Hosing instability in the blow-out regime for plasma-wakefield acceleration.

Abstract

The electron hosing instability in the blow-out regime of plasma-wakefield acceleration is investigated using a linear perturbation theory about the electron blow-out trajectory in Lu et al. [in Phys. Rev. Lett. 96, 165002 (2006)10.1103/PhysRevLett.96.165002]. The growth of the instability is found to be affected by the beam parameters unlike in the standard theory Whittum et al. [Phys. Rev. Lett. 67, 991 (1991)10.1103/PhysRevLett.67.991] which is strictly valid for preformed channels. Particle-in-cell simulations agree with this new theory, which predicts less hosing growth than found by the hosing theory of Whittum et al.

Full text

Page 1

Work supported in part by US Department of Energy contract DE-AC02-76SF00515
Hosing Instability in the Blow-Out Regime for Plasma-Wakefield Acceleration
C. Huang,1W. Lu,1M. Zhou,1C.E. Clayton,1C. Joshi,1W.B. Mori,1P. Muggli,2S. Deng,2E. Oz,2T. Katsouleas,2
M.J. Hogan,3I. Blumenfeld,3F.J. Decker,3R. Ischebeck,3R.H. Iverson,3N.A. Kirby,3and D. Walz3
1University of California, Los Angeles, California 90095, USA
2University of Southern California, Los Angeles, California 90089, USA
3Stanford Linear Accelerator Center, Menlo Park, California 94025, USA
The electron hosing instability in the blow-out regime of plasma-wakefield acceleration is investigated
using a linear perturbation theory about the electron blow-out trajectory in Lu et al. [in Phys. Rev. Lett. 96,
165002 (2006)]. The growth of the instability is found to be affected by the beam parameters unlike in the
standard theory Whittum et al. [Phys. Rev. Lett. 67, 991 (1991)] which is strictly valid for preformed
channels. Particle-in-cell simulations agree with this new theory, which predicts less hosing growth than
found by the hosing theory of Whittum et al.
Recent experiments have shown amazing progress for
both plasma-wakefield acceleration (PWFA) and laser
wakefield acceleration (LWFA) [1–3] in the electron
blow-out regime. In this regime, plasma electrons are
completely evacuated by the space charge force of an
electron beam or the ponderomotive force of a laser pulse,
forming an ion channel on the axis of the system with a
laminar sheath at the channel boundary carrying large
concentrations of relativistic electrons. However, the elec-
tron hosing instability [4–7] of the drive and/or trailing
beam remains a major concern for PWFA/LWFA concepts.
The hosing instability results from the interaction between
the electron sheath and the self-injected or externally in-
jected electron beam. It leads to spatiotemporally growing
oscillations of the beam centroid at each axial slice thus
limiting the useful acceleration length and making it diffi-
cult to aim the beam. Existing standard theory [4,6] pre-
dicts rapid growth for this instability. However, recent
experiments [1,2] have shown little evidence of hosing.
In this Letter, we present a more general hosing theory
based on a perturbation method to the zeroth order trajec-
tory [8] for the ion-channel/electron-sheath boundary. The
initial hosing growth predicted by the linearized coupling
is found to be affected by the nonconstant channel radius,
relativistic mass corrections, and the longitudinal velocity
of electrons in the plasma sheath. We verify this theory
using particle-in-cell (PIC) simulations and compare it to
the standard theory.
The existing work [4,6] focused on the hosing in a long
ion channel with a radius near the charge neutralization
radius, i.e., rc? rneu?
beam and plasma density, respectively, and Rbis the beam
radius. Such a channel is either preformed or adiabatically
formed. The electrons in the sheath layer are assumed to be
at rest, i.e., the nonrelativistic limit; therefore, they do not
generate or feel the magnetic fields. This adiabatic (refer-
ring to the channel formation), nonrelativistic (referring to
the plasma sheath motion) limit is appropriate for a beam
???????????????????
nbR2
b=np
q
, where nb, npare the
with a long bunch length L ? ?pand weak charge per unit
length ? ? k2
!p?
angular frequency, and wavelength of the plasma wave
moving at the speed of light, and Iband IA? 17 kA are
the beam and Alfve ´n current, respectively. Such a beam
creates a long channel with radius rc? rneu? k?1
fore, the plasma sheath is nonrelativistic. In this limit, the
linearized coupled equations for the channel centroid xc
and the beam centroid xbare [4,7]
pr2
neu? 4Ib=IA? 1, where kp? !p=c,
???????????????????????
4?npe2=m
q
, ?p? 2?k?1
p
are the wave number,
p; there-
@2
sxb? k2
?xb? k2
p
?xc;
??????
@2
?xc? !2
???
0xc? !2
, s is the propagation
0xb;
(1)
where k?? kp=
distance into the plasma, ? ? ct ? z is the location within
the beam, and ? is the beam Lorentz factor. For a PWFA,
the ‘‘short-pulse’’ limit of these equations is relevant, i.e.,
k?s ? !0?, and the asymptotic solution for a linear tilt in
xbis xb=xb0?0:341A?3=2eAcos?k?s?A=
where A ? 1:3??k?s??!0??2?1=3. In this short-pulse regime
@2
a wave number k?.
InthisLetter weinvestigate howthe physicsofthe short-
pulse asymptotic behavior is modified when the channel is
formed nonadiabatically and when relativistic mass cor-
rections arise (i.e., L < ?p, ? > ?1, and rc;max> k?1
Under these conditions, the variation of rc along the
beam isnot negligible and the motion inthe electron sheath
can be relativistic. And using a fully explicit 3D simulation
[9], hosing was found to be less severe than the standard
theory prediction. The reasons for the reduced growth rate
were not then clearly identified. Here we identify the
physics that reduces the growth rate and quantify their
effects.
We first illustrate how hosing occurs in the nonadiabatic,
relativistic blow-out regime using a 3D fully nonlinear PIC
code, QuickPIC [10], which uses the quasistatic approxi-
mation as discussed just after Eq. (2) below. Figure 1(a)
2?
, !0? kp=
2
p
???
3
p
??=4? [6],
?xc? !2
0xcand xboscillates nearly resonantly in s with
p).
Submitted to Phys.Rev.Lett.
SLAC-PUB-13122

Page 2

shows the beam after the hosing instability has developed
and saturated in the simulation. In Fig. 1(b) the centroid of
the beam at the location of the left-hand (blue) arrow is
plotted and compared to the asymptotic solution from the
standard theory. The simulation parameters are as follows
[9]: np? 2 ? 1016cm?3, nb=np? 25:9, kp?r? 0:19,
kp?z? 1:2, where ?rand ?zare beam RMS radius and
length, respectively. The beam has an initial linear tilt of
angle ? ? 0:011.
There are three regions along the beam which exhibit
different behavior as indicated in Fig. 1(a) by the (yellow)
box, left-hand (blue) arrow, and right-hand (green) arrow,
respectively. In region I the ion channel has not yet com-
pletely formed and the latter part of the head gradually
aligns with the front of the beam. In region II, the ion
channel is fully formed and the electron sheath is laminar.
Recently, Lu et al. [8] showed that the channel shape in this
region can be represented by the trajectory of an electron at
the channel boundary, defined as r0???. We label the upper
and lower trajectories as r?and r?, respectively. For a
beam without a tilt, r?? r?? r0. The linear (described
by linearized equations) and nonlinear stages of hosing are
seen in the lower red curve in Fig. 1(b) before and after
0:65m, respectively. In region III, the hosing amplitude
becomes comparable to the channel radius and the beam
breaks apart when it hits the channel boundary.
In this Letter, the linear stage of hosing growth in region
II is studied. Clearly any theory for hosing of short pulses
needs to include channel formation. (Our theory should
also be valid when the plasma is created by self-ionization
from the drive beam when the diameter of the ionized gas,
d ? 2rc. The opposite limit where d ? 2rcwas studied in
[11].) A perturbative model to the electron trajectory at the
channel boundary is developed for small hosing ampli-
tudes. To lowest order, the charge and current density
perturbations at the boundary can be approximated as di-
poles by a slightly shifted channel.
We use the coordinates (x, y, s ? z, ? ? ct ? z) with xz
being the hosing plane and assume a quasistatic plasma
response which essentially means that the s dependence
(except for that due to beam evolution) is removed when
calculating the electron trajectory. The perturbation to the
zeroth order trajectory is assumed to be small. Also, the
focusingforce is still assumedto be linear in r, i.e., channel
deformation is omitted. The channel centroid can be de-
fined as xc??? ? ?r???? ? r?????=2. The channel diameter
d ? r?? r?is also assumed to be unchanged. Further-
more, the beam is assumed to be axisymmetric and narrow
compared to r0, so beam deformation can be neglected.
Therefore, the coupling from the channel to the beam
centroid xbis treated the same way as in [4].
Based on these assumptions, we derive a new equation
for the channel centroid from the relativistic radial equa-
tion of motion for a plasma electron, beginning from what
is essentially Eq. (3) of [8] but with no laser field
?
where P?, Vzare the perpendicular (radial) momentum
and longitudinal velocity of a plasma electron, and Erand
B?are the self-consistent radial electric field and azimuthal
magnetic field, respectively. We normalize mass to m,
charge to e, velocities to the speed of light c, lengths to
k?1
quasistatic approximation, ??1 ? Vz? ? 1 ? [12], or
? ? ?1 ? P2
? ? ? Az is the solution to the Poisson equation
?r2
plasma charge and longitudinal current densities, respec-
tively. Thus d=dt ? ?1 ? Vz?d=d? ? ?1 ? ?d=??d?? and
P?? ?dr=dt ? ?1 ? ?dr=d?. So Eq. (2) becomes (we
define d=d? ?0)
dP?
dt
?d
dt
?dr
dt
?
? ??Er? VzB??;
(2)
p, densities to np, and fields to mc!p=e. Under the
?? ?1 ? ?2?=2?1 ? ?, where the potential
? ? 4??? ? Jz? in Lorentz gauge, and ?, Jzare
?1 ? ???1 ? ?r0?0? ???Er? VzB??:
When the beam is straight, i.e., unperturbed, the solution
to Eq. (3) is defined as r0??? with other unperturbed
quantities being E0, B0, 0, V0???, and ?0???, respectively
(subscripts r, z, and ? are dropped). Next, perturbations
with subscript ‘‘1’’ are introduced.
(3)
r ? r0? r1;V ? V0? V1;
(4)
E ? E0? E1;B ? B0? B1;
(5)
?r0? r1? ? 0?r0?; i.e., 1? 0:
Equation (6) follows from the assumption that the chan-
nel is simply displaced by r1without deformation and the
fact that is insensitive to the change in the profile of
(p ? Jz) [8]. Equation (6), together with ? ? ?1 ? P2
?1 ? ?2?=2?1 ? ? and P?? ?1 ? ?r0, leads to ? ?
?0? ?1? ?0? ?1 ? 0?r0
Eqs. (4)–(6) into Eq. (3) and ordering the resulting terms,
we obtain the zeroth order Eq. (7) and first order Eq. (8),
(6)
??
0r0
1. By substituting ? and
?1 ? 0???1 ? 0?r00
0? 0
0r0
0? ? ??0?E0? V0B0?
(7)
Ion Channel
(a)
(b)
0.1 mm
0.05 mm
S=0.52 m
0.0001
0.001
0.01
0.1
1
0 0.2
s (m)
0.40.60.81
|xb| (mm)
Standard Hosing Model
Simulation Result
Region I Region II Region III
r-
r+
x=0
FIG. 1 (color).
and plasma density after the beam has propagated 0.52 m,
illustrating the three regions of behavior. (b) The beam centroid
jxbj (lower red curve) in the simulation (the asymmetry in this
curve is due to beam centroid aligning to the axis at region I) and
from the standard theory prediction (upper blue curve). The
growth in the simulation is 1 order of magnitude smaller than
the standard theory prediction.
(a) A 2D slice of the beam (moving to the left)

Page 3

?1 ? 0?? 0
0r0
1? ?1 ? 0?r00
1? ?E0? V0B0?r0
0r0
1? ? ??0?E1? V0B1? V1B0?:
(8)
After substituting the value for (E0? V0B0) from Eq. (7) into Eq. (8) and dividing by ?1 ? 0?2, one finds
r00
1? f 0
0?1 ? 0??1? ??1
0??1 ? 0?r00
0? 0
0r0
0?r0
0gr0
where cr? nbR2
1=??0?1 ? V0?? now both vary along the trajectory.
Equation (11) is for the upper trajectory r?? r0? r1.
For the lower trajectory r?? r0? r2, r2satisfies a similar
equation. So xc? ?r?? r??=2 ? ?r1? r2?=2 satisfies
c? crc !2
The equation for beam centroid is the same as Eq. (1).
The differences between Eq. (12) and (1) are cr, c . In
the adiabatic nonrelativistic limit, Vx0? 0, r0? rneu,
V0? 1; hence cr? 1, 0? 0, and c ? 1. Therefore,
we recover the result in [4], i.e., Eq. (1). For more intense
beam-plasma interactions: (1) the channel radius varies
along the beam; (2) the relativistic mass changes the
plasma electron resonant frequency and the magnetic field
is important due to large V0. These two effects change the
coefficients crand c , respectively, thereby changing the
1? ??0?1 ? 0??2?E1? V0B1? V1B0?:
(9)
Equation (9) is a second order ordinary differential
equation for r1with coefficients depending on the zeroth
order quantities and their ? derivatives. The first two terms
in the last bracket on the right-hand side ??E1? V0B1? are
the sum of the perturbation force from the displacement of
the beam, Fb1, and the change in the plasma self-force due
to the displacement of the sheath, Fe1. The former can be
expressed as Fb1? ??Eb1? V0Bb1?, where Eb1and Bb1
are the change to the electric and magnetic fields from the
beam when the centroid is shifted by xb, i.e.,
Eb1? Bb1? ?1
Therefore,
?xb? r1?=r2
the actual displacement of the channel would be more
complicated. We argue, however, that to lowest order it
can be neglected. Under the assumption (confirmed by
simulation) that the channel shape does not change signifi-
cantly then ?1? Az1? 0; thus Fe1? Ex1? V0By1?
??1 ? V0?@?Ax1, where Ax1is the perturbation to the x
component of the vector potential, and Ax1 satisfies
?r2
To lowest order, Ax1?y ? 0? ? f?r0????r1Vx0, where Vx0is
the magnitude of the velocity in the ^ x direction for an
electron at the channel boundary. Fe1? ??1 ? V0?@?Ax1
can now be neglected because Vx0? 1 in much of the
channel and because we are interested in the ‘‘short pulse’’
for which @?r0=r0? 1=r0? @?r1=r1.
The last term V1B0in Eq. (9) can be expressed in terms
of r1by noting that if 1? 0 then V1? ?1?1 ? V0???1
?1? ?1 ? 0?r0
giving V1B0? ??
tional to r0
0
Therefore, Eq. (9) can be written as
2nbR2
b??r0? r1? xb??1? r?1
Fb1? ??Eb1? V0Bb1? ?1
0when jxb? r1j ? r0. Evaluating Fe1from
0?:
b?1 ? V0??
(10)
2nbR2
?Ax1? 4?Jx1with Jx1being a first order term in r1.
0,
0,
0r0
1, and B0? ?1
2r?1
1. Nextwe dropall the r0
0= 0, r00
2nbR2
0r0
1terms in Eq.(9)because
1in the ‘‘short-pulse’’ limit.
br?1
0
1which is propor-
? ?1
2?r?1
0?1 ? 0?2??2
0r0
0, 0
0, r0
0=r0
0? r00
1=r0
r00
1? crc !2
0r1? crc !2
0xb;
(11)
b=r2
0? r2
neu=r2
0and c ? 1=?1 ? 0? ?
x00
0xc? crc !2
0xb:
(12)
TABLE I.Simulation parameters, and ? ? 55, 773 for each case.
Simulation (a) Simulation (b)Parameter Simulation (c)Simulation (d)
np(1016cm?3)
Nb(1010e?)
?r(10?3c=!p)
L (c=!p)
? (10?3rad)
rblow-out(c=!p)
c?
c
5.665.66
6=24
89.8
4:91=8:72
7.69
4.3
?1
<1
2.0
0.018
5.3
3.72
4.17
0.24
avary w=?
?1
2.0
0.9
13
3.72
4.17
1.5
0:02=0:12
9.09
4:91=11:4
1.96
0.23
?1
?1
avary w=?
<1
aSee text.
FIG. 2 (color).
plasma (blue) in (a) the adiabatic nonrelativistic regime, (b) the
adiabatic relativistic regime, (c) the nonadiabatic nonrelativistic
regime, (d) the nonadiabatic relativistic regime. Left-hand
(white) and right-hand (gray) arrows denote where tilts are added
and the centroids are measured.
The beams (red, moving to the left) and the

Page 4

hosing growth. Generally, crc < 1 in a nonadiabatic rela-
tivistic channel, therefore reducing hosing growth.
To verify this new hosing theory and to study the effects
of crand c , we conduct QuickPIC simulations in the
(a) adiabatic nonrelativistic regime, (b) the adiabatic rela-
tivistic regime, (c) the nonadiabatic nonrelativistic regime,
and (d) the nonadiabatic relativistic regime, respectively.
Table I summarizes the parameters and the roles of crand
c and Fig. 2 shows the initial beam and the shape of the
channel in these simulations. In case (a) and (b), a non-
evolving beam isusedto create aninitial channel, while the
trailing beam propagates in an adiabatic (long flat) channel
with r0? rneuand cr? 1. In case (c) and (d), a beam with
a triangular longitudinal profile creates its own channel.
The beam density reaches its peak at 0:27c=!pfrom the
head so the ion channel radius varies significantly (non-
adiabatically) along the beam but for most part r0> rneu
and cr< 1. Futhermore, the beams in case (b) and (d) have
more charge than in case (a) and (c), resulting in relativistic
channels with larger r0and c < 1 (see Table I). To trigger
the hosing instability, a small linear tilt with angle ? is
imposed on the second beam for case (a) and (b) and at
0:8c=!pfor case (c) and (d), with the head of the beam on
axis and the tail off axis. The beam centroid oscillations at
3c=!pbehind from where the tilts are added are plotted as
solid red curves in Fig. 3 for all four regimes.
We take zeroth order quantities from simulations and
numerically integrate Eq. (12) and the beam equation from
(1). The results [dashed black curves in Figs. 3(b)–3(d)]
agree well with the simulations, while the standard theory
[dotted blue curves labeled cr? c ? 1 in Fig. 3, not
shown in 3(d)] overestimates the hosing by 1 order of
magnitude except for case (a). Therefore, the relatively
low hosing growth rate is no longer an anomaly, and our
simple model can now be used with confidence to discuss
future projects.
In [8] an approximate but analytic model, resulting in
Eqs. (10), (11) of [8], was created and used. Using an
approximate sheath layer width ?sand linear response
layer width ?L, this model relates to r0, involving a
complicated parameter ??r0;?s;?L? [as defined between
Eqs. (10), (11) of [8]]. From this model one also has the
useful result 0? ?r2
small for large r0, i.e., the relativistic limit. This model
can also be used to directly calculate crand c rather than
going through simulation. The result [dotted green curve
labeled as ‘‘analytic’’ in Fig. 3(d)] works well comparing
with the simulation.
Assuming constant crand c , our theory predicts xb=xb0
has the same asymptotic expression in the ‘‘short-pulse’’
limit as the standard theory result but with A ?
1:3?crc ?k?s??!0??2?1=3. Clearly, reducing the bunch
length is still the most effective way to suppress hosing,
while the nonadiabatic relativistic regime is advantageous
because typically crc ? r2
Therefore, the propagation distance can be made about 10
times longer for the same tolerance of hosing growth. In
the recent E167 experiment [2], the beam had Nb? 1:8 ?
1010electrons, spot sizes of ?r? 10 ?m, a bunch length
(in ?) of ?z? 15 ?m, and ?b? 82, 192. k?s ? 205 for
85 cm long plasma at np? 2:7 ? 1017cm?3. Assuming a
constant blow-out radius, we obtain cr? 0:25, c ? 0:54.
Thus A is reduced from 7.7 to 3.95 and xb=xb0is reduced
from 35 to only 2.3 for !0? ? 1 [13].
Last, our theory indicates that the hosing growth rate de-
pends on the beam parameters. When Ibincreases, crand
c both decrease, thereby reducing hosing growth. Fur-
thermore, the hosing amplitude is not amplified between
the drive/trailing beams where cr? 0. These conclusions
may help designing more stable PWFA experiments.
Work was supported by DOE under Contracts No. DE-
FC02-01ER41179, No. DE-FG02-03ER54721, No. DE-
FG03-92-ER40727, No. DE-FG02-03NA00065, No. DE-
FG02-97ER41039,No. DE-FG02-92ER40745,
No. DE-FC02-01ER41192, and by NSF under Grant
No. PHY0317271. Simulations were performed at local
Dawson cluster and NERSC.
0=4; thus c ? 1=?1 ? ?r2
0=4? is
neur?2
0?1 ? ?r2
0=4??1? O?0:1?.
and
[1] M. Hogan et al., Phys. Rev. Lett. 95, 054802 (2005).
[2] I. Blumenfeld et al., Nature (London) 445, 741 (2007).
[3] W.P. Leemans et al., Nature Phys. 2, 696 (2006).
[4] D.H. Whittum et al., Phys. Rev. Lett. 67, 991 (1991).
[5] J. Krall et al., Phys. Plasmas 2, 1326 (1995).
[6] A.A. Geraci et al., Phys. Plasmas 7, 3431 (2000).
[7] M. Lampe et al., Phys. Fluids B 5, 1888 (1993).
[8] W. Lu et al., Phys. Rev. Lett. 96, 165002 (2006).
[9] E.S. Dodd et al., Phys. Rev. Lett. 88, 125001 (2002).
[10] C. Huang et al., J. Comp. Phys. 217, 658 (2006).
[11] S. Deng et al., Phys. Rev. Lett. 96, 045001 (2006).
[12] P. Mora et al., Phys. Plasmas 4, 217 (1997).
[13] In the experiment, ?ris comparable to rc, which further
reduced the growth rate. See [2].
(a)
(b)
(c)
(d)
0
1
2
3
4
5
01020 30
simulation
varying cr, c
analytic cr, c
0
1
2
3
4
5
0 1020 30
simulation
cr=c =1
varying cr, c
k s
0
1
2
3
4
5
0102030
simulation
cr=c =1
varying cr, c
0
2
4
6
8
10
0816 24
simulation
cr=c =1
|xb/xb0|
k s
|xb/xb0|
k s
k s
FIG. 3 (color).
displacement of the beam centroid. We assume ?s? 0:1r0,
?L? 1c=!pfor the analytic curve in case (d). The slightly
slower hosing growth in simulation (d) is caused by nonlinearity
in beam-channel centroid coupling.
Hosing growth in four regimes. xb0is the initial