Proton beam defocusing in AWAKE: comparison of simulations and measurements
A.A. Gorn, M. Turner, E. Adli, R. Agnello, M. Aladi, Y. Andrebe, O. Apsimon, R. Apsimon, A.-M. Bachmann, M.A. Baistrukov, F. Batsch, M. Bergamaschi, P. Blanchard, P.N. Burrows, B. Buttenschon, A. Caldwell, J. Chappell, E. Chevallay, M. Chung, D.A. Cooke, H. Damerau, C. Davut, G. Demeter, L.H. Deubner, A. Dexter, G.P. Djotyan, S. Doebert, J. Farmer, A. Fasoli, V.N. Fedosseev, R. Fiorito, R.A. Fonseca, F. Friebel, I. Furno, L. Garolfi, S. Gessner, B. Goddard, I. Gorgisyan, E. Granados, M. Granetzny, O. Grulke, E. Gschwendtner, V. Hafych, A. Hartin, A. Helm, J.R. Henderson, A. Howling, M. Huther, R. Jacquier, I.Yu. Kargapolov, M.A. Kedves, F. Keeble, M.D. Kelisani, S.-Y. Kim, F. Kraus, M. Krupa, T. Lefevre, L. Liang, S. Liu, N. Lopes, K.V. Lotov, M. Martyanov, S. Mazzoni, D. Medina Godoy, V.A. Minakov, J.T. Moody, P.I. Morales Guzman, M. Moreira, T. Nechaeva, H. Panuganti, A. Pardons, F. Pena Asmus, A. Perera, A. Petrenko, J. Pucek, A. Pukhov, B. Raczkevi, R.L. Ramjiawan, S. Rey, H. Ruhl, H. Saberi, O. Schmitz, E. Senes, P. Sherwood, L.O. Silva, R.I. Spitsyn, P.V. Tuev, F. Velotti, L. Verra, V.A. Verzilov, J. Vieira, C.P. Welsch, B. Williamson, M. Wing, J. Wolfenden, B. Woolley, G. Xia, M. Zepp, G. Zevi Della Porta
PProton beam defocusing in AWAKE: comparison ofsimulations and measurements
A.A. Gorn , , M. Turner , E. Adli , R. Agnello , M. Aladi ,Y. Andrebe , O. Apsimon , , R. Apsimon , ,A.-M. Bachmann , , , M.A. Baistrukov , , F. Batsch , , ,M. Bergamaschi , P. Blanchard , P.N. Burrows ,B. Buttensch¨on , A. Caldwell , J. Chappell ,E. Chevallay , M. Chung , D.A. Cooke , H. Damerau ,C. Davut , , G. Demeter , L.H. Deubner , A. Dexter , ,G.P. Djotyan , S. Doebert , J. Farmer , , A. Fasoli ,V.N. Fedosseev , R. Fiorito , , R.A. Fonseca , ,F. Friebel , I. Furno , L. Garolfi , S. Gessner , ,B. Goddard , I. Gorgisyan , E. Granados , M. Granetzny ,O. Grulke , , E. Gschwendtner , V. Hafych , A. Hartin ,A. Helm , J.R. Henderson , , A. Howling , M. H¨uther ,R. Jacquier , I.Yu. Kargapolov , , M. ´A. Kedves ,F. Keeble , M.D. Kelisani , S.-Y. Kim , F. Kraus ,M. Krupa , T. Lefevre , L. Liang , , S. Liu , N. Lopes ,K.V. Lotov , , M. Martyanov , S. Mazzoni ,D. Medina Godoy , V.A. Minakov , , J.T. Moody ,P.I. Morales Guzm´an , M. Moreira , , T. Nechaeva ,H. Panuganti , A. Pardons , F. Pe˜na Asmus , ,A. Perera , , A. Petrenko , J. Pucek , A. Pukhov ,B. R´aczkevi , R.L. Ramjiawan , , S. Rey , H. Ruhl ,H. Saberi , O. Schmitz , E. Senes , , P. Sherwood ,L.O. Silva , R.I. Spitsyn , , P.V. Tuev , , F. Velotti ,L. Verra , , , V.A. Verzilov , J. Vieira , C.P. Welsch , ,B. Williamson , , M. Wing , J. Wolfenden , ,B. Woolley , G. Xia , , M. Zepp , G. Zevi Della Porta (The AWAKE Collaboration) Budker Institute of Nuclear Physics SB RAS, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia CERN, Geneva, Switzerland University of Oslo, Oslo, Norway Ecole Polytechnique Federale de Lausanne (EPFL), Swiss Plasma Center(SPC), Lausanne, Switzerland Wigner Research Center for Physics, Budapest, Hungary Cockcroft Institute, Daresbury, UK Lancaster University, Lancaster, UK Max Planck Institute for Physics, Munich, Germany Technical University Munich, Munich, Germany John Adams Institute, Oxford University, Oxford, UK Max Planck Institute for Plasma Physics, Greifswald, Germany UCL, London, UK UNIST, Ulsan, Republic of Korea a r X i v : . [ phy s i c s . acc - ph ] A ug omparison of AWAKE simulations and measurements University of Manchester, Manchester, UK Philipps-Universit¨at Marburg, Marburg, Germany University of Liverpool, Liverpool, UK ISCTE - Instituto Universit´eario de Lisboa, Portugal GoLP/Instituto de Plasmas e Fus˜ao Nuclear, Instituto Superior T´ecnico,Universidade de Lisboa, Lisbon, Portugal TRIUMF, Vancouver, Canada SLAC National Accelerator Laboratory, Menlo Park, CA University of Wisconsin, Madison, Wisconsin, USA Technical University of Denmark, Lyngby, Denmark Accelerator Science and Technology Centre, ASTeC, STFC DaresburyLaboratory, Warrington, UK Belarusian State University, 220030 Minsk, Belarus Heinrich-Heine-Universit¨at D¨usseldorf, D¨usseldorf, Germany Ludwig-Maximilians-Universit¨at, Munich, Germany31 August 2020
Abstract.
In 2017, AWAKE demonstrated the seeded self-modulation (SSM)of a 400 GeV proton beam from the Super Proton Synchrotron (SPS) at CERN.The angular distribution of the protons deflected due to SSM is a quantitativemeasure of the process, which agrees with simulations by the two-dimensional(axisymmetric) particle-in-cell code LCODE. Agreement is achieved for beampopulations between 10 and 3 × particles, various plasma density gradients( − ÷ × cm − and 7 × cm − ). Theagreement is reached only in the case of a wide enough simulation box (at leastfive plasma wavelengths). omparison of AWAKE simulations and measurements
1. Introduction
Acceleration of particles in plasmas, or plasmawakefield acceleration, offers the possibility to reachmulti-GeV and, potentially, TeV range electronand positron energies in facilities that are ordersof magnitude smaller than modern high-energyaccelerators [1–10]. Studies of novel acceleratorssignificantly benefit from numerical simulations thatgo in parallel with them [11]. These simulationscomplement fragmentary experimental data [12] andform a complete picture of physical processes thatoccur at tiny spatial and temporal scales inside theplasma and, therefore, cannot be measured.Recently, the Advanced WAKefield Experiment(AWAKE) at CERN [13–15] demonstrated seeded self-modulation of a long proton beam in the plasma[16, 17] and electron acceleration by the wakefield ofthis beam [18]. This milestone achievement opens theway for using proton beams from modern synchrotronsas drivers for plasma wakefield acceleration [10, 19, 20].Because of the high proton energy, the micro-bunchingand acceleration occurs in a single plasma cell, avoidingdifficulties related to staged acceleration [21–24].In this paper, we show that simulations ofbeam self-modulation agree with related AWAKEmeasurements. The axisymmetric self-modulationprocess is thus well understood in the sense thatsimulations include the most important effects, theycan help in understanding details of the process andcan serve as a valuable starting point for futurepredictions. We also discuss how to conduct thesesimulations properly, and show what the results wouldlook like otherwise. The AWAKE data used for thestudies reported in this paper were taken during the2017 and early 2018 running periods.
2. Methods
AWAKE uses the proton beam from the SPS [25].The proton beam co-propagates with a short laserpulse through a 10-meter-long cylindrical cell filledwith rubidium vapor (Fig. 1). The laser pulse quicklyionizes the vapor, so that the newly appearing plasmainteracts only with the rear part of the beam, asif the beam would be cut at the pulse position.The leading edge of this “sharply cut” beam seeds asmall amplitude wakefield in the plasma that growsin space and time and eventually converts the beaminto a train of short micro-bunches [26, 27]. The self-modulation occurs due to defocusing of protons fromthe space between the micro-bunches. The radialmomentum gained by the defocused protons is anintegral quantitative measure of the self-modulationprocess: the stronger the longitudinal wakefield micro-bunch formation, the larger the transverse momentum of the defocused protons [17, 28, 29].The defocused protons form a halo around themicro-bunched part of the beam. Its time-integratedflux is measured at two imaging stations located 1.66 mand 9.75 m downstream the plasma cell exit [30, 31](Fig. 1). Each imaging station separately measuresthe flux distribution in the halo and in the centralpart (core) of the beam with two cameras of differentsensitivity (Fig. 2). Separate registration of the coreand halo, followed by stitching two parts of theimage, provides a wide dynamic range of measurements(approximately 10 ) and absolute flux calibrationthrough equating the integral flux and the knownbeam charge (Fig. 3). The halo images processed witha contour method [32] yield the maximum radius ofdefocused protons, which is a convenient scalar to becompared with simulations.The simulations are performed with the qua-sistatic two-dimensional (axisymmetric) particle-in-cellcode LCODE [33,34]. The simulation window enclosesthe whole beam and moves with the speed of light c in the positive z direction. The radial window size isalways 30 c/ω p , where ω p = (cid:112) πn e /m is the plasmafrequency, n is the plasma density, e is the elemen-tary charge, and m is the electron mass. The grid cellhas equal sizes 0 . c/ω p in both longitudinal and ra-dial directions. There are 10 radius-weighted plasmamacro-particles of each species (electrons and ions) ineach cell and about 10 or more (proportionally to ω p )equal beam macro-particles in total. The time stepfor the beam is 100 ω − p , and the plasma state is up-dated every 200 c/ω p . The plasma has sharp bound-aries in both longitudinal and radial directions, as sim-ulations [35–37] have suggested. After the plasma cell,the protons propagate to the screens along straightlines. Table 1.
Beam and plasma parameters taken as input forsimulations. Parameters in the first group are the same in allruns. Parameters in the second group (below the line) may varyin different regimes, and the listed values are those for illustrativecases (Figs. 3, 5, 6).Parameter, notation ValuePlasma length, L R p M i
157 000Plasma initial temperature, T e W b
400 GeVBeam energy spread, δW b n . × cm − Beam population, N b . × Beam length, σ z σ r µ mNormalized beam emittance, (cid:15) b ξ s omparison of AWAKE simulations and measurements
10 m Rb PlasmaLaser beam LaserdumpDipole Imaging sta(cid:2)on 1 Imaging sta(cid:2)on 2OTR, CTR screens StreakcameraProton beam Light pathIonising laser pulseProton microbunches Plasma
Figure 1.
Setup of AWAKE experiments on beam self-modulation.
Figure 2.
Images of beam core (a) and halo (b) at thesecond (downstream) imaging station. The red solid line in (b)shows the detected halo boundary used for determination of themaximum deflection radius. Note that the intense light emittedby the core of the beam is blocked by a mask.
Most of the input parameters for simulations(Table 1) are known with sufficient precision andtheir uncertainties have little effect on quantitativecharacteristics of beam self-modulation. Less wellknown are the beam length, radius and emittanceand these are determined from the data used inthis analysis. The proton beam was modelled withGaussian distributions in both the longitudinal andtransverse directions. The standard deviation ofthe proton beam in the longitudinal direction, σ z ,was determined prior to the start of data takingusing a streak camera [39] (Fig. 4) and correlated toparameters measured with beam quality monitors inthe SPS [38]. Measurements performed in the SPSduring data taking were then used to calculate thebunch length on a shot-by-shot basis.The initial beam radius σ r and emittance (cid:15) b werenot measured with the required precision during thediscussed parameter scans, so their input values arefitted to the experimental data in four steps. First, − −
10 0 10 x (mm)10 − − − F l u x d e n s i t y ( a r b . un i t s ) meanoptimum (cid:15) b = 2 µ m σ r = 120 µ mhalocore Figure 3.
Angle-averaged radial distributions of the beam fluxin several individual shots (grey lines), additionally averagedover 30 shots (thin dark gray line), and simulated: with optimumparameters from Table 1 (solid red line), with a reduced angularspread and the same beam radius (dotted line) and with areduced beam radius and the same angular spread (dashedline). The vertical black lines mark the maximum proton radiuscalculated with the contour method [32]. we determine the radial beam flux distributions byseparate averaging of left and right halves of the images(Fig. 3). This allows evaluation of the symmetry ofthe distribution and verifies that the axisymmetricmodel is applicable. The systematic asymmetry thatwe see in the halo is relatively low and can be causedby transverse misalignments between the proton beamand the plasma column. Second, we average themeasured distributions over 30 shots. Third, insimulations, we vary the beam angular spread δα ≈ (cid:15) b / ( γσ r ), where γ is the relativistic factor of thebeam. It determines the width of the beam head that omparison of AWAKE simulations and measurements
50 200 400 600 800 1000time (ps)0 . . . c o un t s ( a . u . ) σ z = 222.68 ps (b) projectiongaussian fit − x ( mm ) (a) Figure 4.
Streak camera image of the proton beam (a) andGaussian approximation of the beam current (b). propagates in the neutral gas and therefore the shapeof the core part of beam flux distribution (comparered solid and magenta dotted lines in Fig. 3). Fourth,we adjust the beam radius to match the height andwidth of the flux profile “shoulders” (compare redsolid and blue dashed lines in Fig. 3). The heightand width of the shoulders depend on the strength ofthe fields produced in the plasma and are thereforesensitive to the initial beam density, which scales as σ − r . Even with the best parameter fit (red line),the simulated beam profile is sharply peaked at theaxis and therefore locally differs from the measuredprofile. This happens because both radial and angulardistributions of beam micro-bunches in the plasma arestrongly peaked in the perfectly axisymmetric case [40],while in the experiment this narrow spike is blurred.The obtained values of σ r and (cid:15) b depend weakly onbeam population. We approximate them as (cid:15) b [ µ m] = 2 . . × − N b , σ r [ µ m] = 105 (cid:112) (cid:15) b [ µ m] , (1)so the beam parameters vary from (cid:15) b = 2 .
75 mm mradand σ r = 175 µ m at N b = 10 to the values listed inTable 1 at higher N b .The simulated maximum radius of deflectedprotons r max depends on the number of beam macro-particles used. The larger the number, the betterwe resolve the “wings” of the Gaussian angulardistribution, the larger the maximum initial transversemomentum beam macro-particles can have, and thelarger the maximum deflection we observe. To avoidthis ambiguity, we consider r max as the radius at whichthe simulated beam flux density equals the noise levelin experiments.The wide simulation window (30 c/ω p ) is necessary 0 . . . r , c m (a) −
10 0 ln( n e /n − −
10 0 z − ct , cm − Φ , k V (b) narrow wide Figure 5. (a) Density of plasma electrons (color map) andtrajectories of individual electrons (lines) after 4 meters of beampropagation in the plasma. The vertical black line shows thecross-section detailed in Fig. 6. (b) Wakefield potential Φ on theaxis after the same propagation distance calculated with wide(30 c/ω p ≈ . . ≈ . c/ω p ) simulationwindows. r , mm0 . . . . h n e i / n , h n i i / n h E r i , M V / m h E r i × h n e ih n i ih n e i R p e p Figure 6.
Radial dependencies of plasma-period-averaged iondensity (cid:104) n i (cid:105) , electron density (cid:104) n e (cid:105) , and radial electric field (cid:104) E r (cid:105) for correct calculation of high-energy electron trajec-tories. These electrons appear as a result of wave-breaking, which often occurs in the simulated regimesafter beam micro-bunching [Fig. 5(a)]. The electronsare ejected from the plasma column radially and leavean unbalanced positive charge behind [41, 42]. Thecharge separation electric field keeps them from travel- omparison of AWAKE simulations and measurements . . . . . . N b , 10 . . . . r m a x , c m (a) baseline R w = 1 . . . . . . . N b , 10 r m a x , c m (b) baseline σ r × . δα × . Figure 7.
Beam radius on the screens for low (a) and high (b)density plasmas and varying beam population. Top and bottomgroups of lines in each panel are for two imaging stations. Blackdots with error bars (“expt”) are experimentally measured values(individual shots), lines are simulation results. The error barswere determined the same way as in [32] and show the standarddeviation of the determined radii along different directions ofthe image. Solid blue lines are for optimum parameter sets. Thegreen dashed line in (a) shows the effect of a narrow simulationwindow ( R w = 1 . σ r reduced by 20% at the same δα , greendashed line) or lower-emittance beam ( δα reduced by 20% atthe same σ r , orange dotted line). Dotted red lines are scalings ∝ N / b in (a) and ∝ N / b in (b). ing far, so these electrons form a negatively chargedhalo around the positively charged plasma column(Fig. 6). The halo electrons move predominantly in thepositive z direction [41] and efficiently interact with thewakefield when crossing the near-axis area: they cantake energy from some regions or release it in others.As a consequence, the wakefield damps when a sub-stantial number of halo electrons returns to the near-axis area. This is seen when comparing trajectoriesof halo electrons [Fig. 5(a)] and the wakefield poten-tial on the axis [Fig. 5(b)]. The wakefield potential Φcharacterizes the force exerted on axially moving ultra-relativistic particles: E z = − ∂ Φ ∂z , E r − B ϕ = − ∂ Φ ∂r . (2)A reduced amplitude of potential oscillations on theaxis evidences reduction of both accelerating anddefocusing properties of the wakefield.
3. Comparison of simulations and experiments
We compare simulations and experiments in threeparameter scans. The first two are beam populationscans at low (1 . × cm − ) and high (7 . × cm − ) plasma densities (Fig. 7). The bestagreement variants (blue lines) correspond to theapproximation (1) and other parameters from Table 1,with the exception that the high density scan was with σ z = 7 cm and ξ s = 1 . r max al-most imperceptibly (orange dotted line). Variation ofother beam parameters also have small effects on pro-ton defocusing, except the seed position discussed inRef. [43].Theory [44] suggests that the maximum wakefieldat AWAKE may be limited by nonlinear elongationof the plasma wave period. The wave goes outof resonance with the bunch train, and electricfield growth saturates at about 0 . mcω p /e . If thismechanism is indeed the main limiting factor, thenthe maximum wakefield amplitude should scale as thecontribution of an individual bunch to the power of1/3, or as N / b (see Eq. (12) of [44]). The dependenceof maximum proton deflection on beam population atthe low plasma density follows this scaling remarkablywell [dotted line in Fig. 7(a)]. At the high density,however, the maximum radius scales as N / b [dottedline in Fig. 7(b)], which indicates that the microbunchtrain is not dense or long enough to excite the nonlinearwakefields at this plasma density. In this case, theradial momentum gained by the deflected protons p max is defined by the depth of the wakefield potentialwell e Φ max that scales as N b . As a result, p max ∝ omparison of AWAKE simulations and measurements √ e Φ max W b /c ∝ √ N b . − −
10 0 10 20density increase, %0 . . . r m a x , c m µ m220 µ m 260 µ mexpt Figure 8.
Beam radius on the screens for differentplasma density gradients (characterized by the relative densityincrease at the downstream plasma end): black dots are theexperimentally measured values; error bars show the standarddeviation of all individual measurements (about 30 shots) at agiven gradient setting, lines are simulation results for differentbeam radii.
For the density gradient scan, the simulationsalso follow the measured trend (Fig. 8). Here theplasma density changes linearly along the cell due to atemperature difference between the two Rb reservoirsattached to the cell at the ends [35]. The density atthe upstream end is always 1 . × cm − . Beamparameters are those from Table 1, except σ z = 7 . ξ s = 2 . σ r ≈ µ m.0 2 4 6 8 10 z , m050100150 Φ m a x , k V +5%0%-19%+20% Figure 9.
Maximum amplitude of the wakefield potentialΦ max versus beam propagation distance z for σ r = 220 µ m anddifferent magnitudes of relative density increase along the plasmacell. Small positive gradients contribute to beamself-modulation by increasing the total charge ofmicrobunches [45]. The effect is similar to that ofthe density step [27, 46]: larger fractions of initiallyformed micro-bunches remain focused by the wavebecause of plasma wavelength shortening. With smallpositive gradients (density increase about 2% over10 meters), the longitudinal wakefield is stronger [45], and externally injected electrons gain a higherenergy [18]. Higher positive gradients, however,are destructive and reduce the maximum wakefield(Fig. 9), which results in weaker deflection of defocusedprotons (Fig. 8). High negative gradients also reducethe wakefield, but not the maximum deflection radius.This happens because the strongest defocused protonsdo not always experience the strongest wakefieldgenerated by the beam. Negative density gradientschange the wavelength so that the micro-bunchesmostly fall into the defocusing wave phase, and moreefficient coupling with the transverse field compensatesthe lower wakefield amplitude.
4. Summary
To conclude, simulations with two-dimensional (ax-isymmetric) particle-in-cell code LCODE agree withthe measurements of the maximum radius of deflectedprotons in three different experimental scans: pro-ton beam population scans (1 ÷ × particles)for high and low plasma densities, and gradient scan( − ÷ ∼ c/ω p forAWAKE parameters) that are necessary to correctlyaccount for ejected plasma electrons that charge theplasma column positively and then return to the axisdestroying the wakefield. This additionally increasesthe computational power required for simulations oflong microbunch trains in a plasma. Acknowledgments
This work was supported in parts by the Founda-tion for the Development of Theoretical Physicsand Mathematics “BASIS”; a Leverhulme TrustResearch Project Grant RPG-2017-143 and by STFC(AWAKE-UK, Cockcroft Institute core, John AdamsInstitute core, and UCL consolidated grants), UnitedKingdom; a Deutsche Forschungsgemeinschaft projectgrant PU 213-6/1 “Three-dimensional quasi-staticsimulations of beam self-modulation for plasmawakefield acceleration”; the National Research Foun-dation of Korea (Nos. NRF-2016R1A5A1013277and NRF-2019R1F1A1062377); the PortugueseFCT—Foundation for Science and Technology,through grants CERN/FIS-TEC/0032/2017, PTDC-FIS-PLA-2940-2014, UID/FIS/50010/2013 andSFRH/IF/01635/2015; NSERC and CNRC forTRIUMF’s contribution; the U.S. National ScienceFoundation under grant PHY-1903316; the Wolfgang omparison of AWAKE simulations and measurements
References [1] A. J. Gonsalves, et al., Phys. Rev. Letters , 084801(2019).[2] I. Blumenfeld, et al., Nature , 741 (2007).[3] S. Corde, et al., Nature , 442 (2015).[4] M. Litos, et al., Plasma Phys. Control. Fusion , 034017(2016).[5] E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev. Mod.Phys. , 1229 (2009).[6] K. Nakajima, Reviews of Accelerator Science and Technol-ogy , 19 (2016).[7] M.J. Hogan, Reviews of Accelerator Science and Technology , 63 (2016).[8] A.Caldwell, K.Lotov, A.Pukhov, and F.Simon, NaturePhys. , 363 (2009).[9] A. Caldwell, K. V. Lotov, Phys. Plasmas , 103101 (2011).[10] E. Adli and P. Muggli, Reviews of Accelerator Science andTechnology , 85 (2016).[11] J.-L. Vay and R. Lehe, Reviews of Accelerator Science andTechnology , 165 (2016).[12] M.C. Downer, R. Zgadzaj, A. Debus, U. Schramm, andM.C. Kaluza, Rev. Mod. Phys. , 035002 (2018).[13] A. Caldwell, et al. (AWAKE Collaboration), Nuclear Instr.Methods A , 3 (2016).[14] E. Gschwendtner, et al. (AWAKE Collaboration), NuclearInstr. Methods A , 76 (2016).[15] P.Muggli, et al. (AWAKE Collaboration), Plasma Phys.Control. Fusion , 014046 (2018).[16] E. Adli, et al. (AWAKE Collaboration), Phys. Rev. Lett. , 054802 (2019).[17] M. Turner, et al. (AWAKE Collaboration), Phys. Rev. Lett. , 054801 (2019).[18] E. Adli, et al. (AWAKE Collaboration), Nature , 363(2018).[19] R. Assmann, et al. (AWAKE Collaboration), Plasma Phys.Control. Fusion , 084013 (2014).[20] A. Caldwell and M. Wing, Eur. Phys. J. C , 463 (2016).[21] S. Cheshkov, T. Tajima, W. Horton, and K. Yokoya, Phys.Rev. ST Accel. Beams , 071301 (2000).[22] T. Mehrling, J. Grebenyuk, F.S. Tsung, K. Floettmann,and J. Osterhoff, Phys. Rev. ST Accel. Beams , 111303(2012).[23] X.L. Xu, et al., Phys. Rev. Lett. , 124801 (2016).[24] C.A. Lindstrom, E. Adli, J. Pfingstner, E. Marin, D.Schulte, Proceedings of IPAC2016 (Busan, Korea), pp.2561-2564. [25] J.S. Schmidt, et al. (AWAKE Collaboration), Proceedingsof IPAC2017 (Copenhagen, Denmark), pp.1747-1750.[26] N. Kumar, A. Pukhov, and K. Lotov, Phys. Rev. Lett. ,255003 (2010).[27] K.V. Lotov, Phys. Plasmas , 103110 (2015).[28] M. Turner, A. Petrenko, B. Biskup, S. Burger, E.Gschwendtner, K.V. Lotov, S. Mazzoni, H. Vincke,Nuclear Instr. Methods A , 314 (2016).[29] M. Turner, A. Petrenko, E. Gschwendtner, K. Lotov, A.Sosedkin, Proceedings of NAPAC2016 (Chicago, IL,USA), p.707-709.[30] M. Turner, B. Biskup, S. Burger, E. Gschwendtner, S.Mazzoni, A. Petrenko, Nuclear Instr. Meth. A , 100(2017).[31] M. Turner, V. Clerc, I. Gorgisyan, E. Gschwendtner,S. Mazzoni, A. Petrenko, Proceedings of IPAC2017(Copenhagen, Denmark), pp.1682-1684.[32] M. Turner, E. Gschwendtner, P. Muggli, Nuclear Inst.Meth. A , 123 (2018).[33] K.V. Lotov, Phys. Rev. ST - Accel. Beams , 061301 (2003).[34] A.P. Sosedkin, K.V. Lotov, Nuclear Instr. Methods A ,350 (2016).[35] G. Plyushchev, R. Kersevan, A. Petrenko, and P. Muggli,J. Phys. D: Appl. Phys. , 025203 (2018).[36] E. Oz, P. Muggli, Nucl. Instr. Meth. A , 197 (2014).[37] G. Demeter, Phys. Rev. A , 063423 (2019).[38] G. Papotti, Beam quality and availability from theinjectors. Proc. 2nd 2010 Evian workshop on LHC beamoperation (Geneva, Switzerland, 2010), pp. 49–54.[39] K. Rieger, A. Caldwell, O. Reimann, R. Tarkeshian, andP. Muggli, Review of Scientific Instruments , 025110(2017).[40] K.V. Lotov, Phys. Plasmas , 023119 (2017).[41] K.V. Lotov, A.P. Sosedkin, A.V. Petrenko, Phys. Rev. Lett. , 194801 (2014).[42] A.A. Gorn, P.V. Tuev, A.V. Petrenko, A.P. Sosedkin, andK.V. Lotov, Phys. Plasmas , 063108 (2018).[43] M. Turner, et al. (AWAKE Collaboration), Phys. Rev.Accel. Beams , 081302 (2020).[44] K.V. Lotov, Phys. Plasmas , 083119 (2013).[45] A. Petrenko, K. Lotov, A. Sosedkin, Nuclear Instr. MethodsA , 63 (2016).[46] K.V. Lotov, Phys. Plasmas18