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We develop a theoretical model for a unidirectional ring laser consisting of an isotropic active medium inside a cavity containing a birefringent Kerr cell. We analyze the dynamical behavior of the system as we modulate the voltage applied to the Kerr cell. We discuss the bifurcation diagram and we study the regions of control parameter space where it becomes possible to observe and predict extreme events.

Lasers have been used as test benches for nonlinear dynamics in many different configurations, some of them requiring a complicated set up or involving a very large number of degrees of freedom. Lasers with optical feedback, laser with saturable absorbers, and lasers with large Fresnel number are typical examples appearing in recent literature. In particular, lasers with a modulated parameter are able to display a large variety of dynamical regimes [

Today there is an increasing interest on the study of the so called optical rogue waves. Optical rogue waves are high intensity pulses much larger than average and therefore rare events [

Here we analyze a theoretical model of a laser in which the modulation is applied to the relative phase between the two components of the linear polarizations of the field. Modulation of such parameter is usually achieved by introducing inside the cavity a birefringent material whose extraordinary refractive index is changed through a sinusoidal voltage. If we assume that the active medium and the cavity are isotropic, the laser may operate in principle at any polarization of the field. We identify the existence of generalized multistability, period doubling transition to chaos, and three types of crises of strange attractors. However the main objective of this work is to show the appearance of optical rogue waves and to identify the physical mechanism at their origin. Special interest is put also in establishing our ability to predict them [

The theoretical model is based on a single mode, Class B unidirectional ring laser with an electro optic modulator (EOM) placed inside the cavity. After applying the rotating wave and slowly varying amplitude approximations and without taking into account diffraction, the set of Maxwell-Bloch equations describing the interaction between a single mode electromagnetic field and a two-level atom are

Figure

Bifurcation diagram showing the Intensity local maxima, I, as a function of the modulation amplitude, m. The parameters corresponding to the figure are

Several facts appear from a direct observation of the bifurcation diagram:

In the region marked by (a) there is a bistable cycle between two periodic solutions. The lower branch is period 1 (T1) and the upper solution is a period two (T2) with respect to the modulation frequency. It is clear that there is a subcritical bifurcation of T1 leading to T2.

In (b) there is coexistence of several attractors of different periods. We identify the previous T2 and then three other branches of period 3, period 5, and period 6. Each of them gives rise to a period doubling bifurcation leading to chaos. Only the chaotic behavior resulting from T2 survives as the amplitude of the modulation is increased. As described in [

In (c) we observe an abrupt expansion of the chaotic attractor. This expansion is produced by a bifurcation called “external crisis" in which the chaotic attractor collides with an unstable orbit generated in the saddle bifurcation of a different preexisting branch. Each collision generates a sudden expansion of the attractor in phase space. This process generates the appearance of high intensity pulses. Close to the bifurcation they are rare but then they become more and more frequent causing a significative increase in the average intensity of the maxima. The first region where such optical rogue waves appear corresponds to the interval between

In region (d) it appears a new branch of period 4. The chaos originated in such branch disappears at a boundary crisis. The unstable period 4 orbit collides with the existing chaotic attractor generating a new expansion in phase space. It is important to note that the bifurcation producing the crisis of the chaotic attractors were in all cases supercritical, and there is no bistable behavior.

As the modulation amplitude increases, the strange attractor expands gradually until region (e). It appears then another periodic solution, of period 5 (T5), in a saddle node bifurcation. The stable T5 solution shows period doubling bifurcations, ending in chaotic behavior. The unstable T5 solution collides with the preexisting strange attractor and this external crisis produces a chaos merging [

Inside every chaotic region there are periodic windows. The transition from the strange attractor to a periodic solution can be done through a process of intermittency and therefore it may alternate high intensity peaks with regular periods of low intensity.

The bifurcation diagram of Figure

Intensity, I, as a function of time measured in units of the modulation period for a modulation amplitude

Histogram of the number of intensity peaks as a function of the intensity maxima for the temporal sequence of Figure

In fact we are thinking that the process we observe here corresponds to the dynamical behavior described in reference [

Figure

Histogram of the maxima of intensity at

Predictability of extreme events in deterministic systems was studied in [

Intensity as a function of time in units of the modulation period using an extreme event as trigger and superposing 322 signals.

It is worthwhile mentioning that optical rogue waves in vector lasers have been observed in Er doped fiber lasers [

The [DATA TYPE] data used to support the findings of this study are available from the corresponding author upon request.

The actual address of Alexis Gomel is Universite de Geneve, Suisse, and Jorge R. Tredicce’s is also at Departamento de Fisica, Universidad de Buenos Aires, CONICET, CABA, Argentina.

The authors declare that they have no conflicts of interest.

The authors wish to acknowledge support from the Project ECOS-Sud A14E03 “Evenements extremes en dynamique non lineaire."