Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity

Abstract

A unidirectional optical oscillator is built by using a liquid crystal light-valve that couples a pump beam with the modes of a nearly spherical cavity. For sufficiently high pump intensity, the cavity field presents a complex spatio-temporal dynamics, accompanied by the emission of extreme waves and large deviations from the Gaussian statistics. We identify a mechanism of spatial symmetry breaking, due to a hypercycle-type amplification through the nonlocal coupling of the cavity field. . Associated with extreme waves are L-shaped statistics, with a probability of large peak occurrence much larger than predicted by Gaussian statistics. Different mechanisms to explain the origin of extreme waves have been proposed, including nonlinear focusing via modulational instability (MI) [4] and focusing of line currents [5]. In optical fibers, numerical simulations of the nonlinear Schrödinger equation (NLSE) [6] have established a direct analogy between optical and water rogue waves. In a spatially extended system, the formation of large amplitude and localized pulses, so-called optical needles, has been evidenced by numerical simulations for transparent media with saturating self-focusing nonlinearity [7]. Similar space-time phenomena, such as collapsing filaments, are also predicted in optical wave turbulence [8]. Recently, the spatio-temporal dynamics of MI-induced bright optical spots was observed in Ref. [9] and an algebraic power spectrum tail was reported due to momentum cascade [10]. Nevertheless, no experimental evidence has been given up to now of extreme waves in a spatially extended optical system.Here, we report, what is, at our knowledge, the first experimental evidence of extreme waves and non-Gaussian statistics in a 2D spatially extended optical system. The experiment consists of a nonlinear optical cavity, formed by a unidirectional ring oscillator with a liquid crystal light-valve (LCLV) as the gain medium. While for low pump the amplitude follows a Gaussian statistics, for sufficiently high pump we observe large deviations from Gaussianity, accompanied by the emission of extreme waves that appear on the transverse profile of the optical beam as genuine spatiotemporal phenomena, developing erratically in time and in space. The observations are confirmed by numerical simulation of the full model equations. Moreover, by introducing a mean-field simplified model, we show that extreme waves in the cavity are generated by a novel mechanism, which is based on a hypercycle-type amplification [11] occurring via nonlocal coupling of different spatial regions. The experimental setup is essentially the one described in [12]. The ring cavity is formed by three highreflectivity dielectric mirrors and a lens of f = 70 cm focal length. The total cavity length is L = 273.3 cm and the lens is positioned at a distance L 1 = 88.1 cm from the entrance plane of the LCLV. The coordinate system is taken such that z is along the cavity axis and x, y are on the transverse plane. A LCLV supplies the gain through a two-wave mixi...

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents

“…(5) The fastness of appearance of the RW and the duration of its stay can be regulated by yet another parameter linked to the ocean current. (6) The proposed solution and MI exhibit broken spatial symmetry as well as a directional preference, which are suspected to be the crucial features in the formation of a twodimensional RW [10,11,17,18]. Note again that these features obtained earlier through observation or numerical simulation found and confirmed in our model through exact analytic result.…”

“…We hope that this breakthrough in describing large ocean RWs by an analytic dynamical lump soliton with adjustable height, inclination and duration would also be valuable for experimental findings of two-dimensional RWs in other systems, such as capillary fluid waves [13] optical cavity waves [11] and basin water waves [10]. Derivation of our exact lump soliton from the breather solution of the integrable two-dimensional NLS equation presented here in a systematic way as well as to find higher order rational lump solitons would be challenging theoretical problems.…”

“…Note however that the two-dimensional NLS equation (1.6) is not an integrable system and gives only approximate numerical solutions with no stable soliton. Nevertheless, this unlikely candidate is found to exhibit RW-like structures numerically, with higher amplitude and modular inclination and with an intriguing directional preference [10,18] with broken spatial symmetry [11,17]. However, though the experimental and theoretical studies on nonlinear systems in twodimensional space have shown promise in describing more realistic situations in the formation of two-dimensional ocean RWs, unfortunately, all of them can give only approximate numerical results and most of these models could not consider the effect of ocean current, which is supposed to play a crucial role in the formation of ocean RWs [17,18].…”

“…This elusive freak wave caught the imagination of the broad scientific community quite recently [8][9][10][11][12][13][14][15], triggering off an upsurge in theoretical [7,[16][17][18] and experimental [5,[8][9][10][11][12][13][14][15] studies of this unique phenomenon. For identifying such extreme waves, the suggested signature of these rare events is a deviation of the probability distribution function (PDF) of the wave amplitude from its usual random Gaussian distribution (GD), by having a long-tail, indicating that the appearance of high-intensity pulses more often has much higher probability than that predicted by the GD [19].…”

“…In extended space dimensional systems, the nonlinear effect owing to the MI in combination with a space-asymmetry directional spectra and broken symmetry owing to nonlocal coupling is suspected to be the major cause of such extreme waves [10,11,18]. The NLS equation (1.1) is a well-known evolution equation with integrability properties, such as having a Lax pair and exact soliton solutions [22].…”