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International audience; We investigate theoretically and experimentally the development of instabilities in passive ring cavities composed of pieces of fiber with different dispersion. We derive an extended version of the Lugiato-Lefever equation that permits to model dispersion oscillating cavity and we demonstrate that this equation is valid well beyond the mean field approximation. We review the theory of Turing (modulational) and Faraday (parametric) instability in inhomogeneous fiber. We report the experimental demonstration of the generation of stable Turing and Faraday pattern in the same device , which can be controlled by changing the detun-ing and/or the input power. Moreover, we experimentally record the round-trip-to-round-trip dynamics of the spectrum, which shows that Turing and Faraday instabilities not only differ by their characteristic frequency but also by their dynamical behavior.

We consider the existence and propagation of nondiffractive and nondispersive spatiotemporal optical wavepackets in nonlinear Kerr media. We report analytically and confirm numerically the properties of spatiotemporal dark line solitary wave solutions of the (2 + 1)D nonlinear Schrödinger equation (NLSE). Dark lines represent holes of light on a continuous wave background. Moreover, we consider non-trivial web patterns generated under interactions of dark line solitary waves,which give birth to dark X solitary waves. These solitary waves are derived by exploiting the connection between the NLSE and a well-known equation of hydrodynamics, namely the (2 + 1)D type II Kadomtsev-Petviashvili (KP-II) equation. This finding opens a novel path for the excitation and control of optical solitary waves, of hydrodynamic nature.

International audience; We experimentally investigate the roundtrip-to-roundtrip dynamics of the modulation instability spectrum in a passive fiber ring cavity presenting an inhomogeneous dispersion profile. By implementing a real-time spectroscopy technique we are able to record successive single-shot spectra, which display the evolution of the system toward a stationary state. We find that the two instability regimes (Turing and Faraday) that compete in this kind of inhomogeneous cavities not only differ by their characteristic frequency but also by their dynamical behaviour. The dynamic transition between those two regimes of instability is also presented.

We address the degree of universality of the Fermi-Pasta-Ulam recurrence induced by multisoliton fission from a harmonic excitation by analyzing the case of the semiclassical defocusing nonlinear Schrödinger equation, which models nonlinear wave propagation in a variety of physical settings. Using a suitable Wentzel-Kramers-Brillouin approach to the solution of the associated scattering problem we accurately predict, in a fully analytical way, the number and the features (amplitude and velocity) of solitonlike excitations emerging post-breaking, as a function of the dispersion smallness parameter. This also permits us to predict and analyze the near-recurrences, thereby inferring the universal character of the mechanism originally discovered for the Korteweg-deVries equation. We show, however, that important differences exist between the two models, arising from the different scaling rules obeyed by the soliton velocities.

We investigate modulation instability in a dispersion modulated passive fiber-ring cavity with a very low local dispersion. In this configuration, an unprecedented number of quasi-phase-matched spectral sidelobes are observed, related to Faraday parametric resonances.

International audience; We present a theoretical and experimental study of the modulation instability process in a dispersion oscillating passive fiber-ring resonator in the low dispersion region. Generally, the modulation of the dispersion along the cavity length is responsible for the emergence of a regime characterised by multiple parametric resonances (or Faraday instabilities). We show that, under weak dispersion conditions, a huge number of Faraday sidebands can grow under the influence of fourth order dispersion. We specifically designed a piecewise uniform fiber-ring cavity and report on experiments that confirm our theoretical predictions. We recorded the dynamics of this system revealing strong interactions between the different sidebands in agreement with numerical simulations.

We report on the design of a hybrid III/V semiconductor/silicon three-port channel drop filter at 1,550 nm. The device is excited through a Si photonic wire positioned below the resonant cavity, which is in turn designed by properly drilling holes on a InP 1D-PhC wire. The output signal is collected by a photonic wire waveguide, laterally coupled to the resonator and terminated with a suitable mirror. To raise the drop efficiency above the theoretical limit of 50 % allowed by a design exploiting a single cavity, a different topology based on a two-cavity configuration is then investigated. In this case, a drop efficiency close to 100 % is obtained.

International audience; We report on the experimental generation of Riemann waves in an optical fiber system, allowing for the controlled formation of shock waves, as analytically described by a seminal equation of fluid dynamics: the so-called Inviscid Burgers’ Equation.

International audience; We investigate the nonlinear stage of modulational instability in dispersion oscillating fibers in normal dispersion regime. We unveil a heteroclinic structure leading to the excitability of superbreathers and parametric amplification outside the linear gain bandwidth.

International audience; We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.

International audience; We report the first observation of Riemann (simple) waves, which play a crucial role for understanding the dynamics of any shock-bearing system. This was achieved by properly tailoring the phase of an ultrashort light pulse injected into a highly nonlinear fiber. Optical Riemann waves are found to evolve in excellent quantitative agreement with the remarkably simple inviscid Burgers equation, whose applicability in physical systems is often challenged by viscous or dissipative effects. Our method allows us to further demonstrate a viable novel route to efficiently control the shock formation by the proper shaping of a laser pulse phase. Our results pave the way towards the experimental study, in a convenient benchtop setup, of complex physical phenomena otherwise difficult to access.

International audience; We study theoretically and numerically the dynamics of a passive optical fiber ring cavity pumped by a highly incoherent wave: an incoherently injected fiber laser. The theoretical analysis reveals that the turbulent dynamics of the cavity is dominated by the Raman effect. The forced-dissipative nature of the fiber cavity is responsible for a large diversity of turbulent behaviors: Aside from nonequilibrium statistical stationary states, we report the formation of a periodic pattern of spectral incoherent solitons, or the formation of different types of spectral singularities, e.g., dispersive shock waves and incoherent spectral collapse behaviors. We derive a mean-field kinetic equation that describes in detail the different turbulent regimes of the cavity and whose structure is formally analogous to the weak Langmuir turbulence kinetic equation in the presence of forcing and damping. A quantitative agreement is obtained between the simulations of the nonlinear Schrodinger equation with cavity boundary conditions and those of the mean-field kinetic equation and the corresponding singular integrodifferential reduction, without using adjustable parameters. We discuss the possible realization of a fiber cavity experimental setup in which the theoretical predictions can be observed and studied.

International audience; Considering the nonlinear Schrödinger (NLS) equation as a representative model, we report a unified presentation of different forms of incoherent shock waves that emerge in the long-range interaction regime of a turbulent optical wave system. These incoherent singularities can develop either in the temporal domain through a highly noninstantaneous nonlinear response, or in the spatial domain through a highly nonlocal nonlinearity. In the temporal domain, genuine dispersive shock waves (DSW) develop in the spectral dynamics of the random waves, despite the fact that the causality condition inherent to the response function breaks the Hamiltonian structure of the NLS equation. Such spectral incoherent DSWs are described in detail by a family of singular integro-differential kinetic equations, e.g. Benjamin–Ono equation, which are derived from a nonequilibrium kinetic formulation based on the weak Langmuir turbulence equation. In the spatial domain, the system is shown to exhibit a large scale global collective behavior, so that it is the fluctuating field as a whole that develops a singularity, which is inherently an incoherent object made of random waves. Despite the Hamiltonian structure of the NLS equation, the regularization of such a collective incoherent shock does not require the formation of a DSW — the regularization is shown to occur by means of a different process of coherence degradation at the shock point. We show that the collective incoherent shock is responsible for an original mechanism of spontaneous nucleation of a phase-space hole in the spectrogram dynamics. The robustness of such a phase-space hole is interpreted in the light of incoherent dark soliton states, whose different exact solutions are derived in the framework of the long-range Vlasov formalism.

We experimentally report the observation of dispersive shock waves from a short pulse superimposed to a small cw background in optical fibers. The visibility of intensity oscillations inherent to shock waves is strongly enhanced.

We observe the dispersive breaking of cosine-type long waves [Phys. Rev. Lett. 15, 240 (1965)] in shallow water, characterizing the highly nonlinear ``multisoliton'' fission over variable conditions. We provide new insight into the interpretation of the results by analyzing the data in terms of the periodic inverse scattering transform for the Korteweg--de Vries equation. In a wide range of dispersion and nonlinearity, the data compare favorably with our analytical estimate, based on a rigorous WKB approach, of the number of emerging solitons. We are also able to observe experimentally the universal Fermi-Pasta-Ulam recurrence in the regime of moderately weak dispersion.

Contrary to conventional coherent shocks, we show theoretically and experimentally that nonlocal turbulent flows lead to the emergence of large-scale incoherent shock waves, which constitute a collective phenomenon of the incoherent field as a whole.

Dispersive shock waves are strongly oscillating wave trains that spontaneously form and expand thanks to the action of weak dispersion, which contrasts the tendency, driven by the nonlinearity, to develop a gradient catastrophe. Here we review the basic concepts and recent progresses made in the description of such nonlinear waves, both in terms of experimental results and modelling. In particular, we discuss the formation of dispersive shocks in shallow water, which can be described in terms of Korteweg-de Vries or Whitham nonlocal equations. We contrast such results with those obtained in the field of nonlinear optics, described in terms of local or nonlocal nonlinear Schrodinger equations. Finally we show that a dispersive shock propagating under the action of small perturbations can radiate. A perturbative approach allows for the accurate prediction of the radiated frequencies.

We report on the experimental observation of inviscid Burgers’ equation dynamics and Riemann wave formation when a properly tailored optical pulse is coupled into a nonlinear fiber. Experimental results clearly show controllable pulse front steepening and shock formation.

International audience; We investigate modulational instability in inhomoge-neous passive cavities modeled by the Ikeda map. The cavity boundary conditions and the modulation of the fiber dispersion force the system to develop paramet-ric instabilities, which lead to the generation of simple as well as period doubled temporal patterns. The analytical results obtained by means of Floquet theory are validated through numerical solution of the Ikeda map, and the limitation of the mean-field Lugiato-Lefever model are highlighted.

International audience; We investigate the nonlinear stage of modulational instability in dispersion oscillating fibers in normal dispersion regime. We unveil a heteroclinic structure leading to the excitability of superbreathers and parametric amplification outside the linear gain bandwidth.

We introduce spatiotemporal optical dark X solitary waves of the (2+1)D hyperbolic nonlinear Schrodinger equation (NLSE), which rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are derived by exploiting the connection between the NLSE and a well-known equation of hydrodynamics, namely the type II Kadomtsev-Petviashvili (KP-II) equation. As a result, families of shallow water X soliton solutions of the KP-II equation are mapped into optical dark X solitary wave solutions of the NLSE. Numerical simulations show that optical dark X solitary waves may propagate for long distances (tens of nonlinear lengths) before they eventually break up, owing to the modulation instability of the continuous wave background. This finding opens a novel path for the excitation and control of X solitary waves in nonlinear optics.

The propagation of a continuous wave in the average anomalous dispersion region of a dispersion oscillating fiber is investigated numerically and experimentally. We demonstrate that the train of solitons arising from modulation instability is strongly affected by the periodic variations of the fiber dispersion. This leads to the emission of multiple resonant radiations located on both sides of the spectrum. Numerical simulations confirm the experimental results, and the position of the resonant radiations is well predicted by means of perturbation theory.

Contrary to conventional coherent shocks, we show theoretically and experimentally that nonlocal turbulent flows lead to the emergence of large-scale incoherent shock waves, which constitute a collective phenomenon of the incoherent field as a whole.

We analyze a microdisk resonator of sub-micrometer radius where confinement is obtained in a vertical sandwich characterized by a central slot of lower refractive index material. The properties of the antiguiding even modes characterized by TM fields peaked in the slot are analyzed with specific reference to the resonant wavelengths, the quality factor, the mode confinement properties, and the Purcell factor. It is shown, by means of a comparative analysis, that, for a small disk radius a full vectorial approach is required in order to properly account for the hybrid nature of the modes. This reflects on the dependence of such figures of merit upon geometrical parameters of the structure.

Numerical integration schemes of time-domain Maxwell equations in optical dispersive and nonlinear media are considered. In this context, we study quantitatively the impact of numerical dispersion on a typical nonlinear parametric conversion process, comparing the widely used finite difference (FDTD) approach and pseudo-spectral (PSTD) methods. Our results show that, unless using very dense grid, only fourth-order PSTD gives results in good quantitative agreement with standard coupled-mode theory.

International audience; We experimentally investigate the interplay of Turing (modulational) and Faraday (parametric) instabilities in a bistable passive nonlinear resonator. The Faraday branch is induced via parametric resonance owing to a periodic modulation of the resonator dispersion. We show that the bistable switching dynamics is dramatically affected by the competition between the two instability mechanisms , which dictates two completely novel scenarios. At low detunings from resonance switching occurs between the stable stationary lower branch and the Faraday-unstable upper branch, whereas at high detunings we observe the crossover between the Turing and Faraday periodic structures. The results are well explained in terms of the universal Lugiato-Lefever model.

We experimentally study modulational instability in a dispersion oscillating passive fiber ring cavity. This dispersion profile leads to new high-frequency gain bands that can be used for the generation of ultra-high repetition rate pulse trains.

International audience; We study, both theoretically and experimentally, modulational instability in optical fibers that have a longitudinal evolution of their dispersion in the form of a Dirac delta comb. By means of Floquet theory, we obtain an exact expression for the position of the gain bands, and we provide simple analytical estimates of the gain and of the bandwidths of those sidebands. An experimental validation of those results has been realized in several microstructured fibers specifically manufactured for that purpose. The dispersion landscape of those fibers is a comb of Gaussian pulses having widths much shorter than the period, which therefore approximate the ideal Dirac comb. Experimental spontaneous MI spectra recorded under quasi continuous wave excitation are in good agreement with the theory and with numerical simulations based on the generalized nonlinear Schr\"odinger equation.

Fundamental physical phenomena such as laser-induced ionization, driven quantum tunneling, Faraday waves, Bogoliubov quasiparticle excitations and the control of new states of matter rely on time-periodic driving of the system. A remarkable property of such driving is that it can induce the localized (bound) states to resonantly couple to the continuum. Therefore experiments that allow for enlightening and controlling the mechanisms underlying such coupling are of paramount importance. We implement such an experiment in a special optical fiber characterized by a dispersion oscillating along the propagation coordinate, which mimics “time”. The quasi-momentum associated with such periodic perturbation is responsible for the efficient coupling of energy from the localized wave-packets (solitons in anomalous dispersion and shock fronts in normal dispersion) sustained by the fiber nonlinearity, into free-running linear dispersive waves (continuum) at multiple resonant frequencies. Remarkably, the observed resonances can be explained by means of a unified approach, regardless of the fact that the localized state is a soliton-like pulse or a shock front.

Understanding turbulent flows arising from random dispersive waves that interact strongly through nonlinearities is a challenging issue in physics. Here we report the observation of a characteristic transition: strengthening the nonlocal character of the nonlinear response drives the system from a fully turbulent regime, featuring a sea of coherent small-scale dispersive shock waves (shocklets) towards the unexpected emergence of a giant collective incoherent shock wave. The front of such global incoherent shock carries most of the stochastic fluctuations and is responsible for a peculiar folding of the local spectrum. Nonlinear optics experiments performed in a solution of graphene nano-flakes clearly highlight this remarkable transition. Our observations shed new light on the role of long-range interactions in strongly nonlinear wave systems operating far from thermodynamic equilibrium, which reveals analogies with, for example, gravitational systems, and establishes a new scenario that can be common to many turbulent flows in photonic quantum fluids, hydrodynamics and Bose–Einstein condensates., Understanding turbulent flows arising from random dispersive waves that interact through nonlinearities is a challenging issue in physics. Here, the authors model and observe experimentally in a nonlinear optics set-up the transition between a sea of small-scale shocklets and a giant collective shock wave.

We investigate multi-wave mixing associated with the strongly pump depleted regime of induced modulation instability (MI) in optical fibers. For a complete transfer of pump power into the sideband modes, we theoretically and experimentally demonstrate that it is necessary to use a much lower seeding modulation frequency than the peak MI gain value. Our experiment shows that, at such optimal modulation frequency, a record 95 % of the output pump power is frequency converted into the comb of sidebands, in good quantitative agreement with analytical predictions based on the simplest exact breather solution of the nonlinear Schrodinger equation. (c) 2015 Optical Society of America

International audience; In this review we present recent theoretical and experimental progresses on modulation instability and parametric amplification processes in dispersion oscillating fibers. These optical fibers are characterized by longitudinal periodic variations of their outer diameter engineered over the meter-long scale, which provides an additional degree of freedom to the system and leads to the generation of multiple MI side band pairs. Main results published in single pass configurations and in passive cavities are summarized in this review.

We investigate wave collapse ruled by the generalized nonlinear Schrödinger (NLS) equation in 1+1 dimensions, for localized excitations with non-zero background, establishing through virial identities a new criterion for blow-up. When collapse is arrested, a semiclassical approach allows us to show that the system can favor the formation of dispersive shock waves. The general findings are illustrated with a model of interest to both classical and quantum physics (cubic-quintic NLS equation), demonstrating a radically novel scenario of instability, where solitons identify a marginal condition between blow-up and occurrence of shock waves, triggered by arbitrarily small mass perturbations of different sign.

We show that the intracavity field evolving in an externally driven passive Kerr resonator subject to weak normal dispersion undergoes wave-breaking, thus forming dispersive shock waves. At variance with the cavity-less propagation, such dispersive wave-breaking turns out to be strongly favored by cavity bistability and coexisting modulational instability.

International audience; We demonstrate that wave-breaking dramatically affects the dynamics of nonlinear frequency conversion processes that operate in the regime of high efficiency (strong multiple four-wave mixing). In particular, by exploiting an all-optical-fiber platform, we show that input modulations propagating in standard telecom fibers in the regime of weak normal dispersion lead to the formation of undular bores (dispersive shock waves) that mimic the typical behavior of dispersive hydrodynamics exhibited, e.g., by gravity waves and tidal bores. Thanks to the nonpulsed nature of the beat signal employed in our experiment, we are able to clearly observe how the periodic nature of the input modulation forces adjacent undular bores to collide elastically.

We study the wave breaking mechanism for the weakly dispersive defocusing nonlinear Schrödinger equation with a constant phase dark initial datum that contains a vacuum point at the origin. We prove by means of the exact solution to the initial value problem that, in the dispersionless limit, the vacuum point is preserved by the dynamics until breaking occurs at a finite critical time. In particular, both Riemann invariants experience a simultaneous breaking at the origin. Although the initial vacuum point is no longer preserved in the presence of a finite dispersion, the critical behavior manifests itself through an abrupt transition occurring around the breaking time.

International audience; Incoherent dispersive shock waves and collapselike singularities have been recently predicted to occur in the spectral evolution of an incoherent optical wave that propagates in a noninstantaneous nonlinear medium. Here we extend this work by considering the generalized nonlinear Schrödinger equation. We show that self-steepening significantly affects these incoherent spectral singularities: (i) It leads to a delay in the development of incoherent dispersive shocks, and (ii) it arrests the incoherent collapse singularity. Furthermore, we show that the spectral collapselike behavior can be exploited to achieve a significant enhancement (by two orders of magnitudes) of the degree of coherence of the optical wave.

We investigate wave collapse in the generalized nonlinear Schrodinger (NLS) equation and in the presence of a non vanishing background. Through the use of virial identities, we establish a new criterion for blow-up.

We predict the existence of incoherent dispersive shock waves and collapse-like singularities that occur in the spectral evolution of incoherent optical waves propagating in a noninstantaneous nonlinear medium.

International audience; We analyze resonant radiation emitted by dispersive shock waves owing to higher-order dispersive corrections of the leading term in the defocusing nonlinear Schrödinger equation. We give criteria for calculating the radiated frequency based on an analytical estimate of the shock velocity and reveal a diversity of scenarios controllable via the corrections, ranging from the radiation-induced transition of the dispersive shock into a classical-type shock to the qualitative modification of the underlying gradient catastrophe or the competition between different breaking mechanisms.

International audience; We investigate nonlinear focusing behavior of light beams propagating in beta-barium-borate (BBO) crystals under mismatched second-harmonic generation. We clearly identify experimentally multiple self-focusing and defocusing regions against the orientation angle, and the condition where competing quadratic and cubic nonlinearities perfectly compensate each other (zero-focusing point).

We investigate the propagation of two parallel polarized pump beams in the framework of a single nonlinear Schrodinger equation in order to thoroughly understand the modulation instability process in the regime of normal group-velocity dispersion. A linear stability analysis based on Floquet theory allows us to account for four-wave mixing and to contrast the results with those arising from incoherent nonlinear coupling due to cross-phase modulation only. Based on the nature of the unstable eigenvectors, we explain why, in the normal dispersion regime, modulation instability is not observed in the scalar configuration. Numerical simulations validate the analysis.

We review different formalisms describing incoherent waves: the wave turbulence kinetic equation, the Vlasov equation in analogy with Gravitation, the weak Langmuir turbulence equation describing spectral solitons and incoherent dispersive shocks.

International audience; We show that the use of a dispersion oscillating fiber in passive cavities significantly extend modulational instability to novel high-frequency bands, which also destabilize the branches of the steady response which are stable with homogeneous dispersion. By means of Floquet theory, we obtain exact explicit expression for the sideband gain, and a simple analytical estimate for the frequencies of maximum gain. Numerical simulations show that stable stationary trains of pulses can be excited in the cavity.

International audience; We investigate the frequency comb spectrum produced in an optical fiber via multiple four-wave mixing pumped in the normal group-velocity region close to the zero-dispersion wavelength. We show that the dynamics is strongly affected by shock formation. In this regime, the resonant radiation emitted by the shock waves correctly explains the enhanced spectral peaks in the comb. The resonant frequencies found by means of perturbation theory accurately fit those observed from the numerical simulation based on the generalised nonlinear Schrödinger equation.

International audience; We show that passive Kerr resonators pumped close to zero dispersion wavelengths on the normal dispersion side develop the resonant generation of linear waves driven by cavity (mixed dispersive-dissipative) shock waves. The resonance mechanism can be successfully described in the framework of the generalized Lugiato-Lefever equation with higher-order dispersive terms. Substantial differences with radiation from cavity solitons and purely dispersive shock waves dispersion are highlighted.

Summary form only given. After successful demonstration of cascading phenomena and solitary waves in quadratic media, we are challenged to fully understand and exploit second-harmonic generation in periodic media. These /spl chi//sup (2)/ nonlinear distributed feedback systems (NLDFBSs) exhibit noteworthy phenomena such as bistability and gap solitons. We analyze the CW field dynamics sustained by different cascading limits in quadratic NLDFBSs. Our analytical approach reveals the existence of spatial instability effects responsible for novel features such as strong frustration (clamping) of NLDFBS transmissivity in the limit of large nonlinear phase mismatch, and the possibility of pronounced multibranch multistable behavior when operating close to nonlinear phase-matching.