We study experimentally and theoretically the dynamics of a spatially extended (along the propagation direction) oscillatory medium with coherent forcing. We observe abnormally high events, responsible for a different statistics of intensity and pulse height, in a regime where solitons and roll patterns are unstable. We focus on the formation of these high-peak events and their connection to the phase dynamics. Each abnormal event can be associated with a change in the slope of the phase time trace. Furthermore, the coexistence of ±2π phase rotations inside the cavity can be associated to the observation of abnormal events, similarly to recent predictions in bidimensional vortex turbulence.
© 2017 Optical Society of America
With the seminal work on optical rogue waves , optical devices have become a system of choice for the analysis of large amplitude and unexpected events occurring in complex wave propagation settings. One possible and fruitful approach puts emphasis on the properties of individual events such as their shape, their reduced spatial extension and temporal duration or the presence of “sister waves” surrounding the main event, in analogy with the so-called “Draupner wave” . From this point of view one can cite the study of the superposition of linear waves [3, 4] and the theoretical and experimental analysis of different kinds of solitons and breathers (recent works include [5–9]).
In a complementary way, other studies focus more on the statistical properties of the field, often aiming at determining the general dynamical mechanisms which favor the emergence of unexpectedly large events or at predicting their occurrence with some anticipation. In conservative systems for instance the regimes of modulation instability [1, 10, 11], optical turbulence  and integrable turbulence [13, 14] have been shown to lead to heavy tailed distributions. In dissipative systems, rogue waves have been observed in lasers with saturable absorber along the propagation direction [15, 16] and also in bidimensional systems, in the plane transverse to propagation [17–19]. In the former case they have been related to chaotic pulsing and in the latter to spatio-temporal chaos  and dissipative solitons . The case of oscillatory media (optical parametric oscillators or lasers) with nearly resonant forcing has also been examined in different numbers of spatial dimensions. In purely temporal systems heavy tailed statistics have been attributed to low dimensional deterministic chaos near a critical transition [20, 21]. In this case noise or deterministic parameter modulation have been shown to heavily impact the presence of rogue events . In two transverse dimensions, the occurrence of rogue events has been associated to vortex turbulence and in particular to the collision of spiral waves [23, 24]. Here we address the emergence of abnormally high events in a complementary case, i.e. the one dimensional case along propagation. We prepare a highly multimode ring semiconductor laser and apply to it a coherent optical injection beam. A wealth of dynamical regimes has been observed in this configuration, including solitary waves , complexes of solitary waves , plane waves, modulational instabilities and front propagation . In contrast with passive nonlinear resonators (see e.g.  in the context of extreme events), the solitary waves observed experimentally to date are not locked to the forcing. Due to the oscillatory nature of the medium, they consist of 2π (or n ∗ 2π in the case of complexes) rotations of the optical phase with respect to the optical phase of the forcing beam . In the propagative geometry with non-instantaneous nonlinearity, only one sign of phase rotation is found to be stable [27, 30]. Here we measure experimentally the statistical distribution of the emitted intensity and relate it to the measured phase dynamics. We analyze our experimental results thanks to numerical simulations of a set of partial differential equations for the electric field envelope and the carrier density. We consider a parametric regime where the locked solution is unstable, and a non stationary irregular roll pattern develops instead. This regime is the same observed experimentally, and it differs from recent observations reported in a similar system , where extreme events are observed in a regime where the locked solution and phase solitons are stable, and very high intensity peaks develop through collisions of localized structures of different nature and velocity.
In section 2 we show the experimental results, their statistical and dynamical analysis, and we focus on the role of the phase in the formation of abnormal events, and on their predictability. In section 3 we present the theoretical model describing our system, and show numerical results about intensity dynamics and its statistics, in excellent agreement with the experiment. We then focus on the phase dynamics and individuate the role of ±2π phase rotations in the formation of abnormal events. Finally, in section 4 we draw some conclusions and perspectives.
2. Experimental results
2.1. The setup
The experimental setup depicted in Fig. 1 is based on a highly multimode semiconductor ring laser under coherent external forcing and has been described extensively in [25, 27, 32]. The ring laser is made of a semiconductor optical amplifier (SOA) acting as gain medium centered around 980 nm and enclosed in a 1 m long ring cavity. The anti-reflection coating on the SOA facets prevents from self-lasing of the semiconductor element. An optical isolator has been placed inside the ring cavity to ensure unidirectional operation and the spatial filtering imposed by its physical aperture forces the ring laser to emit on a single transverse mode. Two beam splitters are inserted in the ring cavity, the first one has 90% reflectivity and allows the injection of the external coherent field while the second one has 10% reflectivity to provide the ring laser output. The ring laser is operated at a bias current of 1.1 times the threshold current value (around 1.6 to 1.8 A depending on precise alignment conditions). The coherent injected field comes from a homemade tunable single mode laser allowing a coarse control of the detuning between the injection and the ring laser. The fine control of the detuning is obtained by adjusting the cavity length by use of a piezo electric actuator placed on one of the mirrors mount. However, due to optothermal nonlinearity [27, 33–35], the value of the detuning can not be kept constant for sufficiently long time to allow meaningful measurements (a situation which may be addressed in the future by temporal engineering of the detuning parameter as in ).
We analyze the output of the ring laser by means of fast photodiodes connected to a 12.5 GHz bandwidth real-time oscilloscope with 100 GS/s sampling rate. Instead of measuring directly the optical power, we use an heterodyne setup based on a 3 × 3 fiber coupler to access both the power and the relative phase between the forcing and the ring laser (see [25, 27] for more details).
The longitudinal extension of the ring laser acts as an effective propagation dimension in space, with mode-defining boundaries, along which the optical field can present many different multimodal behaviors depending on the two main control parameters of the system: the injection strength (optical power) and the frequency detuning between the injection and the closest resonance of the ring cavity. For strong enough injection power (about 5mW measured before the incoupling beam splitter, depending on mode matching conditions), the laser can be fully synchronized (in phase and frequency) to the forcing. In this case, we observe a stable uniform state characterized by a flat intensity profile and constant phase difference between the laser and the forcing. When reducing the forcing strength, the formation of dissipative phase solitons (PSs) has been observed and described as localised 2π phase rotations corresponding to local unlocking of the laser to its forcing. Roll patterns have also been observed and described as the result of modulational instability . These periodic patterns correspond to a phase bounded regime for which the phase circles on a closed loop in a bounded region of the complex plane .
2.2. Statistical analysis
We present here the observation of a dynamical regime in which an unstable roll pattern hosts high intensity peaks appearing randomly in space and time. Figure 2(a) shows a spatio-temporal diagram of this regime, where the horizontal axis refers to the longitudinal spatial dimension of the laser and the vertical axis represents time in units of the cavity roundtrip time τc ≃ 3.6 ns. In this figure, we can clearly identify the unstable roll pattern in background and high intensity pulses nucleated on top of it. In order to get the statistical signatures of this regime, we computed the probability density function (PDF) of the total optical power and the distribution of the peak amplitudes, respectively represented in Figs. 2(b)–2(c). In the PDF of the power, we first distinguish a plateau (up to 10), followed by an exponential decay (up to 17) and a second exponential decay with smaller slope corresponding to a clearly different statistics. The presence of this statistical deviation at high amplitudes implies that high intensity pulses occur more often than if the distribution followed the first exponential decay. Although the shape of this distribution is non trivial, we understand easily that different mechanisms are necessary to produce these three different behaviors.
The statistics of the peak heights is shown in Fig. 2(c). The heights are measured by computing the difference between each maximum and the previous minimum of the full time trace. Note that we disregard events of height below 5, inasmuch we have no way to discriminate the actual field dynamics from the detection noise. Doing so, we do not need to apply any spectral filtering on the data that would modify the distribution. The shape of the distribution is well fitted by a Gaussian up to 16 and presents a huge deviation beyond this point. The Gaussian part of the distribution (at low values) is related to the oscillating background whereas the deviation appearing at high values corresponds to the presence of abnormal events. Since the threshold we use as definition is clearly identified as an inflexion point separating the tail from the rest of the distribution, those events belonging to this tail may be expected to actually differ from others in nature and not only in amplitude.
2.3. Dynamical analysis
To understand the formation of these abnormal events, we focus on one particular event, statistically representative of the relevant regime, and provide a detailed analysis by looking at both phase and intensity dynamics. Figure 3(a) shows a zoom of the spatio-temporal diagram presented in Fig. 2(a) on 150 roundtrips around an abnormal event. The horizontal red lines represent the positions at which intensity profiles shown in Fig. 3(b) have been taken. The first line (roundtrip 40) shows the state of the system before the abnormal event occurs and the associated phase trajectory is presented in Fig. 3(d). At this stage of evolution, the phase remains in a bounded region of the complex plane (Argand plane). Note that the phase measurement we realize gives us the phase difference between the injection and laser relative to an arbitrary reference phase. This means that the position of the bounded state in the complex plane is not relevant, and the whole trajectory could be arbitrarily rotated around the origin without changing its shape. At roundtrip 83, the abnormal event has reached its maximum intensity. By comparing the height of this event with the statistics shown in Fig. 2(c), we can easily understand that this object participates to the strong statistical deviation described before.
As we can see in Fig. 3(c), the high intensity peak is accompanied by a striking topological change of the phase space picture, namely we observe that the trajectory circles around the origin. The system seems to be pushed away from the bounded state and explore a wider region of the complex plane. The process occurs with a phase bounded rotation growing in amplitude and changing shape until it actually includes the origin. Thus, the abnormal event acquires a counterclockwise (+2π) phase rotation. This phase trajectory resembles the one of PSs, that are actually not stable for this set of parameters. PSs have been found to carry only positive chiral charges (counterclockwise phase rotations) due to the strong symmetry breaking caused by the propagative nature of the system . The phase trajectories of the observed abnormal events give a strong argument to describe them as unstable PSs nucleated from the unstable roll pattern. In section 3, a theoretical analysis and numerical simulations are presented to clarify the structure of the phase space and explain this connection in a deeper way.
In Fig. 4, we show the evolution of the unfolded phase around the formation of an abnormal event. The time trace is centered on the abnormal event maximum intensity (t = 0). In this representation, it is clear that the phase stays bounded up to 20 ns before the maximum of the abnormal event. We can clearly identify the phase rotations as a change of slope, that starts before the moment where the abnormal event reaches its maximum. At each roundtrip, the system acquires 2π rotations corresponding in this representation to steps of 2π. The phase and amplitude dynamics leading to the emergence of these abnormal events are reminiscent of the defect mediated turbulence discussed theoretically in the context of forced oscillatory media [24,37].
The model used to describe our system is given by the following set of rate equations [25, 27], which is derived from the effective Maxwell-Bloch equations, when the polarization of the material is adiabatically eliminated:25, 27].
At difference from [25, 27], a diffusion term appears on the left hand side of Eq. (1). While we introduce it phenomenologically, its derivation proceeds from a nonstandard elimination of the polarization which takes into account the finite gain linewidth . Physically this term is weighed by a coefficient which is proportional to the squared ratio of the cavity free spectral range to the gain linewidth.Fig. 5(a) we plotted the stationary intensity of the electric field as a function of the injection amplitude y according with Eq. (3). The values of the pump parameter, detuning and injection amplitude have been chosen to keep the system very close to the left turning point of the stationary homogeneous solution: µ = 1.1, θ = −3.04 and y = 0.004. Points A, B and C indicate the three fixed points for the chosen value of injection amplitude (vertical red dashed line): in particular A is a node, B is a saddle and C is a focus [25, 27].
In Fig. 5(b) the stability scenery of the homogeneous stationary solution is plotted in the (I, n) plane, where n is the order of the longitudinal sidemode. The green domain indicates the unstable region as derived from the linear stability analysis of the locked solutions (3)–(5) in presence of diffusion (d, 0), with the reported parameters, while the blue domain corresponds to the choice d = 0, which amounts to the pure rate equation model without spectral filtering [25, 27]. It is clear that the diffusive term sets a limit to the number of unstable longitudinal modes. Actually, for a cavity length of 1 m and assuming a gain linewidth of 10 THz, the number of modes under the gain line should be much larger, i.e. the diffusion coefficient d should be smaller by some order of magnitude. We chose here d = 10−6 in order to filter the higher order modes, which are not relevant for the dynamics. In that way we could limit the number of grid points in the numerical simulations to a reasonable value (1024 points). Fig. 5(b) shows that our choice is justified because the large value of d affects only slightly the instability domain for the larger intensity values, i.e. in the upper branch of the stationary solution.
From Fig. 5 we can observe that all the three fixed points A, B and C are unstable for the parametric set considered. For a more detailed description on the stability analysis of this system see .
3.2. Numerical simulations
The numerical simulations of Eqs. (1)–(2) confirm that for this system a variety of different regimes is possible. From Eqs. (3)–(5) it is clear that for the free running laser (y = 0) the stationary equation gets reduced to Ds = 1 (which implies |Es |2 = µ −1) and θ + α = 0. Thus θ+α is the frequency mismatch between the driving field and the free running laser . When the injected field is red detuned (θ + α > 0), the whole upper branch of the homogeneous stationary solution is stable and the system can sustain phase solitons (PSs), provided the amplitude of the injected field is such that the fixed points A and B, described above, are sufficiently close, i.e. the system is close to the left turning point of the homogeneous stationary solution [25–27, 32]. Phase solitons are homoclinic trajectories connecting the locked homogeneous state A with itself. We want to stress that unlike purely spatial systems , in a non-instantaneous propagative (temporal) system as in this case, the chiral charge of the phase soliton can only be positive (i.e. corresponding to counterclockwise phase rotations) [27, 30].
In the opposite situation (θ + α < 0), a multimode instability affects a part of the upper branch close to the left turning point, as shown in Fig. 5(b). The PSs are no longer stable and instead a roll pattern arises due to the beating of the fundamental mode with high order sidemodes . Eventually even the rolls become unstable when the injection amplitude is brought too close to the left turning point, and more complex spatio-temporal dynamics can be observed, as in the experiments.
In Fig. 6 we plotted the spatio-temporal diagrams relative to 2000 cavity roundtrips in two different reference frames. The first reference frame (represented in Fig. 6(a)) corresponds to the experimental one, where data are taken at z = L, as one would have from a detector placed right at the output of the cavity (save for the mirror transmission). The second reference frame (represented in Fig. 6(b)) can be used only numerically: at each roundtrip, the entire spatial intensity profile inside the cavity is recorded, and then all the recorded profiles (much like snapshots taken every roundtrip time) are stacked onto one other, in such a way to obtain the spatio-temporal diagram illustrated. It is noticeable that the two spatio-temporal diagrams are mirror images one of the other, as it must be in the case of propagating structures.
From now on, we will only consider the experimental reference frame in order to make easier the comparison with the experimental data.
In Fig. 7 we show a zoom of the spatio-temporal diagram centered on an event of high intensity occurring at the point highlighted by line 3. This picture shows an interval of 2000 roundtrips in a simulation of 3×105 roundtrips where the first 9 × 104 roundtrips were excluded from the following statistical analysis to discharge transient effects.
The four temporal profiles illustrated in Fig. 7(a) correspond to the horizontal cuts highlighted in Fig. 7(b). In particular the temporal profile from section 1 occurs 500 roundtrips before the event and corresponds to a mostly bounded regime, where we can appreciate the remnants of the roll pattern which would be stable for a larger injection amplitude. Section 3 is taken at the maximum of the high intensity event, while sections 2 and 4 are taken, respectively, 40 roundtrips before the event and 160 roundtrips after it.
As for the statistical analysis, we plot (in black) in Fig. 8 the PDF of the temporal peak heights (maximum minus precedent minimum) (a) and of the overall intensity values (b). For comparison we have also included in these plots the corresponding PDFs for a second set of data (plotted in dark yellow), simulated for the same parameters as the data in black but a different value of detuning θ = 3.049: this simulation corresponds to an unstable roll regime, which remains clearly bounded and does not exhibit big excursions in intensity.
As in the corresponding experimental Fig. 2, we observe that the PDF of the heights (a) presents an initial slope that follows a Gaussian distribution (green dashed line), to be associated with the bounded state, while the events in the tail of the distribution appear more frequently than expected by a Gaussian statistics. Furthermore, in the statistics of the total intensity we notice two different negative exponential slopes in the tail of the distribution, as observed in the experiments: the first slope (green dashed line fit), may be be associated with the roll pattern, the second slope (red dashed line fit) highlights a deviation of the tail from the previous exponential decay, where very intense events are more frequent than expected in the bounded regime.
3.3. Phase dynamics
Let us focus now on the role of the phase dynamics in the formation of these abnormal high-peak events. To this purpose, it is useful to set E = ρ exp (iφ) and recast Eqs. (1)–(2) for the field amplitude and phase:Eq. (7), we observe that the sign of the right hand side determines the rotation direction of the electric field in the complex plane for fixed z or t. If the first term θ + αD were null, assuming ρ constant, the equation would be that of a pendulum, with a stable fixed point in 0 and an unstable one in π. Yet the ratio y/ρ in front of the sine is normally very small, because the unstable focus C in the proximity of the origin acts as a repeller that prevents the field amplitude to assume too small values. Therefore the dominant term in Eq. (7) is the first one, which forces the electric field to rotate clockwise or counterclockwise depending on its sign. We can define a critical value of D, Dc = −θ/α which marks the boundary between the two rotation directions. With our choice of the parameters Dc ≈ 1.013. If the dynamics of D were fast enough to follow adiabatically that of the field intensity, we could also define a critical amplitude through the stationary equation . With our parameters we have ρc ≈ 0.293. Although the above assumption is not fully confirmed by the simulation, the critical amplitude ρc plays a role in the phase dynamics, as will be shown in the following.
In order to compare with the experiment, we computed a histogram in the complex plane (Re(E), Im(E)) of all the values assumed by the electric field. Fig. 9(a) is the experimental histogram and Fig. 9(b) is the numerical one. A clear similarity between the two histograms is noticeable, especially regarding the presence of an unstable bounded state (yellow area) from which rarer events get detached, exploring other areas of the histogram (depicted in red). The reason for the different location of the bounded state between the experimental and numerical picture is due to the fact that the relative phase is measured up to a constant, as explained in section 2.
As illustrated in section 2, from the experiments it was already clear that chirality was playing an important role in the formation of high-peak events: in particular the latter seem to be associated with the appearance of a positive (counterclockwise) chiral charge (see Fig. 4). In the numerical simulations, as we will explain later, we noticed that clockwise phase rotations also play an important role in the formation of abnormal events. Unluckily in the experiment it was impossible to clearly identify the role of clockwise rotations in relation to high-peak events due to the low intensity values of the structures carrying the negative charge, which fell within the experimental noise level. Nevertheless, we can now draw a comparison of the phase dynamics between experimental and numerical data as detailed in the following. Eventually this will allow us to suggest a predictor for the abnormal event, associated with the chirality of the propagating structures. In Fig. 10(a) we plotted the evolution of the phase when approaching an abnormally high event, in a situation similar to the experimental case: here we notice that an initial balance between the number of positive and negative chiral charges is broken by the loss of a negative charge, which gives rise to a change in the slope of the phase. A little later, a high-peak event occurs (highlighted by the red vertical dashed line). In the zoom in Fig. 10(b) it is possible to notice that in the initial plateau for the phase there are both a positive and a negative chiral charge, which then get reduced to a single positive charge. This situation is similar to the one illustrated in section 2 in Fig. 4, where an initially balanced situation with no chiral charges becomes unbalanced through the acquisition of a positive chiral charge, right before the occurrence of an abnormal event. This is not nevertheless the only possible configuration regarding the evolution of the phase before or after the occurrence of abnormal high-peak events. As a common thread we can argue that the occurrence of abnormal events may be associated with a global change in the slope of the phase: however, as it has been observed in the experiment, this change is not to be considered as a precursor since it can occur either before or after the high-peak event itself. To better illustrate this point in Fig. 11 we depicted the total evolution of the phase during a time interval of 2×104 roundtrips, the five red vertical dashed lines correspond to five different abnormal events occurring during this time interval. Furthermore from Fig. 11 we can observe that even if sometimes the number of negative chiral charges exceeds the number of positive chiral charges in a single roundtrip (which leads to a negative slope for the phase), the number of positive chiral charges largely exceeds the number of negative ones, as it has already been observed in the experiments: we believe that this strongly asymmetric behavior is to be related to the propagative nature of our system.
It is interesting to analyze also the trajectories in the two phase-space sections (Re(E), Im(E)) and (D, I) to evidence the phase rotation peculiarities. In Fig. 12 we show the trajectories of the four sections in Fig. 7, where the line color changes from dark green to yellow for increasing time (from the left to the right side of each section). The (unstable) locked states A, B and C of Fig. 5 are highlighted in each figure. The presence of two different time scales in the model has an impact on the intensity excursions, similarly to what occurs in systems lacking propagation in space .
In (b) the red vertical dashed line corresponds to the condition D = Dc = −θ/α relevant for the phase dynamics in Eq. (7), while the red dashed circle in Fig. 12(a) has radius ρ = ρc (see the beginning of this Subsection). Although the numerical simulations do not fully support the assumption of adiabatic following for D which led to the definition of ρc, it is interesting to note that, as long as the term (y/ρ) sin ϕ is negligible in Eq. (7), when the trajectory enters the circle (where approximately both fixed points A and B lie) it will start rotating clockwise, and as soon as it exits the circle it will again rotate counterclockwise.
In 12(b), we plotted a red (blue) cross every time a full 2π (−2π) counterclockwise (clockwise) phase rotation was reached. The red and blue circles indicate the starting point for the corresponding full rotation. At first we can observe in column 1 of Fig. 12 a mostly bounded regime, where the trajectory of the system remains in a specific area of the complex plane, also highlighted in Fig. 9, without performing any full phase rotations.
Then, in column 2 of Fig. 12, 40 roundtrips before the event, as the trajectory of the system starts to move towards higher values of the population D, we can observe the occurrence of a first clockwise phase rotation (blue circle → blue cross) and a second counterclockwise rotation (red circle → red cross). We can notice that: i) the clockwise rotation is triggered, for low enough (almost zero) values of the intensity, in the proximity of the condition D = Dc; ii) after performing a full −2π rotation (which ends at the blue cross), the system reaches a maximum value of D and then gets repelled to high values of the intensity I due to the repulsive nature of the point C; iii) thanks to a favorable ϕ the term with sin ϕ becomes more and more important in the determination of the sign of the phase equation and, at the point indicated by the red circle in (b), a counterclockwise phase rotation starts forming, which is reinforced when the trajectory approaches the red vertical dashed line, below which the first term of the phase equation becomes positive as well. Such a rotation reaches 2π at the point highlighted by the red cross.
The excursion of the trajectory toward high values of intensity, due to the interplay between these two opposite phase rotations, reaches then its maximum in column 3 of Fig. 12, and, after that, it starts decreasing, as depicted in column 4 of Fig. 12 where the two phase rotations are still present but the trajectory is slowly moving towards lower values of population, eventually evolving into a bounded state as the one in column 1 of Fig. 12, where phase rotations will disappear (usually not at the same time).
We observed that this mechanism is common to all the high-peak events in the tail of the distribution in Fig. 8(a) and could serve as a predictor of abnormal events. In Fig. 13 we consider a zoom of the spatio-temporal diagram for the intensity I of the electric field (a) and for the population D (b) centered on an abnormal event (highlighted by the green circle). Here we depicted with a light blue (orange) line the clockwise (counterclockwise) phase rotations.
The mechanism illustrated in Fig. 12 is here visible: at first a negative chiral charge (light blue) appears; then, around 100 roundtrips before the event, a positive chiral charge (orange) is formed, but the simultaneous presence of a clockwise phase rotation is necessary to observe the abnormally high event; finally, both charges disappear, at different times. We would like to emphasize that clockwise rotations always occur only for high values of population, when the intensity of the electric field approaches zero (as in Figs. 12(b) and 13(b)), and it is the interplay of clockwise and counterclockwise phase rotations that gives rise to the observed abnormal events. In fact, it may happen that clockwise rotations occur without any positive chiral charge in their proximity: such a configuration does not give rise to high-peak events.
We observe that phase rotations may be associated with a transient exploration of the PS regime, unstable for this choice of parameters: in particular, in a stable PS regime, counterclockwise rotations would correspond to stable PSs. However clockwise rotations cannot be associated with unstable PSs , due to the asymmetric nature of this propagative system.
We have studied both experimentally and numerically the occurrence of abnormal events in a highly multimode semiconductor laser with coherent optical injection, in a configuration close to the roll pattern regime. The experimental and numerical data present an excellent agreement, corroborated by a detailed statistical analysis. In the regime under study, where the roll pattern is modulationally unstable, both the distribution of the field intensity and the peak amplitude are bounded but present a characteristic size separating two types of events. This scale separation emerges as different exponential decays in the distribution of the intensity and as a strong deviation from Gaussianity in the peak amplitude distribution. In this parametric regime phase solitons are unstable but we observed that the phase dynamics of the system plays a crucial role in the formation of the events pertaining to the distribution tail. In particular, the interplay between ±2π phase rotations appears to be related to the observation of abnormally high events and it is absent in other parametric regions, not exhibiting big excursions in intensity. The observed phase and amplitude dynamics is strongly reminiscent of the regime of defect mediated turbulence  or vortex turbulence that has been recently associated with optical rogue waves in an oscillatory two-dimensional transverse spatial system with cohrent forcing . However a full experimental and numerical characterization of the emergence of phase defects and their statistics in the context of this non-instantaneous and propagative oscillatory medium remains to be performed.
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