Quantitative Analysis of Shock Wave Dynamics in a Fluid of Light
T. Bienaimé, M. Isoard, Q. Fontaine, A. Bramati, A. M. Kamchatnov, Q. Glorieux, N. Pavloff
CControlled Shock Wave Dynamics in a Fluid of Light
T. Bienaim´e, M. Isoard,
2, 3
Q. Fontaine, A. Bramati, A. M. Kamchatnov,
4, 5
Q. Glorieux, and N. Pavloff Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS,ENS-PSL Research University, Coll`ege de France, Paris 75005, France Universit´e Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg,Hermann-Herder-Straße 3, D-79104 Freiburg, Germany Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region, 141700, Russia Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia
We report on the formation and evolution of a dispersive shock wave in a nonlinear opticalmedium. The experimental observations for the position and intensity of the solitonic edge of theshock, as well as the location of the nonlinear oscillations are well described by recent developmentsof Whitham modulation theory. Our work constitutes a detailled and accurate bench-mark for thisapproach. It opens exciting possibilities to engineer specific configurations of optical shock wave forstudying wave-mean flow interaction and related phenomena such as analogue gravity.
In many different fields such as acoustics [1], plasmaphysics [2], hydrodynamics [3–5], nonlinear optics [6], ul-tracold quantum gases [7–9], the short time propagationof slowly varying nonlinear pulses can be described dis-carding the effects of dispersion and dissipation. Theprototype of such an approach is given by the systemof equations governing compressible gas dynamics [10].This type of treatment typically predicts that, due tononlinearity, an initially smooth pulse steepens duringits time-evolution, eventually reaching a point of gradi-ent catastrophe. This is the wave-breaking phenomenon,which results in the formation of a shock wave [11, 12].If, after wave-breaking, dispersive effects dominate overviscosity, the shock eventually acquires a stationary non-linear oscillating structure whose width increases for di-minishing dissipation, as was first understood by Sagdeevin the context of collisionless plasmas [13, 14]. The timefor reaching a stationary regime can be long in the caseof weak dissipation. A line of study then concentrates onthe dynamics of the oscillatory nonlinear structure knownas a dispersive shock wave (DSW). This non-stationaryproblem was first formulated and solved for a few typicalsituations by Gurevich and Pitaevskii [15] in the frame-work of Whitham theory of modulations [16].In the present work we study the propagation of anoptical beam in a nonlinear defocusing medium. Wave-breaking and (spatial or temporal) dispersive shocks havealready been observed in such a setting [17–30]. However,all previous theoretical descriptions of experimental op-tical shocks either remained only qualitative or resortedto numerical simulations for reaching accurate descrip-tions. Indeed, a realistic quantitative characterizationof the experimental situation requires to take into ac-count a number of nontrivial effects which sum up toa difficult task. For instance, saturation effects, such asoccurring in semiconductor doped glasses [31] and in pho-torefractive media [32], can only be taken into accountby using a non-integrable nonlinear equation, even for amedium with a local nonlinearity. Besides, both “Rie- mann invariants” typically vary during the pre-breakingperiod and this complicates the description of the non-dispersive stage of the pulse spreading, even in a quasiuni-dimensional (1D) geometry. Moreover, for realisticinitial intensity pulse profiles, the post-breaking evolu-tion corresponds, at best, to a so-called “quasi-simple”dispersive shock [33], the characterization of which re-quires an elaborate extension of the Gurevich-Pitaevskiischeme [15]. Finally, the non-integrability of the waveequation, significantly complicates the post-breaking de-scription of the nonlinear oscillations within the shock.Despite these difficulties, it has recently become possi-ble to combine several theoretical advances to obtain acomprehensive treatment of the nonlinear pulse spread-ing and the subsequent formation of a dispersive shock ina realistic setting [34, 35]. In this letter, we provide forthe first time an unambiguous experimental evidence ofthe accuracy of this theory with a precise description ofthe main features of the shock. This universal and quan-titative benchmark is a major advance for manipulationand engineering of optical shockwaves.We study the propagation of a laser field in a L =7 . Rb va-por (99% purity) warmed up to a controlled tempera-ture of 120 ◦ C to adjust the atomic vapor density. Weuse a Ti:Sa laser detuned by − . F = 3 → F (cid:48) transition of the D2-line of Rb at λ = 2 π/k = 780 nm. For such a large detuning, thenatural Lorentzian shape of the line dominates and theDoppler broadening k v (cid:39)
240 MHz can be safely ne-glected. In these experimental conditions, the system isself defocusing (repulsive photon-photon interaction) andthe transmission through the cell is 60 %. We find thatthis medium is well described by local photon-photon in-teractions, but contrary to previous works [36, 37] wefind it important to take into account the saturation ofthe nonlinearity to quantitatively describe the dynamicsof the shock waves.The input intensity profile is a cross-beam configura- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n tion of two vertically polarized laser beams, both prop-agating along the axis of the cell (denoted as O z ), withtheir respective phase precisely adjusted such that thetwo beams interfere constructively, see Fig. 1. Oneof the beams (which is denoted the hump ) is extendedalong the y -direction and significantly more intense thatthe other one (the background ) which is extended alongthe x -direction. At the entrance of the cell ( z = 0)both beams have an elliptic Gaussian profile. The back-ground (hump) beam has a power P ( P ) and waists w x, > w y, ( w x, < w y, ) [38]. Nonlinearity acts asan effective pressure which favors spreading of the humpin the x -direction along which is initially tighter fo-cused. Conversely, the low intensity background expe-riences almost no spreading and behaves as a pedestalwhich triggers wave-breaking of the hump during itsspreading. Each beam has a maximum entrance inten-sity I α = 2 P α / ( πw x,α w y,α ) ( α = 0 or 1), and we explorethe DSW dynamics for a fixed ratio I / I . We work inthe deep nonlinear regime, with I = 20 I . This largevalue corresponds to a wave breaking distance typicallyshorter than the cell length, and makes it possible, bychanging the total power P tot of the beams, to observeseveral stages of evolution of the DSW. The total powercan be increased up to 700 mW and is limited by the lasermaximum output power.We image the total field intensity I out ( x, y ) at the out-put of the cell on a camera. In order to minimise the ef-fect of absorption and increase the visibility of the DSW,we determine the normalized output intensity (cid:101) I out ( x, y ) ≡ I out ( x, y ) − I ( x, y ) I , (1)where I is the intensity profile at the cell outputwhen only the background beam propagates through themedium (the hump beam is blocked). I is the max-imal value of I ( x, y ). (cid:101) I out ( x, y ) is represented in Fig.2(a). The parameters of the photon-photon interaction,namely the Kerr coefficient, n = 1 . × − mm /W,and the saturation intensity of I sat = 0 . [cf.Eq. (2)] have been determined by comparing the experi-mental results with large-scale 2D numerical simulations[38]. The excellent agreement reached in Fig. 2 indi-cates that two effects –saturable nonlinearity and linearabsorption– are the relevant physical ingredients for atheoretical description of our experiment.In the regime w ,x (cid:28) w ,x and I (cid:29) I , the normal-ized output density (cid:101) I out becomes independent on the pre-cise shape of the background beam. As a result, (cid:101) I out ( x, uniform intensity I . Whitin the cell, the complexfield amplitude at y = 0, denoted as A ( x, , z ) ≡ a ( x, z ),then obeys a 1D nonlinear Schr¨odinger equation wherethe position z along the axis of the beam plays the role of an effective “time” [39]. The equation, once includedthe nonlinearity saturation and the linear absorption [32],reads i ∂ z a = − n k ∂ x a + k n | a | | a | /I sat a − i Λ abs a, (2)where n (cid:39) abs =30 cm, which corresponds to a 60 % transmission for acell of length L = 7 . a ( x,
0) = (cid:112) I + (cid:112) I exp (cid:32) − x w x, (cid:33) , (3)In order to evaluate the accuracy of the mapping to the1D model of Eq.(2), we compare in the upper panel ofFig. 2(b) the corresponding value of | a ( x, L ) | / I − (cid:101) I out ( x,
0) and with the result of2D simulations. The excellent agreement is confirmed inFig. 2(c) for the whole range of beam powers P tot .The mapping to a 1D problem enables us to compareour measurements with recent analytical predictions. Inparticular, if one neglects the linear absorption withinthe cell, for the initial intensity profile (3), wave-breakingoccurs at a propagation distance [40]: z WB = 4 (cid:114) n I ∗ n (1 + I ∗ /I sat ) I ∗ /I sat · (cid:12)(cid:12)(cid:12) dI ( x, dx (cid:12)(cid:12)(cid:12) , (4)where I ( x,
0) = | a ( x, | is the entrance intensity and I ∗ is the value I ( x ∗ ,
0) at point x ∗ where | dI ( x, /dx | reaches its maximum. For low entrance power, no DSW isobserved because z WB is larger than the cell length. Wavebreaking first occurs within the cell for a total power P WB such that z WB = L . For our experimental parameterswe obtain P WB = 48 mW. Numerical tests show thatthe taking into account of absorption does not modifynotably this value.For a total power larger than P WB , the DSW is formedand develops within the cell. The physical phenomenonat the origin of the DSW is the following: large intensityperturbations propagate faster than small ones, so thereexist values of x reached at the same “time” by differentintensities. When this occurs first, the density gradi-ent is infinite. This corresponds to the onset of a cuspcatastrophe [41, 42], the nonlinear diffractive clothing ofwhich is a dispersive shock wave. This takes the form of amodulated oscillating pattern consisting asymptotically(i.e. at large z , or equivalently large P tot ) in a train ofsolitons which, away from the center of the beam, grad-ually evolves into a linear perturbation. The position ofits “solitonic edge” on the y = 0 axis at the cell output( z = L ) is denoted as x s . It is located in Figs. 2(c)by a vertical red bar whose thickness represents the un-certainty on the estimation of x s from the experimental DSW
Hump beam Rb cell
Input intensity(interference) Output intensity z Background beam CameraBeam splitter xyz xyz xyz xyz
Fourierplane
FIG. 1. Sketch of the experimental setup. To create the initial state we overlap the background and the hump beams on abeam splitter with their relative phase precisely adjusted such that they interfere constructively. This state then propagatesinside the nonlinear medium consisting of a hot Rb vapor cell of length L . The insets represent a cut of the relevant intensityprofiles in the plane perpendicular to the direction z of propagation. The output intensity is recorded on a camera by directimaging through two lenses in 4 f configuration. − x (mm) − x (mm) − . − . − . . . . . y ( mm ) (a) − . − . − . . . . . x (mm) e I o u t ( x , ) (b) − . − . − . . . . . y (mm) e I o u t ( , y ) e I o u t ( x , ) (c) P tot =20 mW P tot =40 mW P tot =60 mW P tot =80 mW e I o u t ( x , ) P tot =120 mW P tot =200 mW P tot =360 mW P tot =440 mW − . − . . . . x (mm) e I o u t ( x , ) P tot =500 mW − . − . . . . x (mm) P tot =560 mW − . − . . . . x (mm) P tot =620 mW − . − . . . . x (mm) P tot =680 mW FIG. 2. (a) Left: experimental profile (cid:101) I out taken for P tot = 680 mW. Right: Two-dimensional numerical simulations at the sametotal entrance power. (b) x - and y -profiles along the cuts represented by the two white lines on the two-dimensional profiles(a). The solid blue line represents the experimental data, the dashed green line the two-dimensional numerical simulation. Onthe x -profile, the orange line is a one-dimensional numerical simulation, from Eqs. (2) and (3). (c) (cid:101) I out ( x,
0) for various totalbeam powers. The color code is the same as in (b). The vertical pink and gray bars on the right part of each intensity profileindicate the positions of the solitonic edge of the DSW and of the first maxima of oscillations within the DSW. The thicknessof each bar represents the experimental uncertainty. . . . . . x S ( mm ) N o s a t u r a t i o n S a t u r a t i o n Power (mW) e I o u t ( x S , ) FIG. 3. Characterization of the solitonic edge of the DSW asa function of the beam’s power. The upper panel representsthe position x s of the shock, and the lower one the corre-sponding intensity (cid:101) I out ( x s , (cid:101) I out ( x, x s to powers larger than 120 mW. Thefollowing maxima of oscillations, represented by verticalgray lines, are more precisely determined experimentally.The technique devised in Ref. [35] makes it possible totheoretically determine x s and the corresponding inten-sity (cid:101) I out ( x s , x s essen-tially rely on an approach due to El [43–45] which is validfor any type of nonlinearity, the precise intensity profilewithin the DSW can be computed only for exactly inte-grable systems, i.e., by neglecting saturation effects. Theposition-dependent oscillation period L ( x, z ) was com-puted in this framework in Ref. [34] for a parabolic initialintensity distribution. Fitting the center of the intensityprofile (3) by an inverted parabola, the positions x , x and x of the first maxima of oscillation of the DSW at
100 200 300 400 500 600
Power (mW) − . − . . . . x ( mm ) (cid:54) experiment vs. theory (cid:63) experiment vs. simulationsFIG. 4. Color plot of the experimental intensity profiles (cid:101) I out ( x,
0) as a function of P tot . The purple dot-dashed linerepresents the edge x s of the DSW extracted from Whithamtheory (upper part of the figure: x >
0) and from 1D numer-ical simulations (lower part, x < x ≶
0) the white dashed lines are the corresponding analyticpredictions (5) for the maxima of oscillation. the output of the cell are determined by: x − x s = L (cid:18) x s + x , L (cid:19) , (5)and by similar formulae obtained by replacing x by x (then x ) and x s by x (then x ). The results arecompared with the experimental ones in the upper half( x >
0) of Fig. 4. The small offset in the position ofthe theoretical maxima with respect to the experimentalones observed in the figure is due to an initial small over-shoot in the theoretical position of x s (cf. the green solidline in Fig. 3) which is itself due to the non-taking intoaccount of absorption in the model. Indeed, the 1D nu-merical simulations –which take absorption into account–are in slightly better agreement with the experimental re-sults for x s (cf. Fig. 3). Using the numerical x s in (5)instead of the analytical one yields, for the maxima ofoscillations, an excellent agreement with experiment, cf.the lower half of Fig. 4. Such a good agreement de-spite the non-taking account of saturation in Eq. (5) isnot surprising: the rapid decrease of intensity away fromthe solitonic edge (cf. Fig. 2) significantly reduces theimportance of saturation within the DSW.It thus appears possible to give a detailed description ofprecise experimental recordings of the intensity patternof an optical shock wave, not only thanks to numericalsimulations, but on the basis of Whitham’s modulationtheory. This is an important validation of recent ad-vances in this approach, which is no longer restricted tointegrable systems or idealized initial configurations. Weare reaching a point where these progresses make it pos-sible not only to study DSWs per se , but also as tools forprospecting new physical phenomena, such as the typeof wave-mean flow interaction recently identified in Ref.[46]: our platform is ideally designed to investigate scat-tering of elementary excitations by a DSW, a study whichis also relevant to the domain of analogue gravity. Indeed,as discussed in [38], a dispersive shock can be consid-ered as an exotic model of acoustic white hole, and thegood experimental control and theoretical understandingof this structure demonstrated in the present work opensthe prospect of a detailed investigation of the correspond-ing quantum fluctuations. [1] S. N. Gurbatov, O. V. Rudenko, and A. I. Aaichev, Waves and Structures in Nonlinear Nondispersive Media,General Theory and Applications to Nonlinear Acoustics (Springer-Verlag, Berlin and Heidelberg, 2011).[2] A. Jeffrey and T. Taniuti,
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