factors become the shackles to construct a full-physical layer neural network. In this paper, a neuromorphic regenerative memory scheme is proposed based on a time-delayed broadband nonlinear optoelectronic oscillator (OEO), which enables reshaping and regenerating on-off keying encoding sequences. Through biasing the dual-drive Mach-Zehnder electro-optic modulator in the OEO cavity near its minimum transmission point, the OEO can work in excitable regime, where localized states are maintained for robust nonlinear spiking response. Both simulation and experiment are carried out to demonstrate the proposed scheme, where the simulation results and the experimental results fit in with each other. The proposed OEO-based neuromorphic regenerative memory scheme exhibits long-term response ability for short-term excitation, which shows an enormous application potential for high-speed neuromorphic information buffering, optoelectronic interconnection and computing.

Chaotic microlasers caused by internal mode interaction for random number generation

Chaotic semiconductor lasers have been widely investigated for generating unpredictable random numbers, especially for lasers with external optical feedback. Nevertheless, chaotic lasers under external feedback are hindered by external feedback loop time, which causes correlation peaks for chaotic output. Here, we demonstrate the first self-chaotic microlaser based on internal mode interaction for a dual-mode microcavity laser, and realize random number generation using the self-chaotic laser output. By adjusting mode frequency interval close to the intrinsic relaxation oscillation frequency, nonlinear dynamics including self-chaos and period-oscillations are predicted and realized numerically and experimentally due to internal mode interaction. The internal mode interaction and corresponding carrier spatial oscillations pave the way of mode engineering for nonlinear dynamics in a solitary laser. Our findings provide a novel and easy method to create controllable and robust optical chaos for high-speed random number generation.

Photonic reinforcement learning based on optoelectronic reservoir computing

Reinforcement learning has been intensively investigated and developed in artificial intelligence in the absence of training data, such as autonomous driving vehicles, robot control, internet advertising, and elastic optical networks. However, the computational cost of reinforcement learning with deep neural networks is extremely high and reducing the learning cost is a challenging issue. We propose a photonic on-line implementation of reinforcement learning using optoelectronic delay-based reservoir computing, both experimentally and numerically. In the proposed scheme, we accelerate reinforcement learning at a rate of several megahertz because there is no required learning process for the internal connection weights in reservoir computing. We perform two benchmark tasks, CartPole-v0 and MountanCar-v0 tasks, to evaluate the proposed scheme. Our results represent the first hardware implementation of reinforcement learning based on photonic reservoir computing and pave the way for fast and efficient reinforcement learning as a novel photonic accelerator.

Fast physical repetitive patterns generation for masking in time-delay reservoir computing

Albeit the conceptual simplicity of hardware reservoir computing, the various implementation schemes that have been proposed so far still face versatile challenges. The conceptually simplest implementation uses a time delay approach, where one replaces the ensemble of nonlinear nodes with a unique nonlinear node connected to a delayed feedback loop. This simplification comes at a price in other parts of the implementation; repetitive temporal masking sequences are required to map the input information onto the diverse states of the time delay reservoir. These sequences are commonly introduced by arbitrary waveform generators which is an expensive approach when exploring ultra-fast processing speeds. Here we propose the physical generation of clock-free, sub-nanosecond repetitive patterns, with increased intra-pattern diversity and their use as masking sequences. To that end, we investigate numerically a semiconductor laser with a short optical feedback cavity, a well-studied dynamical system that provides a wide diversity of emitted signals. We focus on those operating conditions that lead to a periodic signal generation, with multiple harmonic frequency tones and sub-nanosecond limit cycle dynamics. By tuning the strength of the different frequency tones in the microwave domain, we access a variety of repetitive patterns and sample them in order to obtain the desired masking sequences. Eventually, we apply them in a time delay reservoir computing approach and test them in a nonlinear time-series prediction task. In a performance comparison with masking sequences that originate from random values, we find that only minor compromises are made while significantly reducing the instrumentation requirements of the time delay reservoir computing system.

Time-delay signature characteristics of the chaotic output from an optoelectronic oscillator by introducing an optical feedback

In this work, via autocorrelation function (ACF) and permutation entropy (PE) methods, we numerically investigate the time-delay signature (TDS) characteristics of the chaotic signal output from an optoelectronic oscillator (OEO) after introducing an extra optical feedback loop. The results demonstrate that, for such a chaotic system, both the optoelectronic feedback with a delay time of T₁ and the optical feedback with a delay time of T₂ contribute to the TDS of generated chaos. The TDS of the chaotic signal should be evaluated within a large time window including T₁ and T₂ by the strongest peak in the ACF curve of the chaotic signal, and the strongest peak may locate at near T₁ or T₂. Through mapping the evolution of the TDS in the parameter space of the optical feedback strength and time, certain optimized parameter regions for achieving a chaotic signal with a relatively weak TDS can be determined.

Contrasting chaotic with stochastic dynamics via ordinal transition networks

We introduce a representation space to contrast chaotic with stochastic dynamics. Following the complex network representation of a time series through ordinal pattern transitions, we propose to assign each system a position in a two-dimensional plane defined by the permutation entropy of the network (global network quantifier) and the minimum value of the permutation entropy of the nodes (local network quantifier). The numerical analysis of representative chaotic maps and stochastic systems shows that the proposed approach is able to distinguish linear from non-linear dynamical systems by different planar locations. Additionally, we show that this characterization is robust when observational noise is considered. Experimental applications allow us to validate the numerical findings and to conclude that this approach is useful in practical contexts.

The Eigenvalue Complexity of Sequences in the Real Domain

The eigenvalue is one of the important cryptographic complexity measures for sequences. However, the eigenvalue can only evaluate sequences with finite symbols—it is not applicable for real number sequences. Recently, chaos-based cryptography has received widespread attention for its perfect dynamical characteristics. However, dynamical complexity does not completely equate to cryptographic complexity. The security of the chaos-based cryptographic algorithm is not fully guaranteed unless it can be proven or measured by cryptographic standards. Therefore, in this paper, we extended the eigenvalue complexity measure from the finite field to the real number field to make it applicable for the complexity measurement of real number sequences. The probability distribution, expectation, and variance of the eigenvalue of real number sequences are discussed both theoretically and experimentally. With the extension of eigenvalue, we can evaluate the cryptographic complexity of real number sequences, which have a great advantage for cryptographic usage, especially for chaos-based cryptography.

Unraveling the decay of the number of unobserved ordinal patterns in noisy chaotic dynamics

In this paper, we introduce a model to describe the decay of the number of unobserved ordinal patterns as a function of the time series length in noisy chaotic dynamics. More precisely, we show that a stretched exponential model fits the decay of the number of unobserved ordinal patterns for both discrete and continuous chaotic systems contaminated with observational noise, independently of the noise level and the sampling time. Numerical simulations, obtained from the logistic map and the x coordinate of the Lorenz system, both operating in a totally chaotic dynamics were used as test beds. In addition, we contrast our results with those obtained from pure stochastic dynamics. The fitting parameters, namely, the stretching exponent and the characteristic decay rate, are used to distinguish whether the dynamical nature of the data sequence is stochastic or chaotic. Finally, the analysis of experimental records associated with the hyperchaotic pulsations of an optoelectronic oscillator allows us to illustrate the applicability of the proposed approach in a practical context.

Advancing Fourier: space-time concepts in ultrafast optics, imaging, and photonic neural networks

The concepts of Fourier optics were established in France in the 1940s by Pierre-Michel Duffieux, and laid the foundations of an extensive series of activities in the French research community that have touched on nearly every aspect of contemporary optics and photonics. In this paper, we review a selection of results where applications of the Fourier transform and transfer functions in optics have been applied to yield significant advances in unexpected areas of optics, including the spatial shaping of complex laser beams in amplitude and in phase, real-time ultrafast measurements, novel ghost imaging techniques, and the development of parallel processing methodologies for photonic artificial intelligence.

Laminar Chaos in Experiments: Nonlinear Systems with Time-Varying Delays and Noise

A new type of dynamics called laminar chaos was recently discovered through a theoretical analysis of a scalar delay differential equation with time-varying delay. Laminar chaos is a low-dimensional dynamics characterized by laminar phases of nearly constant intensity with periodic durations and a chaotic variation of the intensity from one laminar phase to the next laminar phase. This is in stark contrast to the typically observed higher-dimensional turbulent chaos, which is characterized by strong fluctuations. In this Letter we provide the first experimental observation of laminar chaos by studying an optoelectronic feedback loop with time-varying delay. The noise inherent in the experiment requires the development of a nonlinear Langevin equation with variable delay. The results show that laminar chaos can be observed in higher-order systems, and that the phenomenon is robust to noise and a digital implementation of the variable time delay.

Ultrafast physical random bit generation from a chaotic oscillator with a silicon modulator

We demonstrate physical random bit (PRB) generation from two chaotic optoelectronic oscillators (OEOs) with silicon Mach-Zehnder and microring resonator modulators. The carrier-injection modulation and the beam interference provide the nonlinearity for the OEO. We digitalize the chaotic waveforms from the two OEOs at 40 GS/s with 8-bit resolution and use self-delay bitwise exclusive-or operation as a post-processing method. The randomness of the resulting 320 Gbps PRB sequences is verified by the National Institute of Standards and Technology Special Publication 800-22 statistical tests. With the progress of silicon photonic circuits, there is a potential to fabricate a monolithic chaotic OEO chip for compact PRB generation.

Polarization chaos and random bit generation in nonlinear fiber optics induced by a time-delayed counter-propagating feedback loop

In this manuscript, we experimentally and numerically investigate the chaotic dynamics of the state-of-polarization in a nonlinear optical fiber due to the cross-interaction between an incident signal and its intense backward replica generated at the fiber-end through an amplified reflective delayed loop. Thanks to the cross-polarization interaction between the two-delayed counter-propagating waves, the output polarization exhibits fast temporal chaotic dynamics, which enable a powerful scrambling process with moving speeds up to 600-krad/s. The performance of this all-optical scrambler was then evaluated on a 10-Gbit/s On/Off Keying telecom signal achieving an error-free transmission. We also describe how these temporal and chaotic polarization fluctuations can be exploited as an all-optical random number generator. To this aim, a billion-bit sequence was experimentally generated and successfully confronted to the dieharder benchmarking statistic tools. Our experimental analysis are supported by numerical simulations based on the resolution of counter-propagating coupled nonlinear propagation equations that confirm the observed behaviors.

Consistency in experiments on multistable driven delay systems

We investigate the consistency properties in the responses of a nonlinear delay optoelectronic intensity oscillator subject to different drives, in particular, harmonic and self-generated waveforms. This system, an implementation of the Ikeda oscillator, is operating in a closed-loop configuration, exhibiting its autonomous dynamics while the drive signals are additionally introduced. Applying the same drive multiple times, we compare the dynamical responses of the optoelectronic oscillator and quantify the degree of consistency among them via their correlation. Our results show that consistency is not restricted to conditions close to the first Hopf bifurcation but can be found in a broad range of dynamical regimes, even in the presence of multistability. Finally, we discuss the dependence of consistency on the nature of the drive signal.

Finite-time Lyapunov exponents in time-delayed nonlinear dynamical systems

We introduce a method for the calculation of finite-time Lyapunov exponents in time-delayed nonlinear dynamical systems. We apply the method to the Mackey-Glass model with time-delayed feedback. We investigate the standard deviation of the probability distribution of the finite-time Lyapunov exponents when the finite time or the delay time is changed. It is found that the standard deviation decreases in a power-law scaling with the exponent ∼0.5 as the finite time or the delay time is increased. Similar results are obtained for the finite-time Lyapunov spectrum.

Virtual chimera states for delayed-feedback systems

Time-delayed systems are found to display remarkable temporal patterns the dynamics of which split into regular and chaotic components repeating at the interval of a delay. This novel long-term behavior for delay dynamics results from strongly asymmetric nonlinear delayed feedback driving a highly damped harmonic oscillator dynamics. In the corresponding virtual space-time representation, the behavior is found to develop as a chimeralike state, a new paradigmatic object from the network theory characterized by the coexistence of synchronous and incoherent oscillations. Numerous virtual chimera states are obtained and analyzed, through experiment, theory, and simulations.

Photonic nonlinear transient computing with multiple-delay wavelength dynamics

We report on the experimental demonstration of a hybrid optoelectronic neuromorphic computer based on a complex nonlinear wavelength dynamics including multiple delayed feedbacks with randomly defined weights. This neuromorphic approach is based on a new paradigm of a brain-inspired computational unit, intrinsically differing from Turing machines. This recent paradigm consists in expanding the input information to be processed into a higher dimensional phase space, through the nonlinear transient response of a complex dynamics excited by the input information. The computed output is then extracted via a linear separation of the transient trajectory in the complex phase space. The hyperplane separation is derived from a learning phase consisting of the resolution of a regression problem. The processing capability originates from the nonlinear transient, resulting in nonlinear transient computing. The computational performance is successfully evaluated on a standard benchmark test, namely, a spoken digit recognition task.

Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing

Many information processing challenges are difficult to solve with traditional Turing or von Neumann approaches. Implementing unconventional computational methods is therefore essential and optics provides promising opportunities. Here we experimentally demonstrate optical information processing using a nonlinear optoelectronic oscillator subject to delayed feedback. We implement a neuro-inspired concept, called Reservoir Computing, proven to possess universal computational capabilities. We particularly exploit the transient response of a complex dynamical system to an input data stream. We employ spoken digit recognition and time series prediction tasks as benchmarks, achieving competitive processing figures of merit.

Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the system's dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.

Synchronized cluster formation in coupled laser networks

We experimentally investigate the phase dynamics of laser networks with homogenous time-delayed mutual coupling and establish the fundamental rules that govern their state of synchronization. We identified a specific substructure that imposes its synchronization state on the entire network and show that for any coupling configuration the network forms at most two synchronized clusters. Our results indicate that the synchronization state of the network is a nonlocal phenomenon and cannot be deduced by decomposing the network into smaller substructures, each with its individual synchronization state.

Phase transitions of nonlinear waves in quadratic waveguide arrays

We study two-color parametric nonlinear modes in waveguide arrays with a quadratic nonlinear response. We predict theoretically and observe experimentally a new type of phase transition manifested in an abrupt power-controlled change of the mode structure from unstaggered to staggered, due to the interplay of localization and synchronization in parametrically driven discrete systems.

Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback

Incomplete homoclinic scenarios were recently measured in a semiconductor laser with optoelectronic feedback. We show here that such a laser contains cascades of spirals of periodic oscillations and hubs which look identical to the familiar ones observed in complete homoclinic scenarios. This means that hubs are far more general than presumed so far, being not limited by Shilnikov's theorem. Laser hubs open the possibility of measuring complex distributions of non-Shilnikov laser oscillations, and we briefly discuss how to do it.