Topological Properties of Floquet Winding Bands in a Photonic Lattice.

The engineering of synthetic materials characterized by more than one class of topological invariants is one of the current challenges of solid-state based and synthetic materials. Using a synthetic photonic lattice implemented in a two-coupled ring system we engineer an anomalous Floquet metal that is gapless in the bulk and shows simultaneously two different topological properties. On the one hand, this synthetic lattice presents bands characterized by a winding number. The winding emerges from the breakup of inversion symmetry, and it directly relates to the appearance of Bloch suboscillations within its bulk. On the other hand, the Floquet nature of the lattice results in well-known anomalous insulating phases with topological edge states. The combination of broken inversion symmetry and periodic time modulation studied here enriches the variety of topological phases available in lattices subject to Floquet driving and suggests the possible emergence of novel phases when periodic modulation is combined with the breakup of spatial symmetries.

Spatiotemporal observation of higher-order modulation instability in a recirculating fiber loop.

We experimentally investigate higher-order seeded modulation instability in an optical fiber experiment. The recirculating loop configuration with round trip losses compensation enables the observation in single-shot of the spatiotemporal evolution of an initially modulated continuous field revealing intricate yet deterministic dynamics. By tuning the modulation period, a continuous transition between perfectly coherent and purely noise-driven dynamics is observed that we characterize by means of a statistical study.

Single-shot observation of breathers from noise-induced modulation instability using heterodyne temporal imaging.

We report phase and amplitude measurements of large coherent structures originating from the noise-induced modulation instability in optical fibers. By using a specifically designed time-lens system (SEAHORSE) in which aberrations are compensated, the complex field is recorded in single-shot over long durations of 200 ps with sub-picosecond resolution. Signatures of Akhmediev breather-like patterns are identified in the ultrafast temporal dynamics in very good agreement with numerical predictions based on the nonlinear Schrödinger equation.

Nonlinear Spectral Synthesis of Soliton Gas in Deep-Water Surface Gravity Waves.

Soliton gases represent large random soliton ensembles in physical systems that exhibit integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media, their controlled experimental realization has been missing. We report a controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory [inverse scattering transform (IST)] for the one-dimensional focusing nonlinear Schrödinger equation. The soliton gas is experimentally generated in a one-dimensional water tank where we demonstrate that we can control and measure the density of states, i.e., the probability density function parametrizing the soliton gas in the IST spectral phase space. Nonlinear spectral analysis of the generated hydrodynamic soliton gas reveals that the density of states slowly changes under the influence of perturbative higher-order effects that break the integrability of the wave dynamics.

Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics.

We present an optical fiber experiment in which we examine the space-time evolution of a modulationally unstable plane wave initially perturbed by a small noise. Using a recirculating fiber loop as an experimental platform, we report the single-shot observation of the noise-driven development of breather structures from the early stage to the long-term evolution of modulation instability. Performing single-point statistical analysis of optical power recorded in the experiments, we observe decaying oscillations of the second-order moment together with the exponential distribution in the long-term evolution, as predicted by Agafontsev and Zakharov [Nonlinearity 28, 2791 (2015).NONLE50951-771510.1088/0951-7715/28/8/2791]. Finally, we demonstrate experimentally and numerically that the autocorrelation of the optical power g^{(2)}(τ) exhibits some unique oscillatory features typifying the nonlinear stage of the noise-driven modulation instability and of integrable turbulence.

Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics.

We report an optical fiber experiment in which we study the nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as the experimental platform, we show that the initial perturbation evolves into an expanding nonlinear oscillatory structure exhibiting some universal characteristics that agree with theoretical predictions based on integrability properties of the focusing nonlinear Schrödinger equation. Our experimental results demonstrate the persistence of the universal evolution scenario, even in the presence of small dissipation and noise in an experimental system that is not rigorously of an integrable nature.

Bound State Soliton Gas Dynamics Underlying the Spontaneous Modulational Instability.

We investigate the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The statistical properties of the noise-induced MI, observed previously in numerical simulations and in experiments, have not been explained theoretically. In this Letter, using the inverse scattering transform (IST) formalism, we propose a theoretical model of the asymptotic stage of the noise-induced MI based on N-soliton solutions of the focusing one-dimensional nonlinear Schrödinger equation. Specifically, we use ensembles of N-soliton bound states having a special semiclassical distribution of the IST eigenvalues, together with random phases for norming constants. To verify our model, we employ a recently developed numerical approach to construct an ensemble of N-soliton solutions with a large number of solitons, N∼100. Our investigation reveals a remarkable agreement between spectral (Fourier) and statistical properties of the long-term evolution of the MI and those of the constructed multisoliton, random-phase bound states. Our results can be generalized to a broad class of strongly nonlinear integrable turbulence problems.

Optical Random Riemann Waves in Integrable Turbulence.

We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schrödinger equation. This is done theoretically and experimentally, by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. Using a dispersive-hydrodynamic approach, we show that the development of IT can be divided into two distinct stages, the initial, prebreaking stage being described by a system of interacting random Riemann waves. We explain the low-tailed statistics of the wave intensity in IT and show that the Riemann invariants of the asymptotic nonlinear geometric optics system represent the observable quantities that provide new insight into statistical features of the initial stage of the IT development by exhibiting stationary probability density functions.

Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation.

We report experimental confirmation of the universal emergence of the Peregrine soliton predicted to occur during pulse propagation in the semiclassical limit of the focusing nonlinear Schrödinger equation. Using an optical fiber based system, measurements of temporal focusing of high power pulses reveal both intensity and phase signatures of the Peregrine soliton during the initial nonlinear evolution stage. Experimental and numerical results are in very good agreement, and show that the universal mechanism that yields the Peregrine soliton structure is highly robust and can be observed over a broad range of parameters.

Statistics of a turbulent Raman fiber laser.

We report the experimental study of statistical properties of partially coherent waves emitted by a Raman fiber laser operating in the normal dispersion regime. Using an asynchronous optical sampling technique, we accurately measure the probability density function of the optical power of the Stokes wave that exhibits strong and fast fluctuations. As predicted from numerical simulations presented by Randoux et al. [Opt. Lett.36, 790 (2011)], the statistical distributions of the intracavity Stokes power are found to be very different before and after reflection on the cavity Bragg mirrors. In particular, the Stokes wave incident on fiber Bragg grating mirrors exhibits statistics with tails that are much lower than those defined by the normal law.

Optical rogue waves in integrable turbulence.

We report optical experiments allowing us to investigate integrable turbulence in the focusing regime of the one-dimensional nonlinear Schrödinger equation (1D NLSE). In analogy with broad spectrum excitation of a one-dimensional water tank, we launch random initial waves in a single mode optical fiber. Using an original optical sampling setup, we measure precisely the probability density function of optical power of the partially coherent waves rapidly fluctuating with time. The probability density function is found to evolve from the normal law to a strong heavy-tailed distribution, thus revealing the formation of rogue waves in integrable turbulence. Numerical simulations of 1D NLSE with stochastic initial conditions quantitatively reproduce the experiments. Our numerical investigations suggest that the statistical features experimentally observed rely on the stochastic generation of coherent analytic solutions of 1D NLSE.

Intermittency in integrable turbulence.

We examine the statistical properties of nonlinear random waves that are ruled by the one-dimensional defocusing and integrable nonlinear Schrödinger equation. Using fast detection techniques in an optical fiber experiment, we observe that the probability density function of light fluctuations is characterized by tails that are lower than those predicted by a Gaussian distribution. Moreover, by applying a bandpass frequency optical filter, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian. These phenomena are very well described by numerical simulations of the one-dimensional nonlinear Schrödinger equation.

Soliton gas: Theory, numerics, and experiments.

The concept of soliton gas was introduced in 1971 by Zakharov as an infinite collection of weakly interacting solitons in the framework of Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted (rarefied) soliton gas, solitons with random amplitude and phase parameters are almost nonoverlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact. The notion of soliton gas is inherently associated with integrable wave systems described by nonlinear partial differential equations like the KdV equation or the one-dimensional nonlinear Schrödinger equation that can be solved using the inverse scattering transform. Over the last few years, the field of soliton gases has received a rapidly growing interest from both the theoretical and experimental points of view. In particular, it has been realized that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves. The recently discovered deep connections of soliton gas theory with generalized hydrodynamics have broadened the field and opened new fundamental questions related to the soliton gas statistics and thermodynamics. We review the main recent theoretical and experimental results in the field of soliton gas. The key conceptual tools of the field, such as the inverse scattering transform, the thermodynamic limit of finite-gap potentials, and generalized Gibbs ensembles are introduced and various open questions and future challenges are discussed.

Interaction of soliton gases in deep-water surface gravity waves.

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at leading order. We report hydrodynamic experiments in which we investigate the interaction between two beams or jets of soliton gases having nearly identical amplitudes but opposite velocities of the same magnitude. The space-time evolution of the two interacting soliton gas jets is recorded in a 140-m-long water tank where the dynamics is described at leading order by the focusing one-dimensional nonlinear Schrödinger equation. Varying the relative initial velocity of the two species of soliton gas, we change their interaction strength and we measure the macroscopic soliton gas density and velocity changes due to the interaction. Our experimental results are found to be in good quantitative agreement with predictions of the spectral kinetic theory of soliton gas despite the presence of perturbative higher-order effects that break the integrability of the wave dynamics.

Transient buildup of the optical power spectrum in Raman fiber lasers.

We implement an experimental technique enabling to study the transient buildup of the optical power spectrum in a Raman fiber laser. We investigate the way through which the laser optical power spectrum broadens before reaching its shape at steady-state.

Experimental evidence of extreme value statistics in Raman fiber lasers.

We present an experiment in which the intracavity broadened spectrum of a Raman fiber laser is sliced by using a narrow-bandwidth optical filter. High-contrast fast fluctuations of the Stokes power are observed at the output of the optical filter. The statistics of the power fluctuations strongly change with the central wavelength of the filter, and rare extreme events are found to be generated in the far wings of the spectrum.

Nonlinear dispersion relation in integrable turbulence.

We investigate numerically and experimentally the concept of nonlinear dispersion relation (NDR) in the context of partially coherent waves propagating in a one-dimensional water tank. The nonlinear random waves have a narrow-bandwidth Fourier spectrum and are described at leading order by the one-dimensional nonlinear Schrödinger equation. The problem is considered in the framework of integrable turbulence in which solitons play a key role. By using a limited number of wave gauges, we accurately measure the NDR of the slowly varying envelope of the deep-water waves. This enables the precise characterization of the frequency shift and the broadening of the NDR while also revealing the presence of solitons. Moreover, our analysis shows that the shape and the broadening of the NDR provides signatures of the deviation from integrable turbulence that is induced by high order effects in experiments. We also compare our experimental observations with numerical simulations of Dysthe and of Euler equations.

Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers.

We study theoretically, numerically and experimentally the nonlinear propagation of partially incoherent optical waves in single mode optical fibers. We revisit the traditional treatment of the wave turbulence theory to provide a statistical kinetic description of the integrable scalar NLS equation. In spite of the formal reversibility and of the integrability of the NLS equation, the weakly nonlinear dynamics reveals the existence of an irreversible evolution toward a statistically stationary state. The evolution of the power spectrum of the field is characterized by the rapid growth of spectral tails that exhibit damped oscillations, until the whole spectrum ultimately reaches a steady state. The kinetic approach allows us to derive an analytical expression of the damped oscillations, which is found in agreement with the numerical simulations of both the NLS and kinetic equations. We report the experimental observation of this peculiar relaxation process of the integrable NLS equation.

Intracavity changes in the field statistics of Raman fiber lasers.

We present an experiment in which intracavity optical power spectra of a Raman fiber laser are precisely measured both in the forward and in the backward propagation directions near the cavity mirrors. The statistical properties of the intracavity Stokes field are found to be very different before and after reflection on the cavity mirrors. The influence of both the dispersion and the spectral filtering actions of fiber Bragg grating mirrors are discussed.

Solitonic model of the condensate.

We consider a spatially extended box-shaped wave field that consists of a plane wave (the condensate) in the middle and equals zero at the edges, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. Within the inverse scattering transform theory, the scattering data for this wave field is presented by the continuous spectrum of the nonlinear radiation and the soliton eigenvalues together with their norming constants; the number of solitons N is proportional to the box width. We remove the continuous spectrum from the scattering data and find analytically the specific corrections to the soliton norming constants that arise due to the removal procedure. The corrected soliton parameters correspond to symmetric in space N-soliton solution, as we demonstrate analytically in the paper. Generating this solution numerically for N up to 1024, we observe that, at large N, it converges asymptotically to the condensate, representing its solitonic model. Our methods can be generalized for other strongly nonlinear wave fields, as we demonstrate for the hyperbolic secant potential, building its solitonic model as well.