Plasma surface dynamics and smoothing in the relativistic few-cycle regime
S.G. Rykovanov, H. Ruhl, J. Meyer-ter-Vehn, R. Hoerlein, B. Dromey, M. Zepf, G.D. Tsakiris
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a r Plasma surface dynamics and smoothing in the relativistic few-cycle regime
S. G. Rykovanov, ∗ H. Ruhl, J. Meyer-ter-Vehn, R. H¨orlein,
1, 3
B. Dromey, M. Zepf, and G. D. Tsakiris Max-Planck-Institut f¨ur Quantenoptik, D-85748 Garching, Germany Moscow Engineering Physics Institute, 115409 Moscow, Russia Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, D-85748 Garching, Germany Department of Physics and Astronomy, Queens University Belfast, BT7 1NN, UK (Dated: October 9, 2018)In laser-plasma interactions it is widely accepted that a non-uniform interaction surface willinvariably seed hydrodynamic instabilities and a growth in the amplitude of the initial modulation.Recent experimental results [Dromey, Nat. Phys. 2009] have demonstrated that there must be targetsmoothing in femtosecond timescale relativistic interactions, contrary to prevailing expectation. Inthis paper we develop a theoretical description of the physical process that underlies this novelphenomena. We show that the surface dynamics in the few-cycle relativistic regime is dominated bythe coherent electron motion resulting in a smoothing of the electron surface. This stabilization ofplasma surfaces is unique in laser-plasma interactions and demonstrates that dynamics in the few-cycle regime differ fundamentally from the longer pulse regimes. This has important consequencesfor applications such as radiation pressure acceleration of protons and ions and harmonic generationfrom relativistically oscillating surfaces.
PACS numbers: 52.35.Mw, 42.65.Ky, 52.27.Ny
The dynamics of the plasma-vacuum surface are of crit-ical importance to the understanding laser-plasma inter-actions. Until now, the prevailing expectation has beenthat laser-plasma surfaces are inherently unstable enti-ties, since the morphology of the laser-plasma interactionsurface is determined by hydrodynamics for timescaleswhere ion motion is significant and initial perturbationwill grow rapidly due to instabilities. Specifically, the sur-face dynamics play a central role for many applications ofintense laser-plasma interactions. The best known exam-ple is probably the Rayleigh-Taylor instability which lim-its the parameter space that can practically be accessedin Inertial Confinement Fusion [1]. For relativistic pulseswith durations of > et al. showedthat the reflection of high-order harmonics from initiallyrough targets was consistent with the existence of a veryeffective smoothing mechanism [6, 7]. This suggests anew paradigm in laser plasma interactions, in that the ex-act opposite of the generally accepted behaviour occurs − surface smoothing rather than a modulation growth dueto instabilities. Motivated by this surprising behaviour,we have investigated the interaction of a relativisticallystrong laser pulse with an overdense, modulated plasma surface. We show for the first time that the surface mor-phology in the few-cycle relativistic regime is dominatedby coherent electron motion in the laser field. This isa complete departure from physical picture of how theinteraction surface evolves in laser-plasma interactionsand has important consequences for the frontier of ultra-fast science in the relativistic regime. This effect existsin a broad, practically important parameter range span-ning ultrafast pulses with intensities from 10 Wcm − upwards.To gain insight into the particle motion in the over-dense plasma we use a simple one-dimensional model[12–14]. This single-particle model describes the motionof an incompressible electron layer bound to immobileion background via charge-separation fields under the in-fluence of normally incident, linearly polarized electro-magnetic wave. This layer, later referred to simply asthe electron, is initially located on the vacuum-plasmainterface at x = 0. Taking into account that the chargeseparation fields are proportional to the electron longitu-dinal coordinate x , the equations of motion can be readilyobtained as: dp x dt = − β y ∂a y ( t, x ) ∂x + n e x, (1) dp y dt = da y ( t, x ) dt . (2)where x, y are the propagation and transversal coordi-nates respectively, β x,y and p x,y - velocity and momentacomponents respectively, a y - the driving vector poten-tial and n e is the electron density. We work in relativisticunits . The normalized quantities for vector potential a ,time t , length l , momentum p , and density n are obtainedfrom their counterparts in SI-units A , t ′ , l ′ , p ′ , and n ′ via a = eAm e c , t = ω L t ′ , l = ω L c l ′ , p = p ′ m e c , n = n ′ n cr . (3)Here e and m e are the charge and the mass of the elec-tron, ω L is the laser angular frequency, c is the speed oflight in vacuum, and n cr = ε m e ω L /e is the electroncritical density.In equations (2) a y ( t, x ) denotes the driving vectorpotential on the vacuum-plasma interface, which resultsfrom the interference between the incident and reflectedwave. It can be found by imposing the standard bound-ary conditions for the continuity of electromagnetic fields,i.e. of vector potential and its spatial derivative ∂ x a y atthe plasma-vacuum interface. Without losing the gen-erality, the incident a i , reflected a r , and transmitted a t vector potentials can be taken in the form a iy ( t − x ′ ) = − E i · sin( t − x ′ ), a ry ( t + x ′ ) = − E r · sin( t + x ′ + φ r )and a ty ( t, x ′ ) = − E t · sin( t + φ t ) · exp[ − ω p ( x ′ − x )] re-spectively. With x ′ we denote the longitudinal coordi-nate for the electromagnetic field, while keeping the no-tation of x for the coordinate of the electron. Applyingthe boundary conditions (thus setting x ′ = x ) one gets E i = E r = · q ω p · E t and φ r = 2 φ t = 2( α − x ),where α ≃ arctan ω p with ω p = √ n e the plasma fre-quency. One can use the transmitted vector potential at x = x ′ to obtain the driving vector potential: a y = − E i q ω p sin( t − x + α ) · e − ω p ( x ′ − x ) . (4)It is important to notice that the actual vector po-tential driving the electron is approximately ω p / a exceeds ω p / τ F W HM andamplitude a = 10 (corresponding to an intensity of1 . · W/cm for a laser wavelength of λ L = 1 µ m)is shown in Fig. 1a. Plasma density is n e = 400. Fig. 1bshows the transverse coordinate y as a function of time t .In Fig. 1c the solid line shows the longitudinal coordinate x (horizontal axis) as a function of time t (vertical axis).Figure 1c allows us to understand the origin of theharmonic generation process. One can see that duringthe interaction the model-electron (the step-like reflect-ing surface) oscillates in longitudinal direction with twicethe laser frequency. Each time the surface moves towardsthe laser it produces a flash with attosecond duration[15]. This simple and intuitive picture is called the Oscil-lating Mirror model and was proposed by Bulanov et al. FIG. 1: Electron motion obtained using the capacitor modelfor laser pulse with a = 10 with 4 cycles FWHM-durationand n e = 400. Electron is initially located at x e = y e = 0.Subfigure (a) shows the electron trajectory, subfigure (b)demonstrates the behaviour of the transverse coordinate y e intime, on subfigure (c) the dashed line represents the longitu-dinal coordinate x e of the electron (vertical axis) versus time(horizontal axis) obtained from the model, the color coded im-age displays the spatio-temporal picture of the electron den-sity obtained from 1D-PIC simulations with same laser andplasma parameters. [8] and further developed by Lichters et al. [9]. In thispaper we want to pay attention to the transverse motionof the electron, which extends to a considerable fractionof the laser wavelength (see Fig. 1b) and therefore mightbe responsible for the surface smoothing. Indeed if thetransverse motion of the electron exceeds the character-istic size of the modulations on a rough surface, then theroughness is likely to disappear.In assessing the role of the transverse motion as a pos-sible smoothing mechanism a simple expression for itsamplitude is needed. Neglecting longitudinal motion onecan get from Eq. 4 an estimate for the amplitude of thetransverse motion y max : y max ≈ · a q ω p + 4 a (5)The dependence of the transverse electron motion am-plitude y max on the laser pulse amplitude a for plasmadensity n e = 400 is shown on Fig. 2b. The solid lineshows the results obtained by numerically solving themodel equations and the dashed line represents equa-tion (5). The simple estimate (5) works fairly good forthe parameter range studied, and its simplicity makesit convenient for the following estimates. More accurateresults can be obtained by numerically integrating themodel equations.Having estimated the amplitude of the transverse co-ordinate y max one can establish an ad-hoc criterion forsurface smoothing to occur based on the ratio of thisamplitude to the characteristic roughness size h . For in- FIG. 2: Dependance of the amplitudes of longitudinal x max (a) and transverse y max (b) motion on laser amplitude a . In figure (a) the circles represent the results of 1D PICsimulations and the solid line shows the results of the numer-ical integration of the capacitor model. In figure (b) the solidline depicts the numerical integration of the model equationsand the dashed line is obtained from equation (5). stances where the transverse motion is on the order of thecharacteristic roughness size within the interaction area,considerable smoothing can be expected. One can definea dimensionless parameter ξ separating the case whensmoothing takes place from the case when the roughnesssurvives during the interaction: ξ = 2 a q ω p + 4 a · h y · e − ω p h x , (6)where h x and h y are the characteristic roughness size inlongitudinal and transverse directions respectively. Wemake an assumption that the boundary conditions staythe same independent of surface structure and that thefield exponentially decays inside the plasma. In the casewhen ξ ≫ ξ ∼ a . The code allows the simu-lation of the interaction of the intense laser pulses withpre-ionized non-collisional plasma with the beam inci-dent normally onto the target. The typical plasma den-sity used in 1D simulations is n e = 400 and n e = 30in the 2D case. A step-like vacuum-plasma interface isassumed, the ions are immobile. In the 2D case the sur-face is modulated sinusoidally in order to simulate theroughness (see left part of Fig. 4). For convenience, themodulation period and amplitude are linked and the po-sition of the vacuum-plasma interface is given by the law x = h · sin(2 πy/h ). The laser pulse amplitude was variedup to a = 20 in the 1D case and is fixed to a = 10 inthe 2D scenario. Throughout the paper we use FWHMof the electric field as the definition of the laser pulseduration and use pulses with an electric field that has a Gaussian envelope function in both time and space. E y ( t, x, y ) = E · exp (cid:20) − y ρ (cid:21) exp (cid:20) − ( t − x ) τ L (cid:21) , (7)where ρ and τ L are the width of the focus and durationof the laser pulse respectively. The FWHM duration isrelated to τ L by τ F W HM = τ L √ λ L , the time step is T L / T L the period of the driving laser and each plasmacell is initially occupied by 1000 macro-electrons. In the2D case the size of the simulation box is 3.5 λ L in laserpropagation direction and 40 λ L in polarization direction.The time step is T L /
300 and the laser propagation direc-tion spatial step is λ L / a compared to simulations (circles). The factthat simulation results lie on the curve obtained fromthe model and as longitudinal motion is directly corre-lated to transverse motion allows us to claim that themodel works well and gives correct results for both lon-gitudinal (Fig. 2a) and transverse (Fig. 2b) coordinates.The latter are hard to obtain from 1D PIC simulationas the particles leave the interaction region and are veryintricate to trace.In the 2D case we investigate the spatial beaming ofharmonics as a possible indication of smoothing. We an-alyze the propagation of the harmonics emission awayfrom the target using Kirchhoff diffraction theory [17] fol-lowing the approach used in earlier investigations [10, 18].The harmonic beam (from 15th to 25th harmonic, cen-tral wavelength 0 . λ L ) 200 λ L away from the target isshown on Fig. 3. On all four sub-figures the color surfacepresents the distribution of normalized intensity of thefiltered harmonics as a function of both time t and trans-verse coordinate y (the ceiling panel shows the same dataas a color-coded image). The upper-right plane showsthe projection of the beam to the time axis thus thetime structure of the harmonics beam exhibiting a trainof several attosecond pulses. On the upper-left planethe intensity distribution of the harmonics beam as afunction of transverse coordinate y is shown (black solidline). Results presented on Fig. 3 a,b,c,d are obtained fora surface with modulation size h = 0 (smooth surface, ξ → ∞ ), h = 0 . λ L ( ξ ≈ . h = 0 . λ L ( ξ ≈ .
05) and h = 0 . λ L ( ξ ≈ . λ L corresponds to the position of the har- FIG. 3: Farfield distribution of the reflected harmonics beam 200 λ L away from the target for a) smooth surface, b) surfacewith modulation size h = 0 . λ L , c) surface with modulation size h = 0 . λ L , d) surface with modulation size h = 0 . λ L . monics focus due to surface denting as discussed in thepaper by H¨orlein et al. [10]. This can be illustratedfrom the Fig. 3a by the fact that the transverse widthof the reflected harmonics beam (see graph in the upper-left plane) is much less than the initial laser width with ρ = 5 λ L .Secondly for dimensionless smoothing parameter ξ onthe order of unity the spatial and temporal structureof the harmonic beam is not influenced by the surfaceroughness. Figure 3 shows the harmonic orders from the15th to the 25th, which should undergo diffuse reflectionby each of the rough surfaces simulated. Contrary tothe Rayleigh criterion [11], but in agreement with exper-imental observation [6], almost no change in the harmonicbeam structure is observed for ξ ≈ . ad-hoc smoothing criterion. Surfaceswith ξ ≪ ξ ≪
1, the characteristic surface rough-ness was significantly diminished during the interaction.The analysis of the spatial structure of harmonics gener-ated on the corrugated surfaces exhibits collimated beamstructure and serves as an indirect proof of the surfacesmoothing.
FIG. 4: (a)Initial density profile and (b) the smoothed densityprofile in the middle of the interaction process for the surfacewith ξ = 0 . Direct proof of the surface smoothing can be found onFig. 4 where initial density distribution (as function oflongitudinal and transverse coordinates) and the densitydistribution near the moment when the pulse maximum reaches the surface are shown (left and right sub-figuresrespectively). The results here are presented for the sur-face with ξ ≈ .
6. The evolution of the electron densityin time can be traced in the animation made from simu-lation data (see Supplementary material), showing thatthe transverse motion of the electrons leads to rapid (incontrast to the hydrodynamically slow smoothing due toion motion) smoothing of the corrugation.In conclusion we have shown for the first time that co-herent electron dynamics is the dominant effect shapingthe laser plasma interaction surface. This is a paradigmshift from the way that surface dynamics have beenviewed to date - as purely hydrodynamic in nature. Dueto their much smaller mass, electrons can modify thesurface morphology on the time-scale of even the short-est, few-cycle laser pulses and hence must be taken intoaccount when considering intense laser-plasma interac-tions. This effect has important consequences in the fieldof ultrafast pulses and their application (e.g. harmonicgeneration, ion acceleration via radiation pressure) andimplies that surface imperfections on a scale smaller thanthe laser wavelength can be neglected.This work was funded in part by the InternationalMax-Planck Research School-APS, the MAP excellencecluster, and by the Association EURATOM - MPI f¨urPlasmaphysik. ∗ Corresponding author: [email protected][1] J. Lindl, Phys. Plasmas , 3933 (1995).[2] S. Wilks et al., Phys. Rev. Lett. , 1383 (1992).[3] M. Zepf et al., Phys. Plasmas , 3242 (1996).[4] M. Tabak et al., Phys. Plasmas , 1626 (1994).[5] L. Willingale et al., Phys. Rev. Lett. , 125002 (2009).[6] B. Dromey et al., Nat. Phys. , 146 (2009).[7] B. Dromey et al., Phys. Rev. Lett. , 085001 (2007).[8] S. Bulanov, N. Naumova, and F. Pegoraro, Phys. Plas-mas , 745 (1994).[9] R. Lichters, J. Meyer-ter Vehn, and A. Pukhov, Phys.Plasmas , 3425 (1996).[10] R. H¨orlein et al., Eur. Phys. J. D , 475(2009).[11] A. Ishimaru, Wave Propagation and Scattering in Ran-dom Media. Volume II: Multiple scattering, turbulence, rough surfaces and remote sensing (Academic Press, NewYork, 1978).[12] S. Wilks, W. Kruer, and W. Mori, IEEE Trans. PlasmaScience , 120 (1993).[13] D. Zaretsky et al., J. Phys. B: At. Mol. Opt. Phys. ,4817 (2004).[14] P. Mulser, D. Bauer, and H. Ruhl, Phys. Rev. Lett. , 225002 (2008).[15] G. D. Tsakiris et al., New J. Phys. , 19 (2006).[16] S. Rykovanov et al., New J. Phys , 025025 (2008).[17] M. Born and E. Wolf, Principles of Optics (CambridgeUniversity Press, Cambridge, 1999).[18] M. Geissler et al., New J. Phys9