Optical Rogue Waves in Integrable Turbulence

Abstract

We report optical fiber experiments allowing to investigate integrable turbulence in the focusing regime of the one dimensional nonlinear Schrödinger equation (1D-NLSE). Our experiments are very similar in their principle to water tank experiments with random initial conditions (see M. Onorato et al. Phys. Rev. E 70 067302 (2004)). Using an original optical sampling setup, we measure precisely the probability density function (PDF) of optical power of partially coherent waves rapidly fluctuating with time. The PDF 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 reproduce quantitatively the experiments. Our investigations suggest that the statistical features experimentally observed rely on the stochastic generation of coherent analytic solutions of 1D-NLSE such as Peregrine solitons.The field of nonlinear optics has recently grown as a favorable laboratory to investigate both statistical properties of nonlinear random waves and hydrodynamic-like phenomena [1][2][3]. In particular, several recent works point out analogies between hydrodynamics and nonlinear fiber optics in the observation of supercontinuum generation [4], undular bores [2], optical turbulence, laminarturbulent transition [1] or oceanographic rogue waves (RW) [5,6].Rogue waves (RW), also called freak waves, are extremely large amplitude waves occurring more frequently than expected from the normal law [7,8] As stressed out in the recent review [6], there is no obvious analogy between most of the optical experiments on extreme events and oceanography. However a direct correspondence between nonlinear optics and hydrodynamics is provided by the one-dimensional nonlinear Schrödinger equation (1D-NLSE) (see Eq. (1)) that describes various wave systems [6,18]. In particular, the focusing 1D-NLSE describes at leading order the physics of deep-water wave trains and it plays a central role in the study of RW [6-8, 18, 19].Modulational instability (MI) is believed to be a fundamental mechanism for the formation of RW [8,20]. Moreover, analytical solutions of the integrable 1D-NLSE such a Akhmediev breathers (AB), Peregrine solitons or Kuznetsov-Ma solitons (KMs) are now considered as possible prototypes of RW [19,[21][22][23]. These coherent structures have been generated from very specific, carefullydesigned coherent initial conditions in optical fiber experiments [21,23,24].On the contrary, oceanic RW emerge from the interplay of incoherent waves in turbulent systems. The occurrence of RW in wave turbulence has been theoretically studied in optics [25,26] and in hydrodynamics [27]. In hydrodynamical experiments made with one-dimensional water tanks, non gaussian statistics of the wave height has been found to emerge from random initial conditions [20,28].The appropriate theoretical framework combining a statistical approach of random waves together with the 1D-NLSE is integrable turbulence...

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

“…It has also been understood that rogue waves play important role in the characterization of the nonlinear stage of modulational instability [41], [17] and particularly, in the development of integrable turbulence [40], [1], [37], [33]. One of the universal mathematical models for the rogue wave description is the one-dimensional focusing Nonlinear Schrödinger (fNLS) equation,…”

“…Rogue waves are waves of unusually large amplitude Am, whose appearance statistics deviates from the Gaussian distribution by exhibiting "heavy tails" in the probability density function (see [37] and references therein). The conventional amplitude criterion for rogue waves is Am/As > 2, where As is the significant wave height defined in oceanography as the average wave height (trough to crest) of the highest third of waves (see e.g.…”

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

“…With the input pulse well-fitted by a Gaussian profile, the parameter T 0 = ∆τ /1.665 [13]. Simulations were used to select the input power such that the fibre length corresponded to close to the first compression distance, and the pulse intensity profile was measured using a customdesigned optical sampling oscilloscope (see [26] and Supplementary Information). The results obtained in this case are displayed in Fig.…”

Formation of rogue waves from a locally perturbed condensate

“…Later, J.M. Soto-Crespo, N. Devine and N. Akhmediev have demonstrated that in the case of higher condensate disturbances, the tails of the probability density function of wave amplitudes increases [38] (see also the works [38][39][40] where the non-Gaussian statistics was observed for the development of strongly fluctuating initial conditions). They concluded that only initial perturbations of significant amplitude can contain spatially localized breathers, that is consistent with the theory suggested here (the minimal amplitude of SLCP generated by breathers is determined by the perturbation width, see the second paragraph).…”