Spatial rogue waves in photorefractive SBN crystals
C. Hermann-Avigliano, I. A. Salinas, D. A. Rivas, B. Real, A. Mančić, C. Mejía-Cortés, A. Maluckov, R. A. Vicencio
SSpatial rogue waves in photorefractive SBN crystals
C. Hermann-Avigliano, ∗ I.A. Salinas, D.A. Rivas, B. Real, A. Manˇci´c, C. Mej´ıa-Cort´es, A. Maluckov, and R.A. Vicencio Departamento de F´ısica and Millennium Institute for Research in Optics (MIRO),Facultad de Ciencias and Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Santiago, Chile Dept. of Phys., Faculty of Sciences and Math., University of Niˇs, Serbia Programa de F´ısica, Facultad de Ciencias B´asicas,Universidad del Atl´antico, Barranquilla, Colombia Vinca Institute of Nuclear Sciences, University of Belgrade, Serbia
The phenomenon of rogue waves (RWs) dates backfrom observations of isolated large amplitude water waveson the sea surface, appearing out of nowhere and dis-appearing without a trace [1]. These rare events werestatistically associated with long tails of high amplitudewave distributions. Nowadays, they are related to ex-treme events (EEs) arising in the presence of many un-correlated grains of activity, which are inhomogeneouslydistributed in large spatial domains of complex mediaand they are being studied in different fields of science [2].Diversity and particularity of RWs cause many uncer-tainties regarding their definition, origin, predictabilityand statistics [3]. Besides the fact that EEs generationis related to the phenomena with long-tails statistics,there is a well-established approach that relates their ap-pearance with the merging of coherent structures [4, 5].Within optical systems, the study of RWs include lightpropagation in optical fibers [4], nonlinear optical cavi-ties [6], and photorefractive crystals. RWs were observedon a BaTiO :Co crystal [7] on a highly nonlinear regimeshowing spatiotemporal turbulence. Ref. [8] reports op-tical RWs on a KLTN crystal where nanodisorder, giantnonlinearity (NL) and high temperature generate largeintensity events (IEs). The ferroelectric-to-paraelectrictransition also generate RWs due to thermally inducedfocusing and defocusing effects [9], where a transitionfrom linear to highly nonlinear regimes promotes a tur-bulent dynamics [10]. Modulational instability (MI) andpattern formation in SBN photorefractive crystals, usingincoherent [11] and coherent [12] light beams, promotethe appearance of different optical patterns, includingstripes and filaments. This is due to a combined actionof different mechanisms like crosstalk and NL, which arebelieved to be essential for the observation of RWs [7].It was shown recently that the existence of purely linearlarge IEs was also possible due to isolated caustic effects ∗ Corresponding author email: [email protected] on an optical sea [13].Here, we investigate experimentally, and corroboratenumerically, the appearance of large-amplitude eventson a SBN photorefractive crystal. The key elements inour study are the simplicity of the experiment, its re-producibility, and the robust appearance of RWs undersimple controlled conditions, without requiring large NLsneither turbulence phenomena. By injecting a low powerGaussian beam (GB), and ramping an externally appliedvoltage, we are able to distinguish between different dy-namical regimes. We observe that for low applied volt-ages (weak NL) the beam experiences a caustic-like dis-tribution. Amplitudes at the background level are verysmall and, therefore, we observe a mixture of linear andnonlinear waves coexisting and forming different interfer-ence profiles, which resemble caustic-like patterns [13].When increasing the voltage and, as a consequence theNL of the system, we observe a pattern fragmentationinto narrow light spots, where some of them have a hugeintensity. We compare the peak intensities measured inthe experiment and in the numerical simulations to iden-tify the excitation of EEs. Additionally, we numericallystudy the dynamics along the crystal to elucidate theappearance of RWs during propagation. µm MO20xLaser532-nm MM Pλ/2 f CCDMO10xV SBN:75 WL zyx µm
75V 150V50V0V (b) (c) (d) (e)(a)
FIG. 1. (a) Experimental setup for observing RWs. Inset:Input profile. (b)–(e) Output beam profiles for indicated volt-ages. In (b) V = 0. a r X i v : . [ n li n . PS ] M a y We start our investigation by experimentally study-ing the propagation of GBs on a 0 . dopedSBN:75 photorefractive crystal, using the setup sketchedin Fig. 1(a). Our sample has a transversal area of 5 × with a length of 10 mm (propagation coordinate z ).Our crystal has a 20 times larger nonlinear response for apolarization along the vertical axis ( x ), in comparison tothe polarizations perpendicular to it [14]. The nonlinearresponse is controlled indirectly by an external voltageapplied in the crystal vertical axis (smaller crystal di-mension). We expand a 532 nm laser beam by a 20 × microscope objective. We define the beam polarization( x ) and power (10 µ W) with a sequence of a λ/ µ m wide (seeFig. 1(a)-inset). The output beam profile after propagat-ing 10 mm through the crystal without any applied volt-age, is shown in Fig. 1(b). As it is observed, the inputbeam strongly spreads inside the crystal and it spatiallyevolves to a wide gaussian profile, ≈
50 times larger. Thispattern possesses several spatial irregularities, which canbe related either to fluctuations in the nominal linear re-fractive index (due to previous experiments) or to inho-mogeneities on the GB itself. These spatial fluctuationsare important in our experiment because they create spa-tial regions with different light density and, in someway,they initialize the filamentation process. Therefore, wehave a natural spatial symmetry breaking mechanism,which drives the system to a non-homogeneous state.Our experiment shows rich dynamics depending on theparameters used to excite the SBN sample. This is re-lated to the nonlinear response of the crystal, which inour case depends directly on the light intensity. However,as we completely define the light intensity to a constantvalue (fixing the input power and input waist), we ef-fectively modify the nonlinear response of the crystal bymeans of an externally applied voltage. When increasingthis control parameter, we observe two different dynam-ical regimes. If we increase the voltage fast enough ( ∼ seconds), the GB rapidly collapses to a 2D bright soli-ton [15]. After this solution is formed, it is possible toobserve stable or unstable patterns, which strongly de-pend on the input power, external voltage and crystallength. Differently, a slow voltage increment ( ∼ minutes)allows for the light to spread smoothly over the crys-tal, facilitating the creation of several regions of largerlight density, observing some kind of agglomeration dif-fusive process. Interestingly, here light is able to localizeweakly over different spatial domains. Nevertheless, asthe local power is not that high, the radiation of energy between neighboring big spots (crosstalk) is still possi-ble. This smooth dissemination of energy (mediated byan inhomogeneous initial diffraction process), plus a slownonlinear increment, gives us the necessary mechanismsto observe large-amplitude events on our photorefractivesetup, without requiring to work neither on a highly non-linear regime [7] nor close to a thermal crystal transi-tion [8].Typical output profiles for the slow variation regimeare shown in Figs. 1(c)–(e). To statistically analyze thedata, we defined the following protocol: we increase thevoltage from 0 to a maximum of 225 V, in steps of 25 Vevery 15 minutes. This smooth increment allows an adi-abatic transformation of the output spatial profile. Weobserve a rather static (nonlinear stationary-like complexlocalized) pattern that allows us to take a representativeimage of the output facet every 15 minutes, which char-acterizes the state of the system at a given voltage. Weobtain images every 25 V, although we focus on largervoltage values where peaks are spatially more localized,having an average width lower than ≈ µ m. At lowvoltages, we observe a tendency of a macroscopic agglom-eration of energy in wide light spots, as Figs. 1(c) and (d)show. Then, by a further increment of the voltage up to ∼
100 V, we observe that narrow light spots become con-nected by some kind of light currents that are associatedto a caustic-like energy spreading [13]. By increasingfurther the voltage, we observe the appearance of severallarge amplitude peaks, which have a small individual spa-tial extent [see Fig. 1(e)]. We run the same experiment30 times to increase the statistical ensemble. Before ini-tializing every new experimental realization, we apply awhite light source to erase any induced refractive indexpattern, which could be imprinted inside the crystal ina previous experiment. We check this by inspecting theoutput linear profile [Fig. 1(b)] and determine whetherthere is a need to continue erasing the crystal or to sim-ply translate the sample to a more homogeneous region.
125 V150 V175 V200 V225 Vg=6g=8g=10 P ( I E > A I ) [ % ] Abnormality index (AI) s P D F Peak intensity (a.u.) -1 -2 -3 I s FIG. 2. Semi-log plot of the probabilities (in %) of havingan IE above a particular AI for different voltages. All eventsabove AI = 2 are consider as EEs (vertical dotted line). Thenumerical counterpart for g = 6, 8 and 10 is presented as well.Inset: PDF for 200 V . I s is represented by the vertical dashedline. We analyze each obtained image and look for localmaxima above a given defined threshold value (chosento avoid background events). To avoid saturation, weset the exposure time on the beam profiler to 1 ms for125 and 150 V, and to 0 . −
225 V.The intensity scale is defined in the interval 0 −
255 lev-els (typical scale for images), where zero means no lightand 255 represents the largest intensity, depending onthe chosen exposure time. In general, heavy-tailed in-tensity distributions are an indicator of the existence ofEEs [16]. By following a standard criterion on RWs [5],we consider as EEs those with intensities larger thantwice a significant intensity I s , which is defined as theaverage value of the highest intensity tertile of the cor-responding probability density function (PDF) distribu-tion (see inset in Fig. 2). Events with an abnormal-ity index AI ≡ I/I s > P (IE >AI ) = 1 − cumulative PDF. This represent the proba-bility of having an IE with an AI larger than a certainvalue, P (IE > AI ) (see Fig. 2). Hence, the probability ofhaving RWs corresponds to the value at which the datacrosses AI = 2. We detect for { , , , , } voltages, a total number of { , , , , } IE,from where { , , , , } are considered as EEs, respec-tively. Therefore, the percentage of occurrence in ourexperiments is only { . , . , . , . , . } %. We observethat large intensity events are always below ∼
3% of thetotal data, which indicates that our reported RWs arerare and have very low statistics.From the theoretical side, the light propagationthrough photorefractive media can be modeled mathe-matically by a 2D nonlinear Schr¨odinger equation withsaturable nonlinearity [14] i ∂∂z ψ ( x, y, z ) + β ∇ ⊥ ψ ( x, y, z ) − g ψ ( x, y, z )1 + | ψ ( x, y, z ) | = 0 . (1) ψ ( x, y, z ) corresponds to the envelope of the electricalfield, z to the propagation coordinate, β the diffractioncoefficient (fixed to 1), and ∇ ⊥ corresponds to the trans-verse Laplacian operator. The nonlinear coefficient is de-noted by g and it is proportional to the external appliedvoltage in the experiment. A positive g -value implies afocusing regime, while a negative one refers to a defo-cusing case [3]. We focus here on describing phenomeno-logically the observed steady-state patterns, which are ingeneral well described by model (1) (local saturable NL isable to produce long-range phenomena, due to a naturalgeneration of broader spatial patterns [14]). The quanti-tative changes between the two regimes can be associatedwith the slow nonlinear response of the photorefractiveSBN crystal, which is of saturable nature. This responseis determined by the interplay between external voltage,charge dynamics and light interaction. Under the influ-ence of internal and external electric fields, charges startto redistribute, hence a non-homogeneous refractive in-dex change occurs via the electro-optic effect. Then, thedifferent observed dynamical phases could be related todifferent values and rates of the external voltage in theexperiment. We initialize our numerical simulations by injecting anoise seeded GB, with a certain width and intensity (sim-ilar to experimental values), into equation (1). The noiseis added by an uniform random number generator, havinga mean value equal to zero and a maximum value equalto 1% of the beam amplitude. The beam propagationthrough the SBN saturable media is obtained by apply-ing a standard split-step pseudo-spectral procedure [17].We fix the propagation length to 10 mm, while the inputwidth is of the order of 10 µ m. The transversal area is1 × . Different dynamical regimes can be distin-guished in the parameter region above the MI threshold,depending on g (Fig. 3). Randomly fluctuating inten-sity patterns are obtained for low g -values (Fig. 3(a)),which agrees well with the experiments at low voltage(Fig. 1(c)). We identify a wide agglomeration, having anon-homogeneous pattern due to the effects associated toNL during the beam propagation. When g is increased,the beam shows a speckling-like profile due to the initialnon-homogeneous expansion of the light and the corre-sponding self-focusing response of the crystal. This al-lows the clustering of light in very specific regions gener-ating high intensity spots. Figure 3(b) shows a numericalexample of this for g = 8, which is in good phenomeno-logical agreement with the experimental results shown inFig. 1(e).The numerical integration provides the possibility toconstruct a phase diagram to study different dynami-cal regimes, with g as a control parameter. Figure 3(c)shows the averaged full width half maximum (FWHM)of output profiles versus g . We observe in a purely linearregime that the FWHM expands strongly compared tothe input value. In the presence of NL, the beam simplyshrinks due the self-trapping phenomena and the outputFWHM just decreases. However, a transient regime be-tween 1 (cid:46) g (cid:46) . . (cid:46) g (cid:46) . . µ m (shaded area in Fig. 3(c)).After this caustic region, the averaged FWHM is reduced,which is consistent with the appearance of small size lightspots due to the overall filamentation process occurringat larger values of g . We simultaneously plot the max-imum amplitude ( A max ≡ √ I max ) observed for everyrealization, after averaging at different values of g . Weobserve that this amplitude has a slight increment whenthe dynamics change from a caustic regime to a filamen-tation process. This is due to conservation of energy,which indicates that a multi-peak pattern has to splitthe total power in several spots and, therefore, peaks canonly contain a limited amount of energy. Additionally,the saturable nature of the NL produces an upper boundfor A max .By statistically analyzing the output intensity profiles,we compare our numerical simulations with the datain Fig. 2. We check the distribution of high intensitypeaks for 100 numerical realizations for g = 6, 8 and 10 FIG. 3. Numerically output beam intensity profiles for g =1 .
75 (a) and g = 8 (b). (c) Average FWHM (log-linear scale)and A max vs g , for the output profile ψ ( x, y, z end ). Shadedarea indicates a caustic-like regime region. (speckling regime), and compute as well the P (IE > AI ).These results are presented as additional confirmation ofsuitability of a relatively simple mathematical-numericalmodel to our experiment, with the main objective ofdemonstrating a qualitative comparison of the phenom-ena. For g equal to { , , } we find { , , } EEson an ensemble of { , , } IE. We observe thatRWs increase with NL, as expected from the experimen-tal counterpart.The experimental data in our setup is collected fromthe output intensity profiles, hence the dynamics insidethe crystal is ”hidden”. However, numerical simulationsallow us to study the dynamics along z . We compute theintegral intensity distribution P I [18] for different val-ues of g , which takes into account light intensities alongthe whole propagation through the sample (Fig. 4). For g = 0 . . < − g = 1 .
85, we observe a power-law be-havior of P I in the whole range of intensities. We observethat the slope of this distribution changes with I (see thetwo dark green straight lines in Fig. 4). Power-law likedistributions are observed in the filamentation (speck-ling) regime too, with different slope rates ( g = { , } ),commonly related to the emergence of EEs [3]. Differ-ently, the maximum intensities in the caustic-like regimeare smaller compared to those reached in the specklingregime for the chosen g values. For g = 10 we observethat there are more events with smaller intensity thanfor g = 1 .
85, as a consequence of slow and spontaneouslyinitiated clustering of events in the background. Theseclusters represent energy seeds for the formation of fewhigh IEs. Additionally, we can estimate the probabilityof occurrence of EEs, defined as
P ee = (cid:82) I>Ia P I dI [18],obtaining that P ee is of order 0 .
1% to 1% in caustic-likeand speckling cases presented here. This value agreeswell with the experimentally and numerically estimatedratio (EE/IE) extracted from maximum light intensityat the output facet of the crystal. The last points outthe existence and significance of huge amplitude events P I I FIG. 4. Log-Log plot of P I vs I , for g = 0 . , . .
85 (caustic-like regime), 6 and 10 (speckles regime) repre-sented by light blue, purple dashed lines, black circles, redtriangles, and magenta empty. Dark green dashed line corre-sponds to a − during the whole light propagation through the crystal.We plan to continue our research in this direction expect-ing to arrange a new experimental setup to confirm ournumerical findings.We have studied experimental and numerically the ap-pearance of large amplitude waves on a SBN crystal. Bytuning the external voltage, we were able to generate theexperimental conditions to observe different dynamicalregimes. We identified a caustic-like regime, where aneffective energy dissemination through the crystal wasobserved. When increasing the voltage, we observed thegrowth of local clusters with the consequent generationof large amplitude peaks having very low statistics. Thisis always observed in our setup, showing that large IEsare quite a robust phenomena in these kind of nonlinearsystems and, particularly in our experiment, without theneed of any extra mechanism neither any special stimu-lation. The key elements in our study are the simplic-ity of the experiment, its reproducibility, and the robustappearance of RWs under simple conditions. Our obser-vations are well supported by a saturable model, whichcorrectly predicts the dissemination and the specklingregimes. Our numerics give also an estimation of the oc-currence of RWs in the order of 1% of total peaks, whichagrees well with our experimental results. Funding.
Fondo Nacional de Desarrollo Cient´ıficoy Tecnol´ogico (FONDECYT) (1151444, 3180153); Pro-grama ICM Millennium Institute for Research in Op-tics (MIRO); U-Inicia VID Universidad de Chile (UI004/2018); Ministarstvo Prosvete, Nauke i TehnoloˇskogRazvoja, Republike Srbije (III 45010); National labora-tory for high performance computing (ECM-02), Univer-sity of Chile.
Acknowledgment.
Authors acknowledge C. Cantil-lano, M.G. Clerc and G. Gonz´alez for useful discussions. [2019] Optical Society of America. One printor electronic copy may be made for personal useonly. Systematic reproduction and distribution,duplication of any material in this paper for a feeor for commercial purposes, or modifications of the content of this paper are prohibited. [1] C. Kharif and E. Pelinovsky, “Physical mechanisms ofthe rogue wave phenomenon,” Eur. J. Mech. B/Fluids , 603 (2003).[2] S. Albevario, V. Jentsch, and H. Kantz, “Extreme Eventsin Nature and Society”, Springer, Berlin, 2006.[3] M. Onorato, S. Residori, U. Bortolozzo, A. Montina, andF.T. Arecchi,“Rogue waves and their generating mecha-nisms in different physical context,” Phys. Rep. , 47(2013).[4] D. R. Solli, C. Ropers, P. Koonath and B. Jalali, ”Opticalrogue waves”, Nature 450, 1054 (2007).[5] Marcel G. Clerc, Gregorio Gonz´alez-Cort´es, and MarioWilson, ”Extreme events induced by spatiotemporalchaos in experimental optical patterns”, Opt. Lett. ,2711-2714 (2016), and references within.[6] A. Montina, U. Bortolozzo, S. Residori, and F. T. Arec-chi, ”Non-Gaussian Statistics and Extreme Waves in aNonlinear Optical Cavity”, Phys. Rev. Lett. , 173901(2009).[7] N. Marsal, V. Caullet, D. Wolfersberger, and M.Sciamanna, ”Spatial rogue waves in a photorefrac-tive pattern-forming system,” Opt. Lett. , 3690-3693(2014).[8] D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, andE. DelRe, ”Spatial Rogue Waves in Photorefractive Fer-roelectrics”, Phys. Rev. Lett. , 093901 (2015).[9] M. O. Ram´ırez, D. Jaque, L. E. Baus´a, J. Garc´ıa Sol´e,and A. A. Kaminskii, “Coherent Light Generation froma Nd:SBN Nonlinear Laser Crystal through its Ferro-electric Phase Transition,” Phys. Rev. Lett. , 267401(2005).[10] D. Pierangeli,F. Di Mei, G. Di Domenico, A.J. Agranat,C. Conti, and E. DelRe,“Turbulent Transitions in Op- tical Wave Propagation,” Phys. Rev. Lett. , 183902(2016).[11] D. Kip, M. Soljacic, M. Segev, E. Eugenierva, and D.N.Christodoulides, “Modulational instability and patternformation in spatially incoherent light beams,” Science , 495 (2000).[12] C.-C. Jeng, Y. T. Lin, R.-C. Hong, and R.-K. Lee, “Op-tical pattern transitions from modulation to transverseinstabilities in photorefractive crystals,” Phys. Rev. Lett. , 153905 (2009).[13] A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty andJohn M. Dudley, “Caustic and rogue waves in an opticalsea,” Sci. Rep. , 12822 (2015).[14] R. Allio, D. Guzm´an-Silva, C. Cantillano, L. Morales-Inostroza, D. L´opez-Gonz´alez, S. Etcheverry, R. A. Vi-cencio, and J. Armijo, ”Photorefractive writing and prob-ing of anisotropic linear and nonlinear lattices,” J. Opt. , 025101 (2015).[15] S. Lan, M. F. Shih and M. Segev, ”Self-trapping ofone-dimensional and two-dimensional optical beams andinduced waveguides in photorefractive KNbO(3)”, Opt.Lett. , 1467 (1997).[16] E. Louvergneaux, V. Odent, M.I. Kobolov, and M.Taki,”Statistical Analysis of spatial frequency supercon-tinuum in pattern forming feedback system”, Phys.Rev.A , 063802 (2013).[17] F. Chen, M. Stepi´c, C. E. R¨uter, D. Runde, D. Kip,V. Shandarov, O. Manela, and M. Segev, ”Discretediffraction and spatial gap solitons in photovoltaicLiNbO3 waveguide arrays,” Opt. Express
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