We introduce a model that permits the unified description of the
emergence of different regimes of complex temporal structures in
noise-like or quasi-CW fiber lasers. The model is based on the
vector Ginzburg-Landau equation that also permits to reproduce
the experimentally observed polarization antiphase behavior and
the synchronization of spatiotemporal turbulence into polarizations
domain wall solitons

Wabnitz, S.

WABNITZ, Stefan

Archivio della ricerca- Università di Roma La Sapienza

Rev. Cub. Fis. 33, E32 (2016) ARTI´CULOS ORIGINALESSPATIOTEMPORAL CHAOS AND ORDER IN FIBER LASERSCAOS Y ORDEN ESPACIOTEMPORAL EN LA´SERES DE FIBRAS. Wabnitz†Dipartimento di Ingegneria dell’Informazione, Universita` di Brescia; stefan.wabnitz@unibs.it†† corresponding authorRecibido 26/2/15; Aceptado 22/9/15We introduce a model that permits the unified description of theemergence of different regimes of complex temporal structures innoise-like or quasi-CW fiber lasers. The model is based on thevector Ginzburg-Landau equation that also permits to reproducethe experimentally observed polarization antiphase behavior andthe synchronization of spatiotemporal turbulence into polarizationsdomain wall solitons.Se introduce un modelo que permite la descripcio´n unificada dela emergencia de diferentes regı´menes de estructuras complejastemporales en la´seres de fibra noise-like o cuasi CW. El modeloesta´ basado en la ecuacio´n vectorial de Ginzburg-Landau, quetambie´n permite reproducir el comportamiento experimentalmenteobservado de polarizacio´n antifase y la sincronizacio´n de laturbulencia espacio temporal en el dominio de polarizaciones dela barrera de solitones.PACS: 42.65.Sf Dynamics of nonlinear optical systems; 42.60.Mi Dynamical laser instabilities; 42.55.Wd Fiber lasers; 42.55.Ye Ramanlasers.I. INTRODUCTIONFiber lasers have been widely studied as stable sources ofultrashort mode-locked pulses [1]. In recent years, novelregimes of operation of fiber lasers leading to complextemporal dynamics have been experimentally explored.Consider for example noise-like pulse emission [2], antiphasepolarization temporal structures [3] and quasi-CW Ramanfiber lasers [4]. In this work, we provide a unified theoreticaldescription of how modulation instabilities in quasi-CWlocked fiber lasers may evolve into the different regimes ofturbulent or noise-like pulse emission.II. SCALAR GLEWe discuss first the phase transition diagram that rules thedifferent regimes of stable, intermittent or spatiotemporalchaotic emission for fiber ring lasers, passively mode lockedby nonlinear polarization rotation (see Fig.1). In this diagram,the two free parameters of the ruling Ginzburg-Landauequation (GLE) that describes the cavity-averaged slowtemporal evolution of the field in the laser are the cavitydispersion and nonlinearity, respectively [5]. In Fig. 1,phase transition curves separate domains that are associatedto different statistical states of the light field: amplitudeturbulence (AT), phase turbulence (PT), spatiotemporalintermittency (STI), bi-chaos (BC) that is intermediatebetween amplitude and phase turbulence, and non chaotic(NC) emission.III. VECTOR GLENext we introduce nonlinear polarization coupling ina vector fiber ring laser, that is, without polarizationselective components. In this case, it is remarkable that(complex) nonlinear polarization coupling may lead to theanti-synchronization of spatiotemporal chaos into orderedultrashort pulse or soliton trains, and polarization domains.As we shall see, nonlinear polarization coupling also leadsto vector rogue waves in long-cavity Raman fiber lasers.Complex temporal light pattern dynamics in vector fiberlasers may be phenomenologically described in terms of thecoupled GLEs [6],∂tU = U +(1 + iβ)∂2τU −(1 − iγ) |U|2U − (η − iρ) |V|2U∂tV = V +(1 + iβ)∂2τV −(1 − iγ) |V|2V − (η − iρ) |U|2V. (1)Here the time coordinates t and τ denote a slow time(measuring the number of circulations in the cavity) and afast time (measuring the intra-cavity pulse structures), andU, V are the complex amplitudes of ortogonal polarizations.Figure 1. Phase diagram of transition curves between different lightemission states from a scalar fiber laser [1].Next β measures cavity dispersion (β > 0 for anomalousdispersion), γ is the self-phase modulation (SPM), η is thecross-gain saturation, and ρ is the cross-phase modulation(XPM) coefficient, respectively.REVISTA CUBANA DE FI´SICA, Vol 33, No. 1E (2016) E32 ARTI´CULOS ORIGINALESLet us describe now some characteristic regimes of stable orunstable light emission when the parameters of the coupledGLEs (eq. 1) are varied. In Fig. 2, we set β = 0.2, γ = 2,η = 1.05, and ρ = 2.1.As can be seen, although in the uncoupled (or scalar)case described by Fig.1 laser emission exhibits STI, anonlinear cross-gain saturation coefficient larger than unityleads to stabilization of the field in alternating domains ofconstant polarization, separated by sharp switching regionsor domain walls [6]. Polarization domains and domain wallswere indeed recently experimentally observed in a vectorfiber laser [6].Figure 2. Polarization domain emission. Top (a): evolution of intensities fromthe laser; bottom (b): snapshot of intensities vs fast time τ after a largenumber of circulations.Let us consider laser dynamics in the normal dispersionregime: in the top panels of Fig.3, we set β = −0.5, γ = 1.5,and η = ρ = 0 (no nonlinear polarization coupling). Hereuncorrelated STI occurs: laminar regions of CW emission areseparated by the emission of black holes at the collision ofdark soliton pulses with opposite speed.In the middle panels of Fig.3, we increased γ up to 4: as canbe seen, the laminar regions are much reduced in size, andthe black holes exhibit a relatively long lifetime.The bottom panels of Fig.3 show that the presence of XPM(ρ = γ = 4) dramatically changes the nature of the fieldevolution that exhibits now a fully developed turbulence.By further increasing the strength of both SPM and XPM up toγ = ρ = 25, and introducing also a finite cross-gain saturationη = 0.4, one observes a rarefaction of the high-intensity peaksthat emerge from a sea of low level intensity fluctuations (seeFig.4).Figure 3. Transition from vector STI (top) to coexhisting STI and dark solitons(middle) and fully developed AT (bottom).In the top panel of Fig.4 we show the irregular evolution ofthe total laser intensity as a function of both fast and slowtime coordinates. Whereas in the middle panel of Fig.4 weillustrate a zoom of the intensities of the two orthogonalpolarization components, showing a single extreme event.The corresponding shapshot of the polarization resolvedintensities is reported in the bottom panel of Fig.4.We may call the dynamics that is shown in Fig.4 as“near-conservative turbulence”: in this case Eqs.(1) are closeto perturbed coupled nonlinear Schroedinger equations(CNLSEs) since SPM and XPM terms are much larger thannonlinear self and cross-gain saturation terms.It is interesting to note that the irregular dynamics inFig.4 is qualitatively close to the chaotic emission that isexperimentally observed in a quasi-CW long Raman fiberlaser, which also exhibits extreme high-intensity peaks thatREVISTA CUBANA DE FI´SICA, Vol 33, No. 1E (2016) E33 ARTI´CULOS ORIGINALEShave a complex polarization sub-structure [7].Figure 4. Near-conservative turbulence in a Raman fiber laser.In vector fiber ring lasers operating in the normal dispersionregime, it has also been observed that the spatiotemporaldynamics can be modeled by means of a purely conservativeset of CNLSEs, by supposing that linear gain and loss exactlybalance each other, and that the presence of nonlinear gainsaturation terms can be neglected [8]. Quite remarkably, evenin the conservative limit the CNLSEs exhibit polarizationdomain and domain wall solutions that correspond well tothe experimentally observed antiphase dynamics of the twoorthogonal modes of polarization [8].IV. CONCLUSIONSIn conclusion, we discussed how a simple universalphenomenological model, provided by the scalar or thevector complex GLE with cubic nonlinearity and withnonlinear gain saturation (as opposed to nonlinear gainfor describing stable short pulse emission from passivelymode-locked lasers [1]), may be able to qualitatively describea wide range of complex temporal dynamics in fiber lasers.Specific examples of quasi-CW, high intensity laser sourcesof importance for several applications are: noise-like lasers,polarization antiphase domain lasers, and ultralong Ramanfiber lasers.REFERENCES[1] Ph. Grelu and N. Akhmediev, Nature Photonics 6, 84(2012).[2] M. Horowitz, Y. Barad, and Y. Silberberg, Optics Letters22, 799 (1997).[3] Q. L. Williams, J. Garcia-Ojalvo, and R. Roy, Phys. Rev.A 55, 2376 (1997).[4] E.G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov,X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin,G. Falkovich, and S. K. Turitsyn, Nature Photonics 7,783–786 (2013).[5] S. Wabnitz, Opt. Lett. 39, 1362 (2014).[6] C. Lecaplain, Ph. Grelu, S. Wabnitz, Physical Review A89, 063812 (2014)[7] S. Sugavanam, N. Tarasov, S. Wabnitz and D.V.Churkin, Laser Photonics Rev. 9, L35-L39 (2015)[8] C. Lecaplain, Ph. Grelu, S. Wabnitz, J. Opt. Soc. Am. B30, 211 (2013)REVISTA CUBANA DE FI´SICA, Vol 33, No. 1E (2016) E34 ARTI´CULOS ORIGINALES

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Spatiotemporal chaos and order in fiber lasers

Archivio istituzionale della ricerca - Università di Brescia

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Spatiotemporal chaos and order in fiber lasers

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