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We theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schr\"odinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation of the spectrum. In the latter, only the fundamental frequency and the upper and lower sidebands are retained. Plane wave solutions to the NLSE exhibit modulation instability (MI) within a frequency band determined by a linear stability analysis. For modulation frequencies inside the MI-band, we experimentally demonstrate that forcing and damping cause a separatrix crossing during the evolution. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We therefore extend our analysis to this system. However, our conclusions are general. When the system is damped by the viscosity of the water, it is pulled outside the separatrix, which in the real space corresponds to a phase-shift of the envelope and therefore doubles the period of the Fermi-Pasta-Ulam-Tsingou recurrence cycle. When the system is forced by the wind, it is pulled inside the separatrix. Furthermore, for modulation frequencies outside the conventional MI-band, we experimentally demonstrate that contrary to the linear prediction, we do observe a growth and decay cycle of the plane-wave modulation. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution., Comment: 12 pages, 9 figures

The viscosity of water induces a vorticity near the free surface boundary. The resulting rotational component of the fluid velocity vector greatly complicates the water wave system. Several approaches to close this system have been proposed. Our analysis compares three common sets of model equations. The first set has a rotational kinematic boundary condition at the surface. In the second set, a gauge choice for the velocity vector is made that cancels the rotational contribution in the kinematic boundary condition, at the cost of rotational velocity in the bulk and a rotational pressure. The third set circumvents the problem by introducing two domains: the irrotational bulk and the vortical boundary layer. This comparison puts forward the link between rotational pressure on the surface and vorticity in the boundary layer, addresses the existence of nonlinear vorticity terms, and shows where approximations have been used in the models. Furthermore, we examine the conservation of mass for the three systems, and how this can be compared to the irrotational case., Comment: 32 pages, 5 figures

We study statistical properties after a sudden episode of wind for water waves propagating in one direction. A wave with random initial conditions is propagated using a forced-damped higher order Nonlinear Schr\"odinger equation (NLS). During the wind episode, the wave action increases, the spectrum broadens, the spectral mean shifts up and the Benjamin-Feir index (BFI) and the kurtosis increase. Conversely, after the wind episode, the opposite occurs for each quantity. The kurtosis of the wave height distribution is considered the main parameter that can indicate whether rogue waves are likely to occur in a sea state, and the BFI is often mentioned as a means to predict the kurtosis. However, we find that while there is indeed a quadratic relation between these two, this relationship is dependent on the details of the forcing and damping. Instead, a simple and robust quadratic relation does exist between the kurtosis and the bandwidth. This could allow for a single-spectrum assessment of the likelihood of rogue waves in a given sea state. In addition, as the kurtosis depends strongly on the damping and forcing coefficients, by combining the bandwidth measurement with the damping coefficient, the evolution of the kurtosis after the wind episode can be predicted., Comment: 30 pages, 12 figures

We experimentally and numerically investigate the effect of wind forcing on the spectral dynamics of Akhmediev breathers, a wave-type known to model the modulation instability. We develop the wind model to the same order in steepness as the higher order modifcation of the nonlinear Schroedinger equation, also referred to as the Dysthe equation. This results in an asymmetric wind term in the higher order, in addition to the leading order wind forcing term. The derived model is in good agreement with laboratory experiments within the range of the facility's length. We show that the leading order forcing term amplifies all frequencies equally and therefore induces only a broadening of the spectrum while the asymmetric higher order term in the model enhances higher frequencies more than lower ones. Thus, the latter term induces a permanent upshift of the spectral mean. On the other hand, in contrast to the direct effect of wind forcing, wind can indirectly lead to frequency downshifts, due to dissipative effects such as wave breaking, or through amplification of the intrinsic spectral asymmetry of the Dysthe equation. Furthermore, the definitions of the up- and downshift in terms of peak- and mean frequencies, that are critical to relate our work to previous results, are highlighted and discussed., Comment: 30 pages, 11 figures

We study the triggering of single filaments due to turbulence in the beam path for a laser of power below the filamenting threshold. Turbulence can act as a switch between the beam not filamenting and producing single filaments. This 'positive' effect of turbulence on the filament probability, combined with our observation of off-axis filaments suggests the underlying mechanism is modulation instability caused by transverse perturbations. We hereby experimentally explore the interaction of modulation instability and turbulence, commonly associated with multiple-filaments, in the single-filament regime., Comment: 5 pages, 6 figures

We test numerical filamentation models against experimental data about the peak intensity and filament density in laser filaments. We show that the consideration of the higher-order Kerr effect (HOKE) improves the quantitative agreement without the need of adjustable parameters., Comment: 8 pages, 4 figures, 8 panels ; submitted to Opt. Lett

The recent measurement of negative higher-order Kerr effect (HOKE) terms in gases has given rise to a controversial debate, fed by its impact on short laser pulse propagation. By comparing the experimentally measured yield of the third and fifth harmonics, with both an analytical and a full comprehensive numerical propagation model, we confirm the absolute and relative values of the reported HOKE indices., Comment: 3 pages, 2 figures

We show that higher-order nonlinear indices ($n_4$, $n_6$, $n_8$, $n_{10}$) provide the main defocusing contribution to self-channeling of ultrashort laser pulses in air and Argon at 800 nm, in contrast with the previously accepted mechanism of filamentation where plasma was considered as the dominant defocusing process. Their consideration allows to reproduce experimentally observed intensities and plasma densities in self-guided filaments., Comment: 11 pages, 6 figures (11 panels)

We derive the spectral dependence of the non-linear susceptibility of any order, generalizing the common form of Sellmeier equations. This dependence is fully defined by the knowledge of the linear dispersion of the medium. This finding generalizes the Miller formula to any order of non-linearity. In the frequency-degenerate case, it yields the spectral dependence of non-linear refractive indices of arbitrary order., Comment: 12 pages, 1 figure (4 panels)

Based on the recently published generalized Miller formula, we derive the spectral dependence of the contribution of arbitrary-order non-linear indices to the group-velocity index. We show that in the context of laser filamentation in gases all experimentally-accessible orders (up to the $9^{th}$-order non-linear susceptibility $\chi^{(9)}$ in air and $\chi^{(11)}$ in argon) have contributions of alternative signs and similar magnitudes. Moreover, we show both analytically and numerically that the dispersion term of the non-linear indices must be considered when computing the intensity-dependent group velocity., Comment: 10 pages, 3 figures (14 panels)

Based on numerical simulations, we show that higher-order nonlinear indices (up to $n_8$ and $n_{10}$, respectively) of air and argon have a dominant contribution to both focusing and defocusing in the self-guiding of ultrashort laser pulses over most of the spectrum. Plasma generation and filamentation are therefore decoupled. As a consequence, ultraviolet wavelength may not be the optimal wavelengths for applications requiring to maximize ionization., Comment: 14 pages, 4 figures (14 panels)

As a contribution to the ongoing controversy about the role of higher-order Kerr effect (HOKE) in laser filamentation, we first provide thorough details about the protocol that has been employed to infer the HOKE indices from the experiment. Next, we discuss potential sources of artifact in the experimental measurements of these terms and show that neither the value of the observed birefringence, nor its inversion, nor the intensity at which it is observed, appear to be flawed. Furthermore, we argue that, independently on our values, the principle of including HOKE is straightforward. Due to the different temporal and spectral dynamics, the respective efficiency of defocusing by the plasma and by the HOKE is expected to depend substantially on both incident wavelength and pulse duration. The discussion should therefore focus on defining the conditions where each filamentation regime dominates., Comment: 22 pages, 11 figures. Submitted to Laser physics as proceedings of the Laser Physics 2010 conference

Modern laser sources nowadays deliver ultrashort light pulses reaching few cycles in duration, high energies beyond the Joule level and peak powers exceeding several terawatt (TW). When such pulses propagate through optically-transparent media, they first self-focus in space and grow in intensity, until they generate a tenuous plasma by photo-ionization. For free electron densities and beam intensities below their breakdown limits, these pulses evolve as self-guided objects, resulting from successive equilibria between the Kerr focusing process, the chromatic dispersion of the medium, and the defocusing action of the electron plasma. Discovered one decade ago, this self-channeling mechanism reveals a new physics, widely extending the frontiers of nonlinear optics. Implications include long-distance propagation of TW beams in the atmosphere, supercontinuum emission, pulse shortening as well as high-order harmonic generation. This review presents the landmarks of the 10-odd-year progress in this field. Particular emphasis is laid to the theoretical modeling of the propagation equations, whose physical ingredients are discussed from numerical simulations. Differences between femtosecond pulses propagating in gaseous or condensed materials are underlined. Attention is also paid to the multifilamentation instability of broad, powerful beams, breaking up the energy distribution into small-scale cells along the optical path. The robustness of the resulting filaments in adverse weathers, their large conical emission exploited for multipollutant remote sensing, nonlinear spectroscopy, and the possibility to guide electric discharges in air are finally addressed on the basis of experimental results., Comment: 50 pages, 38 figures

We present experimental and theoretical results for the angular dependence of third harmonic generation (THG) of water droplets in the micrometer range (size parameter $62