Two-color optically addressed spatial light modulator as a generic spatiotemporal system.

Nonlinear spatiotemporal systems are the basis for countless physical phenomena in such diverse fields as ecology, optics, electronics, and neuroscience. The canonical approach to unify models originating from different fields is the normal form description, which determines the generic dynamical aspects and different bifurcation scenarios. Realizing different types of dynamical systems via one experimental platform that enables continuous transition between normal forms through tuning accessible system parameters is, therefore, highly relevant. Here, we show that a transmissive, optically addressed spatial light modulator under coherent optical illumination and optical feedback coupling allows tuning between pitchfork, transcritical, and saddle-node bifurcations of steady states. We demonstrate this by analytically deriving the system's dynamical equations in correspondence to the normal forms of the associated bifurcations and confirm these results via extensive numerical simulations. Our model describes a nematic liquid crystal device using nano-dimensional dichalcogenide (a-As 2S 3) glassy thin films as photo sensors and alignment layers, and we use device parameters obtained from experimental characterization. Optical coupling, for example, using diffraction, holography, or integrated unitary maps allows implementing a variety of system topologies of technological relevance for neural networks and potentially Ising or XY-Hamiltonian models with ultralow energy consumption.

Coupled Nonlinear Delay Systems as Deep Convolutional Neural Networks.

Neural networks are transforming the field of computer algorithms, yet their emulation on current computing substrates is highly inefficient. Reservoir computing was successfully implemented on a large variety of substrates and gave new insight in overcoming this implementation bottleneck. Despite its success, the approach lags behind the state of the art in deep learning. We therefore extend time-delay reservoirs to deep networks and demonstrate that these conceptually correspond to deep convolutional neural networks. Convolution is intrinsically realized on a substrate level by generic drive-response properties of dynamical systems. The resulting novelty is avoiding vector matrix products between layers, which cause low efficiency in today's substrates. Compared to singleton time-delay reservoirs, our deep network achieves accuracy improvements by at least an order of magnitude in Mackey-Glass and Lorenz time series prediction.

Delayed dynamical systems: networks, chimeras and reservoir computing.

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays. This architecture allows the design of appropriate filters and multiple time delays, and greatly extends the possibilities for exploring synchronization patterns in arbitrary network topologies. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

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.

Dependence of quality factor on surface roughness in crystalline whispering-gallery mode resonators.

We present an experimental study of the variation of quality factor (Q-factor) of WGM resonators as a function of surface roughness. We consider mm-size whispering-gallery mode resonators manufactured with fluoride crystals, featuring Q-factors of the order of 1 billion at 1550 nm. The experimental procedure consists of repeated polishing steps, after which the surface roughness is evaluated using profilometry by white-light phase-shifting interferometry, while the Q-factors are determined using the cavity-ring-down method. This protocol permits us to establish an explicit curve linking the Q-factor of the disk-resonator to the surface roughness of the rim. We have performed measurements with four different crystals, namely, magnesium, calcium, strontium, and lithium fluoride. We have thereby found that the variations of Q-factor as a function of surface roughness is universal, in the sense that it is globally independent of the bulk material under consideration. We also discuss our experimental results in the light of theoretical estimates of surface scattering Q-factors already published in the literature.

Multi-Gbit/s optical phase chaos communications using a time-delayed optoelectronic oscillator with a three-wave interferometer nonlinearity.

We propose a chaos communication scheme based on a chaotic optical phase carrier generated with an optoelectronic oscillator with nonlinear time-delay feedback. The system includes a dedicated non-local nonlinearity, which is a customized three-wave imbalanced interferometer. This particular feature increases the complexity of the chaotic waveform and thus the security of the transmitted information, as these interferometers are characterized by four independent parameters which are part of the secret key for the chaos encryption scheme. We first analyze the route to chaos in the system, and evidence a sequence of period doubling bifurcations from the steady-state to fully developed chaos. Then, in the chaotic regime, we study the synchronization between the emitter and the receiver, and achieve chaotic carrier cancellation with a signal-to-noise ratio up to 20 dB. We finally demonstrate error-free chaos communications at a data rate of 3 Gbit/s.

Nonlinear Memory Capacity of Parallel Time-Delay Reservoir Computers in the Processing of Multidimensional Signals.

This letter addresses the reservoir design problem in the context of delay-based reservoir computers for multidimensional input signals, parallel architectures, and real-time multitasking. First, an approximating reservoir model is presented in those frameworks that provides an explicit functional link between the reservoir architecture and its performance in the execution of a specific task. Second, the inference properties of the ridge regression estimator in the multivariate context are used to assess the impact of finite sample training on the decrease of the reservoir capacity. Finally, an empirical study is conducted that shows the adequacy of the theoretical results with the empirical performances exhibited by various reservoir architectures in the execution of several nonlinear tasks with multidimensional inputs. Our results confirm the robustness properties of the parallel reservoir architecture with respect to task misspecification and parameter choice already documented in the literature.

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.

Kerr optical frequency comb generation in strontium fluoride whispering-gallery mode resonators with billion quality factor.

We report the fabrication for the first time of a strontium fluoride (SrF(2)) whispering-gallery mode resonator with quality factor in excess of 1 billion. The millimeter-size disk-resonator is polished until the surface roughness decreases down to a root-mean square value of 1.2 nm, as measured with a vertical scanning profilometer. We also demonstrate that this ultrahigh Q resonator allows for the generation of a normal-dispersion Kerr optical frequency comb at 1550 nm.

Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications.

Optical Kerr frequency combs are known to be effective coherent multiwavelength sources for ultrahigh capacity fiber communications. These combs are the frequency-domain counterparts of a wide variety of spatiotemporal dissipative structures, such as cavity solitons, chaos, or Turing patterns (rolls). In this Letter, we demonstrate that Turing patterns, which correspond to the so-called primary combs in the spectral domain, are optimally coherent in the sense that for the same pump power they provide the most robust carriers for coherent data transmission in fiber communications using advanced modulation formats. Our model is based on a stochastic Lugiato-Lefever equation which accounts for laser pump frequency jitter and amplified spontaneous emission noise induced by the erbium-doped fiber amplifier. Using crystalline whispering-gallery-mode resonators with quality factor Q∼10^{9} for the comb generation, we show that when the noise is accounted for, the coherence of a primary comb is significantly higher than the coherence of their solitonic or chaotic counterparts for the same pump power. In order to confirm this theoretical finding, we perform an optical fiber transmission experiment using advanced modulation formats, and we show that the coherence of the primary comb is high enough to enable data transmission of up to 144 Gbit/s per comb line, the highest value achieved with a Kerr comb so far. This performance evidences that compact crystalline photonic systems have the potential to play a key role in a new generation of coherent fiber communication networks, alongside fully integrated systems.

Incoherent resonant seeding of modulation instability in optical fiber.

We report control of the spectral and noise properties of spontaneous modulation instability (MI) in optical fiber using an incoherent seed with power at the 10(-6) level relative to the pump. We sweep the seed wavelength across the MI gain band, and observe significant enhancement of MI bandwidth and improvement in the signal-to-noise ratio as the seed coincides with the MI gain peak. We also vary the seed bandwidth and find a reduced effect on the MI spectrum as the seed coherence decreases. Stochastic nonlinear Schrödinger equation simulations of spectral and noise properties are in excellent agreement with experiment.

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.

Random walks and random numbers from supercontinuum generation.

We report a numerical study showing how the random intensity and phase fluctuations across the bandwidth of a broadband optical super-continuum can be interpreted in terms of the random processes of random walks and Lévy flights. We also describe how the intensity fluctuations can be applied to physical random number generation. We conclude that the optical supercontinuum provides a highly versatile means of studying and generating a wide class of random processes at optical wavelengths.

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.

Understanding and mitigating noise in trained deep neural networks.

Deep neural networks unlocked a vast range of new applications by solving tasks of which many were previously deemed as reserved to higher human intelligence. One of the developments enabling this success was a boost in computing power provided by special purpose hardware, such as graphic or tensor processing units. However, these do not leverage fundamental features of neural networks like parallelism and analog state variables. Instead, they emulate neural networks relying on binary computing, which results in unsustainable energy consumption and comparatively low speed. Fully parallel and analogue hardware promises to overcome these challenges, yet the impact of analogue neuron noise and its propagation, i.e. accumulation, threatens rendering such approaches inept. Here, we determine for the first time the propagation of noise in deep neural networks comprising noisy nonlinear neurons in trained fully connected layers. We study additive and multiplicative as well as correlated and uncorrelated noise, and develop analytical methods that predict the noise level in any layer of symmetric deep neural networks or deep neural networks trained with back propagation. We find that noise accumulation is generally bound, and adding additional network layers does not worsen the signal to noise ratio beyond a limit. Most importantly, noise accumulation can be suppressed entirely when neuron activation functions have a slope smaller than unity. We therefore developed the framework for noise in fully connected deep neural networks implemented in analog systems, and identify criteria allowing engineers to design noise-resilient novel neural network hardware.

Wideband chaos generation using a delayed oscillator and a two-dimensional nonlinearity induced by a quadrature phase-shift-keying electro-optic modulator.

We present a new optoelectronic architecture intended for chaotic optical intensity generation. The principle relies on an electro-optic nonlinear delay dynamics, where the nonlinearity originates from an integrated four-wave optical interferometer, involving two independent electro-optic modulation inputs. Consequently, the setup involves both two-dimensional nonlinearity and dual-delay feedback dynamics, which results in enhanced chaos complexity of particular interest in chaos encryption schemes. The generated chaos observed with large feedback gains has a bandwidth ranging from 30 kHz to 13 GHz and is confirmed by numerical simulations of the proposed dynamical model and bifurcation diagram calculation.

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.

Digital key for chaos communication performing time delay concealment.

We introduce a scheme that integrates a digital key in a phase-chaos electro-optical delay system for optical chaos communications. A pseudorandom binary sequence (PRBS) is mixed within the chaotic dynamics in a way that a mutual concealment is performed; e.g., the time delay is hidden by the binary sequence, and the PRBS is also masked by the chaos. In addition to bridging the gap between algorithmic symmetric key cryptography and chaos-based analog encoding, the proposed approach is intended to benefit from the complex algebra mixing between a (pseudorandom) Boolean variable, and another continuous time (chaotic) variable. The scheme also provides a large flexibility allowing for easy reconfigurations to communicate securely at a high bit rate between different systems.