1. Introduction

Ever since the first confirmed field measurements of a freak wave in the North Sea at the Draupner platform on January 1, 1995, and reported in Refs. [1,2], extreme waves or rogue waves have been of great interest to both researchers and engineers [3–6]. The Draupner wave measured at a crest amplitude of 25.6 m on a sea state with a significant wave height (SWH) of 12 m, and one of the possible theories advanced by Haver [3] involved a self-focusing of energy of a wave group into one majestic wave event. They consist of large-amplitude pulses with a narrow width that appears with low probability. In contrast to typhoons, tsunamis, and storms that can be predicted hours in advance, rogue waves appear unpredictable. The long tail probability distribution is the fundamental characteristic which accounts for the generation of rogue waves. They appear unexpectedly on the ocean’s surface which renders them extremely dangerous even for huge ships, marine structures or submarines. They have also been observed in both deep and shallow waters, and also in deep internal layers of the ocean [3–5].

Rogue waves appear not only in the ocean but also in optical fibers [7]. Indeed in 2007, Solli and collaborators observed extraordinary soliton-like pulses with extremely large intensity using fiber in a supercontinuum (SC) generation regime. More importantly, they established an intriguing link between rogue waves in oceans and optical waves along with fiber. In nonlinear optics and laser physics, coherent structures are generated as extreme events from the propagation of laser beams. In particular, the possibility of engineering the parameters of an optical fiber such as nonlinearity and dispersion renders such systems of particular importance for frequency comb generation, which has potential applications in spectroscopy. In addition, the intensity and the phase of the input light may be tailored with advanced data-processing techniques leading to real-time spectrum measurements such as a dispersive Fourier transformation and time-lens techniques. These powerful real-time measurement techniques allow the speed limitations of traditional optical or electronic measurements to be overcome.

A particular experimental originality of the work of Solli and collaborators is in the use of a time-stretch dispersive Fourier transform to map spectral fluctuations into temporal fluctuations in real time. The dispersive Fourier transformation consists of mapping the frequency spectrum onto the temporal waveform [8,9]. Such time-lensing techniques allow for the measurement of ultrashort pulses with sub-picosecond resolution. This method has been applied to observe rogue waves such as Raman rogue waves [10], soliton explosions in lasers and the generation of rogue waves [11], SC generation [12], and Kerr-lens mode-locked [13] and fiber lasers [10,14,15]. On the other hand, the time-lens technique allows one to measure the intensities of localized peaks with a resolution of the order of picoseconds [9]. This comes from temporal imaging systems which are analogous to spatial imaging systems as the time-lens technique allows one to compress or expand the pulse width of optical waveforms without distortion. It has been applied to investigate optical rogue wave (ORW) formation in relation to integrable turbulence [16], and the spontaneous formation of breathers and rogue waves in optical fiber modulational instability (MI) [17]. The combination of dispersive Fourier transformation with a time-lens technique may lead to considerable advances in real-time measurement of ultrafast soliton dynamics. The dispersive Fourier transformation theory and its applications in physics and photonics is beyond the scope of this contribution, and the interested reader may refer to the review papers [18–21].

Since the seminal paper by Solli et al. [7], rogue waves appear in various nonlinear systems such as mode-locked lasers [11,22], cavity semiconductors [23–26], plasma waves [27,28], chemical reaction–diffusion systems [29–31], quantum optics, and even in finance [32,33]. An example of such behavior is shown in Fig. 1 taken in part from Refs. [7,34]. These extreme waves constitute an interdisciplinary issue and considerable progress has been made toward the understanding of their formation, stability, and interaction, though an important number of questions remain open, such as the effect of noisy sources on their formation or their role in turbulent regimes. In particular, optical systems and lasers display such events which arise due to self-focusing nonlinearities, inducing multi-transverse modes and filamentation, which deteriorates the beam quality. The first experimental evidence of rogue waves in fiber optics [7] has triggered several groups to investigate this intriguing phenomenon. The explosive growth of this field of research, can be witnessed by overviews [31,35–44]. Compared with oceanography, there remains a lot to be learned on the mechanisms leading to the formation of rogue waves in optical systems. Access to experimental data is important to better understand the underlying physics. Most systems exhibiting rogue waves, struggle against experimental characterization because of their size or time scales. In this respect, optical and photonics systems constitute an excellent platform for researchers as they are classic examples of dissipative systems which exhibit extreme events that can be experimentally investigated in a reliable way. In particular, high-sensitivity experimental techniques have emerged very recently and can be adapted to ultra-fast-pulse measurements [45,46].

 

Figure 1. Rogue waves: (a) Japanese artist Katsushika Hokusai’s painting, known as “The Great Wave off Kanagawa”; (b) a rough ocean surface with waves towering over a ship; (c) one-dimensional fiber optics; and (d) two-dimensional broad-area lasers. (c) Reprinted by permission from Macmillan Publishers Ltd: Solli et al., Nature, 450, 1054–1057 (2007) [7]. Copyright 2007. (d) Figure 5(b) reprinted with permission from Gibson et al., Phys. Rev. Lett., 116, 043903 (2016) [34]. Copyright 2016 by the American Physical Society.

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Most of the analytical understanding of rogue waves in conservative systems has been realized in the framework of the nonlinear Schrödinger equation (NLSE). The latter serves as a basic model for the analytical understanding of the formation of rogue waves in conservative systems. More precisely, the link between rogue waves and the rational solutions of the NLSE has been established in the seminal paper by Akmediev and collaborators [47]. It is generally admitted that the origin of rogue waves is related to breather-type solutions of the NLSE [48–50]. Several mechanisms have been suggested to be at the origin of the formation of rogue waves in the literature. In most cases, the MI mechanism together with pulse collisions is considered to be the main mechanism that generates rogue waves as shown by Peregrine [51] and by Akhmediev et al. [47]. Experimental confirmation of Peregrine solitons and rogue waves as a result of pulses collision have been demonstrated in optical fibers [52,53] and in water wave tanks [54–56]. In these systems, small-amplitude pulses may grow to large-amplitude pulses provided that their frequencies belong to a band of unstable modes with a positive gain. Nonlinear interaction between these frequencies can lead to complex wave dynamics. For instance, the case of the two frequency solutions of the NLSE in the form of Akhmediev breathers has been analytically studied in Ref. [57] and collisions of Akhmediev breathers have been experimentally evidenced in an optical fiber system [58].

However, in dissipative systems, the NLSE fails to capture the dynamics because it ignores pumping and dissipation. Moreover, in spatially extended systems, when the dimensionality is higher than one, bounded solutions are not admitted by the NLSE due to beam collapse rendering the formation of rogue waves impossible [59]. To circumvent this problem, several physical mechanisms have been proposed in the literature, which perfectly describe the nonlinear dynamics encountered, for example, optical resonators. Indeed, the mean-field approach described by the Lugiato–Lefever type of models for optical cavities have been of significant interest [60] as they take into account, in addition to the nonlinearity, dispersion and/or diffraction, and they also explicitly include pumping and dissipation. The Lugiato–Lefever equation (LLE) has been shown to support the formation of robust stable localized structures (LSs) in one-dimensional (1D), two-dimensional (2D) [61], and in 3D systems [62]. These LSs, often called dissipative solitons or cavity solitons, are one of the most frequently studied nonlinear objects.

In 1D Kerr resonators, temporal [63] and spatial [64] LSs have been observed experimentally. In these systems, the spectral contents of the LSs are the optical frequency combs, referred to as Kerr combs. The link between the formation of temporal LSs and frequency combs have been established in Ref. [65] (see an excellent review by Lugiato et al. [66] in the theme issue [67,68]). This phenomenon has attracted a lot of interest because it has applications in the generation of ultra-stable light wave and microwave signals which find direct applications in optical communication networks, microwave photonic systems, and in aerospace engineering [69,70]. This is mainly due to the development of laser-based precision spectroscopy including the optical frequency comb technique [71]. Driven optical microcavities have been widely used for the generation of optical frequency combs which can be modeled by the LLE with unprecedented precision. In optical resonators subject to a high intensity of the injected field, temporal LSs may exhibit complex behavior. Statistical analyses have indicated that this phenomenon belongs to the category of extreme events or rogue waves [23].

In 2D dissipative broad-area devices, rogue waves have been predicted using the mean-field approach and experimentally observed as well [25,26,34,72–75]. In addition to 2D LSs predicted in Kerr resonators, another interesting system is presented, namely, semiconductor cavities. Thanks to the mature semiconductor laser technology and the possibility for application of 2D localized pulses, they have attracted significant interest in all-optical control of light, optical storage, and information processing. In particular, vertical-cavity surface-emitting lasers (VCSELs) were first realized in Ref. [76]. Light confinement has been investigated in VCSELs for which we refer the reader to these recent reviews [77–79]. Rogue waves have been experimentally observed in 1D setting for VCSELs [24]. In 2D settings, rogue waves have been numerically demonstrated in dissipative systems by pumping [25] or by optical feedback [26] in broad-area surface-emitting laser with a saturable absorber (SA). They have been also observed using photorefractive materials [80,81].

The present contribution aims to shed some light for a better understanding of the mechanisms underlying the formation of rogue waves in optics and photonics, along with an analogy in hydrodynamics when possible. After an introduction, we present the advances in conservative systems in the first part, such as fiber optics. Following this, we turn our attention to dissipative systems such as optical Kerr and semiconductor cavities, in which we discuss how to control 1D and 2D rogue waves by injection or via delayed feedback. In addition to presenting the state of the art, recent progress, new ideas, and novel techniques, we also present experimental evidence of temporal and spatial rogue waves in advanced conservative and dissipative optical systems. Rogue waves in the above-mentioned dissipative systems are discussed in the second part of this article, including experimental evidence of temporal rogue waves in all fiber cavities, and spatial ones in broad-area lasers such as VCSELs.

2. Analytical Treatment of Rogue Waves in Nonlinear Dispersive Media

One of the famous equations that describes many physical problems in dispersive media is the NLSE. This equation has become famous because it provides, in its linear form, a basic description of a wave function in quantum mechanics. Nevertheless, its main characteristic is the generic description of the evolution (propagation) of the slowly varying envelope of dispersive waves; approximation widely used in the majority of physical systems including quantum mechanical ones.

In this section, we are mainly interested in the two fields of optics and hydrodynamics where, for more than a decade, a sustained research activity on the formation and dynamics of ORWs have played an important role thanks to innovative developments in theory and experience. Although hydrodynamics (surface waves) and optics (optical fiber systems) correspond to different fields with regards to the physics of the underlying the observed phenomena, it is possible, in the leading slowly varying envelope approximation, to model the wave propagation in both media by NLSE. A decisive issue in this regard is the considerable progress made by Solli et al. [7] by observing an optical counterpart of oceanic rogue waves in a photonic crystal fiber, producing a phenomenon named the ORW owing to its similarity to the oceanic rogue wave. The authors of Refs. [7,82,83] have shown that the ORWs appear as solutions of the generalized nonlinear Schödinger equation (GNLSE), which is often used to describe the nonlinear dynamics of water waves as well as light propagation in photonic crystal fibers.

Here we focus on the derivation of the NLSE problem because it is much more generic than its solutions. We use a multiple time and space scales analysis and assume that the envelope of the wave packet that propagates in Kerr media is a slowly varying function of time and space. We begin by discussing the linear regime and the role of MI. However, in the nonlinear regime, analytic results for the NLSE require strenuous efforts, difficult and tiring and far beyond the scope of this tutorial. Different analytical approaches from the literature for solving the NLSE can be found in the book by Akhmediev and Ankiewicz [84]. In this context, we present the exact analytical solutions admitted by all the “rogue wave communities” as the most important prototypes of rogue waves, in both optics and hydrodynamics. Following this, we provide analogies between optics and hydrodynamics quantitatively.

2.1 The NLSE

In this subsection, we review the derivation of the NLSE. This equation describes how a pulse with a slowly varying envelope propagates in a Kerr medium. To simplify the analysis, we focus on a medium devoid of magnetization, current density, and charge density. When an intense optical field propagates in a Kerr medium, it is likely to modify the properties of this medium, for instance by distorting the electron balance of its electron clouds. The susceptibility of the material then becomes dependent on the field. Therefore, the response of the medium to an electrical excitation is no longer linear, and the polarization has an additional nonlinear contribution. The classical formalism expresses the vector polarization as ${\bf {P}}={\bf {P}}^{l}+{\bf {P}}^{nl}$ where ${\bf {P}}^{l}$ and ${\bf {P}}^{nl}$ denote the linear and the third-order nonlinear polarizations, respectively. We assume that the second-order nonlinear polarization is absent because the Kerr media possesses an inversion symmetry. This means that the Kerr material is cubic and centrosymmetric. This property allows both anisotropy and second-order nonlinearities to be avoided. This nonlinear response leads to frequency upconversion or downconversion such as second-harmonic generation or optical parametric oscillators, both are non-centrosymmetric media. From the macroscopic Maxwell equations together with the above-mentioned approximations, the pulse propagation equation reads

2.1a. Order $\zeta ^0$

By replacing the unperturbed wave packet ${\bf {E}}_0$ in Eq. (9), we obtain

2.1b. Order $\zeta ^1$

Equation (10) can be rewritten as

2.1c. Order $\zeta ^2$

The nonlinear polarization contribution appears only at this order

2.1d. The NLSE

By multiplying Eq. (14) by $\zeta$, and Eq. (18) by $\zeta ^2$, then the sum of these two equations together with the use of the relation (6) yields

By using the Leibniz transformation (retarded time)

With the following change of variables and parameters: $(x,y) \rightarrow (x,y)/\sqrt {k}$, $\tau \rightarrow \tau / \sqrt {|\beta _{2}|}$, $E=(n/kn_{2})^{1/2}F$, Eq. (20) yields the dimensionless NLSE

2.2 Linear Regime: MI for Initiating Rogue Waves Formation

We restrict ourselves to a strictly 1D setting, by neglecting the diffraction phenomena when using a guided waves structure in the plane $(x,y)$. Nonlinear systems are well known to manifest a large spectrum of dynamical behavior ranging from regular to chaotic regimes. Both regimes exhibit complex spatiotemporal solutions that have been subject to intensive research [88–90]. The key physical phenomenon characteristic of nonlinear systems is instability, which is one of the main and important aspects that encompasses all domains of nonlinear science. It is out of the scope of this tutorial to report on all types of instabilities appearing in nonlinear systems. However, among the different types of instabilities, MI, is one of the most famous because it naturally appears in diverse physical areas, including hydrodynamics, plasma physics, and optics among others. The first studies of MI date back to the early 1960s in such different fields as hydrodynamics [91], nonlinear optics [92], and plasma physics [93]. MI originates from the perturbation of an intense continuous or quasi-continuous field by a weak signal that is then amplified exponentially [91]. In this linear stage of evolution, amplification of modes leads to an increase in the amplitude (or intensity) followed by the nonlinear regime where the system experiences the Kerr effect that becomes dominant. Different bifurcations (or every scenario) arise from this dynamics, which constitutes the origin of the fascinating regularity and complexity behaviors and related states observed in almost all nonlinear systems, irrespective of the domain of nonlinear science. These states may range from solitons (the most well-known regular nonlinear solution) to highly developed turbulence. Of course, consecutive sequences of bifurcations allow transitions between an impressive number of solutions and their dynamical complexity. Fortunately, strong mathematical methods and tools exist to help find analytically some (important) solutions, though this depends on the nonlinear system at hand. Indeed, Hamiltonian systems (or conservative nonlinear systems) are, in general, more tractable analytically than dissipative systems where there is, intrinsically (because of dissipation), a lack of conservative quantities such as energy. Still, in these systems, perturbative methods are highly useful to obtain many relevant results. In general, for strongly nonlinear dissipative systems, mathematics often finds its limit (except in special cases), but the advent of efficient and powerful computers provides a precious approach to deeply understand highly complex dynamics such as spatiotemporal chaos.

Let us return to the process of MI we are interested in. It is present in many fields of physics and has been studied extensively in optical fibers because it provides a convenient and valuable way for experimental generation and observation of a train of ultrashort pulses. Before going further in the analysis, it is worthwhile to recall linear stability analysis in the ubiquitous case of free light propagation in nonlinear optical fiber systems. Light propagation in a mono-mode nonlinear optical fiber is well described by the NLSE for the slowly varying complex amplitude $E(z,\tau )$ of the electric field as shown in the Sec. 2.1 (see Eq. (20)). Let us rewrite the NLSE in the following 1D form $\partial _{z}E=-i\frac {1}{2}\partial _{t}^{2}F+i\left |E\right |^{2}E$, where the complex function $E = E(z,t)$ describes the time evolution of slowly varying wave envelope along the propagation direction. This equation has the following nonlinear stationary (i.e., $\tau$-independent) solution:

The dispersion relation (25) shows an important feature, namely, the instability critically depends on the sign of $\beta _2$. When $\beta _2>0$ ($s=1$), the term under the square root is positive and the wavenumber $K$ is real for all $\Omega$, meaning that the nonlinear stationary solution is stable in the regime of normal dispersion. However, the same nonlinear stationary solution is inherently unstable to plane wave perturbations in the anomalous regime ($\beta _2<0$) because the wavenumber $K$ becomes a complex number with a non-vanishing imaginary part giving rise to an exponential growth of the perturbation as can be seen from Eq. (25). Indeed the term under the square root can be negative for a band of frequencies that can be calculated from the dispersion relation (25) by determining all frequencies $\Omega$ with $\Im (K)[\Omega ] > 0$. The frequencies of marginal (or neutral) stable modes, often called critical modes, are defined as $\Omega _c$ where $\Im (K)[\Omega _c]=0$, which gives

The gain spectrum of MI is obtained from the dispersion relation (25) by setting $s=-1$, and the gain

 

Figure 2. Numerical simulations: (a) spectrum gain (27), for three values of injected power, showing the sidebands MI; (b) temporal intensity injected as initial condition composed of continuous waves with a noisy source; (c) a pulse train resulting from the spontaneous amplitude modulations induced by MI.

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Figure 3. (a) Experimental frequency spectrum showing the sidebands instabilities. (b) Monochromatic CW plane wave injected at the entrance of the nonlinear optical fiber. (c) Corresponding constant amplitude CW plane wave.

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More generally, the process of MI is present in many fields of physics and has been studied extensively in optical fibers because it provides a convenient and precious way for experimental generation and observation of a train of ultrashort pulses. In optics, the first experimental observation of MI was reported in Refs. [94,95] where an intense quasicontinuous field injected in optical fiber was converted into a train of ultrashort pulses [96] or fiber optical parametric amplifiers (FOPAs) [97,98]. This ground-breaking experiment gave rise to the rapid development of parametric amplification and ultrashort pulse generation that have become some of the most important branches of research in nonlinear fiber optics. In hydrodynamics, MI was shown to be one of the fundamental mechanisms responsible for the formation of the so-called rogue waves [99]. Very recently, optical equivalents of the oceanic rogue waves have been discovered in optical fibers [7] and are called ORWs. In this context, the GNLSE was shown to be successful in describing the formation of rogue waves both in the ocean and in fibers.

Nowadays, even after many years, the study of MI in hydrodynamics and optics remains an attractive research activity. For instance, in optics, MI is the key ingredient for the understanding of SC generation [100] and ORW formation [7,52,101]: both physically originate from MI. From a fundamental point of view, MI results from the balance between nonlinear and linear effects experienced by a field during the propagation in dispersive nonlinear media. In the most usual case in optics, the linear phase mismatch depends only on the group-velocity dispersion of the optical fiber [94,95,97,98]. The contribution of other linear terms, such as the group-velocity difference between polarization modes [102] or transverse ones [103], must be accounted for in birefringent or multimode fibers, respectively. In the anomalous dispersion regime, the broadest frequency shifts are obtained by pumping the fiber as close as possible to the zero-dispersion wavelength as in FOPAs, or in the context of SC/rogue wave generation for instance [7]. As a consequence, even higher-order dispersion terms must be included in the linear phase-mismatch relation [104,105]. This explains the MI observed in the normal dispersion region of an optical fiber [103,104]. In all these studies, the theoretical background is based on the standard linear stability analysis, which shows that the gain curve only depends on even-dispersion terms. This has been experimentally confirmed in the context of FOPAs, for instance, in which excellent agreement between theory and experiment is achieved when a CW seeds the process [98].

Many authors have reported the important impact of higher-order dispersive terms in the GNLSE on the main characteristics of rogue waves [99,106,107] and also on their non-Gaussian statistics [52]. In particular, it was demonstrated that the third-order dispersion is already sufficient to explain the ORW formation and, most importantly, their probability density function [101]. A striking feature when dealing with purely NLSE remains in the transition from linear to nonlinear regimes that can be accounted for in an elegant and instructive way by invoking MI. For instance, Akhmediev Breather which is one of the main prototypes of rogue waves can be obtained as a nonlinear stage of unstable linear plane waves.

2.3 Nonlinear Regime: Akhmediev Breather, Peregrine Soliton, and Bright Soliton As Prototypes of Rogue Waves

Let us consider the dimensionless NLSE (21) in a 1D system where diffraction is neglected by setting $\nabla _{\perp }^2=0$, and we assume that the sign of the second derivative with respect to the retarded time $\tau$ is negative (anomalous dispersion regime). The NLSE is a dynamical system with an infinite number of degrees of freedom and corresponds to an infinite-dimensional Hamiltonian system. As it is an integrable system, it has an infinite number of conserved quantities. The lowest-order integrals are the energy (or the number of particles)

It is far beyond the scope of this contribution to present the analytical derivation of nonlinear solutions of the NLSE (21). This is because mathematics is not simple to handle (see the mathematical aspects of optical solitons [109–111]). In particular, let us mention the inverse scattering method that constitutes a powerful tool in the soliton theory which has been applied to the Korteweg–de Vries equation, a nonlinear partial differential equation that appears in hydrodynamics. However, its applicability to the NLSE is an arduous mathematical problem. In this section, we only use the results obtained by the analytical derivation of nonlinear solutions of the NLSE (21) presented by Akhmediev and Ankiewicz [84]. In this book, a method leading to the derivation of solutions associated with not only soliton but also periodic solutions, and higher-order solitons is described. The 1D solutions of the NLSE reads [84]

 

Figure 4. Three-types of breathers obtained using analytical formulas Eq. (34): (a) Akhmediev breather; (b) Peregrine soliton; (c) KM (parameters provided in the inset).

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Let us now return to Fig. 4 and give more insight into the three displayed solutions. They correspond to a particular value of $a$ in the family of one-parameter solutions of Eq. (34). Solutions over the range $0 < a < 0.5$ are the Akhmediev breather, with MI instability growth rate being maximal at $a = 0.25$. Increasing $a$ further gives the Peregrine soliton in the limit $a \rightarrow 0.5$. When $a > 0.5$ the solution exhibits localization in time $\tau$, but becomes periodic along the spatial direction. Such a soliton is referred to as the KM soliton which is shown in Fig. 4 for $a = 0.75$. Note that, higher-order rogue waves can also be analytically obtained from the NLSE and they are not discussed here (see a recent overview on this issue in Ref. [44]). Typical higher-order rogue waves are displayed in Fig. 5. These analytical solutions have motivated researchers to gain insights into the formation of rogue waves in nonlinear dispersive systems which have been verified experimentally in a 1D setting, both in optics and in hydrodynamics [52–56,113]. In this regard, fiber optics provides an excellent experimental platform as dispersion and nonlinearity effects could be controlled to yield a propagation regime where the NLSE is valid. This is done by injecting a multi-frequency field into a nonlinear fiber, using a method developed for coherent pulse generation in telecommunication [96,114]. Before embarking upon discussing possible mechanisms which lead to rogue waves, we establish similarities between optical and hydrodynamical systems in a quantitative way.

 

Figure 5. Examples of higher-order rogue wave solution of NLSE. (a) Solution of the second order. Figure 4 reprinted with permission from Akmediev et al., Phys. Lett. A 373, 675–678 (2009) [47]. Copyright 2009 by the American Physical Society. (b) Third-order rogue wave solution. Reprinted with permission from Akhmediev , Front. Phys. 8, 631 (2021) [44]. Copyright 2021 by Frontiers Media S.A.

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2.4 Similarities between Ocean Waves and ORWs

Establishing similarities between optics and photonics with fluid mechanics has remained an open challenge ever since the derivation of the Lorentz model by Haken [115]. The Lorentz model was originally derived to describe the Bénard convection. Starting from the Maxwell–Bloch equation which describes a single-mode laser, Haken derived for the first time a one-to-one analogy between optics and fluid mechanics [116]. In this analogy, the transport processes such as dispersion, diffraction, or diffusion are neglected. This implies that the field envelope is assumed to be homogeneous along with transverse and longitudinal coordinates. When transverse and dispersive effects cannot be neglected, for instance, in broad-area optical devices several order parameter equations have been proposed in the literature such as the well-known laser Bernoulli [117], Kuramoto–Sivashinsky [118], and Ginzburg–Landau equations [119,120].

In laser physics, the Kuramoto–Sivashinsky equation has been derived close to the lasing threshold [118]. This equation holds in the limit of small atomic detuning, with the resulting dynamics governed by the phase of the optical signal. While in fluid mechanics, this phase equation has been used to model diffusive instabilities inherent in a laminar flame front [121,122]. In the same vein, the complex Ginzburg–Landau equation has been derived for laser systems [119,120], which also describes a large class of fluid mechanics phenomena. This includes superconductivity, superfluidity, and also extends to other systems such as liquid crystals, Bose–Einstein condensation [123,124], and even to string theory [125]. The Swift–Hohenberg equation, on the other hand, has been used to describe dynamics in laser systems (complex equation), and also in passive systems such as driven optical cavities (real equation) [126–128]. In particular, it has to be noted that the real Swift–Hohenberg equation has been first derived for studying convective patterns generated by the Bénard–Marangoni instability [129].

We discuss the link between optics and hydrodynamics in a quantitative way using the NLSE. For the ease of readability, we rewrite Eq. (33) in a 1D setting:

Rogue waves have been attributed to mechanisms that generate large envelope amplitudes $F$ modeled by the NLSE, which takes into account group velocity dispersion and nonlinearity. When these two effects act together, this leads to the nonlinear focusing scenario corresponding to $\eta = 1$. In this regard, the origin of the nonlinearity in optics and hydrodynamics are both governed by a nonlinear dispersion relation [99,113,132]. It is important to note that optical waves with higher amplitudes of $F$ propagate at reduced velocities whereas in hydrodynamical systems such as in water wave tanks, waves with higher amplitudes, propagate at increased velocity. In optical systems, nonlinearity and dispersion are either self-focusing or defocusing unlike in hydrodynamics where directional focusing of waves or random superposition of independent wave train may lead to the generation of extreme events which are not limited to the self-focusing case as in optics. The time-lens measurement technique has been used to qualitatively compare the experimental results generated from the MI of a CW in a single-mode fiber with the theoretical one obtained from the NLSE [17,37].

We remind the reader that the analogy between optical and hydrodynamical systems is strictly limited to the 1D setting. This is mainly due to the fact that the NLSE supports solutions only in the 1D setting because in two or more dimensions, this model does not support solutions due to collapse which is mathematically demonstrated by Ref. [108]. In the following section, we discuss some possible mechanisms in conservative systems which could explain rogue waves in optical systems (1D setting), and in controlled circular laboratory water wave tanks (2D settings). In this regard, the analytical solutions of breathers via collisions is of particular interest as it is considered to be one of the main mechanisms which generate rogue waves in these nonlinear dispersive systems. This follows in the subsequent subsection where the collision of these breathers is discussed following which we present the scenario from a recent study made in circular wave tanks where the Draupner wave is recreated experimentally.

3. Collision and Wave Breaking

3.1 Rogue Wave Observation in Water Wave Tanks

In this section, we summarize the first experimental observations of the Peregrine soliton in a water wave tank [54]. The evolution of the surface gravity waves in the water wave tank is described by the NLSE, which is known to be valid for capturing weakly nonlinear evolution of narrowband processes. The formation of rogue waves in oceans have been attributed to the underlying breather-type solutions of the governing NLSE, such as the Akhmediev breather, Peregrine soliton, and KM soliton [23,48,133,134]. While modeling deep water waves using the NLSE, the Peregrine soliton is considered as an ideal candidate as it is localized both in space and time, and as such, it describes unique wave events. It is to be noted that the Peregrine solution is a limiting case of an Akhmediev breather when the period of modulations is taken to be infinite.

To the best of the authors’ knowledge, the first observation of an isolated rogue wave in a water wave tank was shown in Ref. [54]. Specifically, Chabchoub et al. observed a Peregrine-type breather solution experimentally in a water wave tank by providing a direct analytical comparison with the solutions of the NLSE. The experiments were performed in a water wave tank of dimensions $15$ m $\times$ $1.6$ m $\times$ $1.5$ m with $1$ m water depth. The schematic of the experimental setup is shown in Fig. 6(a). A flap-paddle activated by a hydraulic cylinder is used at one end of the tunnel to generate water wave dynamics in the tank which produces a periodic wave with about $1$ cm amplitude. The dynamics on the water surface is quantified by measuring the surface elevation across the test section. The deep water waves are described by the NLSE as shown in Refs. [54,108].

 

Figure 6. (a) Schematic setup of the water wave tank used in Ref. [54]. (b) Comparison of the experimentally measured surface height at the position of maximum amplitude corresponding to a rogue wave (shown in blue) with the theoretical Peregrine solution (shown in red). Figure 5 reprinted with permission from Chabchoub et al., Phys. Rev. Lett. 106, 204502 (2011) [54]. Copyright 2011 by the American Physical Society.

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For comparing experimental data with the analytical solution of the NLSE, the Peregrine breather solution has to be written in dimensional units (see [54]). In their study, $a_0$ is the dimensional far-field amplitude selected to be $0.01$ m. The wavelength of the carrier wave is set to be $\lambda = 0.54$ m, corresponding to a wavenumber of about $k = 11.63$ m$^{-1}$ with an angular frequency $\omega = 10.7$ s$^{-1}$. As remarked in Ref. [54] it has to be ensured that the wavelength of the carrier wave should be large enough to ignore surface tension effects, while at the same time still small enough to have sufficient tank length to develop the wave evolution described by the Peregrine solution. By measuring the surface elevation of the water wave across the tank at different streamwise locations allows for a direct comparison with this analytical Peregrine solution. Further details on the experimental procedure can be found in Ref. [54].

Figure 6(b) shows the surface-height measurement at a position close to the maximum envelope amplitude across the water wave tank. It can be seen that while the amplitude of the carrier wave is around $1~$cm, the maximum surface height of the breather is almost exactly three times reaching a value of $3$ cm. This indicates the formation of a rogue wave with amplitudes greater than twice the SWH, confirming the existence of rogue waves. The periodic wave shown in Fig. 6(b) reveals a deep trough next to the breather’s maximum either side of the wave crest. In addition, the return from the state of the extreme wave back to the state of a periodic wave in the experiment is almost symmetric in time, as expected from the underlying theory. Indeed, carefully controlled experimental procedures such as that presented in Ref. [54] allows for a controlled observation of rogue waves in water wave tanks. In addition, they also reveal the role of breather-type solutions of the NLSE in the formation of rogue waves in oceans, where detailed experimental measurements are typically incomplete and rare. We now present another interesting study where the observation of super rogue waves, which are higher-order breathers in water waves, are experimentally shown [130].

3.2 Higher-Order Breather Solutions in Water Waves

The Peregrine soliton discussed in the previous section is characterized by an amplification factor of three. By considering a higher-order rational solution, localized in both space and time, allows one to boost the peak amplitude of the carrier wave on the water surface by a factor of five. These solutions are detailed in Ref. [130]. In order to excite super rogue waves in water tanks, as in the earlier study, one has to fix the initial amplitude of the carrier wave $a_0$ and the wavenumber $k$. In doing so, one fixes the steepness of the carrier wave which is given by $\epsilon = a_0 k$.

Figure 7(a) shows the evolution of the temporal profile of the surface elevation along the downstream direction from the paddle (location of the wavemaker) as a function of time. The amplification of the water surface as the wave propagates along the test section can be clearly seen with significant amplification occurring at a distance of nine units from the flap-paddle. The high-order wave packet generated by the wavemaker’s paddle evolves to a super rogue wave as it propagates along the water wave tank. This is further quantified in Fig. 7(b) which compares the theoretical prediction at the position of maximal amplitude with the experimentally measured surface height elevation. An amplification factor of five has been observed indicating the formation of higher-order super rogue waves. Even higher amplifications were obtained for higher values of the steepness of the carrier wave.

The role of MI in governing the growth of small-amplitude pulses leading to higher amplitudes resulting in ocean waves was first discussed by Peregrine [51]. The presence of multiple frequencies in optical and hydrodynamical systems renders explicit modeling rather complicated. However, the problem can be simplified and effectively reduced to the interaction of two independent frequencies within the positive-gain band. For instance, the collision of Akhmediev breathers have been observed experimentally in nonlinear fiber optics [58], confirming the fact that coherent structures may emerge locally in a turbulent environment when the phase difference between the colliding structures is optimal. The collision of Peregrine solitons has been derived analytically [56,135]. The phase difference between the interacting solitons plays a significant role in deciding the initial conditions required for the formation of rogue waves. These solitons are localized along two directions which are well separated. Such collisions can happen in nonlinear systems frequently leading to the generation of high-intensity pulses of small widths. These bounded states of solitons are unstable to variations of the relative phase difference between them. However, no stable bounded states of two solitons in 1D homogeneous media have been reported in the literature. This is due to the fact that these solitons do not have an oscillatory tail that connects the LS with the homogeneous background rendering bounded states impossible.

 

Figure 7. (a) Evolution of the temporal profile of the water-surface elevation along the wave tank demonstrating a super rogue wave formation further downstream from the paddle. (b) Experimental and theoretical confirmation of higher-order rogue waves showing the measured wave profile at the expected position of maximal carrier amplification. Here the background amplitude $a_0 = 0.001$ m, with a steepness of $\epsilon = 0.03$. Figure 6(a) reprinted with permission from Chabchoub et al., Phys. Rev. X, 2, 011015 (2012) [130]. Copyright 2012 by the American Physical Society.

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3.3 2D Settings

In 2D settings spatiotemporal rogue events have been experimentally observed in the context of optical multiple filamentation in Ref. [136]. In this study the transient appearance of bright spots in the beam profile of optical filaments formed in xenon were investigated experimentally. In this regard, optical filaments are dynamic light structures with an intense core, which propagate over extended distances, which is significantly larger than the usual diffraction length in the given medium. This effect allows the maintenance of a narrow beam size without any external guidance which is governed by two counteracting nonlinear optical mechanisms which are self-focusing and plasma defocusing. There exists a critical power $P_c \approx 0.15 \lambda ^2/n_0 n_2$ above which the breakup of a spatially homogeneous beam profile into one or several highly localized strings, where $\lambda$ is the wavelength and $n_0$ and $n_2$ are the linear and the nonlinear index of refraction, respectively. The experimentally observed filamentation was numerically investigated using simulations of the NLSE. A snapshot of the formation of multiple parallel optical filaments is shown in Fig. 8. The formation of extreme events was quantified using probability distribution functions to quantify the heavy-tail character which represents the presence of extreme events in such optical cavities. Rogue events that exceed the SWH by a factor of $10$ were reported. Of particular interest, mergers between filament strings were identified as a possible microscopic driver mechanism for the observed rogue events. As the system is 2D as in the ocean surface, such filamentation techniques allows for an excellent experimental platform for studying the phenomenon of rogue waves.

 

Figure 8. Multiple parallel filament strings along the longitudinal direction $z$ obtained from numerical simulations of the NLSE with the color coding showing the relative phase differences between the optical filaments. Figure 2 reprinted with permission from Birkholz et al., Phys. Rev. Lett., 111, 243903 (2013) [136]. Copyright 2013 by the American Physical Society.

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We now turn to an example from hydrodynamical settings wherein significant advances have been made in the past two decades in the context of extreme events, using statistical, numerical, and experimental tools [131,133,137–139]. The Draupner wave was recently recreated in controlled laboratory settings by McAllister et al. [137] to gain an understanding of the role of the type of wave breaking for crossing and non-crossing waves. The conditions at the Draupner platform based on hindcasting studies have indicated the presence of wind fields at different angles [137,140]. More pertinently, the question of interest raised by McAllister et al. [137] were on the directional conditions of the waves whose interactions lead to large crest amplitudes, without crest-amplitude-limiting breaking, as the crest amplitude from a single wave train is limited due to breaking. Their study showed that the breaking mechanism can be fundamentally altered, with the crest amplitude not being limited due to breaking, but rather resulting in upward jet-like behavior.

As a brief summary, McAllister et al. [137] performed experiments in a circular wave tank surrounded by $168$ wavemakers, which allows for the creation of waves traveling in all directions. For instance, by sending out waves at different frequencies toward the center of the circular tank that travel at different speeds, they were able to create a “spike” wave which arises due to the constructive interference with the tank. It is to be noted that previous numerical studies could reproduce the Draupner wave only by the crossing of two wave groups, due to the interaction between the wave trains at a specific angle [131,141]. Based on this, McAllister et al. [137] created two different wave groups traveling toward the center of the tank, and varied the angle between them. They found out that a “freak wave” or a “rogue wave” formed only when the difference in angle between the two wave groups was around $\Delta \theta \approx 135^\circ$.

Figure 9 shows a summary of the observations from Ref. [137] for different values of $\Delta \theta$, the angle between the wave groups. They observed that the breaking process when the angle between the wave groups was large changes significantly. For $\Delta \theta = 0^\circ$, one can see that the wave energy is traveling unidirectionally, similar to what can be observed at a beach, with the waves breaking in a “plunging”-type manner, thereby limiting the maximum crest amplitude. When the angle between the wave groups was changed to $\Delta \theta = 120^\circ$, a “spouting”-type wave breaking can be observed in the form of upward projected jets. This type of wave breaking does not limit the wave height under the crossing-sea conditions, and in fact enhances the wave height, which could explain the Draupner wave observed in the North Sea. A summary of the Draupner wave features are provided in Table 1. The study clearly shows that the wave breaking plays a key role in the formation of extreme waves, which are, in turn, governed by the crossing angles between the traveling wave groups.

 

Figure 9. A summary of the observations from [137] where $\Delta \theta$ is the angle between the two wave groups. The still images of the free surface were taken at intervals of 100 ms (0.6 s at field scale), with the time sequence increasing from left toward right. Reprinted with permission from McAllister et al., J. Fluid Mech. 860, 767–786 (2019) [137]. Copyright 2019 Cambridge University Press.

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Table 1. Summary of the Draupner Wave Reconstruction from Ref. [137] ^a

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4. Diffractive and Dispersive Optical Resonators

The above theory of rogue waves in the context of optical systems based on the NLSE holds only in the 1D setting. However, beam collapse prevents the NLSE to have bounded solutions in more than one dimension which renders the formation of high-dimensional rogue waves impossible. There exist several mechanisms leading to the stabilization of 2D solutions such as SAs, non-local coupling, or optical cavities. Due to this reason, we focus on optical cavities filled with a nonlinear material by using the mean-field approach, which was first used to describe Kerr media and semiconductor materials: the LLE. This approach includes pumping and dissipation, in addition to nonlinearity, dispersion, and/or diffraction [60]. Furthermore, the LLE is by now recognized as a very accurate model to capture the physics of Kerr optical frequency comb generation using integrated ring resonators [142], and in whispering gallery mode cavities [143]. In what follows, we focus on dissipative rogue wave formation in nonlinear optical cavities. For this purpose, two dissipative systems are considered: the driven Kerr optical cavities subjected to an optical injection, and the broad-area surface-emitting lasers with a SA.

4.1 The Lugiato–Lefever Model

In free propagation, the evolution of the envelope of the electric field inside the Kerr material can be described by the standard NLSE Eq. (35). In what follows, we consider a Fabry–Perot cavity filled a Kerr medium and driven coherently by an injected beam with an instantaneous Kerr media (see Fig. 10).

 

Figure 10. Schematic setup of an optical cavity filled with a Kerr medium. The cavity is driven by an external field $E_i$. The mirrors $M_{1,2}$ are identical and their reflection and transmission coefficients are denoted by $\rho$ and $\theta$, respectively. Reprinted with permission from Ref. [62].

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The cavity is driven by a coherent plane wave wherein the intracavity field undergoes a coherent superposition with the injected field at the input of the mirror. This process corresponds to the following cavity boundary conditions

The NLSE (35) supplemented by the cavity boundary conditions (36) constitutes an infinite-dimensional map. In the mean-field approach based on a good cavity limit and high Fresnel number, this infinite-dimensional map can be reduced to a single partial differential equation referred to as the LLE [60]. To perform this reduction, we consider cavities with high finesse. In this case, the transmission coefficient $\theta$ is much smaller than unity ($\rho \approx 1-\theta ^{2}/2$). In this limit, the temporal evolution of the field within the cavity is slow compared with the round trip time $t_r$. Therefore, this temporal evolution can be considered as continuous by replacing the map index $p$ by a slow time scale $t$ for the modeling of the field envelope evolution at the input of the cavity, i.e., the point $z = 0$. This can be realized by defining the continuous variable $E(t, x, y, \tau )$ as the intracavity field envelope at $z = 0$ as

Kerr resonators constitute a basic configuration in nonlinear optics and photonics which allows for the generation of 1D and 2D spatial rogue waves described by the LLE. The latter was first derived to describe the dynamics in passive resonators [60], which was later on extended to fiber optics [149], left-handed materials [150], whispering gallery mode cavities [144], integrated ring resonators [69], and chains of coupled silver nanoparticles embedded in glass [151]. A discrete version of the LLE has been established to model coupled-waveguide resonators [152–155], or extended Josephson junction [156]. The LLE constitutes a paradigm for the study and for the understanding of various dynamical properties of laser fields confined in nonlinear optical resonators such as hard-mode symmetry-breaking instability and self-organization either in time and/or space (see a recent overview [157]). The mean-field approach has been compared with the propagation model supplemented by cavity boundary conditions in Refs. [158,159].

The model, Eq. (43), supports stationary localized [61,160] and self-pulsating localized [161] structures. In the conservative limit, where injection and dissipation are both small, LSs have analytical solutions [88,162–164]. LSs can exhibit regular time oscillations leading to complex behavior (see also the recent review on the link between the LLE and the driven damped sine-Gordon and the Lugiato–Lefever model [165]). It is also worth mentioning several interesting works on their bifurcation structures which belong to the class of homoclinic snaking type of instability [166–168]. Their relative stability analysis as well as their snaking bifurcation diagram have been reported [169]. When polarization degree of freedom is taken into account within Kerr resonators, new MIs appear [170–172] and vector LSs are predicted [173–176]. However, when the Kerr resonator is operating in a dispersion regime, another class of LSs resulting from interactions may be generated in scalar [177–182] and vectorial cases [183,184]. The stabilization mechanism of these localized states is attributed to a front locking mechanism in the bistable regime [185,186]. Their bifurcation diagram exhibits a heteroclinic collapsed snaking type of bifurcation.

4.2 Linear Stability Analysis

In the absence of diffraction and dispersion, the homogeneous steady-state solutions of Eq. (43) satisfying $\partial _t E=0$ and $\nabla ^2 E=0$ are

When diffraction and dispersion are taken into account, spatial and/or temporal fluctuations can destabilize the system and allow for the formation of dissipative structures. The spontaneous transition from a homogeneous steady state to a periodic structure is a general feature of many far from equilibrium systems, and has motivated many studies since the pioneering work of Turing [187], followed by the physical interpretation by Prigogine [188,189]. It concerns almost all fields of natural science such as biology, chemistry, ecology, physics, fluid mechanics, optics, and photonics (see the reviews on this issue [189–197]).

The linear stability analysis of the homogeneous steady states can be performed by linearizing Eq. (43) and its complex conjugate around $E_s$ and $E^\ast _s$ as

In the presence of small fluctuations depending on the space $(x,y)$ or the retarded time $\tau$, the homogeneous steady state undergoes a symmetry-breaking instability when the eigenvalue $\lambda$ vanishes for a finite wavenumber. Above this instability point, there exist a finite band of unstable Fourier modes $k_<k<k_+$, with

Note that in hydrodynamics, and in particular in the Bénard cells experiments [199], discovered in 1900, the emerging dissipative structures wavelength is not intrinsic but rather extrinsic. This famous experiment consists of a fluid between two horizontal conducting plates in the field of gravity and heated from below. The fluid will reach the state of thermodynamic equilibrium, in which there is no bulk motion and both temperature and density are constant. In this case fluctuations such as local increase of temperature will be damped out. This is a conduction regime so that the fluid remains at rest. However, if we increase further the bottom plate temperature, and above some threshold, the fluid starts to exhibit a bulk movement leading to spatial order in the form of well-structured convection cells. These cells that emerge in the convective regime generate an extrinsic dissipative structures. This is because their wavelengths are determined by external factors such as the boundaries and the thickness of the fluid layer considered [198].

4.3 Optical Crystals

The LLE supports periodic dissipative structures that emerge spontaneously from MI. When using guided wave structures, the intracavity field is spatially stabilized in the transverse plane-$(x,y)$, and therefore in this case diffraction can be neglected. In a 1D setting, periodic structures have been observed using all-fiber resonators [200]. When, however, diffraction cannot be neglected when using broad-area devices, the periodic patterns such as rolls or hexagons exist as shown in Refs. [61,201]. Analytical analysis based on a pattern selection process and a relative stability analysis have proved that only hexagonal structures of the $H0$ are stable over other possible periodic transverse structures [202]. When diffraction and dispersion have a comparable influence on the system, 3D dissipative structures are formed. These structures consist of self-organized light-bullets traveling in the cavity and have been reported in nonlinear Kerr cavities [145], and optical parametric oscillators [145,203].

At that bifurcation point, the homogeneous steady-state solution of the LLE (44) becomes unstable with respect to modes satisfying the relation, $k_x^{2} + k_y^{2} + k_{\tau }^{2} = 2 -\delta$. Beyond this critical point, there exists a finite band of unstable Fourier modes lying on a sphere with the radius $k_c= \sqrt {2 - \delta }$. These modes are arbitrarily directed in the Fourier space $(k_x,k_y,k_{\tau })$ because the system is isotropic. Although a large number of unstable modes may be excited along arbitrary directions, a periodic optical crystal is selected and emerges due to the nonlinear interaction. In the linear regime, crystals, or 3D periodic patterns, can be approximated by a linear superposition of $n$ pairs of opposite wave vectors $k_j$ lying on the critical sphere of radius $k_c$ as

In a weakly nonlinear regime, normal form analysis in the neighborhood of the 3D MI allows the determination of all the above-mentioned 3D periodic solutions. A relative stability analysis has proved that only the bcc crystals are stable over other possible periodic structures [145]. The results of these analyses are summarized in the bifurcation diagram displayed in Fig. 11(a) taken from Ref. [204]. We plot the amplitude of lamellae, hpc, and bcc lattice as a function of the injected beam with respect to the relative distance from the MI critical threshold. The bcc/hpc and the hpc/lamellae transitions are both excluded for the 3D LLE. Note that in two dimensions, the well-known hexagon/stripe transition studied in 2D transverse systems is also excluded [202]. As mentioned in an earlier report [145] on 3D crystals, the results were obtained in a perturbative way and they need further support from numerical simulations. More recently, numerical simulations have been performed for both optical crystals and 3D confinement of light in Kerr resonators [62]. The delay in simulating 3D LLE is originated from the fact that the 3D pattern contains a strongly nonlinear term. When performing discretization, this leads to large systems of strongly nonlinear stiff ordinary differential equations [205,206]. The finite-difference methods can lead to non-physical solutions [205]. The spatial discretization of the LLE is done using a Fourier spectral method with periodic boundary conditions [206–208], and the time-stepping is carried out with a fourth-order exponential time differencing Runge–Kutta method [206–208]. All the numerical simulations are carried out on a periodic domain of size $80$ units in each direction resolved using $128$ grid points, with a time step of $0.01$. The initial condition consists of a 3D structure with bcc symmetry. The maximum intensity of the intracavity field associated with the bcc structures is plotted together with their 3D Fourier transform in Fig. 11(b).

 

Figure 11. (a) 3D bifurcation diagram of optical crystals obtained for a fixed cavity detuning parameter ($\delta = 0.7$) versus the relative distance from the MI critical point $(E_i-E_{ic})/E_{ic}$. The homogeneous steady state (HSS) is plotted together with optical crystals. The full and the broken curve indicate stable and unstable states, respectively. Lamellae and hpc are denoted by lam and hpc, respectively. The black circles indicate the corresponding intensity off bcc crystals obtained by numerical simulations: (b) the bcc optical crystal in real space and (c) its Fourier transform in $(k_x,k_y,k_{\tau })$ space obtained for $\delta = 0.7$ and $E_i=0.098$. Reprinted from Chaos, Solitons Fractals 152, Tlidi et al., “Optical crystals and light-bullets in Kerr resonators,” 111364, copyright 2021, with permission from Elsevier [204].

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4.4 Complexity in Kerr Resonators: Spatiotemporal Chaos

The stationary periodic solutions of the LLE (44) lose their stability when increasing the injected field $E_i$. Two routes to spatiotemporal regimes of chaos have been identified: period-doubling and extended quasiperiodicity. In the large-intensity regime, complex spatiotemporal dynamics have been reported in driven Kerr cavities [209–212]. Indeed, transitions from either stationary periodic or localized states to the spatiotemporal complex type of behavior are shown in Fig. 12. This figure is obtained by numerical integration of Eq. (43) using periodic boundary conditions. The integration scheme is based on a Runge–Kutta method combined with a pseudo-spectral algorithm. Fig. 12(a) shows a space–time map showing the evolution of the intracavity intensity when changing the injected field amplitude with a step of $0.02$ at each $200$ time units. The initial condition consists of a stationary localized solution with large intensity.

 

Figure 12. Space–time maps obtained from numerical simulation of Eq. (43) showing the evolution of the intracavity field intensity. (a) Along the $z$ direction $E_i$ is increased from 1.3 to 3.3. (b) Here $E_i$ is decreased from 3.3 to 1.3. (c1), (d1), and (e1) Time traces for an isolated peak in the intracavity structure for $E_i= 1.85$, $1.9$, and $2.1$, respectively. (c2), (d2), and (e2) Corresponding power spectra. Reprinted by permission from Macmillan Publishers Ltd: Panajotov et al., Eur. Phys. J. D 71, 176 (2017) [213]. Copyright 2017.

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When further increasing the intensity of the injected field, the localized states become unstable with respect to the 3D pattern forming process at the critical value $I=I_{im}$. For $I_i>I_{im}$, periodic structures emerge. This is because, above the critical value of the intracavity field, the background state or the lower homogeneous steady state is unstable resulting in the breakup of LSs giving rise to periodic structures. When increasing further the injected field intensity, these periodic solutions undergo a Hopf bifurcation leading to the time oscillations of high-intensity peaks of the intracavity field intensity as shown in Fig. 12(c1). The corresponding power spectrum indicates that, indeed, the patterns exhibit harmonic oscillations with the frequency $f_1$ as shown in Fig. 12(c2). When increasing further the injected field amplitude, $E_i$, the time oscillatory pattern exhibits a Torus bifurcation with two frequencies $f_1$ and $f_2$ as shown in Figs. 12(d1) and 12(d2). For large values of the injection, the system develops an extended quasiperiodic route to chaos as shown in Figs. 12(e1) and 12(e2). This complex behavior is illustrated by Fig. 12(a). When starting with an initial condition obtained for large intensity ($E_i$) which constitutes an irregular pattern, by slowly reducing the injected field amplitude with a step of $0.02$ at each $200$ time unit, the intracavity field undergoes interesting dynamics. The intracavity field exhibits a complex behavior and by reducing further the injection, a transition toward a stationary periodic pattern takes place. Such transitions are illustrated in the space–time map of Fig. 12(b).

A number of tools have been developed to characterize and to classify the complexity of the observed patterns such as power spectra, filtering spatiotemporal diagrams, embedding dimensions, or using a time series analysis. Details of these tools can be found in Refs. [209–212]. An interesting question is how to characterize the complexity of the observed intracavity field structure, and how to distinguish between various complex behaviors such as spatiotemporal chaos, low-dimensional chaos, and turbulence. The only reliable tool for the characterization of these complexities is the Lyapunov spectrum. Classification of these complex regimes based on the Lyapunov spectrum is shown in Refs. [213–216]. Lyapunov exponents measure the sensitivity of the system to initial conditions. If the Lyapunov spectrum possesses a continuous set of positive eigenvalues, the complex behavior belongs to the class of spatiotemporal chaos. In the case of low-dimensional chaos, the Lyapunov spectrum possesses rather a discrete set of positive eigenvalues. Finally, turbulence or weak turbulence in system dynamics are characterized by a power-law cascade, such as the celebrated Kolmogorov scaling law [216].

Numerical simulations of Eq. (43) indicate that the complexity in the intracavity field structure belong to spatiotemporal chaos type of behavior [165,213,217]. Indeed, the Lyapunov spectrum formed by a continuous set of positive exponents are plotted in Fig. 13. These exponents are denoted by ${\lambda _i}$, where $i = 0, 1,\ldots, N$ with $N$ being the grid number. For $p > q$, $\lambda _p \leq \lambda _q$. The optical cavity, which is composed of $N$ cells, exhibits a complex behavior seeded by a random initial condition as shown in the space–time map of Fig. 13(a). The corresponding Lyapunov spectrum is plotted in Fig. 13(b), which constitutes the only reliable tool to characterize the spatiotemporal chaotic regime. In addition, we compute the Yorke–Kaplan dimension ($D_{YK}$) introduced by Ott [218]

 

Figure 13. Characterization of spatiotemporal chaos in the LLE with losses $\alpha$ which are not normalized to unity using numerical simulations [213]: (a) space–time map; (b) Lyapunov spectrum; (c) the Yorke–Kaplan dimension as a function of the system size. The linear growth of $D_{YK}$ dimension is fitted by a slope of $1.73$ as shown by a dashed gray line. Parameter settings: $\alpha = 0.16$, $\Delta = 1$, and $E_i= 0.16$. Reprinted with permission from [217]. Copyright 2017 Optical Society of America.

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4.5 Dissipative Rogue Waves in Kerr Resonators

In addition to the above-mentioned complex behavior in the form of spatiotemporal chaos, theoretical and experimental studies have shown the possibility to generate rogue waves in all-fiber resonators. Their formation is attributed in most cases to the MI. In these dissipative systems, small-amplitude pulses may grow to large amplitudes if their frequencies fall in the band of unstable modes with a positive gain. Evidence of rogue waves formation in the framework of LLE has recently been provided in the 1D setting in the absence of delayed feedback [213,223,224] and in the presence of time-delayed feedback [23,26,75].

Recently, experimental observations of rogue waves in a coherently driven all-fiber cavity have motivated further interest in the field of optics and photonics [224]. A schematic setup is shown in Fig. 14(a). In this experiment, an all-fiber ring cavity of length $26.5$ mm long is used to generate ORWs. The temporal and spectral characteristics of the intracavity field are measured using an intensity autocorrelator with a resolution of $10$ fs at a resolution of $2.5$ GHz for the optical spectrum analyzer. The light field circulating in the fiber ring cavity indicated by the yellow box of Fig. 14(a) suffers dispersion and nonlinearity. The nonlinear resonator depicted in Fig. 14(a) constitutes an ideal testbed for the experimental study of rogue wave formation in systems subject to a permanent pumping of energy. In addition, when taking into account injection and dissipation, this part of the setup can be described by the temporal and spatial LLE (43). We refer the reader to Ref. [224] for a detailed description of the experimental arrangement. Numerical simulations of this equation indicate that when increasing the pump power $E_i$, the temporal profile of the intracavity field exhibits complex behavior. In particular, the optical pulses exhibit “spiking” and “bursting” types of behavior. This feature is shown in Fig. 14(b) taken from Ref. [224]. From this figure, we can see that the highest peak intensity pulses are surrounded by smaller ones located at a position $\zeta _2$ which is often called the correlation length. Mathematically, this length is defined as the exponential decay of the equal time two-point correlation:

 

Figure 14. (a) Schematic sketch of an optical fiber cavity modeled by the LLE. (b) Intensities of the optical pulses from experimental observations and numerical simulations showing rogue events. (c) Logarithmic scaled probability density functions of spatiotemporal pulses as a function of the injected power showing the long tail distribution. Reprinted under a Creative Commons 4.0 license [224].

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In 2D settings, stationary 2D LSs become unstable when increasing the intensity of the injected field. In this regime, the system can exhibit spatiotemporal chaos via period-doubling or quasiperiodicity. This transition is similar to that discussed in the 1D setting in Sec. 4.4. Such high-intensity regimes are characterized by the appearance of rogue waves. An example of such behavior in 2D settings is shown in Fig. 15(a) where an extreme event is captured. A statistical analysis documenting the number of events as a function of pulse amplitude is shown in Fig. 15(b). There exists a considerable number of events with spatiotemporal maxima of intracavity intensity more than twice the SWH, and even events with amplitude as high as six times the SWH appear. The non-Gaussian statistics of the wave intensity, with a long tail of the probability distribution, reveal the typical signature of rogue wave formation. Figure 15(c) shows the variation in the number of events when the injected field $F_i$ is further increased.

 

Figure 15. 2D dissipative rogue event in Kerr resonators. (a) Intensity profile of the intracavity field in logarithmic scale. (b) Probability distribution function (PDF) of the pulse amplitudes indicating extreme events with pulse amplitude larger than twice the SWH. Parameter settings: $E_i = 5$, $\theta = 2$. (c) As in (b) with the injected field $E_i$ varied between $1.5$ and $7$. Reprinted by permission from Macmillan Publishers Ltd: Panajotov et al., Eur. Phys. J. D 71, 176 (2017) [213]. Copyright 2017.

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Large-intensity pulses generated in Fig. 15 belong to the class of rogue waves or extreme events. We would like to emphasize that rogue waves are only formed in the LLE model when the spatiotemporal chaos is well developed, i.e., when the neighboring pulses in the oscillating pattern are interacting strongly. For example, Fig. 15 displays the statistics of pulse heights in the quasiperiodic and chaotic regimes discussed in the previous section for the 1D LLE model. Even when the peak amplitude of the pulses display chaotic dynamics, as in the case of $E_i = 2.1$ [Fig. 15(c)], no rogue waves are formed in the system. This is reflected in the tail of the pulse height distribution which stays below the threshold of twice the SWH. Note that the first experimental evidence of 2D rogue waves in nonlinear optics has been realized using a nonlinear optical cavity, formed by a unidirectional ring oscillator with a liquid crystal light valve [225,226].

5. Dissipative rogue waves in cavity semiconductor

We consider the mean-field model describing the space–time evolution of broad-area VCSEL with saturable absorption [227]. We first describe experimental evidence of extreme events in 1D setting without injected field and delayed feedback. Then we investigate the control of rogue waves in two dimensions by taking into account the combined influence of optical injection and delay feedback. The advantage of implementing delayed feedback in optical resonators is that it allows for the reduction of the threshold associated with MI, self-pulsating cavity solitons, and rogue wave generation.

5.1 Model Equations for Cavity Semiconductors

The modeling of temporal–spatial dynamics in semiconductor broad-area lasers is complicated, requiring extensive knowledge of semiconductor physics, laser dynamics, and numerical methods. Many of these techniques have been already developed and used in both small and broad-area VCSEL nonlinear dynamics. In particular, the complicated microscopic physics of semiconductor materials imposes to include all the physically relevant variables, including those having a slow dynamic such as the crystal or carriers lifetime. Therefore, the radiation–matter interaction dynamics cannot be described only by the electric field alone as in the case of the Kerr media. To simplify further, we assume that only the coupling of the electric field with the excitonic resonance is considered in that model. This means that the continuous absorption band that alters the high-frequency side of the excitonic line in the absorption spectrum of the semiconductor material is neglected.

To model the cavity semiconductor, we use the slowly varying approximation and we assume a paraxial approximation of the coupled Maxwell–Bloch equations. We assume the validity of the mean-field approximation which requires high-finesse cavity or high-$Q$-factor resonators such that the semiconductor cavity operates on a single longitudinal mode. We neglect the field polarization degrees of freedom by assuming that the coupled Maxwell–Bloch equations are scalar. To control the 2D rogue waves by time-delayed optical feedback, we modify the model from Ref. [227] by adding delayed optical feedback from a distant mirror in a self-imaging configuration, i.e., light diffraction in the external cavity is compensated for [228,229]:

We are interested in investigating the effect of the time delayed feedback control (TDFC) schemes on the space–time dynamics of the cavity semiconductor. TDFC is based on the use of the difference between systems variables at the current moment of time and their values at some time in the past. The delay feedback is modeled by adding an external mirror located at a distance $L_{ext}$ from the output facet of the VCSEL. The diffraction in the external cavity is fully compensated, and the feedback field is sufficiently attenuated so that it can be modeled by a single delay term with a spatially homogeneous coefficient $\eta e^{i\phi }E(t-\tau )$. The modeling of the delayed feedback has been introduced in an earlier report by Rosanov [230] in solid-state lasers and Lang–Kobayashi in semiconductor lasers [231]. The delay feedback is characterized by the time delay $\tau$, the feedback strength $\eta$, and phase $\phi$.

In a 2D setting, despite significant theoretical advances made in cavity semiconductor devices regarding the formation of rogue waves, there are still some important remaining issues, such as their experimental evidence are still lagging. To some extent, this is because only recently wide-aperture VCSEL technology has attained maturity [232,233]. The first experimental demonstration of LSs in optically pumped semiconductor microresonators was carried out in 2001 [234] and in electrically pumped VCSELs in 2004 [235]. Their properties were studied in 2004 [236]. The possibility to obtain LSs above the lasing threshold was demonstrated in 2006 [237]. Experimental observation of LSs in medium-size VCSELs has been reported [238] as well as their polarization properties [239]. Alternatively, LSs can be realized without holding beam in a laser containing a SA or subject to frequency-selective feedback. The SA scheme was considered theoretically as early as 1992 [240] and realized experimentally in 1997 [241]. Only recently such LSs have been experimentally demonstrated in broad-area VCSELs in Ref. [242], and Ref. [243] for SA systems and frequency selective feedback. However, the monolithic semiconductor laser–SA system has only been realized by optical pumping [244]. When the interaction of several field components of different frequencies or polarizations is considered, vector LSs are predicted theoretically for a dichroic Kerr cavity [245]. Due to the surface emission, the weak cavity anisotropies, and the spin-flip mechanism, VCSELs emit linearly polarized light that may switch between two orthogonal directions [245]. A single orthogonally polarized LS with 44 $\mu$m diameter has been demonstrated experimentally [246]. Control of LSs by periodic forcing in space or time (“rocking”) was suggested in Refs. [247–249]. The inclusion of delayed feedback in the modeling of VCSELs with saturable absorption has been investigated recently. It has been shown that the delay-induced motion of dissipative solitons can be predicted to appear in nonlinear [250], and laser [251] systems. Recent studies on VCSELs [252–254] showed that the phase of the delayed feedback and the carrier decay rate strongly affect the parameter region where LSs experience a drift.

5.2 Experimental Evidence of Dissipative Rogue Waves in a Cavity Semiconductor

The observation of rogue wave formation in Kerr optical fiber has spurred interest in other laser configurations such as semiconductor resonators. In particular, spatiotemporal chaos and extreme events have been demonstrated in an extended planar microcavity laser with a SA [24]. In small-area semiconductor lasers, where the diffraction is neglected, there exist narrow parameter regions where the laser intensity exhibits high-intensity pulses in the time domain [255–257]. When the delay feedback is taken into account temporal rogue waves have also been generated in a semiconductor laser with a short external cavity [258].

In what follows, we consider a 1D problem where the diffraction term is given in Eqs. (51)–(53). In addition, we do not consider the delayed feedback and optical injection. The above-mentioned experimental observations of extreme events were conducted in a microcavity laser which possesses a VCSEL with intracavity SA [24,259]. It consists of two multilayer mirrors for optimized optical pumping and an active zone with two InGaAs quantum wells for the gain section and one InGaAs quantum well for the SA section. The cavity has a large Fresnel number and is pumped along a rectangular aperture. The dynamics are recorded at the same time at two different spatial points. The spatial coupling mediated by diffraction together with light–matter interaction, and dissipation are at the origin of spatiotemporal complexity observed in this system [24,260]. In particular, the transition from regular self-pulsating pulses to a very irregular one has been observed [24]. It has been shown that this simple device undergoes spatiotemporal chaos. This behavior is similar to that observed in Kerr resonators (see Sec. 4.4).

The recorded time traces allow one to obtain a histogram of the intensity heights. This statistical analysis together with the estimation of the SWH that is larger than two, indeed indicates the evidence of extreme events as shown in Fig. 16. This figure shows histograms of the intensity heights as a function of the injected field intensity obtained experimentally and numerically. Numerical simulations are performed by neglecting the diffusion of the carriers, the optical injection field, and delayed feedback. Time has been rescaled to the field lifetime in the cavity, which is calculated to be of the order of $8.0$ ps. Thanks to this short time scale, large recordings and accurate statistics can easily be extracted from the output of this simple device. Space is rescaled to the diffraction length which is of the order $7.4$ $\mu$m. The numerical simulations of the model given in Eqs. (51)–(53) have been realized by using periodic boundary conditions. In the experiments, as the normalized pump power $P/P_0$ is increased, the histograms of the intensity heights develop long tails.

 

Figure 16. Histogram of the intensity heights as a function of the pump intensity obtained (a) from experimental data and (b) from numerical simulation obtained for the parameters $\alpha _1=$, $\beta =0$, $\gamma _1=\gamma _2=0.005$, $\gamma =0.5$, $s=10$, and $\eta =0$. Figures 2 and 3 reprinted with permission from Selmi et al., Phys. Rev. Lett. 116, 013901 (2016) [24]. Copyright 2016 by the American Physical Society.

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6. Control of rogue waves by delayed feedback

Delayed feedback can be used to control the formation of 2D rogue waves in the Kerr cavity [23,26,213]. In this section, we consider the formation of rogue waves in a broad-area surface-emitting laser with a SA and with the delay feedback [see Fig. 17(a)]. It has been shown that when applying time-delayed feedback a single 2D dissipative soliton exhibits a period-doubling bifurcation to spatially localized chaos [228]. In 2D settings, it has been numerically predicted that a broad-area surface-emitting laser with a SA is capable of generating 2D spatial rogue waves either by strong pumping [25] or by optical feedback [26,213]. Note that, in small-area VCSELs, optical feedback induces rogue waves as in small-area semiconductor lasers in the short-cavity configuration. In this case, the round trip time in the external cavity is shorter than the laser relaxation oscillation period.

 

Figure 17. Broad-area surface-emitting laser with a SA and optical delayed feedback. (a) A schematic sketch of the VCSEL with delayed feedback. (b) 2D rogue waves in the transverse plane $(x,y)$. (c) Statistical distributions of pulse height as a function of the delay feedback strength. We vary the strength of the delayed feedback $\eta$. The red curve is obtained for $\eta =0.1$, blue for $\eta =0.3$, and green for $\eta =0.5$. (b) and (c) Reprinted with permission from Panajotov et al., Chaos 30, 053103 (2020). [43]. Copyright 2020, AIP Publishing LLC.

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We consider the parameter range where the VCSEL with saturable absorption exhibits a bistable behavior between the zero homogeneous solution ($E=0$, $N=\mu$, $n=-\gamma$) and the lasing solution ($E=\sqrt {I}e^{iwt}$, $N=\mu /(1+I)$, $n=-\gamma /(1+sI)$). In addition, we set parameters such that the upper lasing solutions exhibit a subcritical MI allowing for the formation of stable dissipative solitons [228]. By increasing the strength of the delay feedback, the output undergoes a period-doubling bifurcation to spatially localized chaos [228].

Figure 17(b) shows an example of complex 2D spatiotemporal type of behavior obtained by numerical integration of Eqs. (51)–(53) with periodic boundary conditions. The parameter settings are $\alpha =2$, $\beta =0$, $b_1=0.04$, $b_2=0.02$, $\gamma =0.5$, $s=10$, $\mu =0.3$, and optical feedback with a time-delay of $\tau =100$ and phase $\phi =0$. The time-delayed feedback parameters are: $\tau =50$, $\eta =0.1$, and $\phi =0$. Statistical distributions of pulse height in Fig. 17(c) show a non-Gaussian statistics with a long tail or L-shape indicating the occurrence of extreme or rare events in 2D settings. To control rogue waves, we fix all the parameters and we vary the strength of the delay feedback. When increasing the feedback strength, the histogram evolves toward an L-shape distribution with a long tail.

7. 3D rogue waves

7.1 Acoustic Rogue Waves

In this section, we summarize recent studies on 3D rogue waves. The first experimental observation of rogue waves in 3D settings has been reported in Ref. [261] where the authors showed the existence of acoustic rogue waves in dusty plasmas [261,262]. Specifically, they addressed the problem of rogue wave formation as a result of wave–particle interactions by correlating rogue wave event and low-amplitude hole filaments with their surrounding 3D waveforms and particle motion. Their study focuses on dust acoustic waves which are fundamental nonlinear acoustic-type density waves governed by modulation-type nonlinear dynamical equations. They are associated with the low-frequency longitudinal oscillation of negatively charged dust particles in gaseous plasmas.

A dusty plasma consists of ions, electrons, and neutral particles with the addition of small particulate matter, which is the “dust” [262]. The dust can be any type of small particulate matter which is present in the plasma medium (carbon, pollen, or manufacture spheres of silica). In the study described in Ref. [261], polystyrene particles of $5$ $\mu$m were used. The dust in the plasma environment interacts with the surroundings, and thereby acquiring a net charge of its own, thereby resulting in a more complex system. The size of the dust present in the plasma environment plays a pivotal role as it determines the net charge acquired by the dust. The interaction between the dust and plasma surroundings can be tuned by adjusting the background properties of the plasma. In the study described in Ref. [261], they could track this interaction by performing a direct video imaging of large-area dust density evolution, and by tracking individual particle motion. This allows one to construct an Eulerian–Lagrangian picture, thereby enabling a direct experimental observation of 3D rogue wave event, which are both localized in space and time.

In the experiment of Ref. [261] the dust acoustic wave mode was created by passing a sinusoidally varying current through a gas at low pressure. An ion-streaming instability results in the downward propagation of a naturally occurring dust acoustic wave, which appears as a modulation in the background dust intensity. This is illustrated in Fig. 18. The interaction of the dust particles with the background plasma results in regions where particles accumulate, and regions where there is particle depletion as it propagates [262]. The 3D perspective of this phenomenon is shown in Fig. 19(a) in the $xt$ and $yt$ planes. The rogue wave event, which is localized in both space and time, is indicated by an arrow. The low amplitude holes (defects) are indicated using crosses, the trajectories of which are shown in Fig. 19(b). It is interesting to note that the rogue wave events are surrounded by low amplitude filament holes, which indicates the role of dust particle interaction with the background plasma in the formation of dust acoustic rogue waves.

 

Figure 18. 3D rogue waves. The spatial distribution of the normalized dust intensity at various time instants along a horizontal slice in the $xy$ plane with the dust acoustic wave propagating in the vertical $z$ direction. The peak wave amplitude is indicated by red arrows clearly depicting that the individual wavefronts travel at different speeds thereby causing an irregular spacing where maximum amplitudes are observed. The rogue wave event arises due to particle focusing near regions which are the low-amplitude holes, which triggers the rapid growth of the wave amplitude and the emergence of a rogue wave in a very short period of time, of the order of $10$ ms. Reprinted by permission from Macmillan Publishers Ltd: Williams, Nat. Phys. 12, 529–530 (2016) [262]. Copyright 2016.

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Figure 19. (a) Distribution of the normalized dust intensity in the $xt$ and $yt$ planes showing the typical undulated waveform. The arrows indicate the position of a rogue wave event (RWE), which is localized on the top $xy$ plane. The locations of the low-amplitude holes are indicated by crosses, which are regions where wave amplitudes are null and phases are undefined. (b) Irregular hole filaments surrounding rogue wave event in the $xyt$ space, with their projections on the floor. Reprinted by permission from Macmillan Publishers Ltd: Tsai et al., Nat. Phys. 12, 573–577 (2016) [261]. Copyright 2016.

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7.2 3D ORWs

In nonlinear optics and photonics, most of the study of rogue waves is limited to 1D or 2D settings. Experimental evidence of 3D structures in Kerr media is still missing in the literature. However, 2D [72] and 3D rogue waves have been observed in optical systems devoid of any nonlinearity [263]. They appear in the speckle distribution of a spatially modulated optical beam that generates inhomogeneity in the system. They have been also observed in microwave experiments where the transport of waves is realized through an arrangement of randomly distributed scatterers [264]. This work applies the microwave techniques to obtain information on the transport of waves in a 2D disordered media. Large deviations from Rayleigh’s law for the distribution of the wave height are observed at lower frequency. In spatial locations, the system developed so-called “hot spots” whose intensity is much higher than expected in a random wave field.

The scarcity of 3D study in nonlinear Kerr media is because in the course of propagation, a single or more light-bullets suffers beam collapse in the case of the NLSE when the dimensionality of the system is larger than one [59]. To avoid beam collapse, a saturable nonlinearity [265] or an optical cavity [144,266,267] are used to stabilize the light-bullet. When taking into account 2D diffraction and 1D dispersion, 3D dissipative structures can be formed in an optical cavity. They consist of self-organized light-bullets traveling within the cavity at the group velocity of the light [62]. Note, however, that stable light-bullets were first predicted in a frequency conversion material [268–273] because quadratic nonlinearity does not provoke collapse in two and more dimensions [274]. In contrast to conservative a Kerr medium where the emergence of light-bullets suffers beam collapse, here the balance between pumping and dissipation allows for their stabilization [62,145]. They have been predicted in cavity nonlinear optics such as optical parametric oscillators [266,275], type II second-harmonic generation [203,276], and wide aperture lasers with a SA [277,278], semiconductor [279,280], in an inhomogeneous array of carbon nanotubes [281], or twisted waveguide arrays [282]. Furthermore, 3D structures in nonlinear resonators have also been predicted from models going beyond the mean-field approximation [283,284]. The formation of light-bullets in nonlinear optics is a well-documented issue (see overviews in Refs. [41,285–287]).

Finally, we discuss recent developments in the formation of light-bullets and 3D rogue waves in Kerr cavities. We have shown in Sec. 4.3 that close to the 3D MI, the output field presents a 3D periodic distribution of light-bullets with a well-defined frequency (in the temporal regime) or wavelength in the spatial domain. The weakly nonlinear analysis allows for the construction of different branches of 3D structures. This analysis is valid in the range of the detuning parameter where $\delta <41/33$. In the strongly nonlinear regime where the MI appears subcritical, i.e., $\delta >41/30$, there exists a so-called pinning zone of parameters in which stable light-bullets can be generated. In this regime, spatial and temporal confinement of light is possible [62]. A single, as well as bounded light-bullets, are generated numerically as shown in Fig. 20(a)–20(f). They are stationary solutions of the 3D LLE (43) in the reference frame moving with the group velocity of the light within the cavity. The number and the positions of light-bullets in the Euclidian space $(x,y,\tau )$ depend on the seeded initial condition. As the amplitudes of light-bullets having different numbers of peaks are close to one another it is convenient to plot the dimensionless “$L_2$ norm,”

 

Figure 20. (a)–(f) 3D isosurface of dissipative light-bullets and clusters of them in a Kerr resonator involving up to six 3D peaks. (g) 3D bifurcation diagram obtained for $\delta = 1.7$. (h) Comparison between the LB obtained by a spherical symmetry (red dotted line) and the cross section along 1D direction represented by a continuous black line. (i) Cross section along the transverse plane of the single LB. Parameters are $\delta = 1.7$ and $E_i =1.21$. Figure 2 reprinted with permission from Gopalakrishnan et al., Phys. Rev. Lett. 126, 153902 (2021) [62]. Copyright 2021 by the American Physical Society.

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The 3D bifurcation diagram of Fig. 20(g) suggests that light-bullets formation belong to the class of the well-known homoclinic snaking type of bifurcation [288–292]. The bifurcation diagram is established by continuation methods and its consists of two snaking curves; one describes LSs with an odd number of peaks and the other corresponds to an even number of peaks. As one moves further along the snaking curve, the LS becomes better localized and acquires stability at the turning point where the slope becomes infinite. Afterward, the LS begins to grow in spatial extent by adding extra peaks symmetrically at either side. This growth is associated with back and forth oscillations across the pinning interval where the system develops multistability [see region C in Fig. 20(g)]. This type of bifurcation is well understood in the 1D setting (see overviews of this issue in Refs. [293–296]).

When increasing the strength of the injected beam further, we observe the transition from stationary light-bullets to complex 3D structures. In this large-intensity regime, the cavity field presents an emission of extreme waves. Indeed, statistical analysis of pulse height distribution of this spatiotemporal complex regime in the 3D LLE is summarized in Fig. 21. The long-tailed statistical contribution is a signature of the presence of rogue waves. Waves with pulse heights more than twice the SWH appear in the output of the Kerr cavity.

 

Figure 21. 3D rogue waves in Kerr cavity. (a) Isosurface of 3D rogue waves obtained for $\delta = 1.7$ and $E_{i}=5$. (b) The cross section along the transverse $(x,y)$. (c) The number of events as a function of the intensity of the pulses in semi-logarithmic scale for two different values of $E_{\textrm {i}}$. The dashed line indicates events of amplitudes at twice the SWH. Figure 2 reprinted with permission from Gopalakrishnan et al., Phys. Rev. Lett. 126, 153902 (2021) [62]. Copyright 2021 by the American Physical Society.

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Finally, we discuss some possible experimental parameter values relevant to the generation of light-bullet and transition to rogue waves formation. We suggest using chalcogenide glass, which is characterized by a very strong Kerr effect and a very fast response to an electrical excitation. Typical physical parameters values are as follows. The nonlinear refractive index $n_2$ coefficient is as high as $n_2\approx 2.3\times 10^{-17}\,\textrm {m}^2~\textrm {W}^{-1}$. This results in a nonlinearity coefficient of $\gamma \approx 0.144 \,\, \textrm {W}^{-1}~\textrm {km}^{-1}$ for an effective (illuminated) area of $A_{\textrm {eff}}=25\times 10^4$ $\mu \textrm {m}^2$. The length of the cavity is $l=1000$ $\mu \textrm {m}$ and the reflectivity of the mirrors $1-\theta =0.95$, so that the optical losses are determined by the mirror transmission as the intrinsic material absorption loss can be as small as $40\,\textrm {dB/km}$. The wavelength $\lambda _0=4\,\mu \textrm {m}$ is chosen close to the zero dispersion wavelength. The choice of these physical parameters are from Refs. [297,298]. The response to an excitation by an optical injection $E_{\textrm {i}}$, which is the amplitude of the driving field incident on the cavity input mirror $\textrm {M}_1$ with an intensity transmission coefficient $\theta$, is $P_{\textrm {in}} = |E_{\textrm {i}}|^2$, which is the driving power. The intracavity power is $P=\vert E\vert ^2$. These two powers are linked through the homogeneous steady states $P_{\textrm {in}} = P [1 + (\delta - P )^{2}]$ With the above-mentioned realistic physical parameters and the dimensionless parameters for which the light-bullet exists $E_{\textrm {i}}=1.21$ and $\delta =1.7$, the intensity of the injected field should be of the order of $10$ $\textrm {MW}~\textrm {cm}^{-2}$. This is reasonable because it is well below the damage threshold of chalcogenide, which can be as high as $1$ $\textrm {GW}~\textrm {cm}^{-2}$. The link relation between the physical parameters and the dimensionless parameters are given by the relation (42). The physical model corresponds to Eq. (41) and the adimensional model is Eq. (43). In these settings, we expect that the spatial width in $(x,y)$ plane will be $\approx 270 \,\,\mu \textrm {m}$ and the temporal width of the LB will be $\approx 0.08\,\textrm {ps}$ for a value of $\beta _2 \approx 20\, \textrm {ps}^2~\textrm {km}^{-1}$ and about $1\,\textrm {ps}$ for a value of $\beta _2 \approx 3000\, \textrm {ps}^2~\textrm {km}^{-1}$.

8. Concluding remarks

The bridge between photonics and the field of extreme events and the generation of ORWs has been established in 2007 by Solli and collaborators. They experimentally observed soliton-like pulses with ultrahigh intensities named ORWs along fibers operating in the SC generation regime. The main characteristic of these pulses is that they undergo a strong localization either in time and/or in space, and their formation is ruled by a long-tailed statistical distribution. Since this pioneering work, evidence of rogue events has been shown, thanks to several investigations in different optical systems: Kerr media, semiconductors, photorefractives, liquid crystals, and xenon gas to name a few. We have discussed some similarities between nonlinear optics and photonics regarding the formation of rogue waves.

Their dynamics are characterized by a complex irregularity due to the sensitivity to the initial conditions (chaotic or turbulent nature). These characteristics render these waves unpredictable. However, they are generated in almost all systems either linear or not, stochastic or deterministic. In spite of the accumulation of important theoretical tools describing rogue waves, still a contentious debate remains about their dynamical origin. Statistically, there is a consensus: a wave that is twice the SWH of the surrounding area can be considered as a rogue wave. However, from a dynamical system theory point of view, there remains no consensus about their origin. Recently, important progress has been realized toward a unified concept and framework for describing the formation of rogue waves not only in oceanography and hydrodynamics, but also in various fields such as optics, laser physics, and biology [299,300]. This new approach is grounded on a mathematical concept called instantons well known in quantum chromodynamics [301,302].

In the first part, we reviewed in a detailed way the derivation of the NLSE describing the propagation of wave packets in a homogeneous, cubic, centrosymmetric with Kerr material. We have used a multiple time and space scales analysis allowing one to derive this fundamental equation assuming the envelope of the wave packet to be a slowly varying function of time and space. We have discussed the most important property of this fundamental equation, namely the integrability and it is considered as the simplest model to describe light propagation in a Kerr medium such as optical fibers.

In this framework, we provided a simple theoretical description of rogue waves in terms of breathers as solutions of the NLSE, and their interaction. We have limited the discussion to the first-order basic breathers such as Akhmediev breathers, Peregrine, and Ma solitons. This allows one to understand the formation of rogue waves in the framework of conservative system where most of the theory has been developed. In addition, we have highlighted similarities between oceanic rogue waves and ORWs in a strictly 1D setting. However, we have pointed out that the analogy in 2D settings between optical and hydrodynamical systems is far from being established. In addition, we did not cover the formation of rogue waves from higher-order solutions and their interactions as soliton collision, soliton explosion, and random lasers.

In the second part, we have presented dissipative rogue waves in nonlinear optical cavities. Two systems have been considered: (i) temporal optical pulses occurring in microresonators with application to frequency comb generation; and (ii) semiconductors in relation with processing and encoding information in all-optical devices. We have shown that in all these dissipative systems rogue events are possible in 1D and 2D settings. To control such rogue events, we have discussed the role of pumping, the strength of injected field, and delayed feedback. It has been shown that delayed feedback can effectively decrease the number of rogue events in these optical cavities.

Finally, we have addressed the formation of dissipative 3D light-bullets and their transition to 3D rogue waves in Kerr optical cavities. The multiplicity of these light-bullets solutions of the LLE are strongly reminiscent of homoclinic snaking. Whether the 3D light-bullets exhibit a homoclinic snaking type of bifurcation or not remains an open question that worth to be investigating in the future. Experimentally, in nonlinear optics and photonics devices, most of the studies are limited to 1D or 2D settings. Although experimental evidence of 3D rogue waves has been realized in inhomogeneous and linear media, still there is no experimental evidence of 3D rogue waves in nonlinear media. In particular, in Kerr media, 3D studies are scarce. In this context, we have shown numerical evidence of 3D rogue waves in optical Kerr cavities and we have provided realistic parameters toward an experimental realization of light-bullets and rogue waves in Kerr optical resonators. The subject of rogue waves has evolved so much in recent years that it currently touches all branches of nonlinear science and even beyond (finance research, for instance). We are convinced that its progress will continue in the future, so fascinating and interdisciplinary this theme is.

Funding

Fonds De La Recherche Scientifique - FNRS; Centre National de la Recherche Scientifique — CNRS.

Acknowledgments

We are grateful to S. S. Gopalakrishnan for helpful interactions on the collision and wave breaking in hydrodynamics and for editing the manuscript. We are grateful to M. G. Clerc, B. Kostet, and Y. Soupart for fruitful discussion. M. Tlidi is a Research Director at the Fonds National de la Recherche Scientifique (Belgium). Majid Taki is a full Professor at the University of Lille and member of the Centre National de la Recherche Scientifique (CNRS).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Mustapha Tlidi was born in Morocco. He received MD and PhD degrees in physics from the science faculty of the Université Libre de Bruxelles (ULB), Belgium, in 1989 and 1995, respectively. He is a research director with the Belgian National Science Foundation and Professor at ULB. His research focus is on systems far from equilibrium, nonlinear physics, laser physics, nonlinear optics, and plant ecology. He is the head of Physics of Dynamical Systems at ULB. He is currently teaching nonlinear optics, fiber optics, and laser dynamics at the master’s level at the Physics Department, and general physics in the faculty of science and the faculty of Pharmacie.

Majid Taki was born in Marrakesh, Morocco. He received the PhD degree in Physics from the University of Sophia Antipolis, Nice, France. His research interests are focused on superconductivity. He had a postdoctoral position at the University of Düsseldorf, Germany, and worked on the physics of plasma. Since 1989, he has been working at the Laboratoire de Physique des Lasers, Atomes et Molècules (PhLAM) and is a full Professor of Physics at the University of Lille. His main research interests deal with nonlinear optics, rogue waves, solitons and spatiotemporal chaos and turbulence in nonlinear spatially extended systems including fiber systems (optical cavities), optical parametric oscillators, and laser systems. He has supervised more than 12 PhDs and published more than 90 articles in peer-reviewed international journals, 80 conferences with 40 as an invited speaker. He is member of Scientific Committee of International Conference (Nonlinear Photonics) and Editor of the review Nonlinear Science Letters B: Chaos, Fractal and Synchronization. He conducts several national and international collaborations in Europe, USA, Australia, Mexico, and Chili. He is Head of the Nonlinear Optics and Photonics group at PhLAM laboratory, which belongs to French National Center of Scientific Research (CNRS).