Transition between Instability and Seeded Self-Modulation of a Relativistic Particle Bunch in Plasma
F. Batsch, P. Muggli, R. Agnello, C.C. Ahdida, M.C. Amoedo Goncalves, Y. Andrebe, O. Apsimon, R. Apsimon, A.-M. Bachmann, M.A. Baistrukov, P. Blanchard, F. Braunmüller, P.N. Burrows, B. Buttenschön, A. Caldwell, J. Chappell, E. Chevallay, M. Chung, D.A. Cooke, H. Damerau, C. Davut, G. Demeter, H.L. Deubner, S. Doebert, J. Farmer, A. Fasoli, V.N. Fedosseev, R. Fiorito, R.A. Fonseca, F. Friebel, I. Furno, L. Garolfi, S. Gessner, I. Gorgisyan, A.A. Gorn, E. Granados, M. Granetzny, T. Graubner, O. Grulke, E. Gschwendtner, V. Hafych, A. Helm, J.R. Henderson, M. Hüther, I.Yu. Kargapolov, 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, K. Moon, P.I. Morales Guzmán, M. Moreira, T. Nechaeva, E. Nowak, C. Pakuza, H. Panuganti, A. Pardons, A. Perera, J. Pucek, A. Pukhov, R.L. Ramjiawan, S. Rey, K. Rieger, O. Schmitz, E. Senes, L.O. Silva, R. Speroni, R.I. Spitsyn, C. Stollberg, A. Sublet, A. Topaloudis, N. Torrado, P.V. Tuev, M. Turner, F. Velotti, L. Verra, V.A. Verzilov, J. Vieira, H. Vincke, C.P. Welsch, M. Wendt, M. Wing, P. Wiwattananon, J. Wolfenden, B. Woolley, G. Xia, M. Zepp, G. Zevi Della Porta
TTransition between Instability and Seeded Self-Modulation of a Relativistic ParticleBunch in Plasma
F. Batsch, P. Muggli, R. Agnello, C.C. Ahdida, M.C. Amoedo Goncalves, Y. Andrebe, O. Apsimon,
4, 5
R. Apsimon,
4, 6
A.-M. Bachmann, M.A. Baistrukov,
7, 8
P. Blanchard, F. Braunm¨uller, P.N. Burrows, B. Buttensch¨on, A. Caldwell, J. Chappell, E. Chevallay, M. Chung, D.A. Cooke, H. Damerau, C. Davut,
4, 13
G. Demeter, H.L. Deubner, S. Doebert, J. Farmer,
1, 3
A. Fasoli, V.N. Fedosseev, R. Fiorito,
4, 5
R.A. Fonseca,
16, 17
F. Friebel, I. Furno, L. Garolfi, S. Gessner,
3, 19
I. Gorgisyan, A.A. Gorn,
7, 8
E. Granados, M. Granetzny, T. Graubner, O. Grulke,
10, 21
E. Gschwendtner, V. Hafych, A. Helm, J.R. Henderson,
4, 22
M. H¨uther, I.Yu. Kargapolov,
7, 8
S.-Y. Kim, F. Kraus, M. Krupa, T. Lefevre, L. Liang,
4, 13
S. Liu, N. Lopes, K.V. Lotov,
7, 8
M. Martyanov, S. Mazzoni, D. Medina Godoy, V.A. Minakov,
7, 8
J.T. Moody, K. Moon, P.I. Morales Guzm´an, M. Moreira,
3, 17
T. Nechaeva, E. Nowak, C. Pakuza, H. Panuganti, A. Pardons, A. Perera,
4, 5
J. Pucek, A. Pukhov, R.L. Ramjiawan,
3, 9
S. Rey, K. Rieger, O. Schmitz, E. Senes,
3, 9
L.O. Silva, R. Speroni, R.I. Spitsyn,
7, 8
C. Stollberg, A. Sublet, A. Topaloudis, N. Torrado, P.V. Tuev,
7, 8
M. Turner,
3, 24
F. Velotti, L. Verra,
1, 3, 25
V.A. Verzilov, J. Vieira, H. Vincke, C.P. Welsch,
4, 5
M. Wendt, M. Wing, P. Wiwattananon, J. Wolfenden,
4, 5
B. Woolley, G. Xia,
4, 13
M. Zepp, and G. Zevi Della Porta (The AWAKE Collaboration) Max Planck Institute for Physics, Munich, Germany Ecole Polytechnique Federale de Lausanne (EPFL),Swiss Plasma Center (SPC), Lausanne, Switzerland CERN, Geneva, Switzerland Cockcroft Institute, Daresbury, UK University of Liverpool, Liverpool, UK Lancaster University, Lancaster, UK Novosibirsk State University, Novosibirsk, Russia Budker Institute of Nuclear Physics SB RAS, Novosibirsk, Russia John Adams Institute, Oxford University, Oxford, UK Max Planck Institute for Plasma Physics, Greifswald, Germany UCL, London, UK UNIST, Ulsan, Republic of Korea University of Manchester, Manchester, UK Wigner Research Center for Physics, Budapest, Hungary Philipps-Universit¨at Marburg, Marburg, Germany 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, USA University of Wisconsin, Madison, Wisconsin, USA Technical University of Denmark, Lyngby, Denmark Accelerator Science and Technology Centre, ASTeC, STFC Daresbury Laboratory, Warrington, UK Heinrich-Heine-Universit¨at D¨usseldorf, D¨usseldorf, Germany Lawrence Berkeley National Laboratory, Berkeley, CA, USA Technical University Munich, Munich, Germany (Dated: December 18, 2020)We use a relativistic ionization front to provide various initial transverse wakefield amplitudes forthe self-modulation of a long proton bunch in plasma. We show experimentally that, with sufficientinitial amplitude ( ≥ (4 . ± .
4) MV/m), the phase of the modulation along the bunch is reproduciblefrom event to event, with 3 to 7% (of 2 π ) rms variations all along the bunch. The phase is notreproducible for lower initial amplitudes. We observe the transition between these two regimes.Phase reproducibility is essential for deterministic external injection of particles to be accelerated. PACS numbers: Valid PACS appear here
I. INTRODUCTION
Accelerators rely on precise control of parameters toproduce high-quality, high-energy particle bunches for numerous applications. A class of novel accelerators us-ing plasma as a medium to sustain large accelerating [1, 2]and focusing [3] fields has emerged and has made remark-able experimental progress over the last two decades [4– a r X i v : . [ phy s i c s . p l a s m - ph ] D ec < n e is on the order of thewave breaking field [7]: E W B = ( m e c/e ) ω pe . Here ω pe = (cid:0) n e e /ε m e (cid:1) / is the plasma electron angularfrequency [8]. Assuming the driver of rms duration σ t fitswithin the structure, i.e., σ t ∼ = 1 /ω pe , one can re-write: E W B = ( m e c/e ) (1 /σ t ). Therefore operating at high ac-celerating field ( > < σ r ≤ c/ω pe ≤ µ m) [9].The system extracts energy from the driver and trans-fers it to a witness bunch, through the plasma. The to-tal energy gain of the witness bunch is limited to theenergy carried by the driver. Short laser pulses andparticle bunches available today and suitable to drive > ∼
100 J ofenergy. Laser pulses and particle bunches carrying muchmore energy are too long, typically >
100 ps, to drivelarge amplitude wakefields when following the above E W B ∝ /σ t scaling. However, long laser pulses [10]and long, relativistic particle bunches [11] propagatingin dense plasma, i.e., σ t (cid:29) /ω pe , are subject to self-modulation (SM) instabilities. These instabilities cantransform them into a train of pulses/bunches shorterthan, and with a periodicity of 2 π/ω pe . The train canthen resonantly excite large amplitude wakefields. Con-trol of the SM process, in particular of the relative phaseof the wakefields, is necessary to deterministically injecta witness bunch shorter than 1 /ω pe into the accelerating and focusing phase of the wakefields.As the first proton-driven plasma wakefield acceler-ation (PWFA) experiment, AWAKE [12, 13] recentlydemonstrated that the SM process does indeed transforma long proton bunch ( σ t >
200 ps) into a train of micro-bunches with period 2 π/ω pe ( <
10 ps) [14]. We alsodemonstrated that the process grows along the bunch and along the plasma, from the initial wakefield amplitude, tosaturate at much larger values [15, 16]. Electrons were ex-ternally injected in the wakefields, though without phasecontrol (electron bunch duration on the order of 2 π/ω pe )and accelerated from ∼
19 MeV to ∼ Letter , we demonstrate experimentally for thefirst time that the SM of a long, relativistic particle bunchcan be seeded by a RIF. We define seeding as the condi-tions leading to a reproducible timing/phase of the SMalong the bunch with respect to the RIF. From time-resolved images of the bunch obtained at two plasma den-sities, we analyze the relative timing/phase of the micro-bunches along the proton bunch, after the plasma. Wecontrol the initial wakefield amplitude through the tim-ing of the RIF along the bunch. When the process is notseeded, we observe randomly distributed phases and thusthe SM instability (SMI) [11]. With sufficiently stronginitial amplitude, the phase of the wakefields varies byonly a small fraction of 2 π from event to event, the char-acteristic of seeded SM (SSM) [13]. This is despite natu-ral variations of the incoming bunch parameters [21]. Wethus observe the transition from SMI to SSM. We alsoobserve phase reproducibility over more than 2 σ t alongthe bunch. Phase reproducibility is essential for futureexperiments [13] with deterministic, external injection ofparticles to be accelerated (e − or e + ) at a precise phasewithin the accelerating and focusing region of the wake-fields [18].Experimental results presented here show that thephase of the self-modulation instability, a fundamentalbeam/plasma interaction mechanism [11], can be con-trolled. It is also a requirement for future accelerationexperiments. II. EXPERIMENTAL SETUP
The CERN Super Proton Synchrotron (SPS) pro-vides a Gaussian bunch with 400 GeV energy per pro-ton, 3 × particles and a rms duration σ t = 250 ps.The bunch enters a 10 m-long vapor source [22, 23],as shown in Fig. 1, with rms waist size σ r =150 µ m.The source contains rubidium (Rb) vapor with density n Rb adjustable in the (0.5 - 10) × cm − range andwith uniform temperature and thus density distributions LASER p + dump SPSp + Laser Reference Line
OTR screen
Delay line
Streak camera
Mirror
Rb vapor, 10m, (0.5-10)x10 cm -3 Laser dump p + VaporVapor p + Plasma
Laser pulse Laser pulse
Plasma t (ps) x ( mm ) FIG. 1. Schematic of the experimental setup showing themain components used for measurements presented here. In-set 1: RIF in the middle of the proton bunch ( t RIF = 0 ps).Inset 2: Streak camera image of a modulated proton bunch,laser reference signal at t = 0 ps (red circle). ( ∆ n Rb n Rb = ∆ TT < ≤
450 mJlaser pulse that can serve two purposes. First, whenpropagating along the vapor column it creates the plasmaat the RIF. The RIF transforms the Rb vapor into a ∼ n e = n Rb ). Second, whenpropagating within the proton bunch, the RIF triggersthe sudden ( (cid:28) /ω pe ) onset of beam plasma interactionthat can seed the SM process. Seeding can occur becausethis onset corresponds to the driving of initial plasmawakefields starting at the RIF and with amplitudes de-pending on the local bunch density [14, 15].The train of micro-bunches resulting from the SM pro-cess leaves the plasma after 10 m and passes through analuminum-coated screen where protons emit optical tran-sition radiation (OTR), 3.5 m from the plasma exit. TheOTR has the same spatio-temporal structure as the mod-ulated proton bunch. A streak camera resolves the in-coming OTR light imaged onto its entrance slit in spaceand in time with resolutions of 80 µ m and ∼ III. RESULTS
We observe that when we use the RIF for plasma cre-ation only, placing it nano- to micro-seconds ahead ofthe proton bunch, SM occurs [27]. In this case SM cangrow from noise present in the system. The wakefieldamplitude driven by shot noise in the proton bunch dis-tribution was estimated at the tens of kV/m level [28].The laser pulse drives wakefields at the <
100 kV/m levelat the plasma densities of these experiments [29]. Fig-ure 2 (a) shows a composite image of the time structureof the center part of the modulated proton bunch (com-pare Fig. 1 Inset 2) for ten events in the 73 ps window,placed 150 ps (0.6 σ t ) ahead of the bunch peak. These -100102030 t (ps) E v e n t nu m b e r t (ps) E v e n t nu m b e r (a) (b) FIG. 2. Composite images of the center part of the streakcamera image (see Fig. 1 Inset2) for ten events with (a) RIF600 ps (2.4 σ t ) and (b) RIF 350 ps (1.4 σ t ) ahead of the pro-ton bunch center. Front of the bunch on the right hand side.Events aligned with respect to LRS ([26], at t = 0, not visi-ble). Both cases: LRS 150 ps (0.6 σ t ) ahead of bunch center, n e = 0 . × cm − . events are aligned in time with respect to the LRS. TheLRS alignment procedure yields a ∼
50 ps-long commonwindow between images. The LRS (not shown) is placedat t =0 ps on each image. The RIF is 600 ps (2.4 σ t ) aheadof the bunch peak (i.e., 450 ps, 1.8 σ t between RIF and t =0 on the image). Each image is normalized to its in-coming bunch population. The figure clearly shows thatfrom event to event micro-bunches appear at no partic-ular times with respect to the RIF. It also shows thatthe measured micro-bunch charge density varies consid-erably. Variations in bunch density on these images canbe attributed to amplitude variations of focusing and de-focusing fields [25]. Variations in timing/phase and am-plitude of the modulation are expected for the occurrenceof a (non-seeded) instability such as SMI [11].Figure 2 (b) shows a similar plot to that of Fig. 2 (a),but with the RIF placed closer, 350 ps (1.4 σ t ) ahead ofthe bunch peak and thus with larger wakefield amplitudeat the RIF, with all other parameters unchanged. It isclear that in this case the micro-bunches appear essen-tially at the same time with respect to the RIF and withmuch more consistent charge density than in the previ-ous case. This data shows the behavior expected from aseeded process such as SSM. From these two plots we con-clude that in the first case the phase of the modulationis not reproducible from event to event (SMI), whereasit is in the second case (SSM).In order to quantify the observed effect, we de-termine the phase/timing (using the modulation fre-quency/period) of the bunch modulation with respect tothe RIF. For this purpose we sum counts of the bunchimage in a ∼ = ± µ m-wide region around the axis ofthe bunch at the OTR screen to obtain a time profileof the bunch SM. At this location the incoming bunchtransverse rms size is ∼ = 574 µ m (see Fig. 4 (a), t< Supplemental Material . For the data set analyzedhere ( n e = 0 . × cm − ), the modulation frequencyis 87.1 GHz.Figure 3 shows the variation in relative phase for sixseries (including the events of Fig. 2) of approximately18 events each, measured with the analysis window (andLRS) 150 ps ahead of the bunch peak, as a function of theRIF timing t RIF along the bunch normalized to the rmsbunch duration. The phase distributions for t RIF ≥ σ t t RIF / t / ( % ) W r , R I F ( M V / m ) FIG. 3. Measured rms (blue circles) and full phase variation(blue diamond), and initial linear transverse wakefield am-plitude (filled red circles) as a function of t RIF normalizedto σ t . The error bars indicate the statistical uncertainty of10.1% (see text). Error bars representing the uncertainty in t RIF due to the 15 ps (0.06 σ t ) rms proton timing jitter arenot plotted. Same LRS timing and n e as in Fig. 2. cover a range (blue diamonds) close to 2 π and their rms(blue circles) approaches the value expected for a uni-form distribution, 29%. This corresponds to a phase ran-domly distributed from event to event, possibly varyingover more than 2 π . On the contrary, for t RIF ≤ σ t ,the ranges are (cid:28) π and their rms is small, ∼ π (Fig. 2 (b)), when theinitial wakefield amplitude increases. We show later, bydelaying the observation window timing for a fixed t RIF ,that when reached in one window, the timing/phase re-producibility occurs all along the bunch, as expected.In the SMI regime, time-resolved images (not presentedhere) of the SM near the seed point show that full SMstarts at different times along the bunch, unlike in theseeded cases, where it starts at the RIF [14]. This ex-plains the ∼ π (modulo) phase variations observed withSMI. In the SSM regime, the observed phase rms vari-ations of ∼
6% (of 2 π ) results from at least three maincontributions. First, the intrinsic phase variations thatare the goal of the measurement. Second, variations ofinitial parameters from event to event originating fromthe bunch or the plasma. We measure rms variations inbunch length, ≈ ≈ < Supplemental Material ).The initial transverse wakefield amplitude (at theplasma entrance) can be calculated as a function of theRIF timings of Fig. 3: W r,RIF ( t = t RIF ). The initialproton bunch density ( n b = 1 . × cm − ) is smallerthan the plasma density ( n e = 0 . × cm − ). Wethus use two-dimensional linear plasma wakefield the-ory [30] to evaluate this amplitude. The modulationperiod ( ∼ =11.5 ps) is much shorter than the rms bunchduration ( σ t =250 ps). We therefore consider the bunchdensity constant over one period behind the RIF and thus W r,RIF = 2 (cid:0) m e c /e (cid:1) ( n b ( t RIF ) /n e ) dRdr | r = σ r . The ra-dial dependence of wakefields through the R ( r ) coeffi-cient [30] is a function of the transverse bunch profile,considered as Gaussian.We plot the amplitude of W r,RIF for each data pointin Fig. 3 (filled red circles). The input parameter varia-tions mentioned above cause a maximum statistical un-certainty of 10.1% on the field calculation, which includesa 15 ps (0.06 σ t ) rms timing jitter between the protonbunch and the laser pulses (RIF and LRS), all addedin quadrature. This uncertainty is indicated by the errorbars. The plot shows that for the parameters of these ex-periments, the transition between SMI and SSM occursbetween ∼ (2 . ± .
3) and ∼ (4 . ± .
4) MV/m. The factthat initial wakefield amplitudes of (2 . ± .
3) MV/m donot seed the SM process may indicate that the bunch hasdensity irregularities driving initial wakefields with am-plitude (between (2 . ± .
3) and (4 . ± .
4) MV/m) muchlarger than those of the shot noise assumed in [28] driving <
100 kV/m fields. We also note here that we interpretthe reproducibility of the bunch modulation as also thatof the wakefields driven towards the end of the plasma,after saturation of the SM process [16]. The wakefieldstructure is intrinsically linked to the distribution of theself-modulated proton bunch.The phase reproducibility can be further confirmed bysimilar phase variation measurements at various delaysbehind the RIF. Sets of approximately ten images withdelay increments of 50 ps between each set were acquiredat a higher plasma density n e = 1 . × cm − anda fixed RIF timing of 125 ps (0 . σ t ). For these parame-ters, W r,RIF = 17 . W r,RIF is approximatelyfour times larger, and the density only two times smaller, x ( mm )
500 400 300 200 100 0 t (ps) ( % / ) (a)(b) FIG. 4. (a) Time-resolved, “stitched” image of the self-modulated proton bunch with t RIF = 125 ps (0.5 σ t ), n e =1 . × cm − . The RIF is at t =0 on the image (not visible). The LRS is visible every 50 ps at the bottom of the im-age. (b) Modulation rms phase variation for each set of images with equal LRS timing. than the values for which SSM is observed in Fig. 3, SSMis also expected in this case. Due to the time overlap be-tween sets, all images can be “stitched” together usingthe LRS as described in [26] (see Fig. 4 (a)). It is im-mediately clear from the figure that micro-bunches of allevents align themselves in time/phase and form a coher-ent modulation of the bunch density over ∼ σ t behindthe RIF. This is only possible when proper seeding isprovided (SSM) for each event, relative phase variationsbetween events are small (i.e., all sequences look simi-lar to that of Fig. 2 (b)), and the modulation phase isreproducible all along the bunch. All features visible inFig. 4 (a) would wash out if phases were randomly dis-tributed as in Fig. 2 (a).Figure 4 (b) shows the result of the phase analysis ap-plied to these events. Over the ∼ σ t measurement range,larger than the delay from the RIF of ∼ σ t typically fore-seen for external electron injection, the phase variationsremain small and in a similar range to those obtainedat lower plasma density. Variations along the bunch aremost likely due to changes in signal that can be seen inFig. 4 (a) and on individual images, which affects theaccuracy of the phase determination. The measured vari-ations remain approximately constant and between 3 to7% (of 2 π ) all along the bunch. IV. SUMMARY
We presented the results of experimental studies ofthe SM phase for different timings of the RIF with re-spect to the proton bunch, measured after the 10 m-longplasma. These results demonstrate that the SM processcan be seeded, i.e., the phase of the modulation can bedefined by the RIF and reproducible from event to event.We observe the transition from phase non-reproducibility and instability (SMI) to seeding and phase reproducibil-ity (SSM) when the transverse wakefield at the RIF ex-ceeds a threshold amplitude, between (2 . ± .
3) and(4 . ± .
4) MV/m for n e =0.94 × cm − . We show thatin the SSM regime variations of the modulation phasealong the bunch ( ∼ σ t ) are small, measured at ≤ ∼ ps timeresolution (Fig. 4 (a)).Based on these results, one can thus expect that forthe studies of electron acceleration during AWAKE RunII [29], the wakefields driven by the bunch train in the sec-ond plasma, will have a timing/phase also reproduciblefrom event to event since they will be driven by the bunchemerging from the first plasma. Phase reproducibility isrequired for deterministic acceleration of electrons exter-nally injected into the wakefields, with a fixed delay withrespect to the seed. ACKNOWLEDGEMENTS
This work was supported by the Wolfgang GentnerProgramme of the German Federal Ministry of Educa-tion and Research (grant no. 05E15CHA); in parts bya Leverhulme Trust Research Project Grant RPG-2017-143 and by STFC (AWAKE-UK, Cockcroft Institutecore and UCL consolidated grants), United Kingdom;a Deutsche Forschungsgemeinschaft project grant PU213-6/1 “Three-dimensional quasi-static simulations ofbeam self-modulation for plasma wakefield acceleration”;the National Research Foundation of Korea (Nos. NRF-2016R1A5A1013277 and NRF-2020R1A2C1010835); thePortuguese FCT—Foundation for Science and Tech-nology, through grants CERN/FIS-TEC/0032/2017,PTDC-FIS-PLA-2940-2014, UID/FIS/50010/2013 andSFRH/IF/01635/2015; NSERC and CNRC for TRI-UMF’s contribution; the U.S. National Science Founda-tion under grant PHY-1903316; and the Research Coun-cil of Norway. M. Wing acknowledges the support of DESY, Hamburg. Support of the Wigner DatacenterCloud facility through the “Awakelaser” project is ac-knowledged. The work of V. Hafych has been sup-ported by the European Union’s Framework Programmefor Research and Innovation Horizon 2020 (2014–2020)under the Marie Sklodowska-Curie Grant Agreement No.765710. The AWAKE collaboration acknowledge the SPSteam for their excellent proton delivery. [1] T. Tajima, J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979)[2] P. Chen et al. , Phys. Rev. Lett. 54, 693 (1985)[3] P. Chen et al. , IEEE Trans. Plasma Sci. 15(2), 218 (1987),P. Chen et al. , Phys. Rev. D 40, 923 (1989)[4] I. Blumenfeld et al. , Nature 445, 741-744 (2007)[5] M. Litos et al. , Nature 515, 92 (2014)[6] A. J. Gonsalves et al. , Phys. Rev. Lett. 122, 084801(2019)[7] J. M. Dawson, Phys. Rev. 113, 383 (1959)[8] Constants have usual meaning: e electron charge, ε vac-uum permittivity, m e electron mass, c speed of light invacuum[9] J. J. Su et al. , IEEE Transactions on Plasma Science 15,192–198 (1987)[10] C. J. McKinstrie, Physics of Fluids B: Plasma Physics 4,2626 (1992)[11] N. Kumar et al. , Phys. Rev. Lett. 104, 255003 (2010)[12] E. Gschwendtner et al. , Nucl. Instr. and Meth. in Phys.Res. A 829, 76 (2016)[13] P. Muggli et al. (AWAKE Collaboration), Plasma Physicsand Controlled Fusion, 60(1) 014046 (2018)[14] E. Adli et al. (AWAKE Collaboration), Phys. Rev. Lett.122, 054802 (2019)[15] M. Turner et al. (AWAKE Collaboration), Phys. Rev.Lett. 122, 054801 (2019)[16] M. Turner, P. Muggli et al. , (AWAKE Collaboration),Phys. Rev. Accel. Beams 23, 081302 (2020)[17] E. Adli et al. (AWAKE Collaboration), Nature 561, 363(2018), E. Gschwendtner, M. Turner et al. (AWAKE Col-laboration), Phil. Trans. R. Soc. A 377, 20180418 (2019)[18] V. K. Berglyd Olsen, E. Adli, P. Muggli, Phys. Rev. Ac-cel. Beams 21, 011301 (2018)[19] S. P. Le Blanc et al. , Phys. Rev. Lett. 77, 5381 (1996)[20] Y. Fang et al. , Phys. Rev. Lett. 112, 045001 (2014)[21] M. Moreira, J. Vieira, P. Muggli, Phys. Rev. Accel.Beams 22, 031301 (2019)[22] G. Plyushchev et al. , J. Phys. D: Appl. Phys. 51, 025203(2017)[23] E. ¨Oz et al. , Nucl. Instr. and Meth. in Phys. Res. A 740,197 (2014)[24] F. Batsch et al. , Nucl. Instr. and Meth. in Phys. Res. A909, 359 (2018)[25] A.-M. Bachmann, P. Muggli, J. Phys.: Conf. Ser. 1596,012005 (2020)[26] F. Batsch, J. Phys.: Conf. Ser. 1596, 012006 (2020)[27] S. Gessner et al. (AWAKE Collaboration), submitted(2020), arXiv:2006.09991[28] K. Lotov et al. , Phys. Rev. ST Accel. Beams 16, 041301(2013)[29] P. Muggli et al. , J. Phys.: Conf. Ser. 1596, 012008 (2020)[30] R. Keinigs, M. E. Jones, Phys. Fluids 30, 252 (1987) V. SUPPLEMENTAL MATERIAL
We detail here the analysis that yields the tim-ing/phase variations plotted in Figs 3 and 4. Figure 5 (a)shows an example of a single time-resolved streak cam-era image of the self-modulated proton bunch in a 73 pswindow and of the LRS placed 150 ps (0.6 σ t ) ahead ofthe bunch center. The laser pulse that creates the RIFis 600 ps (2.4 σ t ) ahead of the bunch center. We obtainthe time profile of the bunch density by summing countsof the images in a ∼ = ± µ m-wide region around thebunch axis at the OTR screen (see Fig. 1). At this loca-tion, the incoming bunch transverse rms size is ∼ =574 µ m.The time profile as shown in Fig. 5 (b) (blue line) consistsof 512 amplitude values for the 73 ps-long window.For the modulation frequency and phase determina-tion, we use a discrete Fourier transform (DFT) analysisof the signal. The size of the frequency bin of the DFTis given by ∆ f DF T =1/73 ps ∼ =13.7 GHz. We decrease thesize of the bin to increase the resolution of the frequencydetermination by padding the time profile with an ar-ray of 50 ×
512 zero amplitudes. This procedure brings∆ f DF T to 13.7 GHz/50 ∼ =0.27 GHz, which is on the orderof the Rb vapor and thus plasma density measurementaccuracy [24]. Zero padding is equivalent to including theeffect of the 73 ps-long square function time window ofthe streak camera image on a longer signal. We note herethat in signal processing, a Gauss- or Hann-like windowfunction can be used to decrease the convolution effectof the very broad spectrum of the square window sinc function on the signal power spectrum. However, with -100102030405060-505 x ( mm ) -100102030405060 t (ps) C oun t s ( a r b . un it s ) (a)(b) FIG. 5. (a) Single 73 ps streak camera image of a self-modulated proton bunch recorded at density n e = 1 . × cm − . (b) Blue line: Bunch modulation profile summinga ∼ = ± µ m-wide region around the bunch axis of (a). Blackline: Summed counts over a − ≤ x ≤ − the only six to nine micro-bunches (or DFT periods) in a73 ps window of this experiment, such window functionsdo not improve the quality or precision of the analysis.In addition, we only determine the frequency and phaseusing the DFT signal and not its width or amplitude.Thus we do not use these window functions. Further wenote that we do not include the data set with t RIF = 0(Fig. 4 (a)) in the analysis because images include thenon-modulated part of the bunch not in plasma ( t< sinc DFT function (13.7 GHz in this case) around theplasma frequency expected from the Rb density measuredin the vapor source [14]: n e =0.94 and 1.81 × cm − or ∼ =87.1 and ∼ =120.8 GHz, respectively. The small num-ber of micro-bunches visible on the streak camera im-ages and the significant amount of noise lead to varia-tions in the frequency determination of up to 1.8 GHz(rms) from event to event. These variations ( > < ∝ n / e , < π
500 ps behind the RIF. This is clearlymuch larger than what is observed here. For the phaseanalysis, we therefore select for all events the phase valuefrom the frequency bin corresponding to the average DFTfrequency of all events.The variations in the frequency determination do notinfluence significantly the phase analysis results, as longas these variations are small when compared to the DFTbin width. Figure 6 shows the result for the phase vari-ation as a function of the DFT frequency for the dataset of Fig. 4 at t =50 ps. The figure shows that for allfrequencies within ± ≤ t ≤ +1 psand x ≤ -3 mm in Fig. 5 (a)) as described in Ref. [26] byfitting a Gaussian function to its time projection. Theexpected response of the streak camera to signals shorterthan its intrinsic time resolution, 120 fs for the LRS, is ∼
116 118 120 122 124 126 f (GHz) / ( % ) FIG. 6. Observed phase variation ∆Φ / π versus frequencychosen for the phase analysis of a data set recorded at1.81 × cm − , corresponding to a modulation frequency of120.8 GHz. sult of the Gaussian fit that represents the LRS. Mea-surements with RIF and LRS laser pulses on the same 73 ps window show that their time difference is measuredwith 0.16 ps precision. The mechanical delay line for theLRS has a position accuracy that corresponds to 0.53 ps(rms). This accuracy only influences the time at whichphase variation results are plotted in Fig. 4 (b), not theirvalue. However, this limited accuracy affects how wellthe image data sets acquired with different LRS timing“stitch” together on 4 (a).From the timing of the LRS, we determine the time dif-ference and phase between the LRS and the next max-imum (micro-bunch) of the periodic function using theaverage DFT frequency. For the case of the event ofFig. 5, the time difference we find is 1.1 ps, correspond-ing to a phase difference of 0.8 rad at the frequency of120.8 GHz.We repeat this procedure for all events in each serieswith the same LRS delay with respect to the RIF. Wecharacterize the results for each delay set by the rmsof the phase distribution as well as by the full range ofphases in the distribution, both in % of 2 ππ