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Machine learning algorithms have opened a breach in the fortress of the prediction of high-dimensional chaotic systems. Their ability to find hidden correlations in data can be exploited to perform model-free forecasting of spatiotemporal chaos and extreme events. However, the extensive feature of these evolutions constitutes a critical limitation for full-size forecasting processes. Hence, the main challenge for forecasting relevant events is to establish the set of pertinent information. Here, we identify precursors from the transfer entropy of the system and a deep Long Short-Term Memory network to forecast the complex dynamics of a system evolving in a high-dimensional spatiotemporal chaotic regime. Performances of this triggerable model-free prediction protocol based on the information flowing map are tested from experimental data originating from a passive resonator operating in such a complex nonlinear regime. We have been able to predict the occurrence of extreme events up to 9 round trips after the detection of precursor, i.e., 3 times the horizon provided by Lyapunov exponents, with 92 % of true positive predictions leading to 60 % of accuracy.

Predicting complex nonlinear dynamical systems has been even more urgent because of the emergence of extreme events such as earthquakes, volcanic eruptions, extreme weather events (lightning, hurricanes/cyclones, blizzards, tornadoes), and giant oceanic rogue waves, to mention a few. The recent milestones in the machine learning framework offer a new prospect in this area [1] , [2] . For a high dimensional chaotic system, increasing the system’s size causes an augmentation of the complexity and, finally, more nodes in the network. Here, we propose a new supervised machine learning strategy to forecast bursts occurring in the turbulent regime of a fiber ring cavity. Indeed, we have recently demonstrated [3] that this system can continuously transform a stable periodic pattern with a well-defined frequency comb into a turbulent state via a spatiotemporal intermittency mechanism. Figure 1a ) shows an illustration of this transition. Even though a turbulent evolutions, an interesting feature is the persistent long-range correlation in the dynamics [3] . The system can then be seen as adjacent subdomains. Owing to this feature, instead of predicting the whole system, we use appropriate tools of chaos theory to identify recurrent picture in the past at the same location of the bursts. Figure 1b ) shows the statistics on the bursts (red) and the profiles detected at a previously computed time lag (blue). We have taken advantage of the apparent causality to make an association precursors-pulses. Knowing when and where the bursts may emerge, a fair question is " What is coming? ". On the other hand, as shown in Figure 1c ), no correlation can be found between the profile (amplitude and size) of the precursors of the corresponding pulse. However, using the pulses/precursors pairs to perform supervised learning with a recurrent neural network, we have obtained a high correlated map between the actual observation and the prediction, as can be seen in Figure 1d ). As our strategy deals with intrinsic characteristic quantities of the dynamical behavior, the pre-trained network can trigger the forecasting even for a more extensive system, provided that all the other parameters are the same.

We employ a polarisation-insensitive fibre optical parametric amplifier (PI-FOPA) to demonstrate a dual-band bi-directional 50-km reach symmetric 10G-PON link with a splitter power budget of 14dB. The PI-FOPA provides >16dB gain for a bursty upstream and non-burst downstream signals at 1533nm and 1586nnm respectively.

We propose a new counterintuitive mechanism of modulation instability: asymmetric, with respect to pump, losses can induce gain of sideband modes in absence of phase matching. The mechanism is experimentally demonstrated to generate frequency combs with tunable frequencies.

International audience; Modulational instability in passive optical resonators, the triggering mechanism of frequency comb and pulse train generation, is shown to exhibit transitions between regimes involving period-one (P1) versus period-two (P2) dynamical evolutions. The latter is a signature of parametric resonance occurring in the system, which can arise either from intrinsic cavity periodicity or from spatial modulation of the cavity parameters. We characterize the P1-P2 transition for both cases employing a fiber resonator where the intra-cavity fiber can be either uniform or dispersion modulated. The key element of our setup is a time lens which we exploit to resolve the temporal dynamics over successive round-trips, allowing crystal clear evidence of the existence of P1-P2 transitions for suitable changes of cavity parameters, as well as for the successful characterization of the relative temporal patterns. Our findings reveal new regimes where the averaged model known as Lugiato-Lefever equation turns out to be inadequate to explain the dynamics, whereas the results are correctly predicted and described on the basis of the full Ikeda map.

Optical frequency combs (OFCs), consisting of a set of phase-locked, equally spaced laser frequency lines, have enabled a great leap in precision spectroscopy and metrology since seminal works of Hänsch et al. Nowadays, OFCs are cornerstones of a wealth of further applications ranging from chemistry and biology to astrophysics and including molecular fingerprinting and light detection and ranging (LIDAR) systems, among others. Driven passive optical resonators constitute the ideal platform for OFC generation in terms of compactness and low energy footprint. We propose here a technique for the generation of OFCs with a tuneable repetition rate in externally driven optical resonators based on the gain-through-filtering process, a simple and elegant method, due to asymmetric spectral filtering on one side of the pump wave. We demonstrate a proof-of-concept experimental result in a fibre resonator, pioneering a new technique that does not require specific engineering of the resonator dispersion to generate frequency-agile OFCs., In normal dispersion, modulation instability, which is a precursor of Kerr combs, is forbidden due to phase mismatch. Here, the authors show the compensation of such phase mismatch by introducing frequency-dependent loss using a notch filter, hence leading to effective parametric gain and comb formation.

Optical frequency combs consist in a set of equally spaced mutually coherent laser frequency lines with a wealth of applications in the most diverse fields of photonics technology including metrology, astrophysics and molecular spectroscopy. Mode-locked lasers and passive driven resonators are the most relevant platforms for their generation. In Kerr resonators, in general frequency combs are generated through four wave mixing triggered by modulation instability (MI) in the anomalous dispersion regime [1]. In this case however the comb repetition rate is not tuneable but it is simply determined by the cavity free spectral range. We present here experimental results about a new method for frequency combs, with tuneable repetition rate, generation in normal dispersion externally driven passive optical resonators with Kerr (cubic) nonlinearity. We exploit the gain-through-losses (GTL) process to excite a modulation instability (MI) which initiates the comb formation. GTL enables energy transfer from a powerful pump field to sidebands frequencies thanks to frequency asymmetric spectral losses for signal and idler waves. The lossy modes themselves are counterintuitively amplified in virtue of the losses presence [2]. We performed an experiment considering a ring fibre resonator of 104 m length, made with a normal dispersion fiber with group velocity dispersion coefficient β 2 =0.5 ps2/km. The resonator was pumped at 1545 nm by a continuous wave laser. A fiber Bragg grating (FBG) with reflection peak located at 400 GHz frequency shift from the pump was used in reflection as a filter with spectral asymmetry with respect to the pump, inducing losses only to the signal (the filter causes no losses to the idler). The GTL MI causes first amplification of frequencies located close to the maximum losses (signal) and to their symmetric with respect to the pump wavelength (idler), which in turn generate higher order harmonics through a cascaded four-wave mixing process hence generating the comb (See Fig. 1(a)). The presence of the unavoidable filter phase contribution is responsible for the shift of the first sideband from the frequency corresponding to the maximum attenuation of the filter. The number of comb lines is limited by the low finesse of our resonator, but could be potentially improved in the future. In the time domain, the waveform corresponds to a train of pulses on the finite background (See Fig. 1(b)). The pulses have a repetition rate related to the inverse of the frequency shift between the first GTL sideband and the pump. Due to the fact that the GTL MI induces amplification of sidebands located close to the filter maximum losses frequency, by changing the detuning between pump and filter it is possible to tune the comb repetition rate. We have indeed verified a tuneability of the comb repetition rate by approximately 100 GHz.

We provide the first longitudinal characterization in phase and amplitude of the nonlinear stage of Modulation instability in optical fibers by means of a heterodyne optical time domain reflectometer. This original setup allows to report the symmetry breaking of the process due to an initial condition change.

International audience; We experimentally demonstrate a novel method based on the gain-through-losses modulation instability process to generate frequency combs with tunable repetition rate in a normal dispersion externally driven passive fibre resonator.

International audience; We provide a direct observation of a period-doubling phenomenon associated to the modulation instability in both uniform and dispersion oscillating passive fiber ring cavities. This has been done by performing roundtrip-to-roundtrip measurements of the temporal pattern generated through the instability.

Modulational instability (Ml) is a nonlinear process which results from the interplay between dispersive and nonlinear effects and leads to the exponential amplification of weak perturbations at the expense of a strong pump. In optical fibers, in large anomalous dispersion regions, MI is affected by the value of the group velocity dispersion (β 2 ) and can only exist in this spectral window. In weak dispersion regions, the contribution of higher order dispersion terms must be accounted for (β 4 ) to accurately model the system. Consequently, it can be shown that MI exists in weak normal dispersion regions [1]. In fiber cavities, most investigations have been carried out in large dispersion regions where the contribution of β 4 is negligible, except in a theoretical study [2].

International audience; We report the experimental observation of modulation instability in the weak normal dispersion region of a passive fiber ring cavity. We show that the fourth-order dispersion strongly modifies the dynamics of the cavity through the generation of new instability bands. Experimental results are in excellent agreement with theoretical predictions and numerical simulations.

International audience; We analyze the interaction between orthogonally polarized solitons and dispersive waves via four-wave mixing in a birefringent fiber. We calculate analytically the efficiency of the phase-sensitive scattering between orthogonally polarized soliton and dispersive wave. Experiments performed by using a photonic crystal fiber perfectly match the analytical predictions.

We predict analytically the generation efficiency of new frequencies produced by the phase-sensitive interaction between a soliton and a dispersive wave. The theoretical results are compared with experiments preformed in a photonic crystal fiber.

We demonstrate amplification of 17x100 GHz-spaced channel in C-band by a PI-FOPA based on Mach-Zehnder architecture with net gain >10 dB and confirmed its robustness over time. A commercial 100G-PDM-QPSK transponder was used to measure/compare the bit-error-rate versus received power with a commercial EDFA.

Modulation instability (MI) mechanism results from the interplay between dispersion, nonlinearity and detuning in passive fiber cavities. This process allows under specific conditions ruled by a phase matching relation to a weak perturbation to grow exponentially. Most investigations have been performed from self starting MI process initiated by noise and only a few focus on coherently seeded cavities [1]. Here, we study a cavity seeded by a strong pump surrounded by two symmetric side lobes (input seeds). We numerically demonstrate that under specific relative phase values between input seed waves and input pump the parametric gain experienced by weak perturbations can vanish. Preliminary experimental results to confirm these predictions are reported.