Special issue on optical rogue waves

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We revisit recent work on the generation of extreme optical events via nonlinear dynamics in silicon waveguides. The underlying processes, modulation instability and stimulated Raman scattering, are able to reshape normally distributed initial conditions into skewed output statistics whose properties can be tailored by controlling experimental variables. While these are both gain processes, they bear fundamental differences: modulation instability is a broadband parametric process, whereas stimulated Raman scattering is a narrowband inelastic process. As a result, they respond to different forms of input noise. Specifically, the extreme events generated spontaneously by modulation instability evidence a strong sensitivity to a particular input noise component. This sensitivity can be controllably seeded to generate coherent supercontinuum radiation, which also offers a means to alleviate conventional free-carrier limitations to chip-scale spectral broadening.

In analogy with ocean waves running up towards the beach, shoaling of pre-chirped optical pulses may occur in the normal group-velocity dispersion regime of optical fibers. We present exact Riemann wave solutions of the optical shallow water equations and show that they agree remarkably well with the numerical solutions of the nonlinear Schrödinger equation, at least up to the point where a vertical pulse front develops. We also reveal that extreme wave events or optical tsunamis may be generated in dispersion tapered fibers in the presence of higher-order dispersion.

The occurrence of rogue waves (freak waves) associated with electromagnetic pulse propagation interacting with a plasma is investigated, from first principles. A multiscale technique is employed to solve the fluid Maxwell equations describing weakly nonlinear circularly polarized electromagnetic pulses in magnetized plasmas. A nonlinear Schrödinger (NLS) type equation is shown to govern the amplitude of the vector potential. A set of non-stationary envelope solutions of the NLS equation are considered as potential candidates for the modeling of rogue waves (freak waves) in beam–plasma interactions, namely in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov–Ma breather. The variation of the structural properties of the latter structures with relevant plasma parameters is investigated, in particular focusing on the ratio between the (magnetic field dependent) cyclotron (gyro-)frequency and the plasma frequency.

We study the features of the optical rogue waves (ORWs) observed in an all-solid-state (Cr:YAG+Nd:YVO₄) passively-Q-switched laser, which is a system of wide practical interest. The extreme events appear as isolated pulses of extraordinary intensity during the chaotic regime of this laser. The standard theoretical description (three-level rate equations for a single mode of the field and a two-level system for the absorber) does predict the existence of many of the observed dynamical features, including chaos, but it fails to predict the existence of ORWs. Faced with the problem of improving the theoretical description, we find that ORWs are observed only when the Fresnel number of the laser cavity and the embedding dimension of the attractor reconstructed from the experimental time series are high, and the laser spot profile has a spatially complex structure. These results suggest that spatial effects are an essential ingredient in the formation of ORWs in this type of laser.

Following the first experimental observation of a new mechanism leading to optical rogue wave (RW) formation briefly reported in Lecaplain et al (2012 Phys. Rev. Lett. 108 233901), we provide an extensive study of the experimental conditions under which these RWs can be detected. RWs originate from the nonlinear interactions of bunched chaotic pulses that propagate in a fiber laser cavity, and manifest as rare events of high optical intensity. The crucial influence of the electrical detection bandwidth is illustrated. We also clarify the observation of RWs with respect to other pulsating regimes, such as Q-switching instability, that also lead to L-shaped probability distribution functions.

We present the most general multi-parameter family of a soliton on background solutions to the Sasa–Satsuma equation. These solutions contain a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits nontrivial limiting cases, such as rogue waves and classical solitons, that are considered in detail.

We show that a rogue wave solution of the nonlinear Schrödinger equation (NLSE) can survive even-parity perturbations of the equation, such as the addition of a quintic term and fourth-order dispersion. We present a solution which is accurate to the first order for such a perturbation. Our numerical simulations confirm the rogue wave existence when the parameter of perturbation |ν| < 0.05.

We provide a simple technique for finding the correspondence between the solutions of Ablowitz–Ladik and nonlinear Schrödinger equations. Even though they belong to different classes, in that one is continuous and one is discrete, there are matching solutions. This fact allows us to discern common features and obtain solutions of the continuous equation from solutions of the discrete equation. We consider several examples. We provide tables, with selected solutions, which allow us to easily match the pairs of solutions. We show that our technique can be extended to the case of coupled Ablowitz–Ladik and nonlinear Schrödinger (i.e. Manakov) equations. We provide some new solutions.

A numerical study on the generation of extreme events in lumped Raman fibre amplifiers is performed. The evolution of continuous or pulsed signals is analysed using cross-correlations, spectra and probability density functions. For pulsed signals, the phase evolution is also explored. Both signal and Stokes cascaded waves are considered.

We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity–time ()-symmetric system of linearly coupled nonlinear Schrödinger equations, which describes a Kerr-nonlinear optical coupler with mutually balanced gain and loss in its cores. Besides the linear coupling, the overlapping cores are coupled through the cross-phase-modulation term too. While the rogue waves, built according to the pattern of the Peregrine soliton, are (quite naturally) unstable, we demonstrate that the focusing cross-phase-modulation interaction results in their partial stabilization. For -symmetric and antisymmetric bright solitons, the stability region is found too, in an exact analytical form, and verified by means of direct simulations.

We investigate the phase profiles of rogue wave solutions to the nonlinear Schrödinger equation, all produced via the Darboux transformation scheme. We focus specifically on the second-order rogue wave, in both origin-centred and fissioned form, and extrapolate the results for higher-order structures. In particular, a rogue wave solution of order n can be decomposed into n(n + 1)/2 Peregrine breathers, and each peak applies an additive phase shift of 2π to the underlying plane wave background. Yet it is evident that no evolution path can be phase shifted beyond 2πn. We show that a fused rogue wave arranges its components to avoid any contradiction in this matter. We also show that the phase profile for any structure in the rogue wave hierarchy can be determined by examining phase bifurcations marked by zero-amplitude troughs.

Higher-order dispersive and nonlinear effects (alias the perturbation terms) such as third-order dispersion, self-steepening, and the self-frequency shift play important roles in the study of ultra-short optical pulse propagation. We consider optical rogue wave solutions and interactions for the generalized higher-order nonlinear Schrödinger (NLS) equation with space- and time-modulated parameters. An appropriate transformation is presented to reduce the generalized higher-order NLS equation to an integrable Hirota equation with constant coefficients. This transformation allows us to relate a certain class of exact solutions of the generalized higher-order NLS equation to the variety of solutions of the integrable Hirota equation. In particular, we illustrate the approach in terms of the two lowest-order rational solutions of the Hirota equation as seeding functions, to generate rogue wave solutions localized in time that have complicated evolution in space, with or without the differential gain or loss term. We simply analyze the physical mechanisms of the obtained optical rogue waves on the basis of these constraints. Finally, the stability of the obtained rogue wave solutions is addressed numerically. The obtained rogue wave solutions may raise the possibility of related experiments and potential applications in nonlinear optics and other fields of nonlinear science, such as Bose–Einstein condensates and ocean waves.