Quasi-stable injection channels in a wakefield accelerator

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by  Mara Wiltshire-Turkay

Quasi-stable injection channels in a wakefield accelerator
Mara Wiltshire-Turkay,1 John P. Farmer,1 and Alexander Pukhov1
Heinrich-Heine-Universit¨at, 40225 Du¨sseldorf, Germany
The influence of initial position on the acceleration of externally-injected electrons in a plasma wakefield is
investigated. Test-particlesimulationsshowpreviouslyunobservedcomplexstructureintheparameterspace,
withquasi-stableinjectionchannelsformingforparticlesinjectedinnarrowregionsawayfromthewakecentre.
Particles injected into these channels remain in the wake for a considerable time after dephasing, and as a
6 result achieve significantly higher energy than their neighbours. The result is relevant to both the planning
1 and optimisation of experiments making use of external injection.
0
2
I. INTRODUCTION electron dephasing with the wake. The stability of the
y
a effectandtheapplicabilitytoexperimentsisdiscussedin
M High-energy electrons have a wide range of appli- Section IV, and conclusions are drawn in Section V.
cations, ranging from fundamental research in particle

6 physics to medical diagnostics and treatment. However,
1 conventionalacceleration methods require long accelera- II. ACCELERATION DEPENDENCE ON INJECTION
tion lengths, which also acts to limit the maximum at- POSITION
]
h tainable energy to some tens of GeV.
p Onepromisingalternativetechniqueiswakefieldaccel- In order to gauge the influence of injection position
- eration, first proposedby Tajima and Dawson1. A short on energy gain, a parameter scan was carried out using
m
driver, either a laser pulse or charged particle beam, is a series of test particle simulations. A slice in the x–y
s
used to excite a plasma wave. The resulting charge im- plane was sampled, intersecting with the centre of the
a
l balance can lead to large electric fields – much larger wake. The results are shown in Fig. 1.
p
thancanbesupportedbyconventionalmedia. Awitness The electromagnetic wakefields were generated using
.
s bunch within the wake may then be accelerated. thefullythree-dimensionalquasistaticversionofthePIC
c The majority of works to date have considered inter- code VLPL10. A section of the associated potential is
i
s nal injection, due to the relative ease with which high- shown in Fig. 1a. A short, pre-modulated proton beam
y
energy electrons can be achieved. The first experiments was used to give a wake with parameters relevant to the
h
p made use of self-injection, in which the plasma wave is AWAKE experiment, with a driver energy of 400 GeV,
[ driven beyond the wavebreaking limit, allowing plasma andaplasmadensityof7×1014 cm−3 (correspondingto
electrons to be trapped within the accelerating phase of λp = 2πc/ωp = 1.26 mm). For simplicity, injection into
3
the wake2. Several schemes have since been proposed a low-amplitudewake,for whichthe plasmaresponsere-
v
to allow better control of the injection process3, with a mains linear, is first considered. A plasma modulation
5
2 view to improve the energy-spread and emittance of the depth of 8.5% is therefore used, giving a maximum ac-
0 acceleratedelectronbeam. Theuseofanexternalsource celerating field of 220 MV/m. Comparison to a larger
6 of electrons may also offer accelerated beams of higher accelerating field, as predicted for AWAKE, as well as a
0 quality4. discussionofthe useofalaserorelectrondriver,ismade
1. The AWAKE project5 will make use of a proton in Section IV.
0 driver, potentially allowing electron acceleration to TeV Thewakeisassumedtoremainstaticintheco-moving
6 energies6,7,ordersofmagnitudehigherthancurrentstate frame, ξ = x − vgt, with vg the group velocity of the
1 of the art. The long protondriver interacts with its own driver, which is valid for a slowly evolving drive beam.
:
v wake via the self-modulation instability, leading to the This is somewhat different to the proposed AWAKE ex-
i creation of a train of bunches8, making efficient wake- periment,inwhichthe electronbeamis injectedinto the
X
field generation possible. In the proposed experiment, wake during self-modulation, leading to a growing wake
r anexternallyinjectedelectronbunchwillbeaccelerated. withanon-constantphasevelocity11,12. Thecasetreated
a
Alternatively, high energies could be achieved by mak- here is instead applicable to injection into the developed
ing use of a staged laser-driven wakefield accelerator9, wake13,14.
with the accelerated electrons from each stage injected In the figure, the driver is located at some position
into the next. In both cases the dynamics of external in the +ξ direction. Electrons were placed in the wake
electron injection will play an important role. with an initial energy of 5 MeV, propagating parallel to
In this paper, we investigate the influence of injection the driver. Experimentally, this could be realised via
positionontheenergygainofelectronsinaplasmawake. injectionbyaco-propagatingelectronbeamenteringthe
InSection II we show thatthis dependence is morecom- plasmabehind the drive beam. Although notconsidered
plexthanpreviouslyrealised,withnarrowfilamentsaway here,itisworthnotingthatsuchinjectioniscomplicated
fromthecentreofthewakeinwhichhighaccelerationcan by edge-effects upon entering the plasma14–16.
be achieved. Section III explains the underlying physics Particles with different initial positions were propa-
of these quasi-stable injection channels, resulting from gated in the wakefield using a Boris push. Calculations
2
show that radiation reaction is negligible in this regime. a) 1 220
The energy attained after a driver propagation distance
of25cm(correspondingto∼200λ )isshowninFig.1b.
p m]
As expected, the largestaccelerationoccurs for particles
p V/
wfohciucshsianrgeainnjdecatcecdeilnertaottinhge.wHakowewevheerr,eitthceanfieilmdsmaerdeiabtoetlhy l/y0 0 0 [M
fp
be seen that there additionally exist narrow filaments -k
away from the centre of the wake in which high energy
can be achieved. -1 -220
Figure1cshowsthefinaltransversepositionofinjected
b) 1 60
electrons. It can be seen that for fixed ξ, there exist re-
gionsinwhichelectronsarealternatelyejectedinthepos-
itiveandnegativey directions,correspondingtothe par- 40
V]
ticles executing a different number of half-oscillations in p e
tnhoenlwinaekaerb,aefroisrienbgedinugeteojetchteedn.oTnlhinisebareihtyavoifouthreisfostcruosnsignlyg l/y0 0 20 [ME
electric fields,andalsodue to the largerforwardacceler-
ation experienced by particles nearer the axis, which in-
-1 0
creases both their inertia and the time taken to dephase

withthewake. Ateachtransitionthereisanarrowband c) 1 25
inwhichthefinaltransversedisplacementremainssmall,
which correlate with the high-energy filaments.
p p
l/y0 0 0 l/y
III. QUASI-STABLE INJECTION CHANNELS
In order to better understand the underlying physics
-1 -25
of the filament structure, we consider the trajectories of
0 0.5 1 1.5
three individual particles. Figure 2a shows the final en- x /l
0 p
ergy for varying initial transverse position y for a fixed
0
ξ =0.5. Wechoosethreeparticleswithinitialpositions,
0
marked on the plot, corresponding to a local maximum FIG. 1. Influence of injection position on acceleration after
in the final energy (y /λ = 0.4963) and two straddling 25 cm (∼ 200λp). a) Wakefield potential, −kpφ, assumed to
0 p
points, taken at ±0.02λ . Figures 2b, c and d show the be static in this frame. Blue areas will act to trap electrons.
p The driver is located in the +ξ direction. b) Final electron
corresponding evolution of ξ, y and the energy.
electron energy against initial position. c) Final transverse
The initial energy of the injected electrons (5 MeV) is
position against initial position.
much lower than that of the driver, and so the particles
rapidly fall backwards in the co-moving frame, as seen
in Fig 2b. As they dephase, they pass through the fo-
tral particle remains in the wake more than 50T longer
cussing/acceleratingphaseofthewakeandbegintogain p
thanitsneighbours,whereT =2π/ω . Werefertothese
energy, as seen in Fig. 2d. p p
trajectories as “quasi-stable” – the particles are globally
The particles are injected away from the centre of
unstable, but are able to climb the potential gradient in
the wake, and so they oscillate in the y plane, and
the defocussing phase, gaining more energy as they do.
as such experience a smaller average accelerating field
than those injected close to the axis. As a result they TheeffectisillustratedinFig.3,whichshowsthepar-
do not gain enough energy to become trapped in the ticle trajectories superimposed on the wakefield poten-
wake,andcontinue to dephase until they reachthe defo- tial. Particles oscillate in the trapping potential (blue),
cussing/acceleratingphase of the wake. The defocussing before dephasing, falling back to the defocussing phase
field then causes the electrons to be ejected from the (red). Depending on the angle with which the particles
wake in either the +y or −y direction. The direction approachthedefocussingpotential,theywillbedeflected
in which particles are ejected depends on their position toonesideortheother. Theangleofthecentralparticle,
andmomentumastheyexitthefocusingphase,whichin however, is such that it climbs the potential gradient in
turn depends on their initial position. On the threshold thedefocussingphase,ratherthanimmediatelybeingde-
between ejection in the positive and negative directions, flected. The transverse velocity of the particle decreases
there exist particles which achieve a temporary equilib- asitapproachesthecentreofthewake,allowingittostay
rium, allowing them to remain in the wake for a longer in the accelerating field for significantly longer, leading
period, and in doing so gain more energy than their to increased energies.
neighbours. For the particles considered here, the cen- These quasi-stable injection channels are a direct con-
3
a)
50 EEE [[[MMMeeeVVV]]]
000 222000 444000 666000
40
V] 30
e
M 220
[E 20
10
m]
0
0 V/
-1 -0.5 0 0.5 1 0 M
y0/l p f [p
k
b) -
0.6 yyy00///lll pp === 000...445791666333 x0/l p 0.5 1 -220
0.4 0 p 0
p
l/ 1 -1 y0/l p
x
0.2
FIG.3. Trajectories andenergyevolutionforthethreeparti-
0 cles in Fig. 2, superimposed on the wakefield potential. The
c) potential is shown in the vertical direction, with blue areas
2
acting to trap electrons. The colour of the electron trajecto-
ries corresponds to theenergy.
1
p
0
ly/
ture can be observed using a simple analytical form for
-1 a purely electrostatic field. Figures 4a,b show the in-
fluence of changingthe initialmomentum of the injected
-2 electrons. OtherparametersarethesameasinFig.1. In
d) ◦
20 Fig. 4a, electrons are injected from below at a 1 angle,
i.e. propgating in the +ξ, +y direction. Although this
15 breaks the symmetry in the y plane, it can be seen that
V] the injection channels remain. Reflection of the beam,
e
M 10 as discussed in Ref. 17, is not observed in this case. Re-
[E flection is expected when the angle relative to the driver
5
is below the optimal angle, which for these parameters
◦
is 0.3 . Modelling the effect would therefore require a
0
0 50 100 150 200 longer propagation distance than the 25 cm used here.
t / Tp Theeffectofincreasingthe energyofthe injectedelec-
trons is shown in Figure 4b. An electron energy of
FIG. 2. Trajectories of individual particles in the wake. a) 16MeVisused,propagatingparalleltothedriver. These
Slice showing final electron energy against initial transverse faster electrons take longer to dephase, and so execute
positionforξ0=0.5. Thethreeinitialpositionstobetracked more oscillations in the trapping potential before being
are marked. b) Evolution of ξ against time. c) Evolution of ejected from the wake. As the quasi-stable channels oc-
y against time. d) Evolution of the energy against time. cur on the threshold between different number of oscil-
lations, this results in the channels being more closely
spaced in y . Changing the velocity of the driver will
0
sequence of the wakefield structure. As the accelerat- have a similar effect – if the driver velocity is decreased,
ing and focussing fields are a quarter-wavelength out of the injected electrons will take longer to dephase. Con-
phase,particlescancontinuetobeacceleratedafterleav- versely, a higher-velocity driver will cause dephasing to
ing the focussing field. occur sooner, and so the injection channels will be more
widelyspaced. Themaininfluenceofthechoiceofdriver,
whetheritbeanion,electronorlaserbeam,istheresult-
IV. STABILITY AND APPLICABILITY TO ingphasevelocityofthewake. Thepropagationdistance
EXPERIMENTS overwhichthedrivercanbeconsidered“slowly-varying”
may also be different.
The emergence of these quasi-static injection channels Figures4c,dshowtheinfluencemodifyingthemodula-
appears to be remarkably robust. Although the wake- tiondepth ofthe wakefield. Again,other parametersare
fieldsusedherearefullyelectromagnetic,thesamestruc- asforFig.1. InFig.4c,a15%modulationdepthisused,
4
a) 1 60 200
100 cm
40
p eV] 150 75 cm
l/y0 0 20 [ME eV]
M 100 50 cm
[E
-1 0
50 25 cm

b) 1 75
0
50 0.2 0.25 0.3 0.35 0.4
l/y0p 0 25 [MeV]E y0/l p
FIG.5. Evolutionoftheenergyagainstinjectionpositionover
propagation distance. 5 MeV electrons are injected parallel
-1 0 tothedriverintoan 8.5% modulated wake. Theenergies are
c) 1 100 shown (from lowest to highest) after 25, 50, 75 and 100 cm.
The energies after 25 cm shown here equivalent to those in
Fig. 2a.
75
V]
p e
l/y0 0 50 [ME sjeoctliitotnlecdhiaffnenreelnsc.e is observed in the structure of the in-
25
The modulation depth is further increased to 40% in
Fig4d. Thisisthemaximummodulationdepthobserved
-1 0
in Ref. 18, and corresponds to a maximum accelerating

d) 1 240 field of ∼ 1 GV/m. It is important to note, however,
thatinthatwork,thewakequicklydecaysafterreaching
a maximum, while here we assume it is constant. Again,
160
V] thelargeracceleratingfieldgivesrisetoanincreasedpeak
p e
l/y0 0 80 [ME eunlaetrigoynfdoerptthhe,atchceelweraakteedisewleecatkrolynsn.oFnolirntehairs,lraersguelrtimngodin-
a wakefield with anharmonic structure. The positively
chargedtroughsofthe wakefieldbecomelargerandshal-
-1 0 lowerthanthenegativelychargedpeaks,whichleadstoa
0 0.5 1 1.5 significantlylargertrappingareathanobservedinFigs.1
x /l
0 p and 4c. However, despite these differences, the injection
channels outside the main trapping region remain.
FIG. 4. Influence of injection position on electron energy For longer propagation distances, additional channels
after 25 cm (∼ 200λp) for different momenta of the injected formcloserto the centreof the wake,asshowninFig. 5.
electrons(Figs.a,b)anddifferentwakeamplitudes(Figs.c,d). This is because electrons closer to the wake centre expe-
a) 5 MeV electrons injected from below at a 1◦ angle to the rience a strongeracceleratingfield, and so take longerto
driver into a 8.5% modulated wake. b) 16 MeV electrons dephase. As particles ejected at later times will have ex-
injected parallel to the driver into a 8.5% modulated wake. ecutedmoreoscillationsinthetrappingpotential,thein-
c) 5 MeV electrons injected parallel to the driver into a 15%
jectionchannelsagainbecomemoretightlypacked. Over
modulated wake. d) 5 MeV electrons injected parallel to the
time, the rate at which particles are ejected from the
driverinto a 40% modulated wake.
wake decreases, but the effect will occur as long as some
particles continue to dephase.
Inthiswork,wehaveconsideredthecaseofawakefield
correspondingto a peak acceleratingfieldof 390MV/m, with a constant phase velocity. As the process relies on
near the saturation limit predicted in Ref. 11 for a self- electrons dephasing, and ultimately being ejected from
modulating proton beam. As expected, the peak energy the wake, it is not itself useful as an acceleration tech-
of electrons is larger than those obtained in Fig. 1 due nique. However,theresultisstillrelevanttoexperiments
to the larger accelerating fields. Although the trapping in which external injection is used. Finding the optimal
potentialislarger,the transversemomentumofthe elec- injectionpositioninsuchschemeswillrequirecarefultun-
trons in the wake is also correspondingly increased, and ing of the angle and offset of the witness bunch relative
5
to the driver. The presenceofnumerous “falsepeaks”in VI. ACKNOWLEDGEMENTS
the parameter space topology makes this process more
difficult than if there were only a single peak. Knowl- JPF and AP would like to acknowledge funding from
edge of this structure is therefore vital to facilitate such DFGTR18,EUCard2 EUFP7andBMBF.MWTwould
tuning. like to thank G¨otz Lehmann and Matthias Dellweg for
helpful discussions.
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