Observation of a gradient catastrophe generating solitons
Claudio Conti, Andrea Fratalocchi, Marco Peccianti, Giancarlo Ruocco, Stefano Trillo
aa r X i v : . [ phy s i c s . op ti c s ] S e p Observation of a gradient catastrophe generating solitons
Claudio Conti, Andrea Fratalocchi, , Marco Peccianti, , Giancarlo Ruocco, , Stefano Trillo , ∗ Research center SOFT INFM-CNR Universit`a di Roma “La Sapienza”, P. A. Moro 2, 00185, Roma, Italy Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89/A, 00184 Rome, Italy INRS-EMT University of Quebec, 1650 Blvd. Lionel Boulet, Varennes, Quebec,Canada Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, P. A. Moro 2, 00185, Roma, Italy Dipartimento di Ingegneria, Universit`a di Ferrara, Via Saragat 1, 44100 Ferrara, Italy (Dated: October 29, 2018)We investigate the propagation of a dark beam in a defocusing medium in the strong nonlinearregime. We observe for the first time a shock fan filled with non-interacting one-dimensional greysolitons that emanates from a gradient catastrophe developing around a null of the optical intensity.Remarkably this scenario turns out to be very robust, persisting also when the material nonlocalresponse averages the nonlinearity over dimensions much larger than the emerging soliton filaments.
Introduction
In many physical systems propagationphenomena are affected primarily by the interplay of dis-persive and nonlinear effects. In this context, solitons(or solitary waves), i.e. wave-packets that stem from amutual balance between the two effects, account success-fully for several phenomena ranging from long-span non-spreading propagation and elastic interactions of beams,to the coherent behavior of ensembles of particles, e.g.ultracold atoms, or coupled oscillators. Studies in thisfield were mainly focused on individual solitons or in-teractions between them. However, several solitons canemerge at once from breaking of large amplitude smoothwaves [1], as for instance observed in oceanography [2].While theoretical studies indicates the phenomenon tobe generic [3, 4], the observation of such multi-solitonregime in reproducible lab experiments has been elusive.In this letter, we report a lab experiment in opticswhich demonstrates that a fan of non-colliding 1D soli-tons emerge, owing to a gradient catastrophe (i.e. aninfinite gradient developing from a smooth input) devel-oping around a zero of the field. Specifically we considera dark-like optical beam (i.e., a dark stripe on a brightbackground) and operate, unlike previous experiments ondark solitons [5, 6], in a regime where nonlinearity out-weighs diffraction (i.e., power of the background largelyexceeding that needed to trap a fundamental dark soli-ton). In this regime, we are able to monitor directlythe evolution along a thermal defocusing medium. Weobserve the formation of a dark focus point which corre-sponds to a gradient catastrophe of the hydrodynamictype around a point of vanishing intensity. The infi-nite gradient of the hydrodynamic stage is regularizedby the presence of weak diffraction, which causes theappearance of fast oscillations in an expanding region(fan), a feature common to the wide class of so-calledcollisionless or dispersive shock waves (DSW) or undu-lar bores, investigated theoretically in several contexts[7, 8, 9, 10, 11, 12, 13, 14, 15, 16].In our setting, the DSW is essentially composed by 1Ddark soliton filaments, which become manifest after thecatastrophe point, and maintain fixed parameters (ve- locity and darkness) as soon as they emerge [16], alsonot exhibiting the rapid decay into vortices characteris-tic of shock waves in superfluids [17]. Our scenario turnsout to be remarkably robust against the nonlocal charac-ter of the nonlinearity, and presents also significant dif-ferences with DSW resulting from bright disturbances[18, 19, 20, 21]. In the latter case the catastrophe oc-curs, indeed, at finite transverse extension, giving rise,in 1+1D, to two symmetric fans (connected by a quasi-flat background) [18, 20], while the relative oscillationschange dynamically (i.e., dark solitons in the train havealways slowly-varying parameters upon propagation).
FIG. 1: (Color online) Sketch of the experimental setup.
Experiment
Our sample consists of a BK7 glass cell ofdimensions 1cm x 4cm x 1cm in the
X Y and Z (propa-gation) direction respectively, containing a solution (con-centration C = 0 . mM ol/dm ) of Rodhamine-B inMethanol. A beam at λ = 532 nm from a diode-pumpedCW Nd-Vanadate laser was focused down to a stronglyelliptical beam (ellipticity 1:30) by means of a cylindricallens of focal length L f = 100 mm and a 20X microscopeobjective. The beam is coupled in the cell at Z = 0 lyingin the cell mid-plane (sufficiently far from bottom, top,and lateral liquid-glass interfaces to avoid the dynamicsto be significantly affected by boundary conditions). Aphase mask is placed on the beam path resulting in anabrupt change of π in the optical phase across the line X = 0. After the mask we let the beam diffract shortlyand then we focus it onto the sample, producing an inputbackground bright beam of dimension 600 × µ m, ontowhich a dark stripe (with zero intensity in X = 0 andhyperbolic-tangent-like X -profile) is nested. The stripeis parallel to the narrower spot-size ( Y direction), whilealong X the dark notch of 25 µm FWHM sees a quasi-constant background due to the 600 µ m width. We de-tect no significant changes along Y over the propagationlengths involved, witnessing that our arrangement mim-ics a strict 1+1D ( X − Z ) setting.The power coupled into the sample is measured bymeans of a beam splitter in the front of the laser outputand a silicon detector. As sketched in Fig. 1 a micro-scope and a charge coupled device (CCD) camera allowsus to collect the light scattered in the vertical (Y) direc-tion above the cell, so to produce a direct planar imageof the relevant beam evolution ( X − Z plane). In Fig. 2we show the field intensity distribution collected in thisway at different input powers P in .At low power (4 mW ) the dark notch diffracts, broad-ening in propagation toward positive Z , whereas athigher power the nonlinear thermal response of the sam-ple counteracts the diffraction leading to exact counter-balance (dark soliton formation, P in ≃ mW ), and sub-sequent overall focusing. By further increasing the powerto P in = 260 mW (Fig. 2c) and P in = 600 mW (Fig. 2d),the dark notch undergoes to a clear focus point (gradi-ent catastrophe or breaking point). Beyond such point,the beam undergoes non-trivial breaking forming a DSWconstituted by narrow dark soliton filaments which fillprogressively a characteristic fan. The number of darkfilaments in the fan grows with power. This is also clearfrom the measured far-field relative to the output of ourcell (i.e., after Z = 1 cm of propagation) displayed inFig. 3. Importantly, these data clearly shows that thefilaments, once seen in the transverse plane, maintain thestripe features of the input (similarly to the bright case[20]), not exhibiting any transverse instability or decayinto vortices characteristic of other superfluidity and op-tical experiments [5, 17, 22], allowing for a description interms of pure 1+1D ( X − Z ) dynamics. Theory
The dynamics observed experimentally can bestudied and understood on the basis of the following gen-eralized nonlinear Schr¨odinger (NLS) model iε ∂ψ∂z + ε ∂ ψ∂x − δnψ = − i α εψ, (1) − σ ∂ δn∂x + δn = | ψ | , (2)where the first equation stands for the paraxial nonlin-ear wave equation for ψ ≡ A/ √ I , i.e. the beam en-velope A = A ( Z, X ) normalized to peak intensity I (inthe experiment I = P in /A e , where A e is the backgroundbeam area). The transverse and longitudinal coordinates x, z = X/w , Z/L are scaled to the waist of the inputdark notch w , and the geometric mean L ≡ √ L nl L d be-tween the length scales L nl = ( k | n | I ) − and L d = kw FIG. 2: (Color online) Transverse distribution of the inten-sity along the cell ( x − z plane), as observed from top scat-tered light for four different input powers. Superimposed lightcurves (yellow) are retrieved intensity profiles at Z = 0 . Z = 2 . X − Y at the sample output for differentinput powers: the intensity is collected after about 1 meter offree-air propagation. characteristic of the nonlinear and diffractive terms, re-spectively ( n is the Kerr coefficient that characterizesan index change of the local type ∆ n = n | A | ), and α = α L is the normalized attenuation constant. Suchscaling allows us to highlight the fact that we operatein the weakly dispersive case, where the model mimicsthe quantum Schr¨odinger equation with the smallnessparameter ε ≡ L nl /L = p L nl /L d playing the role ofPlanck constant. The normalized refractive index change δn = k L nl ∆ n acts as a self-induced potential driven bythe normalized intensity profile | ψ ( x ) | . The free param-eter σ measures the diffusion length and gives the degreeof nonlocality of the nonlinear response. This model de-scribes the nonlocal features of the nonlinear responsewith sufficient accuracy regardless of their physical ori-gin. Specifically, the model was shown to give an excel-lent description of thermal nonlinearity [21, 23], while itallows for a reduction to the integrable semiclassical NLSequation in the local and lossless limit σ = α = 0.The essential physics can be explained indeed by thelatter limit, for which the outcome of numerical compu-tations based on Eqs. (1-2) with input ψ ( x ) = tanh( x ),are displayed in Fig. 4 for two different values of ε (pow-ers). The initially dominant nonlinearity allows us toadopt a description in terms of hydrodynamical vari-ables ρ and u ≡ ∂ x S . This is made by applying theWKB transformation ψ ( x, z ) = p ρ ( x, z ) exp [ iS ( x, z )/ ε ][3, 8, 9, 10, 19], which allows to reduce Eqs. (1-2), atlowest order in ε , to the following system written in theform of hyperbolic conservation laws ∂ a ∂z + ∂ f ∂x = 0; a ≡ (cid:18) ρq (cid:19) ; f ( a ) ≡ q ρ + q ρ ! , (3)where q ( z, x ) ≡ ρ ( z, x ) u ( z, x ). Equation (3), that rulesclassical 1D dynamics of an isentropic gas or shallow wa-ter ( u being, in this case the velocity of the gas or water,and ρ the gas density or the water level), predicts that thedynamics of the input hole in the density ρ ( x,
0) = | ψ | produces a gradient of ”velocity” u , whose sign ( u turnsout to be positive for x <
0, and viceversa) is such togive rise to compressional waves. Equivalently, due to thedefocusing nature of the medium the central dark regionhas a higher index which draws light inwards. As a resultthe input dark notch experiences a dramatic steepeningand focusing around its null, which in turn enforces thevelocity gradient, until eventually a singularity (gradientcatastrophe) develops at a finite distance, consistentlywith the hyperbolic nature of Eqs. (3). Such singularityis characterized by crossing of characteristics associatedwith Eqs. (3) and become manifest as a vertical frontin the variable u and a cusp in the intensity ρ , as dis-played in Fig. 4d (numerical results from Eqs. (3) areexactly coincident to those shown). However, when sucha high (virtually infinite) gradients develop, the hydro-dynamic description breaks down, and diffraction regu-larizes the front through the appearance of an expandingregion of fast oscillations, i.e. a shock fan. The shockfan is progressively filled with non-interacting dark fila-ments, whose angle (transverse velocity) increases as therelative darkness decreases, which is a universal featureof dark (grey) solitons [5]. Furthermore the number offilaments (solitons) increases for smaller ε (higher pow-ers). In particular, for 1 /ǫ = N , N integer, one can showthat the asymptotic state is made by 2 N − FIG. 4: (Color online) Numerical simulations of Eqs. (1-2)with σ = α = 0 and ψ = tanh ( x ): (a-b) Level plots of | ψ | for ε = 0 .
05 (a) and ε = 0 .
02 (c); (b) snapshot of case (a)at z = 4; (d) snapshot of case (c) at the catastrophe point z = 0 .
75: frequency u (solid red curve and intensity (blacksolid curve) compared with the input (blue dashed curve). tion instead of solitons [21]). This is ultimately related tothe persistence of solitons in the presence of terms thatbreak the integrability. We recall that, in our system,the losses and nonlocality are intimately related becausethe index change is due to heating caused by the strongabsorption of the dye, while the nonlocality arises fromthe intrinsic tendency of heat to diffuse [21]. As shown inFig. 5, where we report simulations based on Eqs. (1-2)with σ = 1, the overall dynamics is quite similar to thelocal one, except for a slight adiabatic broadening of thesolitons due to the losses. The post-catastrophe breakingstill occurs in the form of narrow soliton filaments, whichmodel (1-2) support also in the nonlocal case [24]. Thisoccurs despite the fact that the induced potential δn thattrap them becomes, owing to diffusion, a smooth function(see Fig. 5b, red curve) that no longer follows the deeposcillations of the intensity as in the local case. Simu-lations in transverse 2D with elliptical input reminiscentof the experiment confirm the validity of this 1D picture,allowing us to conjecture that nonlocality stabilizes thesoliton stripes in the fan against transverse instabilities.Importantly, the breaking distance (point in Z of max-imum intensity gradient) turns out to be significantly af-fected by the attenuation and the degree of nonlocality,an issue that we have investigated in detail. In the lo-cal and loss-less case, the hydrodynamic limit yields aconstant normalized breaking distance z b = 0 .
75, corre-sponding to a physical distance Z b = z b L that scales withpower as P − / in . This is confirmed by numerical solutionsof Eqs. (1-2) with σ = α = 0, performed for ε = 1 /N , N integer (i.e. P in /P s = N , P s being the fundamental FIG. 5: (Color online) Evolution in the nonlocal and lossycase: (a) Level plot of intensity (the color scale is adapted tocompensate for losses); (b) snapshots at z = 7, of intensity(black solid curve) and relative index change δn (red solidcurve). The input is the dashed blue curve. Here ε = 0 . P ≃ . W in our experiment), σ = 1, α = 0 .
3. The corre-sponding physical breaking (catastrophe) distance turns outto be Z b = Lz b = 0 . mm .FIG. 6: (Color online) Breaking distance Z b vs. inputpower P in (log-log scale): (a) local case, numerical results(dots) vs. behavior Z b ∝ P − / in (solid line), characteristicof the hydrodynamic limit; (b) nonlocal case: experimentalresults (dots) vs. data extrapolated from numerical simula-tions (dashed line) performed different values of powers P in ( ε ). Here σ = 0 . dark soliton power [5]). As shown in Fig. 6a, Z b ap-proaches the law P − / in for high enough values of N . Asimilar trend is confirmed by numerical simulations per-formed in the nonlocal case by employing the parametersof our experiment (we measured by means of Z-scan ap-paratus α = 1 .
17 mm − , n = − × − m /W ), asreported in Fig. 6b. As shown, the expected breakingdistance agrees reasonably well with the measured dataexcept for very low and high powers, the latter show-ing a saturation effect not accounted for by our model.Here we have used σ as a free parameter, finding thebest agreement for σ ≃ .
3, which is consistent with ourindependent estimate σ = h D T ρ c p | n | α | dn/dT | w i / ≃ .
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