
Akhmediev, N., Kibler, B., Baronio, F., Belic, M., Zhong, W. P., Zhang, Y. Q., et al. (2016). Roadmap on optical rogue waves and extreme events. J. Opt., 18(6), 37 pp.
Abstract: The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 14). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.



Baudin, K., Fusaro, A., Garnier, J., Berti, N., Krupa, K., Carusotto, I.:, Rica, S., et al. (2021). Energy and waveaction flows underlying RayleighJeans thermalization of optical waves propagating in a multimode fiber((a)). EPL, 134(1), 14001.
Abstract: The wave turbulence theory predicts that a conservative system of nonlinear waves can exhibit a process of condensation, which originates in the singularity of the RayleighJeans equilibrium distribution of classical waves. Considering light propagation in a multimode fiber, we show that light condensation is driven by an energy flow toward the higherorder modes, and a bidirectional redistribution of the waveaction (or power) to the fundamental mode and to higherorder modes. The analysis of the nearfield intensity distribution provides experimental evidence of this mechanism. The kinetic equation also shows that the waveaction and energy flows can be inverted through a thermalization toward a negative temperature equilibrium state, in which the highorder modes are more populated than loworder modes. In addition, a Bogoliubov stability analysis reveals that the condensate state is stable.



Baudin, K., Fusaro, A., Krupa, K., Garnier, J., Rica, S., Millot, G., et al. (2020). Classical RayleighJeans Condensation of Light Waves: Observation and Thermodynamic Characterization. Phys. Rev. Lett., 125(24), 244101.
Abstract: Theoretical studies on wave turbulence predict that a purely classical system of random waves can exhibit a process of condensation, which originates in the singularity of the RayleighJeans equilibrium distribution. We report the experimental observation of the transition to condensation of classical optical waves propagating in a multimode fiber, i.e., in a conservative Hamiltonian system without thermal heat bath. In contrast to conventional selforganization processes featured by the nonequilibrium formation of nonlinear coherent structures (solitons, vortices, ...), here the selforganization originates in the equilibrium RayleighJeans statistics of classical waves. The experimental results show that the chemical potential reaches the lowest energy level at the transition to condensation, which leads to the macroscopic population of the fundamental mode of the optical fiber. The nearfield and farfield measurements of the condensate fraction across the transition to condensation are in quantitative agreement with the RayleighJeans theory. The thermodynamics of classical wave condensation reveals that the heat capacity takes a constant value in the condensed state and tends to vanish above the transition in the normal state. Our experiments provide the first demonstration of a coherent phenomenon of selforganization that is exclusively driven by optical thermalization toward the RayleighJeans equilibrium.



During, G., Picozzi, A., & Rica, S. (2009). Breakdown of weakturbulence and nonlinear wave condensation. Physica D, 238(16), 1524–1549.
Abstract: The formation of a largescale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrodinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with smallscale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution Of the cumulants of the random wave as originally formulated by DJ. Benney and P.G. Saffman [D.J. Bentley, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a selfconsistent weakturbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a nonvanishing firstorder cumulant in the hierarchy of the cumulants' equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the offdiagonal second order cumulant, as described by the usual weakturbulence theory. Conversely, in the presence of a highamplitude condensate, the diagonal secondorder cumulants no longer vanish in the long time limit, which signals a breakdown of the weakturbulence theory. However, we show that all asymptotic closure of the hierarchy of the cumulants' equations is still possible provided one considers the Bogoliubov's basis rather than the standard Fourier's (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov's offdiagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cutoff. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H << Hc) the Bogoliubov's regime is established, whereas for H less than or similar to Hc the smallamplitude condensate regime is described by the weakturbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases. (C) 2009 Elsevier B.V. All rights reserved.



Picozzi, A., & Rica, S. (2012). Condensation of classical optical waves beyond the cubic nonlinear Schrodinger equation. Opt. Commun., 285(24), 5440–5448.
Abstract: A completely classical nonlinear wave is known to exhibit a process of condensation whose thermodynamic properties are analogous to those of the genuine BoseEinstein condensation. So far this phenomenon of wave condensation has been studied essentially in the framework of the nonlinear Schrodinger (NLS) equation with a pure cubic Kerr nonlinearity. We study wave condensation by considering two representative generalizations of the NLS equation that are relevant to the context of nonlinear optics, the nonlocal nonlinearity and the saturable nonlinearity. For both cases we derive analytical expressions of the condensate fraction in the weakly and the strongly nonlinear regime. The theory is found in quantitative agreement with the numerical simulations of the generalized NLS equations, without adjustable parameters. (C) 2012 Elsevier B.V. All rights reserved.



Sun, C., Jia, S., Barsi, C., Rica, S., Picozzi, A., & Fleischer, J. W. (2012). Observation of the kinetic condensation of classical waves. Nat. Phys., 8(6), 469–473.
Abstract: The observation of BoseEinstein condensation, in which particle interactions lead to a thermodynamic transition into a single, macroscopically populated coherent state, is a triumph of modern physics(15). It is commonly assumed that this transition is a quantum process, relying on quantum statistics, but recent studies in wave turbulence theory have suggested that classical waves with random phases can condense in a formally identical manner(69). In complete analogy with gas kinetics, particle velocities map to wavepacket kvectors, collisions are mimicked by fourwave mixing, and entropy principles drive the system towards an equipartition of energy. Here, we use classical light in a selfdefocusing photorefractive crystal to give the first observation of classical wave condensation, including the growth of a coherent state, the spectral redistribution towards equilibrium, and the formal reversibility of the interactions. The results confirm fundamental predictions of kinetic wave theory and hold relevance for a variety of fields, ranging from BoseEinstein condensation to information transfer and imaging.

