Non-linear Ion-Wake Excitation by the Time-Asymmetric Electron Wakefields of Intense Energy Sources with applications to the Crunch-in regime
NNon-linear Ion-Wake Excitation by theTime-Asymmetric Electron Wakefields of Intense Energy Sourceswith applications to the Crunch-in regime
Aakash A. Sahai ∗ Department of Physics, Blackett Laboratory and John Adams Institute forAccelerator Sciences, Imperial College London, London, SW7 2AZ, UK& Department of Electrical Engineering, Duke university, Durham, NC 27708, USA
A model for the excitation of a non-linear ion-wake mode by a train of plasma electron oscillationsin the non-linear time-asymmetric regime is developed using analytical theory and particle-in-cellbased computational solutions. The ion-wake is shown to be a driven non-linear ion-acoustic wavein the form of a cylindrical ion-soliton. The near-void and radially-outwards propagating ion-wakechannel of a few plasma skin-depth radius, is explored for application to “Crunch-in” regime ofpositron acceleration. The coupling from the electron wakefield mode to the ion-mode dictates thelong-term evolution of the plasma and the time for its relaxation back to an equilibrium, limitingthe repetition-rate of a plasma accelerator. Using an analytical model it is shown that it is the timeasymmetric phases of the oscillating radial electric fields of the nearly-stationary electron bubble thatexcite time-averaged inertial ion motion radially. The electron compression in the back of the bubblesucks-in the ions whereas the space-charge within the bubble cavity expels them, driving a cylindricalion-soliton structure with on-axis and bubble-edge density-spikes. Once formed, the channel-edgedensity-spike is sustained over the length of the plasma and driven radially outwards by the thermalpressure of the wake energy in electrons. Its channel-like structure is independent of the energy-source, electromagnetic wave or particle beam, driving the bubble electron wake. Particle-In-Cellsimulations are used to study the ion-wake soliton structure, its driven propagation and its use forpositron acceleration in the “Crunch-in” regime. ∗ [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] D ec I. INTRODUCTION
FIG. 1:
Laser driven non-linear ion-wake at early time (t = ω − pe = f − pi , where f pi is the plasma ion frequency) in m i = m p = m e plasma . (a) Electron bubble wakefields in cartesian coordinates (fixed-box) with ω ω pe =
10 driven by a matchedlaser pulse (vector potential a = ω ) with R B (cid:39) c ω pe . (b) Non-linear ion-wake in the form of a cylindricalion-soliton of radius (cid:39) c ω pe excited behind the bubble electron wake in a proton plasma. (c) Transverse ion-density profile at z = c /ω pe . Notice that the ion density perturbation in this excitation phase is still building up and is a fraction of thebackground ion density, δ n i n < Plasma ions are generally assumed to be stationary in the theory of ultra-relativistic non-linear plasma electronwaves [1]. Such electron waves are regularly excited as wakefields of high-intensity energy sources such as anultra-short laser or particle beams and have proved to be promising for accelerating and transporting beams withunprecedented field strengths [2][3][4][5][6]. Important exceptions to the fundamental assumption of stationary ionsoccur as the intensities of the energy sources become high enough to lead to significant ion motion within a periodof the electron wave. Ion motion also invariably becomes important over several periods of the electron wake trainfurther behind the driver as the energy left-over in electron oscillation modes couples to the ion modes.The motion of ions has significant implications for plasma acceleration as high-intensity conditions occur whenthe drive beam (or an accelerated witness beam) has fields that lead to ion trajectories that are a considerable fractionof the electron oscillation trajectory [7][8]. These conditions are predicted to arise in the final-stage of ultra-lowemittance future plasma-based collider designs at the TeV energy scale. The subject of this paper however is theion motion at longer timescales, understanding the long-term ion behavior is important to determine the state of theplasma for succeeding bunches in a high repetition rate future plasma-based collider [10][11]. The work presentedhere shows that a long-lived ion-mode is leftover in the plasma, establishing an upper-limit on the repetition-rate ofthe plasma-based accelerators.In this paper, using theoretical analysis and computational modeling, the excitation of a nonlinear ion-wake in thetrail of a non-linear “bubble” plasma electron, is shown. The electron wake may be driven by either an intense laseror particle beam energy source [12][17]. We show that the time asymmetry of the focusing fields of the bubble leadsto the excitation of non-linear ion-acoustic modes in the form of a cylindrical ion-soliton.The application of the non-linear ion-wake for plasma-based accelerators in “crunch-in” wakefield regime isexplored. The “crunch-in” regime of plasma wakefields in an ideal hollow-channel was introduced in [13][14] andCh.8 of [15]. In this wakefield regime, it was shown that hollow-channel is driven by energy-sources such that thechannel-wall electrons collapse to the axis, driving strong wakefields of the order of channel-wall cold-plasma wave-breaking fields. Importantly, it was also shown that the focusing fields excited in this regime have a linear radialdependence of magnitude (with direction favorable for positron transport) and are of the order of accelerating fields[13] (Fig.2 there-in). The excitation of strong focusing fields in this regime is completely opposite to the conventionallyestablished conclusion that relativistic particles have zero focusing fields in hollow-channels [16]. This regime isenabled by the ion-wake channel because it is shown to have an initial radius close to the electron wake transversesize while uniquely its length is as long as the acceleration length. Here we show that the ion-wake channel-wallelectrons collapse towards the energy-propagation axis resulting in a non-linear on-axis electron density compressionmany times the near-void background density. The optimal compression is shown to be only possible if the drivingbeam properties are matched to the channel radius [13], a strong dependence on the excitation which is a signatureof non-linearity. The choice of appropriate channel radius is enabled by launching the driver at an appropriate time,resulting in excitation at an appropriate channel radius during the expansion of the ion-wake channel.The ion-wake model shows two distinct phases of the non-linear ion-wake: inertial and thermalized phase. In theinertial or excitation phase the time-asymmetry between the attractive and repulsive radial fields of the bubble onthe ions excites them into a soliton-like structure. We show that in this phase the inertial response of the ion ringsis dictated by an equilibrium or separatrix radius. The ion rings located within this radius collapse towards the axiswhereas rings outside are driven outwards. The outward propagating rings are only driven up to the bubble radiusbeyond which the force of the bubble radial fields rapidly falls o ff , resulting in the accumulation of the ion rings atthe bubble radius.At later times, the non-linear radial electron oscillations undergo phase mixing [18] leading to coherent electronmotion becoming thermalized. The thermalized phase is shown to be a driven non-linear ion-acoustic wave in theform of a cylindrical ion-acoustic soliton. Its characteristics are similar to the solutions of the cylindrical Korteweg -de Vries equation (cKdV) [19][20][21][22]. However, the ion-wake shown here di ff ers from a cylindrical-KdV solitonin several aspects: (a) The bubble wake electron oscillations do not thermalize into an isothermal plasma, so theion-wake soliton is driven (or forced) by the electron temperature gradient; (b) the ion-wake soliton breaks up intoN-solitons as it evolves and (c) at early times there is an ion-density spike on the axis which collapses at a later time.The soliton propagates radially outwards leaving behind a flat residue resulting in a near-void ion-wake channel.Representative PIC simulation results in Fig.1 and Fig.2 illustrate the salient features of the non-linear ion-wake.Figure 1 shows the excitation phase at an early time when the bubble wake-train is still executing orderly oscillationsand its fields have begun to excite inertial ion motion resulting in a soliton-like ion-wake structure ( δ n i / n (cid:39) .
2) asseen in Fig.1(b),(c). At later times as shown in Fig.2 the radial oscillations sustaining the bubble train have phase-mixed, converting much of the wave energy into electron thermal energy. The resulting electron thermal pressuredrives the ion-soliton ( δ n i / n >
1) outwards. The time evolution of the radial dynamics of the ion rings driven bythe time-asymmetric nonlinear electron is shown in a movie in supplementary material [9]. We show below, it isthe longitudinal or time asymmetry of the radial electron wakefields that excites the ion soliton which propagatesleaving behind a near-void channel shown in the PIC simulations in Fig.1 and Fig.2.The paper is organized into the following sections. In section II using the linearized fluid equations for iondynamics we show the two distinct phases of the ion-wake: the excitation phase and the propagation phase. Usingthe fact that the thermalizing electron wakefield is non-isothermal with radial electron temperature gradients, wemodel the non-linear ion-acoustic waves as a driven cylindrical ion soliton. We use an analytical model based onfields of a non-linear plasma wave and simulations to demonstrate the inertial phase of the ion-wake in section III. Insection IV the propagation phase of the ion-wake is analytical modeled with simulations verifying the propagation ofFIG. 2:
Electron beam-driven non-linear ion-wake at late time (t = ω − pe = . f − pi ) in m i = m p = m e plasma . (a)Beam-driven ion-wake electron density in cylindrical coordinates (fixed-box). The beam parameters are n b = n , σ r = . c /ω pe , σ z = . c /ω pe , γ b = , c /ω pe ≤ z ≤ c /ω pe . The later times in thetime-evolution of the ion-wake is also inferred from density snapshots farther behind the beam. (c) Radial electron and iondensity profile at z = c /ω pe . A full movie of radial electron and ion density dynamics is presented in supplementary material[9]. the cylindrical ion-soliton driven by the radial temperature gradient of the phase-mixed electrons. Finally, in sectionV we introduce and analyze the properties of “crunch-in” wakefield regime in an ion-wake channel, using analyticalmodel and simulations. In appendix A we present considerations and assumptions made to derive the ion-wakemodel. II. NON-LINEAR ION-WAKE:AS A DRIVEN PLASMA ION-WAVE
To develop insight into the ion wake physics, we consider the 1-D simplified dispersion relation of the ion-acousticplane waves, ω = c s k + ( c s /ω pi ) k (1)where, ω pi = ω pe √ m e / m i and c s = √ Υ k B T wk / m i under the collision-less condition, T iwk (cid:28) T ewk and Υ = + / f is theadiabatic index with f being the degrees of freedom of the ions.At early times the ion motion is dominated by inertia, thus ions move over the plasma-ion timescales when drivenby time varying and asymmetric fields of non-linear electron plasma-wave. As the ion inertia leads to very smallspatial displacement scales k → ∞ , the term k ( c s /ω pi ) (cid:29) c s /ω pi = λ De = (cid:113) k B T e π e n is the Debye wavelength).Thus, ω (cid:39) ω pi and the ion-soliton density spikes grow over the plasma-ion frequency timescales, 2 πω − pi . The radialelectron oscillations sustaining the bubble undergo phase-mixing, the electron trajectories lose orderly motion andthermalize. As the electrons thermalize, the ion motion is driven by thermal pressure of electrons.FIG. 3: Bubble-wake train behind an ultra-relativistic electron beam with bubble: β g (cid:28) β φ (cid:39) β beam . (a) electron density in 2Dcylindrical real-space, (b) corresponding longitudinal electric field profile and (c) corresponding radial-field profile. Here thebeam is located between 170 and 180 c ω pe . The bubbles just behind the driver in Fig. 3a undergo phase-mixing over several cycles.The intermediate stages of the extent of phase-mixing can be inferred from the bubbles that are closer to the beam. Thebeam-plasma parameters are the same as in Fig.2 but the electron-wake is shown at an earlier time t = ω pe . When the ions gain significant momentum and start oscillating over larger spatial scales in response to the electrondynamics then k λ De (cid:28)
1. In this thermally driven phase, the acoustic wave propagation becomes dispersion-lesswith ω = kc s .An ion-acoustic wave growing in amplitude undergoes self-steepening, forming a density spike over much smallerspatial scales; k becomes large while dispersion becomes important. The ion-wake modeled here is non-linear,thus the large k dispersion relation retaining the higher-order terms in k in the Taylor series expansion of eq.1, is, ω = c s k − c s λ De k .Also, at much later times, the ions undergo heating; T i increases and modifies the sound speed to c s = (cid:112) k B ( Υ e T e + Υ i T i ) / m i . A. Time-scale separation of Ion-dynamics:a simplified driven linearized ion-fluid model
In this section, we derive the wave equation for the ion-wake in the linear fluid approximation driven by twoterms: the electron wakefields and the electron thermal pressure. The linear ion-acoustic wave can be obtained byperturbative expansion of ion density, n i and ion fluid velocity v i in the zeroth-order ion fluid continuity equation, n ∇ · v (1) i + ∂ n (1) i ∂ t =
0. Taking a partial derivative with time, ∇ · ∂ v (1) i ∂ t + ∂ ∂ t n (1) i n =
0. The ion-fluid equation of motionwhere the electron temperature ( T e ) has a spatial gradient and electron wakefields ( E wk ) still persist is, m i ∂ v (1) i ∂ t = eZ i E wk − Υ k B T e ∇ n (1) i n − Υ k B n (1) i n ∇ T e . The assumption of spatial gradient of electron temperature has been used becauseelectron plasma wave oscillations phase-mix into non-isothermal plasma (this is substantiated through numericalresults in the simulations section in Fig.5). Upon substituting the equation of motion in the time-derivative of thelinearized continuity equation, ∇ · (cid:18) eZ i m i E wk − Υ k B T e m i ∇ n (1) i n − Υ k B m i n (1) i n ∇ T e (cid:19) + ∂ ∂ t n (1) i n =
0. Thus, a driven ion-acoustic wavelinearized to the first-order in density perturbation has the form, (cid:32) ∂ ∂ t − c s ∇ (cid:33) n (1) i ( r , t ) n = − eZ i m i ∇ · E wk ( r , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) wake + Υ k B m i n (1) i n ∇ T e (cid:12)(cid:12)(cid:12) thermal (2)In this first-order approximate ion-fluid model the right-hand side of eq.2 shows two separate timescales of theion-wake.At earlier times, the first term on the right-hand side dominates. This is the formation or inertial phase of the ion-wake where the bubble electron oscillations undergo ordered radial motion and the bubble radial electric field excitesthe inertial response of the ions. The group velocity of the electron bubble wake ( β g ≈ v th / c , in the 1-D limit, where, v th (cid:39) √ k B T e / m e is the mean electron thermal velocity [25]) is much smaller than the phase velocity so the bubblefields interact with the background plasma over several oscillations. Fig.3 shows the non-linear electron-wake train(electron density in real space in 3(a)) and its time-asymmetric fields (longitudinal 3(b) and radial 3(c)) driven by anear speed-of-light energy-source of high-intensity. The fields lead to the formation of the on-axis and R B ion densityspikes. At later times after the phase-mixing between radial oscillators the electrons thermalize and E wk ( r , t ) ∼
0. Thisis the propagation or thermal phase where the electron thermal pressure gradient drives the cylindrical soliton around R B radially outwards to many times R B .Eq.2 is not directly solved as it can be separated over the two di ff erent timescales. In the inertial or excitationphase when the plasma is cold ( T e (cid:39) c s (cid:39) ff of the electron wakefields to nearly zero as the electrons thermalizeis later shown using PIC simulations over a much longer time-scale in sec. IV (see Fig.6(b) where the radial electricfield goes to zero around 200 ω − pe and the ion-soliton is seen moving outwards radially in 6(a)). III. EXCITATION PHASE:ION INERTIAL RESPONSE TO THE BUBBLE FIELDS
Since the characteristic time of ion-motion is much longer than the electron oscillations, the longitudinal field E wk · ˆ z averages out over the full bubble electron oscillation. So, the ions gain relatively small net longitudinal momentum.However, atypical radial ion-dynamics arise because the radial fields, E wk · ˆ r are asymmetric in time as shown in Fig.4and do not average to zero, driving an average radial ion-momentum. A. Ion-ring analytical model:interaction with time-asymmetric bubble radial fields
The first stage of the ion-wake formation is controlled by the di ff erent time-asymmetric phases of ion dynamicsinertially responding to the bubble radial field impulses shown in Fig.4 namely, “suck-in” due to the electroncompression in the back of the bubble F back during τ back shown in Fig.4(c), and the “push-out” due to the mutual-ionspace-charge Coulomb repulsion force F sc during τ cav shown in Fig.4(d). The crunch-in force is spatially-periodic atnon-linear plasma wavelength, λ Np ≈ R B with a duty-cycle D = τ back τ back + τ cav (cid:28)
1. In addition to the plasma wake, thepropagating energy sources themselves impart impulses such as the laser ponderomotive force F pm τ las ( τ las is laserpulse duration) where F pme ( r , z ) = − m e c γ e ∇ r , z | a ( r ) | ( γ e is the plasma electron Lorentz factor) and the radial force of thedrive beam F b τ b where F b ( r ) = − π e n b r . The short driver impulses are neglected (below threshold intensity fordirect non-linear ion excitation [7][8])) because they act on the ions over their sub-wavelength short duration. This isunlike the slowly-propagating wake-plasmon bubbles that undergo continual interaction over many plasma periods.The validity of this assumption is evident from the laser ion-wake in Fig.1. Since the ponderomotive force of a laserdriver is an outward force for both the electrons and ions, the on-axis density-spike cannot be from this direct forceFIG. 4: Ion dynamics in longitudinally asymmetric phases of the radial forces in an electron bubble . (a) electron density of abubble in 2D cylindrical real-space. (b) longitudinal on-axis profile of the electron density (black), longitudinal field (blue),focusing field (red). (c) radial-field profile close to the back of the bubble. This is the focussing “crunch-in” phase for the ions. (d)the fields at the center of the ion-cavity of the bubble. This is the defocussing “push-out” phase for the ions. from the laser. Similarly the ion-density-spike at the radial wake-edge in an electron beam driven ion-motion cannotbe excited directly by the space-charge force of the beam, and is caused by the electron wake’s radial-edge densitycompression.The Lagrangian fluid model of the ions in a bubble consists of ion-rings under cylindrical symmetry with m i d r i / dt = Σ F wk (where F wk is the force of the electron wake on the ions). The bared-ion region inside the bubble isassumed to be a positively charged cylinder under steady-state approximation ( R B > r Be , back of the bubble electroncompression radius). The force on the ions from the non-linear electron compression δ n e = n Be (cid:29) n in the back ofthe bubble and radius r Be , pulls the ion rings towards the axis; and within the bubble, the mutual space-charge forceof the ion-rings leads to the ion-rings being driven outwards, away from the axis. The “suck-in” force on the ions is F back = − Z i π e n Be r Be r i . The space-charge force on the ions in the cavity is F sc = Z i π e n r i . The equation of motion is m i d r i / dt − c β φ λ Np ( F sc τ cav − F back τ back ) = ω pi = Z i π e n / m i , we have, d r i dt + β φ ω pi n Be n τ back τ cav r Be r i − r i = c τ cav /λ Np (cid:39)
1. Therefore the ion dynamics is dictated by an equilibrium or a separatrixion-ring radius, where the inward and the outward impulses balance out, r eqi = r Be (cid:113) n Be n D . The ion-rings at r i ≤ r eqi collapse inwards towards the axis resulting in an on-axis density spike. Whereas the ion-rings at r i ≥ r eqi move outaway from the axis. For m i / Z i > m p the ion-response is slower but similar.When the radially outward moving ion-rings reach beyond R B , there is excess net negative charge of the wakeelectrons within the bubble-sheath. As a result the radially propagating ion rings get trapped and start accumulatingjust inside the bubble-sheath and cannot freely move beyond, forming a density compression at R B . So, the cylindricalion soliton is formed around R B . This accumulation of the moving ion-rings is shown in Fig.1, where it is seen thatthe ion and electron density start forming a peak at R B .The radial location of the excitation of the ion-soliton in the non-linear electron-wave regime is much greaterthan a skin depth, c /ω pe ; thus the ion-wake starts o ff with a spatial-scale which is over several c /ω pe . This is dueto the balance of opposing radial forces on the plasma electrons from the driver and the ion cavity, resulting intheir radial accumulation at R B [5]. In the laser-driven case - the outward ponderomotive force is balanced by theevacuated ion-cavity: F pmlas = − m e c γ e ∇ r | a ( r ) | (cid:39) F cav = π e n R B gives R B ∼ ( c /ω pe ) γ e ∇ r | a ( r ) | when simplified using ∇ r | a ( r ) | (cid:39) a / R B and γ e (cid:39) a , R B (cid:39) √ a c ω pe (computationally, (cid:39) √ a c /ω pe [6]). In the electron beam-driven bubblethe outward force of the beam on the plasma electrons is balanced by the inward pull of the evacuated ion-cavity: F b ( R B ) = π e n b r b / R B (cid:39) F cav = π e n R B . This gives, R B (cid:39) (cid:112) Λ b / ( π n ), where Λ b = n b π r b is the line charge densityof the beam, where r b is the beam-radius computed here as 2 . σ r to account for 95% of beam particles for a radiallyGaussian beam profile. B. Bubble field time-asymmetry driven ion-soliton:simulation results
The above ion-ring model is verified using 2 D OSIRIS PIC simulations [29] of the ion-wake in the bubbleregime by simulating various energy-sources - laser-pulses in cartesian coordinates and electron-beams in cylindricalcoordinates. The laser pulse is circularly polarized with radially Gaussian and longitudinally polynomial profile (asin [37]) with a = a = . . ω , matched focal spot-size radius of 40 c ω , andlaser frequency ω = ω pe (the pulse dimensions are in the FWHM of the field). The electron beam is initializedwith γ b ∼ , n b = n (not shown n b = . n to 50 n ) and spatial Gaussian-distribution with σ r = . c ω pe and σ z = . c ω pe (the beam spatial dimensions are 5 σ in both the dimensions). The smallest spatial scale, c /ω pe is resolvedin the beam case and c /ω in the laser case (laser frequency ω ), with 20 cells in the longitudinal direction and 50 cellsin the transverse direction. Each of the plasma grid cell has 36 particles. The beam is initialized with 64 particles percell. The plasma is initialized in the Eulerian specification (non-moving window) and pre-ionized with Z i =
1. At thelongitudinal boundaries we initialize vacuum space of 50 c /ω pe followed by density ramps of 20 c /ω pe sandwichingthe homogeneous plasma. Absorbing boundary conditions are used for fields and particles.The electron-beam driven ion-wake soliton structure in theory is compared to the simulations in Fig.4(a) and 3(a).The observed R B = . c /ω pe (just behind the beam) whereas the estimated bubble radius is R B = (cid:112) n b / n (2 . σ r ) = . c /ω pe ( r b = . σ r = . c /ω pe , the assumption r b (cid:28) R B is not strictly satisfied). In Fig.2 which is in the propagation-phase, the observed ion-wake soliton is located at r (cid:39) . c /ω pe at 460 ω − pe which is about 1 . πω pi . The ion-soliton isexcited at an early time around R B and in the snapshot in Fig.2 it has propagated outwards. The on-axis densityspike drops to a minimum at r eqi ≈ . c /ω pe in Fig.2 whereas the estimated r eqi = . c /ω pe ( n Be / n (cid:39) D (cid:39) . r Be (cid:39) . c /ω pe ). The radial ion momentum p r − r phase-space in Fig.7(b) shows the ions accumulate at the axis andthe channel edge, at a time corresponding to Fig.2(b). The ions at the channel edge are seen to have a drift velocityand a thermal spread. The radial electron momentum p r − r phase-space in Fig.7(a) shows that a large density ofthermalized electrons are trapped within the ion soliton which is confirmed from the density plots in Fig.2(a).In the laser-driven bubble simulations the expected and observed R B (cid:39) c /ω pe as shown in Fig.1(a). In Fig.1(c) theion-wake soliton is created at r = . c /ω pe . The expected and observed on-axis density-spike radius is r eqi = . c /ω pe ( n Be / n (cid:39) D (cid:39) . r Be (cid:39) . c /ω pe ). The model for the excitation of this structure of the non-linear wake has beenverified for a range of laser and beam parameters from quasi-linear to strongly non-linear electron wake regime. IV. PROPAGATION PHASE:SOLITON DRIVEN BY ELECTRON THERMAL PRESSURE GRADIENT
As described in section III the electron bubble-wake train fields excite a cylindrical ion soliton. Eventually,the electron oscillations phase-mix and thermalize as electron thermal energy on the time-scale of about an ionplasma period. In this section we model the propagation of the cylindrical soliton radially outwards driven by thetemperature gradient as shown in eq.4. This soliton propagation is modeled using a modified cKdV equation in anon-equilibrium condition such that an electron temperature gradient sustains and drives the cylindrical ion soliton.
A. Thermally-driven ion-acoustic soliton:
Analytical model
In the linear regime the homogenous ion-acoustic wave equation predicts sinusoidal radial ion oscillations thatsupport the wave. However, the linearized ion-acoustic wave equation is inadequate to describe the propagatingsolitary density spike at the ion-wake edge, with ion density accumulation many times the background density.When the density in the ion perturbation begins to rise to the order of the background density, the electrostaticpotential due to charge-separation between the ions and the thermal electrons correspondingly rises. This leads to wave-steepening due to the preferential acceleration of ions in the direction of the ion-acoustic wave velocity. Whenthe potential of the wave is large enough the ions get trapped and co-propagate at the ion-acoustic wave phasevelocity, this non-linearity is the basis of the soliton. It should be noted that the linearized kinetic theory does notformally incorporate the trapping of particles at the wave phase-velocity. In this limit the density perturbation shapeis therefore not sinusoidal as the co-propagating background ions accumulate and their density perturbation takesthe form of an ion-soliton. The co-propagating ion velocity in the soliton can therefore exceed the ion-acoustic phasevelocity, v i > c s and M − > M = v i / c s is the Mach number. Therefore, non-linear acoustic waves are in theform of a soliton and propagate faster than the ion-acoustic velocity.To second-order, the non-linear ion-density spike n i ( r , t ) > n propagation is governed by the Korteweg-de Vries(KdV) equation [22] which has propagating solutions of the form U ( r − M c s t ) [21] where U is the ion-acousticwaveform, a soliton solution and M is the Mach number ( = v i / c s ) of the propagating solution. Higher-ordercontributions to the KdV equation have also been considered by earlier works. However, the more important andrelevant here is that the standard form of the cKdV equation assumes an isothermal plasma whereas the bubble-wake phase-mixes into a plasma with a radial electron temperature gradient, whereas the ions are initially cold. Ina non-isothermal plasma the e ff ect of trapped electrons in the ion-soliton have been considered using the Bernstein-Greene-Kruskal (BGK) model at the ion-acoustic velocity [23].It is also known that a single ion-soliton under the appropriate conditions can break-up into multiple solitonsleading to a N-soliton solution. This is also a phenomenon we observe in the simulations shown in the ion densityof the beam-driven case at z = c ω pe in Fig.2.We consider a description of the non-linear cylindrical ion-acoustic waves with a radial temperature gradient. Weassume that the background electron trapping does not significantly modify the distribution function. We assumethat the temperature changes slowly in vicinity of the ion soliton. This assumption is validated by the PIC simulationsin Fig.5.FIG. 5: Radial profile of the root-mean-square radial electron momentum (proportional to the square-root of the electrontemperature, √ T e ) at 460 ω − pe for the beam-driven ion-wake in Fig.2 . The blue curve shows the root-mean-squared radialelectron momentum, p eth ( r ) = (cid:112)(cid:2) Σ k p r ( k , r ) 2 π r n e ( k , r ) (cid:3) / Σ k π r n e ( k , r ), profile of the wake electrons corresponding to the time inFig.2 at 460 ω − pe . This represents the square-root of the electron temperature, p eth ∝ √ T e . The radial gradient of the temperature, ∂∂ r T e is thus computed at the peak of the soliton (red) and in its vicinity (green). It is interesting to note that ∂∂ r T (1) e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) peak = To obtain the KdV equation [15] in cylindrical coordinates with radial temperature gradient we normalize withrespect to the local electron temperature, the radius: ˆ r = r λ D , time: ˆ t = ω pi t = t (cid:113) π e n m i , electric field: ˆ E = e λ De k B T e E ,potential: φ = ek B T e Φ , ion-density perturbation: ˆ n i = n i / n , electron-density perturbation: ˆ n e = n e / n , ion-fluid0velocity: ˆ v = v i c s . Under this normalization the cylindrical coordinate equations transform as: electron Boltzmanndistribution equation ∂ ˆ n e ∂ ˆ r = − ˆ n e ˆ E − ˆ n e φ ∂∂ ˆ r ln T e , ion-fluid continuity equation ∂∂ ˆ t ˆ n i + ˆ n i ˆ v ˆ r + ∂∂ ˆ r ˆ n i ˆ v =
0, ion-fluid equationof motion ∂∂ ˆ t ˆ v + ˆ v ∂∂ ˆ r ˆ v = ˆ E and the Poisson equation ∇ Φ = r ∂∂ ˆ r (ˆ r ˆ E ) = ˆ n i − ˆ n e . The electric field ˆ E is both due to thethermal pressure and the radial fields of the wake, ˆ E wk + ˆ E th . But, in the following analysis the propagation of anon-linear ion-acoustic wave is considered, so we assume that the electron oscillations are thermalized and thus thee ff ect of the fields of the wake is negligible, ˆ E wk → n e , ˆ n i , ˆ v and ˆ E in a stationary background plasma with uniform backgrounddensity n . We consider weakly non-linear ion-acoustic wave and expand all the wave quantities in the powers of δ = M −
1. We perturbatively expand ˆ n i , ˆ n e , ˆ E , φ , T e and ˆ v i and retain all terms up to the order of δ . Note that wehave assumed that before the electron wake excitation the plasma is cold, T (0) e (cid:39) ξ = δ / (ˆ r − ˆ t ) and τ = δ / ˆ t . Using this, ˆ r = δ − / ( ξ + δ − τ ) and ∂ξ∂τ = ∂ξ∂ ˆ t ∂ ˆ t ∂τ = − δ . We renormalize the electric field as,˜ E = δ − / ˆ E . Note that in the moving frame the potential gradient is, E = − ∂∂ ˆ r Φ = − δ / ∂∂ξ Φ , so ˜ E is a more appropriatequantity.Under the assumption that in the moving-frame the quantities of the disturbance change with small δ = M − δ are collected. From the δ order terms of all the equations above, we infer Φ (1) = n (1) e = v (1) = n (1) i ≡ U and ∂∂ξ U = − ˜ E (1) .By collecting the δ terms from the Boltzmann’s equation we obtain ˜ E (2) = − ∂∂ξ n (2) e + U ∂∂ξ U − U ∂∂ξ T (1) e . Similarly,collecting the δ terms from the ion-fluid equation of motion we obtain ∂∂ξ ˆ v (2) − ∂∂ξ n (2) e = ∂∂τ U + U ∂∂ξ T (1) e and fromthe Poisson equation we obtain ∂ ∂ξ U = − ∂∂ξ ( n (2) i − n (2) e ). Taking the δ -order terms of the continuity equation andsubstituting U we obtain, U + τ (cid:16) ∂∂τ U + U ∂∂ξ U + (cid:104) ∂∂ξ v (2) − ∂∂ξ n (2) e (cid:105)(cid:17) − δ (cid:16) U + v (2) (cid:17) =
0. Neglecting quantities with δ times the second-order terms and using the ∂∂τ U result above, U τ + ∂∂τ U + U ∂∂ξ U + (cid:104) ∂∂ξ n (2) e − ∂∂ξ n (2) i (cid:105) = −U ∂∂ξ T (1) e .Using the δ terms of the Poisson equation in the above result and using the self-similarity property of the ion-soliton, we obtain the driven Korteweg-de Vries equation in cylindrical coordinates [19] (a more detailed derivationof this modified cKdV model can be found in [15]), Φ (1) = n (1) e = v (1) = n (1) i ≡ UU τ + ∂∂τ U + U ∂∂ξ U + ∂ ∂ξ U = −U ∂∂ξ T (1) e (4)It di ff ers from the cartesian-KdV equation by the term U τ and the temperature-gradient based driver term −U ∂∂ξ T (1) e .The cartesian KdV equation can be analytically solved to obtain two classes of solutions: (a) self-similar solutionswhich are shown in [21] to be Airy functions and (b) soliton solutions. A “soliton” is a single isolated pulse whichretains its shape as it propagates at some velocity, v soliton . This means that for a soliton-like solution the U onlydepends upon the soliton-frame variable, ζ = ξ − M c s τ and not on space-like ξ and time-like τ variables separately.The solution of the cartesian KdV equation in this co-moving frame is U ( ζ ) = v s sech ( (cid:113) v s ζ ) [21].The cKdV equation and the driven cKdV equation obtained here cannot be solved analytically. However, thenumerical analysis and experimental verification [20] of the cylindrical-KdV (cKdV) equation show that it supportsfunctions of the form U ∝ sech ( r − M c s t ) in the form of a cylindrical ion-soliton. But, the amplitude of the cylindricalsoliton changes as it propagates. Here, we show that the wake electron temperature drives the ion soliton for muchlonger distances than possible in an isothermal plasma. The velocity of the soliton in the cylindrical case is higherthan in cartesian case [19]. Since the ion-wake is excited in a non isothermal plasma its velocity changes as it is driven.The mean electron temperature reduces as the soliton propagates radially outwards because the electron thermalenergy is distributed over a larger volume. The cKdV equation is also known to support an N-soliton solution, andsimulations show N-soliton forming during the propagation phase. We computationally seek the dependence of thenon-linear ion-density spike on ( r − M c s t )-coordinate.It should be noted that such soliton solutions are supported under certain limiting condition on the Mach number, M . The strict condition on the existence and stability of ion-soliton arises from a threshold limit on the magnitude ofsoliton potential to continue trapping the background ions.Here we find that the speed of ion soliton is nearly equal to and only slightly higher that the ion-acoustic speedcalculated using the mean temperature. As this is not an isothermal plasma, there is no well-defined ion-acousticspeed. So, the ion-acoustic wave is phase-mixed and its velocity also changes as it propagates.1The local electron temperature of the ion soliton as shown in Fig.5, is used to calculate the Mach number, M andthus a stability criterion can be derived. This problem is represented using the condition on the Sagdeev psuedo-potential, V ( φ ) = − (cid:16) exp( φ ) − + M ( M − φ ) / − M (cid:17) that it has to be a real number. This condition is satisfiedwhen M − φ ≥ φ < M / φ max = M /
2. Using this we find the well-known condition,1 < M < . , v i < . c s Φ < M max = .
28 (5)As will be shown later, we find from simulations that the Mach number calculated using the mean temperature iswell within these bounds, and thus the soliton is stable.
B. Thermally-driven ion-acoustic soliton:simulation results
The channel-edge density spike, with a form similar to the cKdV-solution in the r − M c s t frame as shown in Fig.1and Fig.2 is seen to be propagating radially outwards. The propagation phase starts around t = ω − pe as the radialelectric fields E wk → r soliton (460 ω − pe ) = . c /ω pe (also seen in Fig.2) to r soliton (1100 ω − pe ) = . c /ω pe which corresponds to an average speed of (cid:104) v soliton (cid:105) = . c .We compare the time-averaged soliton speed (cid:104) v soliton (cid:105) to the average speed of sound, c s / c = p eth (cid:113) Υ m e m i where theaverage p eth (cid:39) .
06 from the electron phase-space (not shown). This gives c s (cid:39) . c ( Υ =
M (cid:39) . c s ( t ) = (cid:112) k B (cid:104) T e ( t ) (cid:105) / m i . The plasmais not in thermal equilibrium and its temperature varies radially as shown in Fig.5. The root-mean-square radialmomentum is used to estimate the temperature at an instant of time, and is calculated over radial dimension fromthe p r − r phase-space. The mean temperature is calculated by taking the average of the rms radial momentum - overthe entire channel: channel- (cid:104) p eth (cid:105) = (cid:104) Σ r sol r = p th ( r ) 2 π r n e ( r ) (cid:105) / Σ r sol r = π r n e ( r ) ∝ channel − √(cid:104) T e (cid:105) or in the vicinity of thesoliton: soliton- (cid:104) p eth (cid:105) = (cid:104) Σ r sol + (cid:15) r sol − (cid:15) p th ( r ) 2 π r n e ( r ) (cid:105) / Σ r sol + (cid:15) r sol − (cid:15) π r n e ( r ) ∝ soliton − √(cid:104) T e (cid:105) . The instantaneous sound speed, c s ( t ) computed with channel- (cid:104) p eth (cid:105) is in the green curve in Fig.6(c) and c s ( t ) computed with soliton- (cid:104) p eth (cid:105) is in the bluecurve in Fig.6(c). The extent of the vicinity ( (cid:15) ) around the soliton peak is shown in Fig.5.We compare the curves in (i) red: v soliton ( t ) (from the 3rd-order polynomial curve-fit of the radial position of theion-density peak as a function of time), (ii) green: c s ( t ) from channel- (cid:104) p eth ( t ) (cid:105) and (iii) blue: c s ( t ) from soliton- (cid:104) p eth ( t ) (cid:105) in Fig.6(c). From the comparison it is observed that they are in good agreement. It can be seen that the velocity ofthe soliton estimated using the location of the ion-density peak (red) lies between c s ( t ) calculated using the averagetemperature over the channel (green) which is the upper limit and c s ( t ) calculated using the average temperatureover the soliton (blue) which is the lower limit.We also present the radial gradient of the electron temperature, ∂∂ r T e ( r , t ) in Fig.6(d). It is interesting to note fromthe blue curve in Fig.6(d) that the temperature gradient at the peak of the ion-soliton is zero, ∂∂ r T e ( r , t ) (cid:12)(cid:12)(cid:12) peak =
0. Inthe vicinity of the soliton peak we see that the gradient of the temperature follows the variation in the ion solitonvelocity, this follows from eq.4. The vicinity of the soliton peak is shown as the green curve overlaid on the thermalmomentum curve in Fig.5.In Fig.2(b) N-soliton formation is observed in the ion-density at around z (cid:39) c ω pe . The single-ion soliton is seensplitting into several solitons. The N-soliton solution can explained by the seeding of di ff erent initial momentum ofthe ion-rings because ion-rings driven in the “push-out” phase have a radial position dependent defocussing forceacting on them, F sc ( r i ) = Z i π e n r i . This is shown in Fig.4(d). Thus the ion-rings originating at a larger radii from the2FIG. 6: Time evolution of the cylindrical ion soliton . (a) electron (black) and ion (red) spike radial positions (in terms of c ω − pe )with time and a third-order fit (green) for the position of the ion density-spike of the soliton. (b) radial wakefields of the electronbubble oscillations (in terms of m e c ω pe e − ) at the electron density spike (magenta) and at the ion density spike (blue). (c) radialvelocity of the ion density spike of the soliton calculated from the third-order fit curve (red). An estimate of the sound speed(green) using the mean temperature, between the axis & the soliton location (green) and in the vicinity of the soliton peak (blue),in the expression c s = √ k B (cid:104) T e (cid:105) / m i . Since the plasma is not isothermal the mean temperature is calculated by averaging thetemperature of electrons over the indicated spatial region. (d) gradient of the electron temperature at the soliton ion density peak(blue) and in the vicinity of the peak (red). The vicinity of the ion density peak of the soliton is defined as shown in Fig.5. axis are pushed outwards with a force of a higher magnitude and the rings originating at a smaller radii just outside r eqi are pushed outwards by a smaller force. So, over a longer time the set of ion-rings with a higher initial momentumpropagate radially outwards at a larger radial velocity. This break-up of a single ion-soliton into N-solitons occursover a longer time-scale because the di ff erence in momentum is small compared to the average momentum.The thermal momentum, p the at this time is less than one-tenth of the peak wake quiver momentum. There areseveral reasons for the cooling, such as, transfer of the wake energy to the ions and the trapped electrons [38], escape3of the highest energy electrons and un-trapped ions from the channel edge, energy loss to the bow-shock and there-distribution of the energy over an expanding volume. The peak radial ion-momentum is (cid:39) .
005 which shows thatnot all the radially propagating ions are trapped. The un-trapped free-streaming ions at (cid:39) c /ω pe can be distinguishedfrom the ions at the channel-edge in p r − r phase-space.FIG. 7: Radial phase-space snapshots of the electron and ion density in Fig.2 . (a) electron p r − r radial momentum phase-spaceshowing the accumulation of thermalized electrons within the ion-soliton. (b) ion p r − r radial momentum phase-space showingthe on-axis and ion-wake edge ion accumulations. It should be noted that the long-term stability of the on-axis ion-density spike of the non-linear ion-acoustic waveis not fully modeled here. The on-axis ion-density spike will disintegrate due to mutual Coulomb repulsion of theions over the sub skin-depth spike radius. This e ff ect of the collapse of the on-axis density-spike will be addressedin future work. We expect that the disintegration of the central structure to be further by azimuthal asymmetriesnot included in the cylindrically symmetric simulations. Earlier disintegration is seen in cartesian simulations not4shown.In summary, the ion-wake is a near-void channel with sub-skin-depth density-spikes on-axis and at the bubble-edge located at the bubble-radius, R B [5] of several c /ω pe . The ion accumulation in both the density-spikes is manytimes the background density, and the outre spike propagates outwards as a solitary structure at slightly above thespeed of sound.The time-scale of dissipation of ion-wake and relaxation of the plasma distribution to v th / c ∼ V. POSITRON ACCELERATION:“CRUNCH-IN” REGIME IN THE ION-WAKE CHANNEL
We explore the use of the ion-wake channel for positron-beam driven positron wakefield acceleration in a novel andrelevant “crunch-in” regime where the channel radius is of a few c /ω pe as is the case for the ion-wake channel. Suchchannels are also promising [31][39][40] for exciting the well-studied purely electromagnetic electron-wakefields.These pure electromagnetic fields driven in a hollow-channel have proven to have zero focusing forces when drivenby relativistic particles [39]. Here we show that in the Crunch-in regime driven even in an ion-wake channel, strongaccelerating and focusing fields of electrostatic nature are excited by the electron rings crunching in from the channelwall.The ion-wake enables the “Crunch-in” regime because as it slowly propagates radially outwards the channel radiiscans over a variety of c /ω pe , while its length is the energy-source plasma interaction length. Meter-scale propagationof electron beams and few centimeter-scale propagation of laser beams in plasmas while exciting nonlinear electron-waves has been well characterized in experiments. The theoretical model presented above thus provides a mechanismto generate long channels of several skin-depth radii. As we show below, the non-linear “crunch-in” regime requiressuch channels to optimally match with the driving energy-source.It is well known [24] that in a homogeneous plasma positron beam driven wakes have two major problems [13]- (i) The plasma electrons collapsing to the axis from di ff erent radii arrive at di ff erent times, preventing optimalcompression. This is because the radial force of the positron beam driving the “crunch-in” decreases with the radii.(ii) The plasma ions located in the path of the positron beam result in a de-focussing force on it. The transport ofthe positron in a positron-beam driven wake is thus not ideal in a homogenous plasma and has to rely on externalfocusing optics ahead of the plasma. The use of hollow plasma channels with a few c /ω pe is shown here to providepossible pathways to overcome these fundamental problems.The formation of much shorter plasma channels excited by significantly di ff erent processes have been shownpreviously. These processes including using a collimated laser with annular profile [34][35], using a hollow capillarydischarge [36], among others [33][32] [30].As the ion-wake channel is a practical realization of the hypothesized ideal hollow-channel plasma [24] of a fewskin-depth channel radius, we examine its excitation by a positron-beam and possible use for positron acceleration[13]. In this section we analyze whether the positron-beam driven wake-fields excited in the ion-wake can be usedfor the acceleration and transport of a positron beam. A. Non-linearly driven Ion-wake channel:analytical model
Positron acceleration using the ion-wake channel is explored in the non-linear “crunch-in” regime of perturbedelectron oscillation radii, δ r e ≥ r ch under the condition that the peak beam density n pb > n .We use the analytical model of the radial electron “crunch-in” based excitation of a positron beam wake in theplasma . The equation of motion of the plasma electron rings at r from the axis, under the positron beam crunch-in force but neglecting the space-charge force of the collapsing electron rings is: d d ξ r ∝ − r n bp ( ξ ) r bp ( ξ ), where ξ = c β pb t − z is the space just behind the positron beam with velocity c β pb driving the collapse. This is a non-linearsecond-order di ff erential equation of the form, r (cid:48)(cid:48) = f ( r , r (cid:48) , ξ ) where f is not linear in r . Under the assumption aboutthe positron-beam properties, n bp ( ξ ) and r bp ( ξ ) being constant during the entire interaction of the positron-beam withthe hollow-channel over its full length. So, upon dropping the dependence on ξ the equation simplifies to its special case which has analytical solutions, r (cid:48)(cid:48) = f ( r , r (cid:48) ). The solution to this equation is [13], r ch √ π erf (cid:16) (cid:112) ln( r ch / r ) (cid:17) = − √ C ξ ,where C = πβ b n bp n π (cid:18) r bp c /ω pe (cid:19) . Therefore, the collapse time-duration is ξ coll = − r ch (cid:112) π C . We note that there is an anomalythat exists in our problem formulation and the solution because we have not taken into account the space-charge forceof the compressing electrons as they collapse to the axis and this force balances the crunch-in force of the positronbeam. Under these approximations the collapse time in a homogeneous plasma is [24]: τ c = √ π r ch ω pe (cid:112) n bp / n r pb (6)This expression shows that the collapse-time even in a homogeneous plasma depends strongly on the properties ofthe beam and the radius from which the rings are collapsing in.Also, note that we have neglected the initial expansion velocity of the channel, dr ch / dt ). For optimal compressionavoiding phase-mixing, the electron rings should collapse over, τ c (cid:39) D λ Np / c where λ Np is the non-linear wavelengthof the positron-driven wake and D is the duty-cycle of compression phase. So, the optimal channel radius is r optch (cid:39) √ π D λ Np λ pe ω pb ω pe r pb . The scaling of the r optch with positron beam parameters is shown in [13]. B. Non-linearly driven Ion-wake channel:simulation result
Using 2- D PIC simulations in a moving window we study the positron beam driven wakefields in cylindricalgeometry. We compare positron acceleration in an ideal (Heaviside density function, n H ( r − r ch )) and an ion-wakechannels (on-axis and channel-edge density-spike, channel minimum density of 0 . n ) with r ch = . c /ω pe . Fornon-linear wake parameters r pb = . c /ω pe , n pb = . n and r optch (cid:39) . c /ω pe ( D λ Np λ pe = . . m e c ω pe e − for an ideal channel and 0 . m e c ω pe e − for the ion-wake channel. Fig.8also shows that the focussing potential (normalized to 27.6 m e c e − ) is similar and overall focussing in both cases.However, in the ion-channel the radial field is defocussing around the on-axis ion-spike.Thus the non-linearly driven ion-wake channel is useful for accelerating and transporting positrons despite thelower accelerating and focusing fields in comparison with the ideal channel. We also note that the on-axis densityspike has detrimental e ff ect on the focussing fields near the axis. However, the on-axis density spike is unstableover longer time-scales and collapses [15]. The cylindrical simulations used in the current work to model theion-wake ignore any azimuthal asymmetries in the distribution of electrons and ions in the on-axis density spike.Exploring the collapse of the on-axis density spike will be addressed in future-work. Ideal channels of a few c /ω pe aretechnologically challenging whereas the ion-wake channel of radius r ch (cid:38) R B is formed behind every bubble-wake.6FIG. 8: .The radial profile of the normalized electron density (black) in an ion-wake channel (normalized to the maximum electroncompression) at longitudinal location of the peak accelerating wakefield ( r pb = . c /ω pe , γ pb = n pb = . n ). The radialprofile of the accelerating-wakefield and normalized focussing-wakefield potential (radial field integrated from the edge of thebox to a radius). VI. CONCLUSION
In conclusion, using theory and PIC simulations we have shown the dynamics of the formation and evolution of anon-linear ion-wake excited by the well-characterized time-asymmetric electron bubble-wakefields independent ofthe type of energy-source. We have shown that the non-linear ion-wake has a characteristic cylindrical ion-solitonsolution and evolves to an N-soliton solution over longer time as described by a driven cKdV equation. Thus over theperiod of persistence of the ion-soliton, a second electron bunch cannot be accelerated in the plasma. This establishesan upper limit on the repetition rate of a plasma collider. We have also shown the feasibility of using the ion-wakechannel for positron acceleration in the positron-beam driven “crunch-in” regime within an experimentally relevant7parameter regime.
ACKNOWLEDGMENTS
Work supported by the US Department of Energy under de-sc0010012 and the National Science Foundation underNSF-PHY-0936278. I acknowledge the hospitality of the Dept. of Physics at the Imperial College London and theJohn Adams Institute, while making corrections to the manuscript. I acknowledge the OSIRIS code [29] for PICsimulations presented here. I acknowledge support for experiments by FACET group at Stanford Linear AcceleratorLaboratory and Prof. M. Downer’s group at University of Texas at Austin. I acknowledge the 256-node
Chanakya server at Duke university.
Appendix A: Considerations in the Non-linear ion-wake model
There are several considerations and assumptions that underlie the non-linear ion-wake model. Here we brieflydescribe these and seek to di ff erentiate the ion-wake from other phenomena. Primarily, we establish that the ion-wakeis a collision-less phenomena and it is significantly di ff erent from di ff usion. Secondly, as the ion-wake is formedbehind the high phase-velocity non-linear electron plasma waves that are excited as the wakefields of near speed oflight energy sources, it is significantly di ff erent from hole-boring which occurs in a plasma where the energy sourcehas nearly zero group velocity.We recognize that to study the time evolution of a wake-excited plasma, for establishing the duration over whichit relaxes to thermal equilibrium, both collisional and collision-less dynamics have to be considered along with thephysics of recombination modes such as electron-ion recombination. However, in this work the dynamics of plasmais modeled under the collision-less approximation. Thus, di ff usion is not important during the timescales over whichthe ion-wake is studied. We do not discuss recombination except mentioning that the “afterglow” is dominated byvolume recombination while localized e ff ects cannot be ruled out.In order to formally establish the di ff erence between the density wave processes that occur over collision-lesstimescale in contrast to the ones that start dominating under collisions, we show the assumptions made to arrive atthe dynamics of di ff usion. The process of di ff usion is modeled with a parabolic partial di ff erential equation which isdeduced from the ion-fluid equations under the assumption that the inertial response of the ions is much faster thanthe collisional timescales.The e ff ect of collisions is introduced as a drag force, m i n ν coll (cid:104) (cid:126) v i (cid:105) . The collisional drag force modifies the ion-fluidequation of motion as, m i n i d (cid:126) v i d t = mn (cid:16) ∂ (cid:104) (cid:126) v i (cid:105) ∂ t + (cid:104) (cid:126) v i (cid:105) (cid:126) ∇ · (cid:104) (cid:126) v i (cid:105) (cid:17) = ± e n (cid:126) E − (cid:126) ∇P e − m i n i ν coll (cid:104) (cid:126) v i (cid:105) where ν coll is the averageelectron-ion collision frequency and is obtained from the mean free path. Di ff usion of plasma is thus driven bythe charge-separation field, (cid:126) E and the thermal pressure, P e while being impeded by the collisional drag. Uponignoring the inertia of the ions, ∂(cid:126) v i ∂ t =
0, the equation for the ion velocity by di ff usion in an isothermal plasma is, (cid:104) (cid:126) v i (cid:105) = ± em ν coll (cid:126) E − k B T e m ν coll (cid:126) ∇ n i n . The characteristic parameter of di ff usion is the di ff usion coe ffi cient or di ff usivity D = k B T e m ν coll and mobility µ = em ν coll which depend upon the collision frequency. Using the gradient of the velocity in the continuityequation and ignoring the mobility µ , leads to a Fick’s law di ff usion equation, ∇ n i n ∝ ∂∂ t n i n ; characteristic of a parabolicequation.When the mobility is retained, the fluid equation is a moment of the Fokker-Planck equation which is the kineticmodel of the collision-driven drift and di ff usion. The di ff usion equation thus cannot support wave-like solutionbecause such solutions are characteristic of a hyperbolic partial di ff erential equation.The solutions of linear and non-linear di ff usion equations show the evolution of density profile by di ff usion andcan be obtained using the self-similar formulation. The self-similar solutions show the spatial and temporal evolutionof the density to be exponentially decaying. In the non-linear case, the density can have a sharp-front as it decays.However, a soliton-like propagating solution cannot be described with di ff usion equation. Hence, the cylindricalion-soliton presented here is not di ff usion but a wave phenomenon.The electron bubble wake is excited by a sub-wavelength impulse of an ultra-short driver. In contrast, the ion-wake is excited as the ions undergo sustained interactions with the bubble fields within the spatial extent of the wakeover several plasma electron oscillations. This happens because the electron wake-plasmon oscillations [2] have anear speed-of-light phase-velocity ( β φ (cid:39)
1) but negligible group-velocity [25] β g ≈ v th / c (in the 1-D limit), where, v th (cid:39) √ k B T e / m e is the mean electron thermal velocity of the background plasma. Therefore a slowly-propagating trainof coupled electron plasmons is excited in a cold collision-less plasma [25]. A large di ff erence between phase-velocityand the group velocity of the electron oscillations allows sustained field-ion interactions. It should be noted that high8phase-velocity plasma electron waves are possible only in a cold plasma with appropriate density, n that allows nearspeed-of-light propagation of the energy sources, β es (cid:39) (cid:117) β φ . Ion-soliton modeled here is assuming a significantdi ff erence between the phase-velocity and the group velocity of the plasma-electron waves.A time symmetric electron wakefield would excite time symmetric ion oscillations where the ion velocities averageto zero. However, the bubble wake is asymmetric in time as the back of the bubble electron compression is a smallfraction of the length of electron cavitation. The electron oscillations become non-linear at high driver intensities asall the interacting electrons are displaced radially, δ n e / n >
1, forming a non-linear bubble-shaped electron spatialstructure enclosing ions in its cavity. The wakefields excited in the bubble are useful for accelerating electrons[4][12][17]. High intensities also lead to fields that can directly drive the plasma electrons to velocities near thespeed-of-light. This occurs when for a laser pulse a ≥ n b n (cid:18) r b c /ω pe (cid:19) ≥ a is the peaknormalized laser vector potential, n b , r b the peak beam density and radius. The radially expelled electrons oscillateradially under the force of the plasma ions. These oscillations are excited over plasma electron oscillation timescales,2 πω − pe ( ω pe = (cid:112) π n e /γ e m e ) where γ e β e m e c is the temporally anharmonic relativistic electron quiver momentum.The normalized quiver momentum of the electrons in the bubble-oscillations is relativistic γ ⊥ β ⊥ ≥ ω ⊥ = ω pe (cid:18) β φ γ (1 − β φ ) (cid:19) / [1].We show that non-linear ultra-relativistic electron wakefields interacting with the plasma ions lead to the excitationof a non-linear ion-wake. The non-linear ion-wake δ n i / n > (cid:29) πω − pe in the trail of a bubble-wake train. By shaping the energy source it can be matched or guided to excite a long trainof nearly identical plasmons, Fig.3. Since it is the electric field E wk of a nearly stationary bubble plasmon that excitecollective ion-motion we model the ion dynamics in a single bubble. Using the single bubble ion dynamics, Fig.4 wemodel the ion-wake over the whole bubble-train spanning several hundred plasma skin-depths ( c /ω pe ).The wake-plasmon energy density ( E wk = . e | E p | / ( m e c ω pe )) m e c n , where E p is the wakefield amplitude) iscontinually partitioned between the field energy and the coherent electron quiver kinetic energy. In our modelwe do not include heavy beam-loading of the bubble electron wake. Under heavy beam-loading the bubble fieldenergy is e ffi ciently coupled to the kinetic energy of the accelerated beam. In this scenario the bubble collapsesand the magnitude of the ion-wake is smaller. The decoherence of the ordered electron quiver to random thermalenergy, E wk → k B T wk due to the phase-mixing [18] of individual electron trajectories caused by the non-linearitiesand inhomogeneities is further stimulated by the ion motion. The details of the thermalization of the wake electronsunder ion motion is beyond the scope of this paper. It is over these timescales upon thermalization that the steepenedion-density expands outwards radially as a non-linear ion-acoustic wave driven by the electron thermal pressure.The energy transfer process observed here is a coupling from the non-linear plasma electron-mode to a non-linearion-acoustic mode [27]. We also observe energy coupling to the bow-shock which is formed behind the bubble, Fig.4.9 [1] Akhiezer, A. I. and Polovin, R. V., Theory of wave-motion of an electron plasma , Zh. Eksp. Teor. Fiz, 30, 915 (1956) [Sov. Phys.JETP 3, 696 (1956))[2] Tajima, T., Dawson, J. M.,
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