Resource-constrained narrowband optoelectronic oscillator-based reservoir computing for classification of modulated signals.

We experimentally investigate the performance of narrowband optoelectronic oscillator (OEO) reservoir computers using the standard 10th-order nonlinear autoregressive-moving-average (NARMA10) task. Because comparing results from differently parameterized photonic time-delay systems can be difficult, we introduce a new, to the best of our knowledge, metric that accounts for system size, computational accuracy, and training effort overhead in order to provide an "at-a-glance" method to holistically determine a reservoir computer's performance. We then demonstrate the first experimental effort of narrowband OEO-based reservoir computing for the RADIOML dataset, which consists of recognizing and classifying IQ-modulated radio signals including analog and digital modulations. Our results indicate that narrowband OEOs are capable of achieving reasonable accuracies with exceptionally small training sets, thereby paving the way to real-time machine learning for radio frequency signals.

Multistability, relaxation oscillations, and chaos in time-delayed optoelectronic oscillators with direct laser modulation.

We investigate the nonlinear dynamics of an optoelectronic oscillator that is implemented with a laser diode (LD) with time-delayed feedback. In this system, electrical-to-optical conversion is directly implemented using the direct modulation of the laser diode itself, instead of an electrooptical modulator as in conventional architectures. Moreover, we consider the cubic nonlinear saturation of the characteristic laser power-intensity (P-I) transfer function far above threshold, instead of its simplified piecewise linear counterpart. We perform the stability analysis of the oscillator, and we show that it displays a rich dynamics that includes quasi-harmonic, relaxation oscillations, and chaos. We also show that the oscillator is strongly hysteretic and displays a wide variety of multistable behaviors, including the rare case of bistability between chaotic attractors. Our analytical and numerical results are found to be in good agreement with the experimental measurements.

Subharmonic instabilities in Kerr microcombs.

We report experimental observation of subharmonic mode excitation in primary Kerr optical frequency combs generated using crystalline whispering-gallery mode resonators. We show that the subcombs can be controlled and span a single or multiple free spectral ranges around the primary comb modes. In the spatial domain, the resulting multiscale combs correspond to an amplitude modulation of intracavity roll patterns. We perform a theoretical analysis based on eigenvalue decomposition that evidences the mechanism leading to the excitation of these combs.

Ultra-broadband Kerr microcomb through soliton spectral translation.

Broadband and low-noise microresonator frequency combs (microcombs) are critical for deployable optical frequency measurements. Here we expand the bandwidth of a microcomb far beyond its anomalous dispersion region on both sides of its spectrum through spectral translation mediated by mixing of a dissipative Kerr soliton and a secondary pump. We introduce the concept of synthetic dispersion to qualitatively capture the system's key physical behavior, in which the second pump enables spectral translation through four-wave mixing Bragg scattering. Experimentally, we pump a silicon nitride microring at 1063 nm and 1557 nm to enable soliton spectral translation, resulting in a total bandwidth of 1.6 octaves (137-407 THz). We examine the comb's low-noise characteristics, through heterodyne beat note measurements across its spectrum, measurements of the comb tooth spacing in its primary and spectrally translated portions, and their relative noise. These ultra-broadband microcombs provide new opportunities for optical frequency synthesis, optical atomic clocks, and reaching previously unattainable wavelengths.

Nonlinear dynamics of continuously tunable optoelectronic oscillators based on stimulated Brillouin amplification.

We present a theoretical analysis for tunable optoelectronic oscillators (OEOs) based on stimulated Brillouin scattering (SBS). A pump laser is used to generate a Brillouin gain which selectively amplifies a phase-modulated and contra-propagating laser signal. The radiofrequency beatnote generated after photodetection is filtered, amplified and fed back to the phase modulator to close the optoelectronic loop. Tunability is readily achieved by the adjustable detuning of the pump and signal lasers. OEOs based on stimulated Brillouin scattering have been successfully demonstrated at the experimental level, and they feature competitive phase noise performances along with continuous tunability for the output radiofrequency signal, up to the millimeter-wave band. However, the nonlinear dynamics of SBS-based OEOs remains largely unexplored at this date. In this article, we propose a model that describes the temporal dynamics of the microwave envelope, thereby allowing us to track the dynamics of the amplitude and phase of the radiofrequency signal. The corresponding nonlinear and time-delayed differential equation is then analyzed to reveal the underlying bifurcation behavior that emerges as the feedback gain is increased. It is shown that after the primary Hopf bifurcation that triggers the microwave oscillations, the system undergoes a secondary Neimark-Sacker bifurcation before fully developed chaos emerges for the highest gain values. We also propose a model for the chipscale version of this SBS-based OEO where the delay line is replaced by a highly nonlinear waveguide. The numerical simulations are found to be in excellent agreement with the analytical study.

Nonlinear dynamics in an optoelectronic feedback delay oscillator with piecewise linear transfer functions from the laser diode and photodiode.

We investigate the nonlinear dynamics of a recent architecture of an optoelectronic oscillator, where the emitting laser and the receiving diode are connected in a head-to-tail configuration via an optical fiber delay line. The resulting nonlinear transfer function is a piecewise linear profile, and its interplay with the delay leads to many complex behaviors such as relaxation oscillations and deterministic chaos. This system belongs to a recent class of optoelectronic oscillators where the nonlinearity does not originate from the sinusoidal transfer function of an imbalanced interferometer, and, in particular, it is a simple optoelectronic oscillator configuration that is capable of displaying a chaotic behavior. The results of the analytic study are confirmed by numerical simulations and experimental measurements.

Nonlinear dynamics of miniature optoelectronic oscillators based on whispering-gallery mode electrooptical modulators.

We propose a time-domain model to analyze the dynamical behavior of miniature optoelectronic oscillators (OEOs) based on whispering-gallery mode resonators. In these systems, the whispering-gallery mode resonator features a quadratic nonlinearity and operates as an electrooptical modulator, thereby eliminating the need for an integrated Mach-Zehnder modulator. The narrow optical resonances also eliminate the need for both an optical fiber delay line and an electric bandpass filter in the optoelectronic feedback loop. The architecture of miniature OEOs therefore appears as significantly simpler than the one of their traditional counterparts and permits us to achieve competitive metrics in terms of size, weight, and power. Our theoretical approach is based on the closed-loop coupling between the optical intracavity modes and the microwave signal generated via the photodetection of the output electrooptical comb. The resulting nonlinear oscillator model involves the slowly-varying envelopes of the microwave and optical fields, and its stability analysis permits the analytical determination the critical value of the feedback gain needed to trigger self-sustained oscillations. This stability analysis also allows us to understand how key parameters of the system such as cavity detuning or coupling efficiency influence the onset of the radiofrequency oscillation. Our study is complemented by time-domain simulations for the microwave and optical signals, which are in excellent agreement with the analytical predictions.

Fluctuations and correlations in Kerr optical frequency combs with additive Gaussian noise.

We investigate the effects of environmental stochastic fluctuations on Kerr optical frequency combs. This spatially extended dynamical system can be accurately studied using the Lugiato-Lefever equation, and we show that when additive noise is accounted for, the correlations of the modal field fluctuations can be determined theoretically. We propose a general theory for the computation of these field fluctuations and correlations, which is successfully compared to numerical simulations.

Machine learning based on reservoir computing with time-delayed optoelectronic and photonic systems.

The concept of reservoir computing emerged from a specific machine learning paradigm characterized by a three-layered architecture (input, reservoir, and output), where only the output layer is trained and optimized for a particular task. In recent years, this approach has been successfully implemented using various hardware platforms based on optoelectronic and photonic systems with time-delayed feedback. In this review, we provide a survey of the latest advances in this field, with some perspectives related to the relationship between reservoir computing, nonlinear dynamics, and network theory.

On the transition to secondary Kerr combs in whispering-gallery mode resonators.

We demonstrate that extended dissipative structures in Kerr-nonlinear whispering-gallery mode resonators undergo a spatiotemporal instability, as the pumping parameters are varied. We show that the dynamics of the patterns beyond this bifurcation yield specific Kerr comb and sub-comb spectra that can be subjected to a phase of frequency-locking when optimal conditions are met. Our numerical results are found to be in agreement with experimental measurements.

A normal form method for the determination of oscillations characteristics near the primary Hopf bifurcation in bandpass optoelectronic oscillators: Theory and experiment.

We propose a framework for the analysis of the integro-differential delay Ikeda equations ruling the dynamics of bandpass optoelectronic oscillators (OEOs). Our framework is based on the normal form reduction of OEOs and helps in the determination of the amplitude and the frequency of the primary Hopf limit-cycles as a function of the time delay and other parameters. The study is carried for both the negative and the positive slopes of the sinusoidal transfer function, and our analytical results are confirmed by the numerical and experimental data.

Spontaneous generation of orbital angular momentum crystals using a monolithic Nd:YAG nonplanar ring laser.

We report the emission of localized orbital angular momentum (OAM) crystals in a millimeter-size monolithic Nd:YAG nonplanar ring laser. Narrow-linewidth single-frequency lasing in the kilohertz level featuring crystal-like vortices is obtained via phase locking of Laguerre-Gaussian modes in the cavity. It is found that the spatially degenerate OAM of high-order LG modes can be easily broken by superimposing a low-order mode, leading to crystal-like vortices. Our theoretical analysis is found to be in agreement with the experimental results for both intensity and interference patterns.

A taxonomy of optical dissipative structures in whispering-gallery mode resonators with Kerr nonlinearity.

In this paper, the research related to the formation of optical dissipative structures in Kerr-nonlinear whispering-gallery mode resonators pumped with continuous-wave lasers is reviewed. Pattern formation in these systems can be analysed using the paradigmatic Lugiato-Lefever model, which is a partial differential equation ruling the dynamics of the intra-cavity laser field. Various dissipative structures such as Turing rolls, solitons, breathers and spatio-temporal chaos can emerge in the resonator depending on the laser power and frequency. The bifurcation analysis enables a classification of these patterns, and also permits identification of their basins of attraction.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Dynamical complexity and computation in recurrent neural networks beyond their fixed point.

Spontaneous activity found in neural networks usually results in a reduction of computational performance. As a consequence, artificial neural networks are often operated at the edge of chaos, where the network is stable yet highly susceptible to input information. Surprisingly, regular spontaneous dynamics in Neural Networks beyond their resting state possess a high degree of spatio-temporal synchronization, a situation that can also be found in biological neural networks. Characterizing information preservation via complexity indices, we show how spatial synchronization allows rRNNs to reduce the negative impact of regular spontaneous dynamics on their computational performance.

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.

Effect of crystalline family and orientation on stimulated Brillouin scattering in whispering-gallery mode resonators.

Ultra-high Q whispering-gallery mode resonators pumped by a continuous-wave laser are known to enhance stimulated Brillouin scattering when optimal resonance and phase-matching conditions are met. In crystalline resonators, this process depends critically on the crystal orientation and family, which impose the elastic constants defining the velocity of the acoustic waves. In this article, we investigate the effect of crystalline orientation and family on this velocity which is proportional to the Brillouin frequency down-shift. In particular, the study is based on the development of a model and numerical simulations of acoustic wave velocities that propagate along the periphery of four fluoride crystals, namely calcium, magnesium, lithium and barium fluoride. We find that depending on the crystal and its orientation, the frequency excursion around the Brillouin offset can vary from few tens of kHz to more than a GHz.

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.

Optimization of primary Kerr optical frequency combs for tunable microwave generation.

We analyze the condition under which Kerr combs generate the highest microwave output power after photodetection. These optimal comb states correspond to configurations in which the sidemode-to-pump ratio is the highest possible. For the case of primary combs, we show how the interplay between the power and frequency of the pump laser critically influences this ratio, which has a direct influence on the phase noise performance of the generated microwaves. We also experimentally demonstrate primary combs with a sidemode-to-pump ratio as high as -2 dB, thereby leading to efficient energy conversion from the lightwave to the microwave frequency range.

Laser-based optoelectronic generation of narrowband microwave chaos for radars and radio-communication scrambling.

This Letter proposes an optoelectronic oscillator architecture for narrowband microwave chaos generation. In the time domain, the microwave signal features a slowly varying envelope with amplitude and phase chaos while, in the frequency domain, it is quasi-indistinguishable from a band-limited white noise. A full theoretical analysis is performed to investigate the stability properties and route to chaos for the microwave oscillations. We experimentally generate the narrowband microwave chaos with a central frequency of 3 GHz and a bandwidth of only 16 MHz, and we discuss the applications for radar engineering and radio-communication scrambling.

On the phase noise performance of microwave and millimeter-wave signals generated with versatile Kerr optical frequency combs.

We investigate the phase noise performance of micro- and millimeter-wave signals generated using a ultra-high Q whispering gallery mode disk-resonator with Kerr nonlinearity. Our study focuses on the stability of the optical spectra and on the performances of the corresponding microwave and millimeter-wave beat notes in terms of power and phase noise. The blue slope of an optical mode of the resonator, allowing for the generation of optical frequency combs, is accurately explored in order to identify various comb patterns. Each of these patterns is characterized in the optical and radio-frequency domains. Phase noise levels below -100 dBc/Hz at 10 kHz offset have been achieved for beat notes in the radio-frequency spectrum at 12 GHz, 18 GHz, 24 GHz, 30 GHz, and 36 GHz with the same resonator.