Deterministic Optical Rogue Waves

Abstract

Experimental observations of rare giant pulses or rogue waves were done in the output intensity of an optically injected semiconductor laser. The long-tailed probability distribution function of the pulse amplitude displays clear non-Gaussian features that confirm the rogue wave character of the intensity pulsations. Simulations of a simple rate equation model show good qualitative agreement with the experiments and provide a framework for understanding the observed extreme amplitude events as the result of a deterministic nonlinear process. DOI: 10.1103/PhysRevLett.107.053901 PACS numbers: 42.65.Sf, 05.45.Àa, 42.60.Mi, 42.55.Px According to fishermen tales from a pub in Ireland, rogue waves (RWs) like solid walls of water, higher than 30 m, are more or less common phenomena in deep ocean waters. This fact is in contradiction with the Gaussian models often used to describe fluctuations of the wave height in the sea [1,2]. A recent experience of a luxury ship close to Antarctica is an example that seems to give credit to such tales. Scientific interest on extremely high waves increased substantially during the past decade not only in oceanography but also in other systems such as capillary waves [3], acoustic waves [4], and optical waves [5][6][7][8][9][10]. Both from the theoretical and from the experimental points of view there are several issues still unclear, such as the physical mechanisms that originate the RWs, the way they develop [11], the probability to observe them [12], and the type of system able to generate such extreme events [13].A first problem is defining quantitatively an extreme event. The oceanography community often employs the abnormality index, which is the ratio between the height of the wave and the average wave height among one-third of the highest waves in a time series [14]. Every event whose abnormality index is larger than 2 is considered a rogue wave. An alternative definition is in terms of the standard deviation of the ocean surface variations: any wave whose height is higher than the mean surface value plus 8 is considered a rogue wave [14].These definitions have the advantage of being precise and the drawback of being quite arbitrary. Since they imply that a rogue wave is highly improbable if the probability distribution function (PDF) of the wave amplitude is a Gaussian, the optical community has employed the general criterion that non-Gaussian and long-tailed PDFs signature the existence of RWs [5][6][7][8][9][10].A second problem is determining which type of system might exhibit these rare extreme events. The intrinsic characteristics of a rogue wave (high amplitude, fast rise and fast fall) imply that the system must be highly nonlinear. A mechanism that has been shown to be directly connected to RWs is the existence of a modulational instability [15], as in the nonlinear Schrödinger (NLS) equation. From the theoretical point of view, the formation of ocean rogue waves has been studied using as a framework the NLS equation (see, e.g., Refs. [16,17] and references therei...

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

“…The higher rational solutions having free parameters [26,27] become parameterless for a single-peak RW solution [27]. On the other hand, in our two-dimensional RW model the maximum amplitude A rog (c) can be varied continuously and increased as required, by tuning an arbitrary parameter c in the same single-peak, first-order solution (3.1) or its dynamical extension (3.8) (as shown in figure 4a-d), making the model suitable for RWs with a diverse range of heights, anywhere in the range 17-30 m in calm sea [1][2][3][4][5][6], as observed in deep-sea two-dimensional RWs.…”

“…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].…”

“…(b) Need for a rogue wave model on a two-dimensional plane Therefore, though the well-accepted class of PB or other solutions of the generalized NLS equation could fit into the working definition of the ocean RW, saying any wave with height more than twice the nearby significant height (average height among one-third of the highest waves) could be treated as the RW [19], they perhaps, with their restricted characteristics, can explain successfully only fixed and moderately intense RW-like events on a one-dimensional line, as observed in water channels [14], optical fibres [12,15] or optical lasers [5,8], but seem to be not satisfactory for modelling the ocean surface RWs.…”

“…The mysterious ocean rogue waves (RWs) are reported to be observed in a relatively calm sea, where they, as a localized and isolated surface waves, apparently appear from nowhere, make a sudden hole in the sea just before attaining surprisingly high amplitude and disappear again without a trace ( [1][2][3][4][5][6][7], http://news.bbc.co.uk/2/hi/8548547.stm). 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.…”

Triggering extreme events at the nanoscale in photonic seas

“…These events seem ubiquitous and reported in a large number of systems, which share diverse degrees of randomness, noise, unpredictability, linear and nonlinear responses [15][16][17][18][19][20][21][22][23][24][25][26][27]. A key issue is the active role of randomness [26,27], and the question whether RW can be observed in fully deterministic structures with no intrinsic disorder. Deterministic systems are more easily controlled, they can provide a simpler platform for studying the physics of these phenomena, and they can challenge us with an intriguing problem: can we embed the exceptional amount of energy of a rogue wave into a simple integrated structure?…”