All-optical density downramp injection in electron-driven plasma wakefield accelerators
D. Ullmann, P. Scherkl, A. Knetsch, T. Heinemann, A. Sutherland, A. F. Habib, O. S. Karger, A. Beaton, G. G. Manahan, A. Deng, G. Andonian, M. D. Litos, B. D. OShea, D.L. Bruhwiler, J. R. Cary, M. J. Hogan, V. Yakimenko, J. B. Rosenzweig, B. Hidding
AAll-optical density downramp injectionin electron-driven plasma wakefield accelerators
D. Ullmann , , P. Scherkl , , A. Knetsch , T. Heinemann , , , , A. Sutherland , , , A. F. Habib , , O. S.Karger , A. Beaton , , G. G. Manahan , , A. Deng , G. Andonian , , M. D. Litos , B. D. O’Shea , D.L. Bruhwiler , J. R. Cary , , M. J. Hogan , V. Yakimenko , J. B. Rosenzweig , and B. Hidding , SUPA, Department of Physics, University of Strathclyde, UK. The Cockcroft Institute,Daresbury, UK. Deutsches Elektronen-Synchrotron DESY, Hamburg,Germany. Department of Experimental Physics, University of Hamburg,Hamburg, Germany. SLAC National Accelerator Laboratory,Menlo Park, California, USA. Department of Physics and Astronomy,University of California Los Angeles, USA. Center for Integrated Plasma Studies,Department of Physics, University of Colorado, Boulder, Colorado,USA. Radiabeam Technologies, Santa Monica, CA 90404,USA. RadiaSoft LLC, Boulder, CO 80301, USA. Tech-X UK Ltd.,Daresbury, UK. Tech-X Corporation, Boulder, USA. (Dated: July 27, 2020)Injection of well-defined, high-quality electron populations into plasma waves is a key challengeof plasma wakefield accelerators. Here, we report on the first experimental demonstration of plasmadensity downramp injection in an electron-driven plasma wakefield accelerator, which can be con-trolled and tuned in all-optical fashion by mJ-level laser pulses. The laser pulse is directed across thepath of the plasma wave before its arrival, where it generates a local plasma density spike in additionto the background plasma by tunnelling ionization of a high ionization threshold gas component.This density spike distorts the plasma wave during the density downramp, causing plasma electronsto be injected into the plasma wave. By tuning the laser pulse energy and shape, highly flexibleplasma density spike profiles can be designed, enabling dark current free, versatile production ofhigh-quality electron beams. This in turn permits creation of unique injected beam configurationssuch as counter-oscillating twin beamlets.
I. INTRODUCTION
In electron beam-driven [1–6] and laser-driven [7–12]plasma wakefield accelerators, transient charge separa-tion of plasma electrons and ions can provide ultra-strongaccelerating and focusing electric fields, whose propertiescan be controlled by the plasma density n e . For example,the on-axis accelerating peak electric field E x in the limitsof classical wave-breaking [13, 14] scales with the plasmadensity as E x ∝ n / e just as the plasma frequency ω p ;this wakefield can reach tens to hundreds of GV/m ampli-tude at plasma densities between n e ≈ − m − .As such field levels are orders of magnitude stronger thanthose in metallic accelerator cavities, plasma acceleratorsdo not only represent an alternative to the unsustain-ably growing footprint of conventional particle accelera-tors [15], but also offer generation of ultra-high qualityelectron beams [16–19] since the rapid acceleration lim-its space charge-based growth of emittance. As plasma-produced electron beams are also ultra-short in duration,down to the fs-level, no further beam compression is re-quired. Such compression is necessary in conventional ac-celerators and may strongly increase the emittance, e.g.due to coherent synchrotron radiation. The potentialof plasma-based electron sources and accelerators withhigh initial and preserved beam quality therefore fuelsa wide range of prospective applications, including com-pact light sources based on free-electron lasers, inverseCompton scattering and betatron radiation [20–22]. Fur- ther applications extend to unique strong field and highenergy physics scenarios [23, 24].The injection of electrons into the plasma wave is a cru-cial challenge, however, as this process determines the ob-tainable electron beam quality. Accordingly, beam injec-tion has represented an intensely researched topic sincethe conception of plasma wakefield accelerators. The keygoals of this effort are high beam quality, tunability, re-liability and stability. Various injection concepts havebeen developed for both laser-driven wakefield acceler-ators (LWFA) as well as particle-beam-driven plasmawakefield accelerators (PWFA). These include the useof colliding laser pulses [12, 25], the generation of addi-tional electrons via ionization of available plasma compo-nents [16, 17, 26–29], and the generation of plasma den-sity downramps. The latter relies on tailoring the plasmadensity profile directly – the underlying medium whichprovides the accelerating and focusing fields – such that aprecise subset of plasma electrons enter the acceleratingphase of the wakefield in order to be captured.Both for LWFA and PWFA, gentle [30] and steeper[31] density downramps have been proposed to achievecontrolled injection. On these density transitions, thephase velocity of the wakefield reduces as v ph = c (1 +
12 1 n e ( x ) ∂n e ( x ) ∂x ξ ) − [32] at a position ξ = x − ct behind thedriver in the co-moving frame, with n e ( x ) being the lon-gitudinal electron density distribution in the laboratoryframe. Consequently, the density downramp alters tra- a r X i v : . [ phy s i c s . p l a s m - ph ] J u l FIG. 1. Representation of a plasma torch injector based on 3Dparticle-in-cell simulations. The electron drive beam (blue)excites a plasma wave (green). At t ≈ x ≈ t ≈
33 ps. Selected tra-jectories of trapped electrons are displayed in the laboratoryframe (black lines). See also [51] for the corresponding video. jectories of ambient plasma electrons, and thereby warpsand elongates the wakefield structure. This can facilitateinjection of plasma electrons, and the spatial distributionof n e defines the injection rate together with the resultingelectron beam phase space.Density downramp injection has been demonstratedfor LWFA, where the density gradient can be generatedby plasma expansion [33–35], gas flow [36–38], or shockfronts in gas jets [39–43]. While downramp injection is anexperimentally established method in LWFA, and experi-mental evidence suggests it can provide even better emit-tance than e.g. LWFA ionization injection methods [41],successful experimental realization of downramp injec-tion in the dephasing-free PWFA has not been achieveduntil very recently [17]. This is despite that a large frac-tion of seminal downramp injection theory work was de-livered in context of PWFA [31, 44], its potential as high-brightness electron beam source [45] had been discovered,and many further theoretical and simulation-based stud-ies with gentle [46–49] and steeper [50] ramps have sincebeen carried through.Further theoretical work suggests improved controlover injection in density downramp schemes by uti-lizing additional magnetic fields [52], and downrampscan be used to facilitate trapping for plasma photo-cathodes in low-current plasma wakefield accelerators[53]. Furthermore, plasma density ramps are crucial el-ements for external injection, extraction and staging ofplasma accelerators to address the challenges in beamquality preservation [54–59]. Tailored plasma slabs canalso act as plasma lenses and (re-)focus electron beamsfrom plasma-accelerators [60–62] and linear accelerators (linacs) [63, 64].In order to realize plasma density downramp injec-tion in PWFA, we have developed the so called ’plasmatorch’ approach [65, 66]. The scheme exploits the – com-pared to LWFA – rather modest peak electric fields of thePWFA drive beam by adding a low-intensity laser pulse.This pulse provides higher electric fields than the driverbeam, which allows production of well-defined, tunableand ’cold’ plasma density regions via tunnelling ioniza-tion from ambient atoms or ions, that are otherwise unaf-fected by the PWFA process. This all-optically generatedplasma torch offers flexible tailoring of the associatedplasma spike and downramp distribution. In contrastto hydrodynamic approaches, which rely on re-arranginggas or plasma volumes, this method locally adds an ex-tra plasma component n T decoupled from the mediumsustaining the PWFA.Figure 1 visualizes the plasma torch scheme basedon a 3D particle-in-cell (PIC) simulation usingVSim/VORPAL [67] (further see [51]). The electrondrive beam (blue) propagates from left to right throughplasma, e.g. generated from low-ionization threshold gassuch as hydrogen, thereby exciting an intense trailingplasma wave (green) in the blowout regime [2]. The laserpulse generating the plasma torch density spike (orange),e.g. from high-ionization threshold media such as helium,has already crossed the electron beam propagation axisand left the simulation box. The snapshot on the left at t ≈ t ≈
33 ps, where this beam (red)’witnesses’ the accelerating and focusing wakefields andgains energy.In the following, we report on the first experimentalrealization of this all-optical plasma density downrampinjection scheme, and explore its further potential andtunability with theory and simulations.
II. EXPERIMENTAL DEMONSTRATION OFDOWNRAMP INJECTION AT SLAC FACET
We developed capabilities required to explore anddemonstrate plasma torch density downramp injectionat the Facility for Advanced Accelerator ExperimentalTests (FACET) at the SLAC National Accelerator Lab-oratory within the E-210 collaboration. The linac pro-vided electron drive beams with charge Q D ≈ . W D ≈
20 GeV, length σ rms ,x ≈ − µ m and typi-cal widths of σ rms ,y ≈ − µ m and σ rms ,z ≈ − µ m,respectively. This drive beam was focused into an exper-imental chamber filled with a pre-mixed 50/50 hydro-gen/helium gas mixture at ∼ . ∼ ∼ µ m[17], that varies substantially in width as shown in Fig. 2a). Similarly tailored preionization setups were also ex-ploited in other plasma wakefield acceleration experi-ments at FACET [17, 70, 71].The plasma size was limited by the available spatialfootprint and laser energy budget. This constrainedthe choice of the plasma density as the channel had tofully enclose the blowout for a sufficiently long accel-eration distance. Under these circumstances, the op-timum condition was chosen by employing a hydrogenplasma channel density of n ch ≈ . × m − . Thedrive beam thus extended substantially into the acceler-ating phase of the blowout, i.e. k p σ rms ,x > / , where k p = ( n ch e /(cid:15) m e c ) / represents the plasma wave num-ber. Here, e denotes the elementary charge, (cid:15) the vac-uum permittivity and m e the electron mass.In addition, a separate laser arm was split off from themain laser path and individually compressed to a FWHMduration of τ L ≈
64 fs. This pulse was focused perpen-dicularly to the electron beam path with an off-axis in-vacuum parabola (f/22.9) to a spot size of w , L ≈ µ mr.m.s. at the interaction point x = 0 in the laboratoryframe. An attenuator allowed adjusting the torch laserenergy up to the maximum energy E L ≈ . I L ≈ . × W / cm .Motorization of the focusing optics allowed for shiftingthe laser focus position along the laser propagation axis z and for rotations in the yz -plane. This motorizationfacilitated versatile positioning of the plasma torch rela-tive to the electron beam axis and the hydrogen plasmachannel. At the same time, varying the laser energychanged the corresponding intensity profile and associ-ated tunneling ionization rates [72–77], which altered thevolume and shape of the plasma torch density distribu-tion n T . Generally, larger E L can produce steeper – andsteplike – plasma density gradients due to the genera-tion of additional He + and He at higher intensities. Insome experimental configurations, this has fully depletedthe ionization levels of helium and has generated verysteep plasma density ramps. The relative time-of-arrival(TOA) between this torch-generating laser pulse and theelectron drive beam was quantified by an electro-opticsampling (EOS) setup [78, 79] upstream of the interac-tion point with an accuracy of τ EOS ≈ . ± . ±
12 fs (r.m.s.)obtainable at FACET. An optical delay stage was usedto vary the nominal TOA. The charge and energy dis-tribution of the generated electron witness beams were measured with beam position monitors (BPMs) and animaging spectrometer.The conditions in the experiment are recaptured with3D PIC simulations using a simulation box size of500 µ m × µ m × µ m in x , y and z with cu-bic cells extending over 2 µ m. Figure 2 a) shows theshape of the background plasma (blue) implemented inthe simulations, resembling the preionized plasma chan-nel in the experiment with 8 particles per cell (PPC). Theinitial simulation starts at the beginning of the plasma at x = − . n T ( x, y, z ) and, consequently, the in-jected witness beam. In Fig. 2 b)-j), simulations for threedifferent torch laser energy levels that were experimen-tally realized [17] are shown, namely E L ≈ . , . . E L ≈ . E L ≈ + , thus in-creasing the peak plasma density and providing steeperdensity ramps. Consequently, the local deformation ofthe plasma blowout is more pronounced, and plasmaelectrons can be captured. The selected trajectories oftrapped electrons (black lines in Fig. 2 f) and g)) cross theblowout sheath further to the front of the blowout, suchthat these electrons gain more energy from the elongat-ing wakefield than in the previous case. The trajectories FIG. 2. Modeling of the FACET experiment in PIC simulations. a) shows the calculated density profile of the plasma channelgenerated by the axilens-focused preionization laser. The drive beam (black) propagates to the right through the plasma(color-coded) and excites a plasma wave. Interactions with different torch distributions outlined by red lines are displayed inb) to j). Simulation snapshots for 0.5, 1 and 5 mJ torch laser energy (first, second and third row) are shown at different timesteps t sim = 0, 0.5 and 12.5 ps (left, center, right column), respectively. The plasma torch modulates the blowout structure(center column) and triggers sheath crossing of electrons as shown by selected trajectories in the co-moving reference frame(black lines). For torch laser energies E L = 1 mJ and 5 mJ, Q ∼
94 pC and Q ∼
498 pC are trapped (i.e. they exceed 5MeV) and form the witness beam (red dots) shown in g) and j), respectively. also indicate the region of origin of trapped electrons,which will be investigated in detail later. In this con-figuration, the formed beam consists of Q ≈
94 pCtrapped charge. At ∼ . W ≈
105 MeV (Fig. 2 f)), corresponding to an average accel-erating field of 28 GV / m along the plasma channel.Further increased torch laser energy E L ≈ I L ≈ . × W / cm , whichalso ionizes the second atomic level of helium and thusincreases the peak plasma density to n T + n ch ≈ . × m − . The resulting torch distribution is much widerand provides steeper density gradients, which intensifiesthe deformation of the plasma blowout as shown in Fig. 2i). This configuration injects a large amount of charge Q ≈
498 pC as represented in Fig. 2 j). After in-jection, the blowout structure is significantly lengtheneddue to beam loading [80–82], which also manifests as re-duced peak energy of the injected electrons of W ≈ (cid:28) -0.15 ps) or after drive beam ar-rival (TOA (cid:29) ∼ FIG. 3. Electron witness beams observed at FACET. a), inte-grated electron line spectra for laser-early (left, TOA < -0.15ps) and laser-late mode (right, TOA > time-of-arrival reveals further details of the plasma torchinjection process and the resulting impact for the pro-duction of witness beams.Electron spectrometer and BPMs were used in con-junction with the EOS to record TOA scans over a 5 pswide time window. Figure 3 shows a consecutive TOAscan for the experimental case of E L ≈ ∼ ±
52 pC, whereas the average charge of the plasmatorch-injected electron population on top of the dark cur-rent amounts to ∼ ±
97 pC. The rate of dark currentproduction is influenced by shot-to-shot jitter variationsof the drive beam on the one hand, and preionized plasmachannel composition on the other hand, but is indepen-dent of the plasma torch laser pulses and the associatedjitters. Analysis of the dark current therefore allows mon-itoring of the impact of these major sources of fluctua-tions as a subset of the contributions relevant for plasmatorch injection. This may become usable to fine-tune thePWFA process in the future, in particular when torchinjection becomes independent of jitters related to theplasma channel generation (see below).We conduct PIC simulations analogous to Fig. 2 h)-j)and include the torch laser in the envelope approxima-tion to reproduce the dynamics of the plasma torch pro-duction. The witness beam charges (cf. Fig. 3 b), bluecrosses) obtained from simulation reproduce the transi-tion region − .
15 ps < TOA < W DC ≈ ± W T ≈ ± σ DC ≈ . ± .
05 GeV in contrast to the narrower spec-tra produced by torch injection with σ T ≈ . ± . ∼ .
25 ps that connects theplasma torch and dark current modes consistent withPIC-simulations. Within this narrow timing window, thelaser generates plasma as the blowout passes, such thatmost electrons are released directly inside the wakefield.This regime is thus dominated by ionization injection[17], which gradually transforms into torch injection forTOA ≤ Q M = − π (cid:15) λ p (cid:114) n T , n ch (cid:20)(cid:90) (cid:90) (cid:90) E ξ n T n ch d ξ d y d z (cid:21) min t (1)where E ξ ( ξ, y, z ) denotes the (quasi-static) acceleratingwakefield before interacting with the torch, and λ p ≈ π/k p describes the plasma wavelength. n T ( ξ − ct, y, z, t )represents the plasma torch density distribution withpeak density n T , . Here, t is the interaction timewhich describes the motion of the plasma spike throughthe wakefield and any temporal dependency of n T suchas its generation by the laser pulse. The integral inEq. (1) exhibits a convolution behavior and its minimumwith respect to t is proportional to the injected charge. E ξ ( ξ, y, z ) can be obtained either from models such as[87, 88] or PIC simulations.In this work, we model n T ( ξ, y, z, t ) based on tun-neling ionization calculations determined by the torchlaser intensity profile. Consequently, the local andmomentary torch density also depends on the TOA ofthe laser pulse. Using E ξ obtained from PIC simulationsfor the experimental situation in Fig. 2, we evaluateEq. (1) across the TOAs in Fig. 3. The result of thisphenomenological model is shown in Fig. 3 b) by theorange line. This simple geometric approach reproducesthe two plateaus as well as the transition region. Itprovides a direct, alternative view on the injectionprocess compared to the established description ofphase velocity retardation as an indirect consequence of the density profile. Furthermore, it correlates the 3Ddistributions of the wakefield with the density spike andmay be applied to efficiently find a specific torch densityprofile for a target witness beam charge.As discussed earlier, plasma torch injection can be de-coupled from TOA jitter when operating in the distinctlaser-early mode. Then, major contributions to the mea-sured output charge jitter originate from the distribu-tion of the preionized plasma channel, its position rel-ative to the electron drive beam, and shot-to-shot fluc-tuations of the corresponding preionization laser pulse[17]. In addition, the radial extent d ch , ( x ) of the plasmachannel was periodically narrowing along the accelera-tion section as shown in Fig. 2 a). This compromisesthe wake excitation and renders the PWFA susceptibleto various jitter sources. Particularly the width of theplasma channel per shot is highly sensitive to fluctua-tions of the preionization laser pulse parameters, whichimpacts on the injection yield significantly. To explorethe effects of this jitter source on the injection process,we model different transverse channel width distributions d ch ( x ) = κ × d ch , ( x ) based on the experimental baselinecase d ch , ( x ) (cf. Fig. 2) in PIC simulations. κ is var-ied from 0.6 to 3.0, where the former corresponds to anarrower channel as shown in Fig. 4 a) and the latter re-sembles a much wider channel as illustrated in Fig. 4 b).For κ < κ ≤
1. One is rele-vant for the plasma torch injection process itself, and theother one for the subsequent evolution of the blowoutthroughout the acceleration process. The former can beseen in Fig. 2 for κ = 1, where trapped electrons originatefrom regions close to the plasma channel edges. Here, thetorch laser can ionize neutral hydrogen in addition to he-lium. The resulting longitudinal density gradient ∂n e /∂x is therefore largest outside and at the edges of the thinplasma channel, where the density drops from maximumplasma torch density to zero. Inside the preionized chan-nel, in contrast, the plasma density decreases along theelectron driver beam propagation direction only from thepeak torch density to the hydrogen plasma backgrounddensity. Here, the longitudinal density gradient is thusmuch softer than outside the channel. This transversemodulation of the longitudinal torch gradient can there-fore increase the trapped charge, which further ampli-fies for even narrower channel widths. In addition, thewidened blowout caused by the narrow channel changesthe injection process. The combination of these effectsimpact on injected charge levels as shown in Fig. 4 c).Counter-intuitively, for thinner channels, the injectedwitness beam charge increases to a level of Q ≈
616 pCfor d ch = 0 . × d ch , . For wider channels d ch > × d ch , , onthe other hand, the blowout shrinks and gets increasinglyenclosed by the plasma channel such that the trappedcharge level saturates at 330 pC. A further widened chan-nel does not change the injected charge, as the regular,uncompromised blowout size is reached.The second consequence from narrow and varyingchannel configurations affects the wakefield evolutiondownstream of the plasma torch and has been discussedextensively in [17]. After injection, the subsequent ac-celeration phase depends on the evolution of the plasmachannel width along the plasma wave propagation axis x . The blowout size, structure and associated wakefieldschange when the local plasma channel width narrows be-low the regular blowout size, which can be approximatedby the plasma wavelength λ p ≈ µ m. Consequently,injected witness beams experience various acceleratingand focusing fields depending on local d ch ( x ). The exper-imentally observed output energies of witness beams thusfluctuate with jittering channel generation as reflected bythe range of spectra shown in Fig. 3 a). We concludethat for thin plasma channels with d ch smaller than theunperturbed blowout radius, stable witness beam gener-ation and acceleration requires precise control over thepreionization laser profile for the plasma channel aroundthe electron beam propagation axis. Alternatively andpreferably, using channel radii wider than encountered inthe experimental proof-of-concept situation at FACET,e.g. d ch (cid:29) d ch , , can effectively resolve these adverse in-fluences. The setup then converges to the ideal textbookPWFA configuration, which provides a blowout struc-ture independent of shot-to-shot plasma channel varia-tions. Wider channels with uniform density profile thusstabilize the injection process against the aforementionedpeculiarities caused by transversally varying longitudinaldensity gradients e.g. at the edge between the channeland the torch filament, and they also stabilize the subse-quent acceleration process.Another source for experimentally observed witnessbeam fluctuations are transverse shot-to-shot fluctua-tions of the torch laser propagation axis e.g. as resultof limited pointing stability. In combination with a finitesize of the produced plasma torch, this entails variation oflongitudinal density gradients ∂n e /∂x across the blowoutdiameter. To study this effect, we perform simulationscans with plasma torches shifted relative to the electrondrive beam propagation axis by ∆ y . Complementaryto the partially generated, but spatially centered plasmatorch occurring in the TOA transition region (see Fig. 3b)), wave-breaking and injection for an off-centered torchonly occurs in the reduced overlap volume of wakefieldand plasma torch as indicated by Eq. (1). For the exper-imental channel width d ch = d ch , , this spatial asymme-try reduces the injected witness charge with increasing FIG. 4. PIC studies modeling jitter sources present in theFACET experiment. a) and b) depict snapshots of blowoutformations containing torch-injected witness beams for re-duced ( d ch = 0 . d ch , ) and increased channel width ( d ch =3 d ch , ). d ch , ( x ) denotes the experimental baseline case shownin Fig. 2. c) shows the injected witness charge Q as func-tion of the channel width d ch ( d ch , ). d) depicts the simulatedtrapped charge Q for varying torch laser misalignment ∆ y forthe experimental configuration (black dots) compared withthe charge Q M obtained from Eq. (1) (orange). Further shownis the trapped charge obtained from a wider torch with flat topradius r flat ≈ µ m and wide channel ( d ch = 3 d ch , ) (blackcrosses) and the expected behavior of a wide slab-shaped torch(dashed green line). misalignment as can be seen in Fig. 4 d) (black dots). Asimilar trend results from the phenomenological model(orange dots). This effect can vary the injected witnessbeam charge over hundreds of pC, consistent with exper-imental observations as shown in Fig. 3. While misalign-ment of the torch laser can be detrimental to injectionstability especially in case of thin plasma channels andsmaller blowouts, asymmetric injection on the other handoffers the possibility to deliberately produce asymmetricwitness beams and to steer betatron oscillations as dis-cussed later.Any relative spatial shot-to-shot jitter between torchlaser and driver beam propagation axis can be eliminatedin a similar fashion as for the plasma channel: by in-creasing the transverse extent of the plasma torch untilit exceeds the blowout diameter significantly. To showthis and rule out any effect arising from limited channelwidth, we increase the latter to d ch = 3 d ch , in another∆ y -scan. Further, the plasma torch flat top radius (cf.Appendix A) is changed to r flat ≈ µ m and now fullycovers the unperturbed blowout. This improves the re-silience of the injector against torch laser misalignmentsubstantially, as shown in Fig. 4 d) (black crosses). Forexample, a misalignment of 20 µ m ( ∼
33 % of the blowoutdiameter) reduces the injected witness charge by only ∼ . ∼ III. SIMULATION-BASED EXPLORATION OFTHE PLASMA TORCH PARAMETER SPACE
After establishing the findings from the experimentalstudy, we now investigate plasma torch injection withoutlimitations of the plasma channel width or spatial jitter.The wide plasma medium therefore now fully containsthe PWFA, and increased plasma density n ch = 6 × m − corresponding to λ p ≈ µ m allows for higher ac-celerating gradients and the exploration of benefits forthe witness beam quality [45]. The preionized channelconsists of fully ionized hydrogen and the first level ofhelium, He + , to mitigate dark current from field ioniza-tion (see Appendix B). The plasma torch density spikeresults from ionizing helium a second time, thus yieldingHe . For the purpose of this study, we omit detailed in-vestigations of the required laser intensity distributions.Instead, we study the capabilities of the plasma torchmethod more fundamentally by exploring the effect of different three-dimensional plasma torch density profilesin a systematic manner.In the following simulations and as outlined in Ap-pendix A, various plasma torches are modeled as cylin-ders extending in z -direction with a central flat top ra-dius r flat corresponding to full ionization at peak densities n T , superimposing the plasma channel. Cosine-shapedramps of total length l ramp connect the flat top densityand the background plasma density n ch in radial direc-tion ( r = x + y ) around the torch laser propagationaxis. Similar profiles are obtained for the laser-generateddistributions in Fig. 2, which describe the FACET exper-iments.The simulations employ electron drive beam parame-ters obtainable at SLACs FACET-II facility [89], tunedfor dark current free PWFA [83] for the given plasmadensity. For instance, the Gaussian driver beam con-tains charge of Q D = 0.6 nC within a length σ rms ,z =7.5 µ m and has an energy W D = 10 GeV correspond-ing to a Lorentz factor γ D ≈ × and energy spread∆ W/W D = 0.02, and normalized emittance (cid:15) rms , n = 100mm mrad in both planes. To avoid envelope oscilla-tions of the drive beam, the transverse size is matched to σ rms ,x = σ rms ,y = ( (cid:15) rms , n λ p / π (cid:112) /γ D ) / ≈ µ m. Themaximum blowout radius observed from simulations is r b ≈ µ m. These simulations employ a moving windowand cell sizes of 0.25 µ m and 0.5 µ m in longitudinal andtransverse directions, respectively, as well as a modifiedYee solver [90]. The electron beam consists of 16 PPCand the plasma is modeled by 8 PPC (16 PPC in theplasma torch region).Figure 5 presents results from injection studies withfixed peak torch density n T , = 0 . × n ch superimpos-ing the plasma channel. At this torch density, no wit-ness beam generates dark current via tunnelling ioniza-tion (see Appendix B). The following simulations varythe torch ramp length l ramp for two different cases: torchdensity columns with i) wider and ii) narrower radial flattop extent than the blowout radius. The first case, shownin Figure 5 a), covers plasma torches with flat top radius r flat = 40 µ m (cid:29) r b . In contrast to our experiments atFACET where the torch diameter was barely as large asthe blowout diameter, this allows wave breaking acrossthe full cross section of the blowout. The cylindrical ra-dius of curvature of this torch distribution is large com-pared to the plasma wave’s transverse extent, such thatthe injection process is approximately symmetric in the yz -plane. We scan ramp lengths from l ramp = 10 µ m– highlighted by the dashed circle and the 3D-inset inFigure 5 a) – up to l ramp = 400 µ m, and present thetrapped witness beam charge (black dots) along withvalues obtained from the phenomenological model (or-ange dots, cf. Eq. (1)). This wide scan range com-prises different regimes of injection: with long ramps l ramp > λ p , injection occurs predominantly from therear of the blowout, while with short ramps l ramp < λ p FIG. 5. Injected charge Q in dependence of torch geometry.a), wide plasma torches with r flat (cid:29) r b and b), narrow plasmatorches with r flat (cid:28) r b for different ramp lengths. Black: PIC-simulations, orange: Q M modeled via Eq. (1). The insetsvisualize the short ramp scenarios for the cases highlightedby the dashed circles. the blowout collapses off-axis closer to the drive beamand sudden re-phasing provides injection as in Fig. 2.These different injection mechanisms are explored in the-ory ever since [30] (gentle ramps, in context of LWFA)and [31] (steep ramps, in context of classical hydrody-namic downramp-based PWFA) and have fueled differentapproaches for optimization of injected beam quality forhydrodynamic downramp injection for PWFA and LWFA[46, 50, 91, 92].We repeat the same scan for wide, uniform plasmaslabs approximating idealized conventional density down-ramp configurations and find similar injection dynamics,along with almost identical amounts of trapped chargeas for the wide plasma torch. For both geometries, morecharge is trapped for steep ramps, and less for softerramps in agreement with [93]. We thus infer that wideplasma torches can inherently mimic conventional den-sity downramp configurations with both steep and gentleramps as a subset of their range of capabilities.As wide torches with short ramps feature maximal in-jected charge for a given plateau density, only steep gradi-ents provided by the plasma torch scheme may facilitateinjection in PWFA driven by comparatively low-currentelectron beams. Further amplification can be achieved byincreasing the plateau density as discussed in AppendixB. These strategies enable PWFA applications in accel-erators lacking the intense electron beams available at FACET.Additionally, the plasma torch technique facilitatesunique plasma density spikes thinner than those in ourproof-of-concept experiments described in Section II. Infact, the flat top radius can be significantly smaller thanthe blowout radius. In a corresponding simulation scanshown in Fig. 5 b), it is set to r flat = 2 . µ m (cid:28) r b .Increasing the torch ramp length in this mode of opera-tion changes the radial torch extent substantially, andthus effectively enhances the overlap volume betweentorch and blowout in x and y until the blowout inter-acts with the plasma spike similarly to a wide torch.The trapped witness beam charges therefore convergeto those associated with wide torches as in Fig. 5 a)for ramps l ramp > ∼ µ m. Reducing the ramp length,on the other hand, increases the trapped charge simi-larly to the steep ramps in the wide-torch case. How-ever, this scenario exhibits a global maximum of injectedcharge at l ramp ≈ µ m, formerly not observed in den-sity downramp schemes. In this range, the typical impactof shorter ramps – namely injecting ever higher charge –is overpowered by the narrowness of the created plasmatorch profile. Then, r flat + l ramp < ∼ r b and the plasmatorch only covers the full extent of the blowout alongthe main torch axis in z , i.e. in the propagation direc-tion of the plasma torch-generating laser pulse. In theperpendicular radial direction y , the narrow torch den-sity profile reduces the overlap volume with the blowout.Wake deformation and wave breaking therefore happenasymmetrically, only affect a subset of the blowout andthus change the injection process substantially. Even forthis regime, the phenomenological model through Eq. (1)reproduces the overall distribution of Q ( l ramp ) obtainedfrom simulations.Both ramp scans shown in Fig. 5 highlight the capabil-ity of plasma torch PWFA to realize a wide range of dif-ferent density downramp physics and seamlessly switchbetween them, e.g. by changing the intensity profile ofthe torch laser. In contrast to typical downramp ap-proaches, however, plasma torch injectors prove highlyversatile as their density spike freely superimposes thePWFA medium. For example, they can be located arbi-trarily along the PWFA and even multiple instances canbe created in short succession. A double-torch injectoris presented Appendix C and demonstrates generation ofmulti-color beams, e.g. applicable in pump-probe exper-iments.In addition to freely locate plasma torch injectors,we show in the following sections that shaping theplasma torch density profile n T ( x, y, z, t ) in space andtime facilitates control not only over injected charge,but also over the resulting witness beam distributionand quality. The initial density distribution of trappedplasma electrons is of critical importance, as it representsthe initial conditions for the injection process and thusgoverns the properties and evolution of the witness0beam. By back-tracking all trapped plasma electronsin PIC-simulations, we reconstruct and visualize this trapping volume . SYMMETRIC INJECTION
As shown in the previous sections, wide torches exhibitsimilar features as conventional downramp schemes. Thisparticularly results from radially symmetric interactionsbetween the blowout and the plasma torch, which im-plies a symmetric drive beam and isotropic torch densitydistribution across the blowout extent. Figure 6 visual-izes simulations where the electron drive beam (orange)propagates to the right through the plasma representedas color-coded slice through the simulation box center.The plasma torches shown here combine the plateau re-gion with radius r flat = 40 µ m with short l ramp = 10 µ mand long ramps l ramp = 200 µ m, respectively. Thesecases were already shown in Fig. 5 a). The projectedtrapping volumes n trap ( ξ, y, z ) associated to the subse-quently trapped witness beams are color-coded in yellow-red. The first column in Fig. 6 shows the projection inplasma torch laser propagation direction ( xy -plane), thecenter column shows the projection in the propagationdirection of the electron beam driver ( yz -plane), and thethird column shows the side-view of the plasma torch( xz -plane). These densities are normalized across eachcolumn of the figure. For the given interaction, the lon-gitudinal projection of the trapping volume (central col-umn) displays a characteristic annular shape similar to[50, 94] around the drive beam propagation axis. Weobserve this effect for all interactions between radiallysymmetric blowouts and wide plasma torches and callthis fundamental structure trapping doughnut .The longitudinal extent of the trapping volume of thewide torch with short ramp length l ramp = 10 µ m in Fig. 6a)-c) slightly exceeds the downramp length. It broadensradially towards the downstream end of the ramp (seeFig. 6 a) and c)) due to the expanding blowout on thedensity downramp. The trapping doughnut as depictedin Fig. 6 b) in head-on view is perfectly symmetric andregular due to the likewise symmetric interaction. Asalready shown in Figure 5 a), the corresponding injectedand trapped charge amounts to 256 pC and agrees withthe phenomenological model Eq. (1) Q M ≈
251 pC.The trapping volume for the wide torch with long ramp l ramp = 200 µ m remains circular, but extends over muchlonger distance than for short ramps. In contrast to thetrapping volume in the previous case, however, the longramp generates an irregular density pattern consisting ofmultiple injection filaments. We attribute this inhomo-geneity to the sensitivity of the adiabatic injection pro-cess, as the soft gradient reduces the efficacy of the den-sity ramp. Similar irregularities were reported for LWFAself-injection [94], outlining comparable injection dynam-ics as for long downramps in PWFA. Trapping is there-fore susceptible to inhomogeneities of the driver beam, wakefield and plasma profile – and is computationallyprone to noise and resolution [95]. Repeating the simu-lation at refined grid resolution of 0.1 µ m in the longi-tudinal direction resembles a similarly irregular trappingpattern (cf Appendix D). The witness beam generated onthis gentle ramp has a charge 82 pC ( Q M ≈
68 pC) andexhibits slices with particularly low emittance as shownin Fig. 7, consistent with findings in [46].As a side note, the annular trapping doughnut shaperemains similar for a case with equally long ramp butshorter, 2.5 µ m-long radius (see Appendix D), since l ramp + r flat (cid:29) r b is still fulfilled. However, the injectedcharge reduces slightly to 80 pC ( Q M ≈
68 pC) due tothe weak torch density modulation at the edges of theblowout.We now investigate witness bunch parameters pro-duced by the two trapping volumes shown in Fig. 6 inmore detail. Figure 7 a)-c) shows key slice parameters(50 nm bin size) of the witness beam resulting from theshort ramp visualized in Fig. 6 a)-c). Figure 7 d)-f) con-trasts them with the witness beam injected on the longramp shown in Fig. 6 d)-f). The short ramp case with in-jected charge 256 pC ( ∼ . × macro particles) formsa beam with σ x ≈ . µ m r.m.s. length, whereas the longramp case is formed of 82 pC ( ∼ . × macro particles)within σ x ≈ . µ m r.m.s. length. The average obtainedwitness electron energies amount to ∼
103 MeV and ∼ I p ≈
20 kA. The current profiles are color-coded for eachslice of the witness beam by the average longitudinal ori-gin position within the trapping doughnut, shown as xy trapping region in the top-left inset. For the gentle ramp,this mapping reveals a strictly linear relation, expressingthat electrons residing at the plateau-end of the densityramp form the head of the witness beam and vice versa.The beam generated by the steep torch gradient, in con-trast, lacks a clear correlation, as electrons injected frommultiple initial positions cross trajectories and producemixed slices within the formed witness beam.With regard to beam quality, both torch configura-tions produce low projected emittances, e.g. (cid:15) n < ∼ . (cid:15) n ≈ . . B = 2 I p / ( (cid:15) n , y (cid:15) n , z ) ≈ A / (m rad) – many orders ofmagnitude larger than obtainable in conventional accel-erator systems. Figure 7 c) and f) present the absoluteslice energy spreads of both beams, which are, except1 FIG. 6. Visualization of trapping doughnuts obtained from PIC-simulations in the plane transverse to the axial dimension ofthe torch (left column), along electron drive beam axis (center column) and torch side view (right column). The two rowscorrespond to Fig. 5 a), the wide torch, with short ramp l ramp = 10 µ m, and long ramp l ramp = 200 µ m, respectively.FIG. 7. Comparison of witness beam slice parameters in-jected by wide plasma torches ( r flat = 40 µ m) with short( l ramp = 10 µ m) a)-c) and long ( l ramp = 200 µ m) d)-f) ramps.a) and d) show current profiles of respective witness beams,color-mapped to the longitudinal position within the trappingvolume (insets). Slice normalized emittances in both planes(black) and brightness (red) are shown in b) and e). Sliceenergy spreads are given in c) and f). for a few slice positions and in particular for the gentleramp, below 1 MeV. The latter further exhibits low sliceenergy spreads below ∆ W < ∼ . ASYMMETRIC INJECTION
The plasma torch process allows injection from asym-metric interactions of the plasma wave with the torchdensity distribution. Modification of torch profiles andthe corresponding trapping doughnut can be used to pro-duce unique witness beam modalities.The first case shown in Fig. 8 revisits a configurationwith narrow spatial extent of the plasma torch shownin Fig. 5 b): r flat + l ramp = 12 . µ m < r b . Here, the trapped charge reduces significantly to 88 pC ( Q M ≈ Q M ≈
251 pC) trapped on thewide torch shown in Fig. 6 a)-c). Figure 8 a)-c) visualizesa substantially diminished overlap of the thin torch withthe plasma blowout in different planes: as shown in Fig. 8a) and b), the reduced transverse overlap in y -directioneffectively crops the trapping volume and removes partsof the typical annular shape (cf. Fig. 6 b). Additionally,the shorter longitudinal extent of the torch reduces theamount of plasma available for trapping. This uniqueinteraction geometry precipitates non-isotropic injectionpredominantly in the xz -plane. As we shall see later, thiscauses the formation of twin populations of injected elec-trons, which originate from the pronounced asymmetryalong the z -direction of the trapping doughnut depictedin Fig. 8 c).Next to spatially asymmetric torch density distribu-tions, the formalism in Eq. (1) also predicts that ra-dial asymmetries in the blowout can change the injectedcharge substantially. This occurs for example for trans-versely asymmetric drive beams. Those are the normrather than the exception both for linacs, due to beamcompression and focusing techniques, as well as for laser-plasma accelerators, where linear laser pulse polarization[94, 96, 97] produces transversely elliptical electron beamprofiles.In Fig. 8 d)-f), a drive beam with σ z = 1 . × σ y ( (cid:15) n ,y = (cid:15) n ,z ) generates an asymmetric blowout with radii r z ≈ . × r y . We study the injection into this wakeformation with a wide torch and long ramp identical toFig. 6 d)-f) and find that such asymmetric blowouts de-form the trapping doughnut substantially: instead of afull 360 ◦ doughnut, trapping is allowed only in a highlyconfined angular range around the z -axis, as shown inFig. 8 e). Remarkably, this angular selection does notresult from cropping the circular trapping doughnut ob-served in the symmetric drive beam case Fig. 6 d)-f), butfrom re-arranging the original distribution. Injection inthe xy -plane, where the blowout is thinner, is suppressed,while it is promoted in the xz -plane, where the blowoutis wider. In fact, the total injected witness charge in- FIG. 8. Visualization of trapping doughnuts obtained from PIC simulations analogous to Fig. 6. The first row corresponds tothe narrow torch with short ramp l ramp = 10 µ m shown in Fig. 5 b). The second row displays an identical configuration as inFig. 6 d)-f) but with an asymmetric drive beam σ z = 1 . × σ y yielding an asymmetric blowout with radius r b ,z ≈ . × r b ,y .The third row repeats the former, extended by an additional torch density modulation n = n T × [1 + 0 . φ ( z, y )) ]. creases slightly by ∼ ∼
10 % to Q M ≈
76 pC). Enhanced injection rates in the xz -planethus (over-)compensate the missing parts of the trappingdoughnut. The strongly planar injection driven by anasymmetric driver beam produces two distinct witnessbeam populations, even more pronounced than for thecropped trapping volume in Fig. 8 a)-c) conversely aris-ing from a thin torch and a symmetric driver beam, andwithout the loss of charge. However, the witness beamemittances produced from the more realistic, asymmet-ric driver beam exceed the particularly low emittanceproduced for the perfectly symmetric drive beam sub-stantially. Similar emittance increase has been observedin [48] for asymmetric driver beams. This configurationand the analysis of resulting witness beam emittance indifferent planes is discussed further in Fig. 10.We also investigate the effect of the same asymmet-ric drive beam on injection from a wide torch with shortramps (see Appendix D). This variation results in a ho-mogeneous doughnut that is, however, stretched in z -direction, where the drive beam is wider, and compressedin the y -direction, both by approximately 20 %. Com-pared to the symmetric driver beam shown in Fig. 8 a)-c), the trapped charge reduces by 10 % ( Q M reduces by5 %) and the emittance does not change substantially.This indicates that steep downramp injection is, in someaspects, less sensitive to drive beam asymmetries thangentle downramp injection.Since the trapping process in the gentle downramp caseis sensitive to asymmetries in the wakefields, we conjec-ture that the asymmetry of the driver can be compen-sated to some extent by a suitable modulation of theplasma density. The plasma torch technique in princi-ple allows such modulations, e.g. by employing specifi-cally shaped, or multiple laser pulses. In a first explo-ration, the torch is modulated by a radial density func- tion n = n T × [1+0 . φ ( z, y )) ] with φ being the polarangle in the zy -plane, to partially counteract the effectof the asymmetric drive beam. Indeed, this can partiallycompensate the imbalance of trapped charge and pre-vent generation of beamlets: as shown in Fig. 8 g)-i),increasing the torch density in the narrow dimension ofthe asymmetric drive beam recovers a more symmetrictrapping doughnut (albeit elliptic, similar to those forasymmetric driver on a short ramp shown in AppendixD) and forms a single beam with more symmetric emit-tance. Such plasma torch based modulation may indicatea potential path to produce small slice emittances fromgentle downramp injection even in case of more realistic,namely asymmetrically shaped driver beams.Finally, the two cases presented in Fig. 9 recall theexperimental conditions at FACET discussed in sectionII. The scenario shown in Fig. 9 a)-c) involves a nar-row plasma torch that is transiently generated while theblowout passes the ionization front of the laser pulse.This simulation corresponds to a TOA ≈ − . z -direction. Its corresponding ionization front,indicated by the dashed line, just traversed the driverbeam axis. Consequently, the arising plasma torch fila-ment is only partially formed along the z -direction whenthe drive beam arrives at the interaction point. The laserpulse continues to ionize across the radial extent of theblowout during the subsequent interaction, partially in-side the blowout, which leads to injection of Q ≈ Q M ≈
486 pC). The resulting trapping doughnutis cropped in the y -direction and partially resembles thecase in Fig. 8 a)-c), since the torch radius is smaller thanthe blowout radius. This effect is overlayed by the sig-nificant asymmetry of the drive beam and the tempo-3 FIG. 9. Visualization of trapping doughnuts analogous to Fig. 6 and Fig. 8 for simulations modeling experiments at FACET.The first row corresponds to the timing scan shown in Fig. 3 with TOA = − . y = 30 µ m as in Fig. 4 is shown in the second row. ral dependence of the torch density spike in z -direction.Furthermore, direct ionization injection via the plasmaphotocathode mechanism [17] contributes charge to thewitness beam. The signature of this mixed-mode injec-tion manifests in the charge originating from the center ofthe trapping doughnut shown in Fig. 9 b), which does notoccur for fully formed torches (cf. Fig. 6 and Fig. 8). ForTOA values > − . z -direction. This likely results fromthe combination of the larger drive beam extent in z andthe cropped trapping doughnut in y -direction. Duringthe experiment at FACET, however, the experimentalsetup did not allow conclusive observation and evidenceof beamlets, likely due to limited resolution of the elec-tron spectrometer in the witness beam energy range, andscattering elements in the beamline. Further observa-tion, study and exploitation of the twin beamlet forma-tion thus requires suitable experimental conditions anddiagnostics.The second simulation shown in Fig. 9 d)-f) corre-sponds to the experimental situation of fully evolved,but transversely offset plasma torch configuration withrespect to the electron driver beam axis. Here, the torchis shifted in y -direction by ∆ y = 30 µ m as already pre-sented in Fig. 4 d), and yields injection of Q ≈ Q M ≈
74 pC). This specific kind of asymmetry fa-cilitates injection only in regions y > xy -plane, e.g. in the plane perpendicular to previ-ously discussed beamlets. The similarly large transverse momentum of the whole injected population manifestsin large-amplitude oscillations of the beam centroid wellsuited for betatron radiators.All presented simulations show modified trapping vol-umes when the spatiotemporal overlap volume betweenthe torch density profile n T ( x, y, z, t ) and the plasmawave deviates from ideal radial symmetry. These regionsof origin determine the initial phase space distribution ofthe formed witness beam and enable the production of awide range of electron beams with exceptional properties.Of those, planar injection and the controlled produc-tion of counter-oscillating beamlet twins represents a par-ticularly interesting capability. We investigate the sliceproperties of produced witness beams and their emit-tance evolution for two different pathways to generatesuch beamlets. Videos presenting the corresponding realspace and phase space evolution of both beams can befound in [51].Figure 10 a)-d) describe the beam produced from thenarrow torch with short ramp, fulfilling r flat + l ramp < r b already shown in Fig. 8 a)-c). The cropped trappingdoughnut is given as inset in Fig. 10 a), and is the ori-gin of injection of two counter-oscillating witness beampopulations as depicted by macroparticles in the 3D real-space snapshot. This beam is extracted ∼ ∼
100 MeV. Electron macroparticles colored in redoriginate from the upper half of the trapping doughnut,corresponding to z >
0, and blue ones from the lower halfwhere z <
0. The transverse projection in the xz -planehighlights the two clearly separated beamlets. The slicecurrent and energy spread profiles are given in Fig. 10b), showing a significantly reduced length of the witnessbeam population(s) compared to the witness beam ana-lyzed in Fig. 7.Figure 10 c) and d) depict the evolution of the pro-jected emittance along the plasma accelerator of the in-dividual beamlets as well as of the combined beam inboth transverse planes. In y -direction, both beamlets4populate a congruent transverse phase space area yy (cid:48) asshown in the inset of Fig. 10 c). The full beam’s emit-tance in this plane thus equals the emittance of individualbunchlets. In z -direction, on the other hand, each beam-let occupies a separate phase space area zz (cid:48) as shown bythe inset in Fig. 10 d). The emittance of both beamletscombined hence amounts to ∼√ z -direction. The projected emittanceboth of individual beamlets, as well as of the combinbedbeam, is larger in the zz (cid:48) -plane than in the yy (cid:48) -plane asconsequence of the thin torch orientation in z -direction.Figure 10 e)-h) present the analysis of beamlet pro-duction from a wide torch with gentle ramps l ramp > r b ,but asymmetric drive beam with σ z = 1 . × σ y , i.e.the configuration shown in Fig. 8 d)-f). Here, theinjection of the two distinct counter-oscillating electronpopulations in the z -direction is based on a re-arrangedrather than a cropped trapping doughnut, as discussed.Figure 10 e)-f) summarize the real space, slice currentand energy spread profile of the produced witnesspopulation(s). Here, the beam reaches ∼
100 MeV laterat approximately 1 . y -direction, the combined emittance ofbeamlets in the plane of oscillation ( zx ) is significantlylarger than the emittance of each individual beamlet.However, in contrast to the thin torch case which cropsthe trapping doughnut, the beamlet emittances in the z -direction are substantially lower than in the y -direction.The very distinct pair of beamlets is visualized by thehigh transverse phase space density shown in the inset inFig. 10 h), and manifests in projected beamlet emittancevalues (cid:15) n ,z < ∼ . y -scan in Fig. 4 we identified thepotential to eliminate one beamlet and excite planarbetatron motion of a single electron population. Asshown by the twin beamlets in Fig. 10, different methodscan be applied to achieve planar injection both on shortas well as long ramps. This flexibility in combinationwith high currents and low emittance represents excel-lent prospects for the generation of polarized betatronradiation [94, 98] with a high degree of tunability. SUMMARY AND CONCLUSIONS
Plasma density downramp injection has been proposedas a method for high-quality electron beam productionmany years ago. While highly successful in laser-driven plasma accelerators, it had eluded realization in electronbeam-driven plasma wakefield accelerators so far. We re-alized such long sought-after density downramp injection– by an advanced, all-optical, flexible generalization ofdownramp injection at the SLAC FACET facility. Thisfirst experimental proof-of-concept of the plasma torchapproach demonstrates an electron beam injector con-trolled by a mJ-class laser pulse. We show that the in-jected beams can be tuned by the laser pulse energy,relative temporal delay and spatial alignment betweenplasma torch laser and plasma wave. Furthermore, wedevelop strategies towards stabilization of witness beamgeneration even under unfavorable experimental condi-tions with regard to stability of incoming beams. Amongthese approaches, plasma torches wider than the blowoutradius prove to generate witness beams that are particu-larly resilient against shot-to-shot jitter. This result of-fers pathways towards stable and reliable plasma-basedinjectors.Based on the experimental observations, we have de-veloped a simple, yet powerful phenomenological modelthat predicts the trapped charge from torch injection.It agrees well with simulations for arbitrary spatiotem-poral torch density distributions and may be combinedwith analytical wakefield theories to design plasma torchinjectors.Additional simulations are used to explore the furtherpotential of the scheme and reveal the influence of thecharacteristic trapping volume on witness beam proper-ties. It can be manipulated by various torch density dis-tributions, including steep and soft downramps, as wellas by specific blowout configurations. In future studies,the trapping volume, representing the initial conditionsof the subsequently formed witness beam, may be ex-ploited for precise tailoring of plasma torches that yieldoptimized witness beam phase space distributions. Par-ticularly scenarios breaking the – typically radial – sym-metry of downramp injection provide unique pathwaysto shaping the resulting witness beams, including wit-ness beams performing large-amplitude betatron oscilla-tions and counter-oscillating beamlet twins. These maybe particularly interesting for generating polarized x-rayradiation.Plasma torch-based electron beam properties canreach charges up to hundreds of pC, tens of kA-levelcurrents, and emittances in the sub-0.1 mm mrad rangeconcomitant with low slice energy spreads. The corre-sponding slice brightness values exceed state-of-the-artof conventional accelerators by orders of magnitude.These features have promising implications for futureaccelerators and applications such as in photon scienceand high energy and high field physics. We antici-pate that plasma torch based injectors – due to theircapability combined with high degree of experimentalfeasibility – will be adapted and further exploitedby other linac-driven as well as hybrid laser-plasma5
FIG. 10. Twin beamlet structures obtained from PIC simulations. a)-d) display the beam obtained from the thin torchwith short ramp shown in Fig. 8 a)-c), and e)-h) present the beam generated on a gentle ramp traversed by an asymmetric( σ z = 1 . × σ y ) drive beam shown in Fig. 8 d)-f). a) and e), 3D real space snapshots with projections, showing beamlettwins, counter-oscillating in the xz -plane. b) and f), slice current and energy spreads. c), d) and g), h) show the evolution ofthe transverse witness beam emittance for the full beam and individual beamlets in both planes. The insets show transversephase space snapshots corresponding to a) and b). The dashed vertical line marks the labframe position where quantities wereextracted from the simulations, i.e. when the beam reaches approximately 100 MeV energy. Two videos present the overallsimulated evolution of real space and phase space for both beam configurations [51]. wakefield accelerators [99, 100]. ACKNOWLEDGEMENTS
The FACET E-210 plasma wakefield accelerationexperiment was built and operated with support fromUCLA (US Department of Energy (DOE) contract no.DESC0009914), RadiaBeam Technologies (DOE contractno. DE-SC0009533), and the FACET E200 team andDOE under contract no. DE-AC02-76SF00515. B.H.,P.S., A.S., F.A.H., T.H., A.B. were supported by theEuropean Research Council (ERC) under the EuropeanUnions Horizon 2020 research and innovation programme(NeXource, ERC Grant agreement No. 865877). Thework was supported by STFC ST/S006214/1 PWFA-FEL, EPSRC (grant no. EP/N028694/1). D.L.B.acknowledges support from the US DOE Office of High Energy Physics under award no. DE-SC0013855.J.R.C. acknowledges support from the National ScienceFoundation under award no. PHY 1734281. M.D.Lacknowledges support from the US DOE Office of HighEnergy Physics under award no. DE-SC0017906. Thiswork used computational resources of the NationalEnergy Research Scientific Computing Center, which issupported by DOE DE-AC02-05CH11231, and of theSupercomputing Laboratory at King Abdullah Univer-sity of Science & Technology (KAUST) in Thuwal, SaudiArabia.D.U. and P.S. contributed equally to this work.
APPENDIXAppendix A: Plasma torch density distributions
The plasma torch technique allows all-optical shaping of various plasma density distributions that superimpose theplasma facilitating the PWFA. In this work, the plasma torch profile is generated either directly in the VSim PIC code,or by mapping tunneling ionization rates [72–77] corresponding to the intensity profile of the laser externally beforeloading into VSim. Here, we concentrate on a subset of possible shapes, namely cylindrical and slab-like profiles.The cylindrical shape corresponds to the FACET E-210 experimental case and is generally valid for symmetricGaussian laser pulses with soft focusing. We consider configurations where the core cylinder of radius r flat along thelaser propagation axis z consists of fully ionized plasma. This core is surrounded by cosine-shaped ramps of length l ramp . The radially symmetric distribution with r = x + y reads n T n T , ( r, z ) = r < r flat cos (cid:16) π r − r flat l ramp (cid:17) : r flat ≤ r < r flat + l ramp r flat + l ramp < r. (2)The second density distribution considered in this work is a slab. A suitable laser pulse configuration which generatessuch a shape could for example be a combination of two crossed cylindrical lenses. The distribution reads:6 n Slab n Slab , ( x, y, z ) = | x | < r flat cos (cid:16) π | x |− r flat l ramp (cid:17) : r flat ≤ | x | < r flat + l ramp r flat + l ramp < | x | . (3) Appendix B: Plasma torch density scan
The peak torch density n T , has a fundamental impacton the injection process. For a fixed intensity distribu-tion of the torch laser, this changes the density gradientand the amount of plasma present in the correspondingtrapping doughnuts. When dealing with mixtures of lowionization threshold (LIT) and high ionization threshold(HIT) gases, the HIT component is independently tun-able from the LIT component and can be adjusted sim-ply by changing partial pressures in the gas mix reser-voir. In case of the experiments at FACET, for exam-ple, a mixture of hydrogen (LIT) and helium (HIT) wasused. However, at elevated LIT densities, the associatedwakefield amplitude increases, and electric field hot spotsat the wake vertex and/or the compressed drive beamcan exceed the tunneling ionization threshold of the HITmedium [83], which can produce dark current.To overcome this limitation and to operate at higherLIT densities, one can switch to using a HIT mediumwith even higher ionization threshold. Then, muchhigher electric fields can be tolerated without danger ofdark current production. For our simulation study insection III we thus choose the combination of fully ion-ized molecular hydrogen and the first helium level He + asamalgamated LIT medium, forming the plasma channeldensity n ch . The He + species is then the HIT mediumand the transition He + → He is exploited for gener-ation of density spikes. The tunability of LIT vs. HITdensities is then limited, but still accessible by varyingthe partial pressure ratio of hydrogen and helium in themixture. Switching off dark current thus comes at theprice of coupled LIT and HIT densities n ch = n H + n He and n T = n He . This composition allows for tuning thetorch density in the range n T = 0 to 1 × n ch while main-taining the LIT density. This can be achieved by adjust-ing the hydrogen component n H accordingly.To explore the effect of HIT density variation, we con-duct simulations with constant plasma density n ch =6 × cm − , and neutral helium densities n He varyingin a range from 0 .
25 to 1 × n ch . The plasma torch radius r flat = 2 . µ m and the ramp length l ramp = 40 µ m (cf.Fig. 5 b) are kept constant. Results of this HIT densityscan are presented in Fig. 11, where panels a)-c) showexemplary blowout structures after injection taken fromsimulations with n He = 0 . × n ch , 0 . × n ch and 1 . × n ch ,and d) shows injected charge as function of n He . Increas-ing n He corresponds to higher peak plasma torch den-sities and steeper density gradients, which consequentlyyields larger injected witness beam charges. This is re- FIG. 11. Scan of the neutral helium density n He at constant n ch = 6 × m − . a) to c) show PIC snapshots after injec-tion, where c) displays charge densities that lead to furtherwitness beam induced ionization and trapping of He . d)shows trapped witness charge Q as function of n He for witnessbeams including (black) and excluding ionized He (green). flected by increasing beam-loading and increasingly de-formed blowout structures. While the torch-based in-jected charge reaches a maximum when n He > . × n ch (Fig. 11 d), green), an additional injection mechanismsets in. Then, the fields of the torch-generated witnessbeam begin to exceed the threshold for tunnel ionizationof the remaining He + ions. This accumulates a secondwitness beam component [101] that is added to the ini-tial torch-injected beam. Figure 11 d) (black) shows thefull witness beam charge, which, after onset of witnessbeam self-ionization, linearly increases with n He . Thisrepresents another method for production of high-charge,multi-color beams in addition to [102, 103]. Appendix C: Multiple beams from multiple torches
The plasma torch scheme allows for generation of wit-ness beams of various energies. This can be accommo-dated by varying the plasma accelerator length as in con-ventional plasma acceleration, but also by moving theposition of the plasma torch within the plasma acceler-ator stage seamlessly. Furthermore, it is possible to em-7
FIG. 12. PIC study of multiple torch injectors spaced by ≈ r flat , = 40 µ m, l ramp , = 300 µ m followed by r flat , = 40 µ m, l ramp , = 10 µ m with reduced peak density n T , = 0 . × n ch .b), results for first torch with r flat , = 40 µ m, l ramp , = 80 µ mfollowed by the same configuration as in a). Red and bluecoloring reflect the origin of charge from the first and secondinjector stage, respectively. ploy multiple plasma torches to trigger multiple injectionprocesses consecutively. To demonstrate this, we con-duct simulations including a second density spike down-stream of the first plasma torch. Figure 12 presents twoPIC simulations employing this double-torch configura-tion, each with flat-top radius r flat = 40 µ m as shownin Fig. 5 a) for a single torch. In both simulations,these plasma torch injectors are separated by ∆ x ≈ n T , = 0 . × n ch and a long ramp l ramp , = 300 µ m, which injects 46 . n T , = 0 . × n ch and ashort ramp l ramp , = 10 µ m, and injects the second wit-ness beam with 64 . n T , < n T , , but its shorter ramp l ramp , < l ramp , , in this example produces a beam withhigher charge, but shorter length and thus higher currentthan the beam from the long ramp injector.The second case shown in Fig. 12 b) varies the ramplength of the first torch to l ramp , = 80 µ m, leavingother parameters unchanged. In this configuration, thetrapped beams contain charge of ≈ . Appendix D: Additional trapping doughnuts
Figure 13 presents further PIC simulations with similarsetup as in Figs. 6 and 8. The first simulation repeatsthe one shown in Fig. 6 d)-f) (40 µ m-wide flat-top radiusand 200 µ m-long ramps) with reduced longitudinal gridsize to 0 . µ m. The trapping volume displays a similar,irregular pattern.In the second row, the torch consists of a 2 . µ m-wideflat-top radius and 200 µ m-long ramps in radial direction.Since l ramp + r flat (cid:29) r b , the trapping volume resemblesthe closely related case shown in Fig. 6 d)-f), which em-ploys a 40 µ m-wide flat-top radius. Also, the trappedcharge is comparable.The following simulations highlight the effect of anasymmetric drive beam ( σ z = 1 . × σ y ) resulting in asym-metric blowout formation. In Fig. 13 g)-i), the torchconsists of a 40 µ m-wide flat-top radius and 10 µ m-longramps. Due to the asymmetric drive beam, the trappingdoughnut in the zy -plane gets squeezed along the widerplane of the drive beam (blowout), and it gets compressedin the perpendicular direction, both by a factor of ∼ . zy -plane.Figure 13 j)-l) and Fig. 13 m)-o) present the effect ofelectron drive beams asymmetric in z ( σ z = 1 . × σ y ) andin y -direction ( σ y = 1 . × σ z ), respectively, for a thintorch l ramp + r flat (cid:28) r b . Here, the trapping doughnutgets cropped like shown in Fig. 8 a)-c) due to the narrowtorch and, additionally, stretched (squeezed) along thewider (narrower) extent of the asymmetric drive beam.Furthermore, these two simulations emphasize that mul-tiple effects can simultaneously be applied to the trappingvolume. [1] P. Chen, J. M. Dawson, R. W. Huff, and T. Katsouleas,Phys. Rev. Lett. , 693 (1985). FIG. 13. Additional trapping doughnuts with similar PIC setting as in Figs. 6 and 8. a)-c): 40 µ m-wide flat-top radius and200 µ m-long ramp, reduced longitudinal grid size to 0 . µ m. d)-f): 2 . µ m-wide flat-top radius and 200 µ m-long ramp. g)-i):40 µ m-wide flat-top radius and 10 µ m-long ramp, asymmetric drive beam wider in z . j)-l): 2 . µ m-wide flat-top radius and 10 µ m-long ramp, asymmetric drive beam wider in z . m)-o): 2 . µ m-wide flat-top radius and 10 µ m-long ramp, asymmetric drivebeam wider in y .[2] J. B. Rosenzweig, B. Breizman, T. Katsouleas, and J. J.Su, Phys. Rev. A , R6189 (1991).[3] J. Rosenzweig, IEEE Trans. Plasma Sci. , 186 (1987).[4] I. Blumenfeld, C. E. Clayton, F.-J. Decker, M. J. Hogan,C. Huang, R. Ischebeck, R. Iverson, C. Joshi, T. Kat-souleas, N. Kirby, et al., Nature , 741 (2007).[5] E. Kallos, T. Katsouleas, W. D. Kimura, K. Kusche,P. Muggli, I. Pavlishin, I. Pogorelsky, D. Stolyarov, andV. Yakimenko, Phys. Rev. Lett. , 074802 (2008).[6] M. Litos, E. Adli, W. An, C. I. Clarke, C. E. Clayton,S. Corde, J. P. Delahaye, R. J. England, A. S. Fisher,J. Frederico, et al., Nature , 92 (2014).[7] T. Tajima and J. M. Dawson, Phys. Rev. Lett. , 267 (1979), URL https://link.aps.org/doi/10.1103/PhysRevLett.43.267 .[8] A. Pukhov and J. Meyer-ter Vehn, Appl. Phys. B ,355 (2002).[9] S. P. D. Mangles, C. D. Murphy, Z. Najmudin, A. G. R.Thomas, J. L. Collier, A. E. Dangor, E. J. Divall, P. S.Foster, J. G. Gallacher, C. J. Hooker, et al., Nature ,535 (2004).[10] J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gor-dienko, E. Lefebvre, J. P. Rousseau, F. Burgy, andV. Malka, Nature , 541 (2004).[11] C. G. R. Geddes, C. S. Toth, J. Van Tilborg, E. Esarey,C. B. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, andW. P. Leemans, Nature , 538 (2004).[12] J. Faure, C. Rechatin, A. Norlin, A. Lifschitz, Y. Glinec,and V. Malka, Nature , 737 (2006).[13] J. M. Dawson, Phys. Rev. , 383 (1959).[14] A. I. Akhiezer and R. V. Polovin, Sov. Phys. JETP ,696 (1956).[15] W. K. Panofsky, SLAC Beam Line , 36 (1997). [16] B. Hidding, G. Pretzler, J. B. Rosenzweig,T. K¨onigstein, D. Schiller, and D. L. Bruhwiler, Phys.Rev. Lett. , 035001 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.108.035001 .[17] A. Deng, O. S. Karger, T. Heinemann, A. Knetsch,P. Scherkl, G. G. Manahan, A. Beaton, D. Ullmann,G. Wittig, A. F. Habib, et al., Nature Physics , 1156(2019).[18] F. Li, J. F. Hua, X. L. Xu, C. J. Zhang, L. X. Yan, Y. C.Du, W. H. Huang, H. B. Chen, C. X. Tang, W. Lu, et al.,Phys. Rev. Lett. , 015003 (2013).[19] L. L. Yu, E. Esarey, C. B. Schroeder, J. L. Vay,C. Benedetti, C. G. R. Geddes, M. Chen, and W. P.Leemans, Phys. Rev. Lett. , 125001 (2014).[20] S. Corde, K. T. Phuoc, G. Lambert, R. Fitour,V. Malka, A. Rousse, A. Beck, and E. Lefebvre, Rev.Mod. Phys. (2013), ISSN 00346861, 1301.5066.[21] B. Hidding, G. G. Manahan, O. Karger, A. Knetsch,G. Wittig, D. A. Jaroszynski, Z. M. Sheng, Y. Xi,A. Deng, J. B. Rosenzweig, et al., J. Phys. B: At. Mol.Opt. Phys. , 234010 (2014), URL https://doi.org/10.1088%2F0953-4075%2F47%2F23%2F234010 .[22] F. Habib, P. Scherkl, G. G. Manahan, T. Heinemann,D. Ullmann, A. Sutherland, A. Knetsch, M. Litos,M. Hogan, J. Rosenzweig, et al., Proc. SPIE ,9 (2019), ISSN 1996756X.[23] E. Adli, J.-P. Delahaye, S. J. Gessner, M. J. Hogan,T. Raubenheimer, W. An, C. Joshi, and W. Mori, arXivpreprint arXiv:1308.1145 (2013).[24] E. Adli, W. An, N. Andreev, O. Apsimon, R. Ass-mann, J. Babiegeon, R. Bingham, T. Blackburn,C. Brady, M. Bussmann, et al. (ALEGRO Collabora-tion), Tech. Rep. arXiv:1901.10370 (2019), URL https: //cds.cern.ch/record/2661806 .[25] E. Esarey, R. F. Hubbard, W. P. Leemans,A. Ting, and P. Sprangle, Phys. Rev. Lett. ,2682 (1997), URL https://link.aps.org/doi/10.1103/PhysRevLett.79.2682 .[26] E. Oz, S. Deng, T. Katsouleas, P. Muggli, C. D. Barnes,I. Blumenfeld, F. J. Decker, P. Emma, M. J. Hogan,R. Ischebeck, et al., Phys. Rev. Lett. , 084801 (2007).[27] D. Umstadter, J. K. Kim, and E. Dodd, Phys. Rev. Lett. , 2073 (1996), URL http://link.aps.org/doi/10.1103/PhysRevLett.76.2073 .[28] M. Chen, Z. M. Sheng, Y. Y. Ma, and J. Zhang, J. Appl.Phys. (2006), ISSN 0021-8979.[29] N. Vafaei-Najafabadi, K. A. Marsh, C. E. Clayton,W. An, W. B. Mori, C. Joshi, W. Lu, E. Adli, S. Corde,M. Litos, et al., Phys. Rev. Lett. , 025001 (2014).[30] S. Bulanov, N. Naumova, F. Pegoraro, and J. Sakai,Phys. Rev. E , R5257 (1998).[31] H. Suk, N. Barov, J. B. Rosenzweig, and E. Esarey, in The Physics Of High Brightness Beams (World Scien-tific, 2000), pp. 404–417.[32] G. Fubiani, E. Esarey, C. B. Schroeder, and W. P. Lee-mans, Phys. Rev. E , 026402 (2006).[33] T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang,and S. Y. Chen, Phys. Rev. Lett. , 115003 (2005).[34] J. Faure, C. Rechatin, O. Lundh, L. Ammoura, andV. Malka, Phys. Plasmas , 083107 (2010).[35] P. Brijesh, C. Thaury, K. T. Phuoc, S. Corde, G. Lam-bert, V. Malka, S. P. D. Mangles, M. Bloom, andS. Kneip, Phys. Plasmas , 063104 (2012).[36] C. G. R. Geddes, K. Nakamura, G. R. Plateau, C. Toth,E. Cormier-Michel, E. Esarey, C. B. Schroeder, J. R.Cary, and W. P. Leemans, Phys. Rev. Lett. , 215004(2008).[37] A. J. Gonsalves, K. Nakamura, C. Lin, D. Panasenko,S. Shiraishi, T. Sokollik, C. Benedetti, C. B. Schroeder,C. G. R. Geddes, J. Van Tilborg, et al., Nat. Phys. ,862 (2011).[38] M. Hansson, B. Aurand, X. Davoine, H. Ekerfelt,K. Svensson, A. Persson, C. G. Wahlstr¨om, andO. Lundh, Phys. Rev. ST Accel. Beams , 071303(2015).[39] K. Schmid, A. Buck, C. M. S. Sears, J. M. Mikhailova,R. Tautz, D. Herrmann, M. Geissler, F. Krausz, andL. Veisz, Phys. Rev. ST Accel. Beams , 091301(2010).[40] A. Buck, J. Wenz, J. Xu, K. Khrennikov, K. Schmid,M. Heigoldt, J. M. Mikhailova, M. Geissler, B. Shen,F. Krausz, et al., Phys. Rev. Lett. , 185006 (2013).[41] S. K. Barber, J. van Tilborg, C. B. Schroeder, R. Lehe,H. E. Tsai, K. K. Swanson, S. Steinke, K. Nakamura,C. G. R. Geddes, C. Benedetti, et al., Phys. Rev. Lett. , 104801 (2017).[42] K. K. Swanson, H. E. Tsai, S. K. Barber, R. Lehe,H. S. Mao, S. Steinke, J. Van Tilborg, K. Nakamura,C. G. R. Geddes, C. B. Schroeder, et al., Phys. Rev.Accel. Beams , 051301 (2017).[43] M. Burza, A. Gonoskov, K. Svensson, F. Wojda,A. Persson, M. Hansson, G. Genoud, M. Marklund,C. G. Wahlstr¨om, and O. Lundh, Phys. Rev. ST Ac-cel. Beams , 011301 (2013).[44] R. J. England, J. B. Rosenzweig, and N. Barov, Phys.Rev. E , 016501 (2002), URL https://link.aps.org/doi/10.1103/PhysRevE.66.016501 . [45] M. C. Thompson, J. B. Rosenzweig, and H. Suk, Phys.Rev. ST Accel. Beams , 011301 (2004).[46] X. L. Xu, F. Li, W. An, T. N. Dalichaouch, P. Yu,W. Lu, C. Joshi, and W. B. Mori, Phys. Rev. Ac-cel. Beams , 111303 (2017), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.20.111303 .[47] C. Joshi, E. Adli, W. An, C. E. Clayton, S. Corde,S. Gessner, M. J. Hogan, M. Litos, W. Lu, K. A.Marsh, et al., Plasma Phys. Control. Fusion , 034001(2018), URL https://doi.org/10.1088%2F1361-6587%2Faaa2e3 .[48] C. Zhang, C. K. Huang, K. A. Marsh, X. L. Xu,F. Li, M. Hogan, V. Yakimenko, S. Corde, W. B.Mori, and C. Joshi, Phys. Rev. Accel. Beams ,111301 (2019), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.22.111301 .[49] J. Grebenyuk, A. M. de la Ossa, T. Mehrling, andJ. Osterhoff, Nucl. Instrum. Meth. A , 246 (2014),ISSN 0168-9002, URL .[50] A. M. de la Ossa, Z. Hu, M. J. V. Streeter, T. J.Mehrling, O. Kononenko, B. Sheeran, and J. Osterhoff,Phys. Rev. Accel. Beams , 091301 (2017).[51] See supplemental material at insert-link for a video rep-resentation of a plasma torch injector shown in fig. 1,and videos presenting the real space and phase space evo-lution of the two beamlet twins shown in fig. 10. [52] Q. Zhao, S. M. Weng, Z. M. Sheng, M. Chen, G. B.Zhang, W. B. Mori, B. Hidding, D. A. Jaroszynski, andJ. Zhang, New J. Phys. , 063031 (2018), URL https://doi.org/10.1088%2F1367-2630%2Faac926 .[53] A. Knetsch, O. Karger, G. Wittig, H. Groth,Y. Xi, A. Deng, J. B. Rosenzweig, D. L. Bruhwiler,J. Smith, D. A. Jaroszynski, et al., arXiv preprintarXiv:1412.4844 (2014).[54] R. Assmann and K. Yokoya, Nucl. Instrum.Meth. A , 544 (1998), ISSN 0168-9002, URL .[55] P. Antici, A. Bacci, C. Benedetti, E. Chiadroni, M. Fer-rario, A. R. Rossi, L. Lancia, M. Migliorati, A. Mostacci,L. Palumbo, et al., J. Appl. Phys. , 044902 (2012),URL https://doi.org/10.1063/1.4740456 .[56] T. Mehrling, J. Grebenyuk, F. S. Tsung,K. Floettmann, and J. Osterhoff, Phys. Rev. STAccel. Beams , 111303 (2012), URL https://link.aps.org/doi/10.1103/PhysRevSTAB.15.111303 .[57] M. Migliorati, A. Bacci, C. Benedetti, E. Chiadroni,M. Ferrario, A. Mostacci, L. Palumbo, A. R. Rossi,L. Serafini, and P. Antici, Phys. Rev. ST Accel. Beams , 011302 (2013), URL https://link.aps.org/doi/10.1103/PhysRevSTAB.16.011302 .[58] I. Dornmair, K. Floettmann, and A. R. Maier, Phys.Rev. ST Accel. Beams , 041302 (2015), URL https://link.aps.org/doi/10.1103/PhysRevSTAB.18.041302 .[59] X. L. Xu, J. F. Hua, Y. P. Wu, C. J. Zhang, F. Li,Y. Wan, C. H. Pai, W. Lu, W. An, P. Yu, et al., Phys.Rev. Lett. , 124801 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.116.124801 .[60] S. Kuschel, D. Hollatz, T. Heinemann, O. Karger,M. B. Schwab, D. Ullmann, A. Knetsch, A. Seidel,C. R¨odel, M. Yeung, et al., Phys. Rev. Accel. Beams , 071301 (2016), URL https://link.aps.org/doi/ .[61] J. van Tilborg, S. Steinke, C. G. R. Geddes, N. H.Matlis, B. H. Shaw, A. J. Gonsalves, J. V. Huijts,K. Nakamura, J. Daniels, C. B. Schroeder, et al., Phys.Rev. Lett. , 184802 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.115.184802 .[62] C. Thaury, E. Guillaume, A. D¨opp, R. Lehe, A. Lifs-chitz, K. T. Phuoc, J. Gautier, J. P. Goddet, A. Tafzi,A. Flacco, et al., Nat. Commun. , 1 (2015).[63] J. B. Rosenzweig and P. Chen, Phys. Rev. D ,2039 (1989), URL https://link.aps.org/doi/10.1103/PhysRevD.39.2039 .[64] C. E. Doss, E. Adli, R. Ariniello, J. Cary, S. Corde,B. Hidding, M. J. Hogan, K. Hunt-Stone, C. Joshi,K. A. Marsh, et al., Phys. Rev. Accel. Beams ,111001 (2019), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.22.111001 .[65] G. Wittig, O. Karger, A. Knetsch, Y. Xi, A. Deng, J. B.Rosenzweig, D. L. Bruhwiler, J. Smith, G. G. Manahan,Z. M. Sheng, et al., Phys. Rev. ST Accel. Beams ,081304 (2015).[66] G. Wittig, O. S. Karger, A. Knetsch, Y. Xi, A. Deng,J. B. Rosenzweig, D. L. Bruhwiler, J. Smith, Z. M.Sheng, D. A. Jaroszynski, et al., Nucl. Instrum. Meth.A , 83 (2016).[67] C. Nieter and J. R. Cary, J. Com-put. Phys. , 448 (2004), ISSN 0021-9991, URL .[68] N. Davidson, A. A. Friesem, and E. Hasman, Opt. Lett. , 523 (1991), URL http://ol.osa.org/abstract.cfm?URI=ol-16-7-523 .[69] S. Z. Green, E. Adli, C. I. Clarke, S. Corde, S. A. Ed-strom, A. S. Fisher, J. Frederico, J. C. Frisch, S. Gess-ner, S. Gilevich, et al., Plasma Phys. Control. Fusion , 084011 (2014).[70] S. Corde, E. Adli, J. M. Allen, W. An, C. I. Clarke,C. E. Clayton, J. P. Delahaye, J. Frederico, S. Gessner,S. Z. Green, et al., Nature , 442 EP (2015), URL https://doi.org/10.1038/nature14890 .[71] S. Gessner, E. Adli, J. M. Allen, W. An, C. I. Clarke,C. E. Clayton, S. Corde, J. P. Delahaye, J. Frederico,S. Z. Green, et al., Nat. Commun. , 11785 (2016).[72] A. I. Nikishov and V. I. Ritus, Sov. Phys. JETP , 145(1967).[73] A. M. Perelomov and V. S. Popov, Sov. Phys. JETP (1967).[74] A. M. Perelomov, V. S. Popov, and M. V. Terentev,Sov. Phys. JETP , 924 (1966).[75] A. I. Nikishov and V. I. Ritus, Sov. Phys. JETP , 168(1966).[76] M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov.Phys. JETP , 2008 (1986), ISSN 0044-4510.[77] D. L. Bruhwiler, D. A. Dimitrov, J. R. Cary, E. Esarey,W. Leemans, and R. E. Giacone, Phys. Plasmas ,2022 (2003), https://doi.org/10.1063/1.1566027, URL https://doi.org/10.1063/1.1566027 .[78] X. Yan, A. M. MacLeod, W. A. Gillespie, G. M. H.Knippels, D. Oepts, A. F. G. van der Meer, and W. Sei-del, Phys. Rev. Lett. , 3404 (2000).[79] P. Scherkl, A. Knetsch, T. Heinemann, A. Suther-land, A. F. Habib, O. Karger, D. Ullmann, A. Beaton,G. Kirwan, G. Manahan, et al., arXiv preprint arXiv:1908.09263 (2019).[80] S. Wilks, T. Katsouleas, J. M. Dawson, P. Chen, andJ. J. Su, IEEE Trans. Plasma Sci. , 210 (1987).[81] M. Tzoufras, W. Lu, F. S. Tsung, C. Huang,W. B. Mori, T. Katsouleas, J. Vieira, R. A. Fon-seca, and L. O. Silva, Phys. Rev. Lett. ,145002 (2008), URL https://link.aps.org/doi/10.1103/PhysRevLett.101.145002 .[82] G. G. Manahan, A. F. Habib, P. Scherkl, P. Delinikolas,A. Beaton, A. Knetsch, O. Karger, G. Wittig, T. Heine-mann, Z. M. Sheng, et al., Nat. Commun. , 1 (2017).[83] G. G. Manahan, A. Deng, O. Karger, Y. Xi, A. Knetsch,M. Litos, G. Wittig, T. Heinemann, J. Smith, Z. M.Sheng, et al., Phys. Rev. Accel. Beams , 011303(2016).[84] A. M. de la Ossa, J. Grebenyuk, T. Mehrling,L. Schaper, and J. Osterhoff, Phys. Rev. Lett. ,245003 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.111.245003 .[85] C. G. Durfee and H. M. Milchberg, Phys. Rev. Lett. ,2409 (1993).[86] R. J. Shalloo, C. Arran, L. Corner, J. Holloway, J. Jon-nerby, R. Walczak, H. M. Milchberg, and S. M. Hooker,Phys. Rev. E , 1 (2018), ISSN 24700053, 1801.00695.[87] W. Lu, C. Huang, M. Zhou, W. B. Mori,and T. Katsouleas, Phys. Rev. Lett. , 165002(2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.96.165002 .[88] A. A. Golovanov, I. Y. Kostyukov, A. M. Pukhov, andJ. Thomas, Quantum Electron. , 295 (2016), ISSN1063-7818.[89] V. Yakimenko, L. Alsberg, E. Bong, G. Bouchard,C. Clarke, C. Emma, S. Green, C. Hast, M. J.Hogan, J. Seabury, et al., Phys. Rev. Accel. Beams , 101301 (2019), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.22.101301 .[90] B. M. Cowan, D. L. Bruhwiler, J. R. Cary, E. Cormier-Michel, and C. G. R. Geddes, Phys. Rev. ST Accel.Beams , 1 (2013), ISSN 10984402.[91] A. V. Brantov, T. Z. Esirkepov, M. Kando, H. Kotaki,V. Y. Bychenkov, and S. V. Bulanov, Phys. Plasmas , 073111 (2008), https://doi.org/10.1063/1.2956989,URL https://doi.org/10.1063/1.2956989 .[92] H. Ekerfelt, M. Hansson, I. G. Gonz´alez, X. Davoine,and O. Lundh, Sci. Rep. , 1 (2017).[93] Method of trapping accelerating electrons in plasma (2003), uS patent No. US7049736B2.[94] A. D¨opp, B. Mahieu, A. Lifschitz, C. Thaury, A. Doche,E. Guillaume, G. Grittani, O. Lundh, M. Hansson,J. Gautier, et al., Light-Sci. & Appl. , e17086 (2017),ISSN 2047-7538, URL https://doi.org/10.1038/lsa.2017.86 .[95] T. Silva, A. Helm, J. Vieira, R. Fonseca, and L. O. Silva,Plasma Phys. Control. Fusion , 024001 (2020), ISSN0741-3335.[96] S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza,O. Lundh, F. Lindau, A. Persson, F. S. Tsung, Z. Naj-mudin, W. B. Mori, C. G. Wahlstr¨om, et al., Phys. Rev.Lett. , 215001 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.96.215001 .[97] J. P. Couperus, R. Pausch, A. K¨ohler, O. Zarini, J. M.Kr¨amer, M. Garten, A. Huebl, R. Gebhardt, U. Hel-big, S. Bock, et al., Nat. Commun. , 487 (2017),ISSN 2041-1723, arXiv:1011.1669v3, URL nature.com/articles/s41467-017-00592-7 .[98] K. T. Phuoc, S. Corde, R. Fitour, R. Shah, F. Albert,J.-P. Rousseau, F. Burgy, A. Rousse, V. Seredov, andA. Pukhov, Phys. Plasmas , 073106 (2008).[99] B. Hidding, T. Koenigstein, J. Osterholz, S. Karsch,O. Willi, and G. Pretzler, Phys. Rev. Lett. (2010),ISSN 0031-9007.[100] M. F. Gilljohann, H. Ding, A. Doepp, J. Goetzfried,S. Schindler, G. Schilling, S. Corde, A. Debus, T. Heine-mann, B. Hidding, et al., Phys. Rev. X (2019), ISSN2160-3308.[101] L. D. Amorim and N. Vafaei-Najafabadi, Plasma Phys. Control. Fusion , 105015 (2019), URL https://doi.org/10.1088%2F1361-6587%2Fab4005 .[102] N. Vafaei-Najafabadi, L. D. Amorim, E. Adli, W. An,C. I. Clarke, C. E. Clayton, S. Corde, S. Gessner, S. Z.Green, M. J. Hogan, et al., Philos. T. R. Soc. A ,20180184 (2019).[103] M. Zeng, M. Chen, L. L. Yu, W. B. Mori, Z. M. Sheng,B. Hidding, D. A. Jaroszynski, and J. Zhang, Phys.Rev. Lett. , 084801 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.114.084801https://link.aps.org/doi/10.1103/PhysRevLett.114.084801