Beam Breakup Mitigation by Ion Mobility in Plasma Acceleration

A. Burov, S. Nagaitsev and V. LebedevFermilab, PO Box 500, Batavia, IL 60510-5011

(Dated: August 11, 2018)

Moderate ion mobility provides a source of BNS damping in the plasma wakefield acceleration,which may serve as an effective remedy against the transverse instability of the trailing bunch.The ion-related BNS parameter κ, proportional to the beam brightness, is introduced as a singleparameter of the partial integro-differential equation of the bunch collective motion. This equationis further reduced to an ordinary differential equation, which solutions are shown versus its singletime-space argument and the BNS parameter. It is demonstrated that conditions of the instabilitysuppression and emittance preservation at energy efficient plasma acceleration leave for the ion BNSparameter κ about an order of magnitude of its possible variation along the acceleration line.

Plasma wakefield acceleration (PWA) suggests ex-tremely high acceleration fields, so it is no surprise thatthis area of research attracts interest of groups workingon future colliders, giving rise to many publications, tar-geted at resolution of multiple interrelated problems inthis challenging area. A special subset of these problemsis associated with stability of both driving (accelerating)and trailing (accelerated) bunches. The latter problemappears to be harder than the former, since mismatchesat every change of the driving bunch between positions ofthe two bunches produce initial kicks for the instabilitydevelopment along the acceleration line for one and thesame trailing bunch. From a very general point of view,the PWA trailing bunch instability belongs to the fam-ily of similar effects in linacs. Due to interactions withthe surroundings, dipole perturbations at the head of thebunch leave electromagnetic wake fields behind, thus act-ing on the bunch tail. The kick felt by the test particleat the unit trajectory length by a unit dipole momentof the leading particle is known as the wake function,W⊥(ξ), where ξ is the separation between the particles,see e.g. Ref. [1]. As a result, the tail dipole oscillationsmay grow more and more, leading to the emittance degra-dation. This sort of unbounded convective instability [2]is known as the beam breakup in linacs [1]. Here we areconsidering the acceleration of a short electron bunch inthe blowout regime, a regime in which the fields of thedriver (laser or an electron bunch) are so intense thatthey expel all plasma electrons, creating a cavity filledwith pure ion plasma [3]. The longitudinal and trans-verse electric fields inside this cavity are used to accel-erate and focus the trailing electron bunch. The excitedtransverse wake fields are very sensitive to the apertureradius, which is the plasma bubble radius at the bunchlocation, rb, for the PWA case: for the short bunches ofthe interest, the wake function is inversely proportionalto the fourth power of this radius, W⊥(ξ) ' 8ξΘ(ξ)/r4b ,where Θ(ξ) is the Heaviside theta-function. Many de-tails on that can be found in the recent Refs. [4, 5]. Toget the desired high acceleration, the plasma bubble hasto be small, typically rb ' 50 − 100µm, compared with1− 2cm for conventional colliders; thus, with the fourthpower of the aperture in the transverse wake, the trans-verse instability is by necessity one of the main obsta-

cles for the PWA colliders. From this, one may correctlyconclude that there must be a relation between energyefficiency and beam stability for PWA: while the formerrequires smaller bubbles, the latter is lost with them.Such efficiency-instability relation has been recently for-mulated and proved in Ref. [6]; here we reproduce thisstatement for the reader’s convenience.

Let ηP < 1 be the PWA energy efficiency, i.e. the ra-tio of the power of the trailing bunch acceleration to thepower of the driving bunch deceleration. Further, let thewake parameter ηt be the ratio of the bunch-averaged de-focusing by the transverse wake fields to the main focus-ing of the bunch electrons by the ions inside the bubble.The efficiency-instability relation of Ref. [6] states that

ηt ≈η2P

4(1− ηP ). (1)

Thus, an increase in the efficiency indeed brings thebunch closer to the instability threshold. To push backthis limitation for a given wake parameter ηt, BNS damp-ing [7] can be used. The main idea of this method isbased on a compensation of the wake deflecting force inevery position of the bunch by additional focusing. If thetwo terms cancel each other for the bunch deflection asa whole, the instability would be effectively suppressed.Let us assume that the relative focusing strength δω⊥/ω⊥varies along the bunch by one or another reason; then,neglecting, for simplicity sake, the variation of the bunchline density, the equation of motion for the bunch nor-malized local offsets X(ξ, µ) can be presented as

∂2X

∂µ2+

(1 + 2

δω⊥ω⊥

)X =

2ηtL2t

∫ ξ0

X(ξ′)(ξ − ξ′)dξ′ . (2)

Here Lt is the full bunch length, µ is the normalized time,dµ = kpds/

√2γ with ds as a differential length along the

bunch motion, γ as the relativistic factor, kp =√

4πn0reas the relativistic Debye length; n0 is the plasma den-sity and re = e

2/(mc2) is the electron classical radius. Itfollows from Eq. 2 that the bunch deflection as a wholewould evolve as stable oscillations if the BNS compensa-tion condition is fulfilled:

2δω⊥ω⊥

= ηtξ2

L2t, (3)

FERMILAB-PUB-18-388-AD-APC

This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

2

where the longitudinal position ξ is measured from thebunch head. In conventional linear accelerators, this con-dition can be fulfilled by momentum modulation δp/palong the bunch [7], using that 2δω⊥/ω⊥ = −δp/p. ForPWA though, there is an additional mechanism of thefocusing variation, associated with the ion mobility inthe Coulomb field of the trailing bunch. Indeed, plasmaions move in this field, causing variation of the ion 3Ddensity δni/ni. Assuming this variation to be small, it isestimated as

δnini

= 2πntriξ2 , (4)

where ri is the ion classical radius and nt = Nt/(πLtb2)

is 3D density of the trailing bunch. Here we assume asimple model of a homogeneous 3D density of the bunchparticles, with the full transverse radius b and the fulllength Lt.

The ion density perturbations (4) entail similar varia-tions of the electron focusing:

2δω⊥ω⊥

=δnini

= 2πntriξ2 . (5)

Note that both the sign and position dependence of thisfocusing variation are the same as BNS compensation re-quires (3). Thus, the latter can be presented as position-independent:

κ ≡ 2NtriLtb2ηt

=µ2iηt

= 1 , (6)

where µi =√

2NtriLt/b2 � 1 is the phase advance ofion’s oscillations in the field of the trailing bunch.

The equation of motion (2) can be further simplifiedwith the slow amplitudes x = X exp (iµ), measuring thepositions inside the bunch as fractions of its full length,ζ = ξ/Lt and using slow time τ = µηt. Then, it is trans-formed to the following equation on the slow amplitudesx(ζ, τ ), with the constant initial condition,

∂x

∂τ= −iκζ2x+ 2i

∫ ζ0

x(ζ − ζ ′)ζ ′dζ ′ ;

x(ζ, 0) = 1 ; 0 ≤ ζ ≤ 1 .(7)

It is straightforward to show that solution of this equa-tion x(ζ, τ ) with the specified initial condition has a scal-ing invariance: it depends on its space and time argu-ments ζ and τ as x(ζ, τ ) = x(1, ζ2τ) ≡ f(ζ2τ). In otherwords, the complex amplitude of the oscillations at po-sition ζ and time τ is the same as at the bunch tail andearlier time ζ2τ . This means that the partial integro-differential equation (7) with the specified initial condi-tion is equivalent to an ordinary one. Apparently, thisordinary integro-differential equation has the simplestform with the space-time argument u ≡ ζ

√2τ . With

x(ζ, τ ) = g(u), Eq. (7) reduces then to the following form

d g

d u= −iκug + 2i

u

∫ u0

g(u− u′)u′du′ ; g(0) = 1 , (8)

At κ = 0, i.e. without any damping, the solution atlarge argument, u � 1, asymptotically tends to g(u) 'exp(3 i1/3(u/2)2/3), omitting the pre-exponential factor.

FIG. 1. Natural logarithm of the bunch oscillation amplitudef(y) ≡ x(1, y), versus the scaled time y ≡ ζ2τ and the BNSparameter κ.

Since the problem is reduced now to finding a func-tion of just two parameters, gκ(u), from a linear ordinaryintegro-differential equation (8), it can be easily solvednumerically for all interesting cases; Fig. 1 presents theamplitude modulus x(1, τ) as a 3D plot for 0 ≤ κ ≤ 1.2and 0 ≤ τ ≤ 100. Patterns of oscillations x(ζ) along thebunch for a case of slight BNS overshooting, κ = 1.2,and more overshooting, κ = 2.0, for τ = 100 are shownin Figs. 2 and 3. A couple of things are worth noting in

0.2 0.4 0.6 0.8 1.0ζ

-0.6-0.4-0.2

log(x)log(x)arg(x)

FIG. 2. Natural logarithm and argument of the perturbationat time τ = 100 and slightly overshooting BNS, κ = 1.2.

relation to Figs. 1 – 3. First, the instability is dramat-ically weakened even with a moderate BNS parameterκ ' 0.2 − 0.5. Secondly, for overshooting κ > 1, thebunch is stable; its initial perturbation decoheres, cer-tainly contributing to the emittance growth.

During acceleration, the ion-driven BNS parameter in-

3

0.2 0.4 0.6 0.8 1.0ζ

-3-2-1123log(x)

log(x)arg(x)

FIG. 3. Same as Fig. 2, but for more BNS overshooting,κ = 2.0.

creases, κ ∝ √γ, so it cannot be the same at the begin-ning and the end of the acceleration. Assuming the initialand final energies differ by the factor of 100, it meansthat the ion-BNS parameter κ = µ2i /ηt increases by afactor of 10. With acceleration efficiency ηP = 50%, thewake parameter is given by Eq. (1), resulting in ηt = 0.1 .Thus, full BNS damping would occur if the ion’s phaseadvance is not too small, µ2i ≥ 0.1. On the other hand,if this phase advance is not at all small, µ2i ≥ 1, the ionscollapse inside the bunch, and that leads to a dramatic

emittance growth. Thus, for the specified parameters,there is an order of magnitude of possible variation of theion-BNS parameter κ compatible with both stabilizationand emittance preservation. This, in turn, supports thepossibility for the acceleration by 100 times, say, from10 GeV to 1 TeV. Keeping in mind the significant helpof an incomplete BNS compensation, demonstrated byFig 1, as well as approximations of this model, one mayhope that this acceleration range may be significantlylarger.

Transverse stabilization with the ion mobility takeninto account was observed in simulations of Ref. [8] forNt = 4 · 109 cm−3, rms bunch length σz = 6.4µm, rmstransverse size σ⊥ = 0.5µm, and the energy efficiencywas ηP = 0.5. According to the efficiency-instability rela-tion (1), it corresponds to the wake parameter ηt = 0.13.For these parameters, the proton phase advance is com-puted as µi = 0.56 rad, and the ion BNS parameterκ = 2.6, assuming Lt ≈ 4σz and b ≈ 2σ⊥. Thus, theobserved stabilization is in agreement with the suggestedion-driven BNS theory.

Fermilab is operated by Fermi Research Alliance, LLCunder Contract No. DE-AC02-07CH11359 with theUnited States Department of Energy.

[1] A. W. Chao, Physics of collective beam instabilities in highenergy accelerators (Wiley, 1993).

[2] A. Burov, (2018), arXiv:1807.04887 [physics.acc-ph].[3] J. B. Rosenzweig, B. Breizman, T. Katsouleas, and J. J.

Su, Phys. Rev. A 44, R6189 (1991).[4] G. Stupakov, Phys. Rev. Accel. Beams 21, 041301 (2018).[5] P. Baxevanis and G. Stupakov, Phys. Rev. Accel. Beams

21, 071301 (2018).

[6] V. Lebedev, A. Burov, and S. Nagaitsev, Phys. Rev. Ac-cel. Beams 20, 121301 (2017).

[7] V. E. Balakin, A. V. Novokhatsky, and V. P. Smirnov,Proceedings, 12th International Conference on High-Energy Accelerators, HEACC 1983: Fermilab, Batavia,August 11-16, 1983, Conf. Proc. C830811, 119 (1983).

[8] W. An, in FACET-II Science Workshop, SLAC, 2017(2017).

http://arxiv.org/abs/1807.04887http://dx.doi.org/10.1103/PhysRevA.44.R6189http://dx.doi.org/10.1103/PhysRevAccelBeams.21.041301http://dx.doi.org/10.1103/PhysRevAccelBeams.21.071301http://dx.doi.org/10.1103/PhysRevAccelBeams.21.071301http://dx.doi.org/10.1103/PhysRevAccelBeams.20.121301http://dx.doi.org/10.1103/PhysRevAccelBeams.20.121301https://conf.slac.stanford.edu/facet-2-2017/

Beam Breakup Mitigation by Ion Mobility in Plasma AccelerationAbstractReferences