3-D Simulations of Plasma Wakefield
Acceleration with Non-Idealized Plasmas and
Beams
S. Deng, T. Katsouleas, S. Lee, P. Muggli
University of Southern California, Los Angeles, CA 90089
W.B. Mori, R. Hemker, C. Ren, C. Huang, E. Dodd, B.E. Blue, C.E.
Clayton, C. Joshi, S. Wang
University of California at Los Angeles, Los Angeles CA 90095
F.-J. Decker, M. J. Hogan, R.H. Iverson,
C. O'Connell, P. Raimondi, D. Walz
Stanford Linear Accelerator Center, Stanford, CA 94025
Abstract. 3-D Particle-in-cell OSIRIS simulations of the current E-162 Plasma Wakefield
Accelerator Experiment are presented in which a number of non-ideal conditions are modeled
simultaneously. These include tilts on the beam in both planes, asymmetric beam emittance,
beam energy spread and plasma inhomogeneities both longitudinally and transverse to the beam
axis. The relative importance of the non-ideal conditions is discussed and a worstcase estimate
of the effect of these on energy gain is obtained. The simulation output is then propagated
through the downstream optics, drift spaces and apertures leading to the experimental
diagnostics to provide insight into the differences between actual beam conditions and what is
measured. The work represents a milestone in the level of detail of simulation comparisons to
plasma experiments.
INTRODUCTION
Particle-in-cell simulations have become a powerful tool for modeling plasma
experiments including plasma accelerators. A number of recent works have
highlighted the ability of PIC simulation to provide exquisite insight into the physics
of plasma wakefield accelerators in 2-D [1-3] and 3-D [4, 5]. These have been used to
design new experiments [1], interpret old experiments, and test new regimes and
scaling laws [6 ]. Typically, such models have used input conditions that approximate
the idealized experimental conditions, such as homogenous plasmas, Gaussian beams,
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
592
etc. In this report, we extend the way PIC models are used to a new level; namely we
perform detailed modeling of the imperfections of a real experiment and an
assessment of their importance. In addition, we use the simulations to provide insight
into the diagnostics of the experiment. To do this, we propagate the model results out
of the plasma and downstream to the diagnostic to assess the differences between what
is the true state of the beam at the plasma exit and what is measured at the diagnostic.
The model we use is OSIRIS, described in Ref. 2. Except where noted, the
physical parameters are that of the E-162 experiment and can be found elsewhere in
these proceedings [7]. The simulation parameters were typically as follows: a
computational mesh of 400x100x100 grids of size .05 c/(flp (approximately 15
microns), 9 plasma and 9 beam particles per cell, a moving window with a time-step
of .02/0)p run for 100,000 steps (1.4 m) on 64 processors at NERSC. In the next
section we briefly review some modeling of non-ideal conditions of the beam and
plasma separately [8]. Then we turn to the case in which all the non-ideal conditions
are present at once. This is in some sense a worstcase scenario for a plasma wakefield
experiment. Finally, we discuss the impact of various conditions on the performance
of the wakefield accelerator.
SIMULATIONS ISOLATING SINGLE NON-IDEAL EFFECTS
Non-Ideal Beams
Figure 1 depicts the wakefields in a 3-D OSIRIS simulation for the case of round
and asymmetric beams. Fig. la corresponds to a round beam with 4xl010 electrons
with a spot size of 75 microns in a plasma of density 2xl014 cm"3. In Fig. Ib, the x2
(or x) spot size has been doubled and the x3 (or y) spot size has been halved. As can
be seen, the amplitude of the wake is smaller when the beam is asymmetric. The
reason for this is that the blow out forces [9] are different in each plane, leading to a
different time for the return of electrons to the axis. The phase mixing or blur in the
time for the electrons to converge to the axis reduces the amplitude of the density
compression there and hence the amplitude of the wakefield spike. Despite the
reduction in peak wakefield (i.e., the spike), the shape of the wake is relatively
unchanged up until the spike. Thus if the wake were beamloaded ahead of the spike
[6], we would expect the asymmetry to have little adverse effect. Also, the effect of
asymmetry becomes less and less if the beam is made much smaller than the plasma
skin depth (in both dimensions).
593
Symmetric Beam ( aspect ratio 1 : 1 )
Asymmetric Beam ( aspect i
0.25
0.00
MneoutsofE1(Ez)> _ 0 25
along x1 (z) * __
-0.25
-0.50
-0.75
9.2mm
2.75 mm
x1
1.8 mi
1.8 mi
color contour
plots of E1 in
x1-x2-plane
for x3 at center
color contour
plots of E1 in
x1-x3-plane
for x2 at center
2.75 mm
x1
9.2mm
color code for E1
FIGURE 1. Comparison of wakes from beams with round (left) and elliptical tranverse crosssections.
Non-Ideal Plasma
Figure 2 depicts a worst case image of the profile of the uv laser used to ionize the
Li plasma in E-157. The hot spots lead to transverse plasma density gradients that are
modeled as shown in Fig. 2b. Note the characteristic sizes of the beam and blowout
channel with respect to the scale of the hot spots. The result of the asymmetric plasma
is to both reduce the peak wakefield and to shift slightly the location of the peak away
from the higher density plasma side. The reason for this is that the restoring force on
the blown out electrons is greater on the high density side, so these electrons return to
the axis first and overshoot it before meeting with the electrons from the low density
side. This shift of the peak accelerating contour is apparent in Fig. 3. We note also
that the area of this contour and hence the acceptance of the accelerator is less than in
the homogeneous case (not shown). Finally, we mention that the focusing force
becomes asymmetric in the density gradient, and this results in a deflection force on
the beam. The stronger focusing on the high density side causes the beam to be
deflected toward lower plasma density. Interestingly, this is opposite to the sense of
the deflection of a beam near a plasma boundary (refraction) [5 ].
594
- 0.75
- 0.5
"1:ttZ5—--0----1IT2S""il
UV profile
y Cnm]
el 57(05715/00)
Simulation Modal
0.5
FIGURE2.
2. Gradients
Gradients in
in laser
laser profile
profile in
in experiments
experiments (left)
(left) and
and model
model density
density
FIGURE
(left)
and
model
density
FIGURE
2.
Gradients
in
laser
profile
in
experiments
gradientsused
used in
in simulations.
simulations.
gradients
gradients
used
in
simulations.
FIGURE 3. Contours of Ez in a transverse density gradient. Inset: line outs of
FIGURE 3.
3. Contours
Contours of
of Ez
Ez in
in aa transverse
transverse density
density gradient.
FIGURE
gradient. Inset:
Inset: line
line outs
outs of
of
Ez on
on axis
axis at
at (black)
(black) and
and at
at +-0.2
+-0.2σ
σ.. The
The peak acceleration
acceleration volume is
is off axis
axis
Ez
Ez
on axis
at (black)
and at
+-0.2a.
The peak
peak acceleration volume
volume is off
off axis
toward the
the low
low density
density side.
side.
toward
toward
the low
density side.
595
0.75
SIMULATION OF SIMULTANEOUS NON-IDEAL EFFECTS:
E162
Each of the non-ideal effects discussed in the previous section yield modest
reductions in the net energy gain of particles in the tail of the beam - typically 15-20%
for the cases shown. When taken together, one might then be concerned that the
reductions could add, resulting in negligible energy gain. To address this we present
next simulation results for a case with worstcase conditions on the beam tilt,
asymmetry, transverse plasma density gradient and longitudinal plasma density
gradient.
For this case, the beam is tilted in both planes
n nn-<———,—————————————————^~™«»,
by ~ Imrad (.2 ox per oz), had an uncorrelated
energy spread of .5% and unequal transverse
emittances of 1571 and 6071 mm-mrad in the y and
x
directions, respectively. The initial spot size
S -3.75was round but evolves asymmetrically due to the
different emittances. The plasma was taken to
have a longitudinal density on axis that decreased
-also -ilas o.oo i.'as a,so
from 2.3 to 1.9 xlQ•U41 __-3
cm" over the 1.4 m length
™™T«^ „ nn
, •
(corresponding to the uncompensated pump
FIGURE 4. Transverse density
\ . *
*
.
,
,
T,U
profile in simulation.
depletion of a non-converging laser).
The
transverse density profile had a linear ramp
increasing by 33% over the 2mm system size in both planes with a "cold spot" of 10%
lower density in the middle 25 microns of the simulation box (see lineout in Fig. 4).
x
1
Figure 5 shows sample output for the beam and plasma wake at early and late times
in the run. The initial beam tilt is visible in Fig. 5a. At later times, a portion of the tail
of the beam has been completely blown out of the simulation box. These particles
were sitting just behind the electric field spike of the wake and are strongly defocused;
since there is a tilt in the beam, this defocusing is asymmetric. Further back there is
another group of electrons riding in the next accelerating bucket. Slices of the plasma
wake show that later in the run, the location of the spike has slipped backward in the
beam frame due to the longer wavelength of the wake in the low density region at the
end of the run.
In Fig. 6 we show the initial and final longitudinal phase space of the beam; the
energy loss of the center of the beam and the gain of the tail in relation to the energy
spread can be appreciated.
596
FIGURE 5.
5. Beam
FIGURE
Beam profiles
profiles (top)
(top) in
iny-z
y-zplane
planeatatearly
early(left)
(left)and
andlate
latetimes
timesinin
FIGUREin5.units
Beam profiles
(top) in y-z plane
at early (left)and
and latetimes.
times in
simulation
ωp and
simulation
in units of
of c/
c/oop
and wakefields
wakefields on
onaxis
axisatatearly
early andlate
late times.
simulation in units of c/ωp and wakefields on axis at early and late times.
FIGURE 6. Phase space Pz/mc-ωpz/c of the beam before and after the
FIGURE 6.Phase
Phasespace
spacePP/mc-(OpZ/c
of the beam
beam before and
z/mc-ωpz/c of
FIGURE
and after
afterthe
the
z
plasma in 6.
OSIRIS simulation;
momentum the
increases before
downward.
plasma
OSIRISsimulation;
simulation;momentum
momentum increases
increases downward.
plasma
ininOSIRIS
downward.
597
InIn Fig.
Fig. 77 we
we
In
Fig.
we
show
show 7 the
the
show
the
result
ofof
result
result
simulating
simulating of
simulating
the
the
the
experimental
experimental
experimental
diagnostic
diagnosticfor
for
diagnostic
for
the
the non-ideal
non-ideal
the
non-ideal
simulation
simulation
simulation
data.
The
data.
The
data.
The
from
data
data
from
data 6 from
Fig.
FIGURE
Fig. 6 has
has
FIGURE7.7. Simulated
Simulatedexperimental
experimentaldiagnostic
diagnosticof
ofy-z
y-z(energy-z)
(energy-z)particle
particle
Fig. 6 has
FIGURE
7.noplasma
Simulated
experimental
diagnostic
of y-z
space
and
Plots
are
generated
by
spacefor
forno
plasma(left)
(left)
andwith
withplasma.
plasma.
Plots
are(energy-z)
generatedparticle
byusing
using
been
beenexported
exported
space
for no plasma
(left)
and with plasma.
Plots are generated
by using
been exported
OSIRIS
OSIRISdata
datafrom
fromthe
theplasma
plasmaexit
exitas
asinput
inputtotoTURTLE
TURTLEto
topropagate
propagate the
the
to
to the
the code
code
OSIRIS
data
from
the
plasma
exit
as
input
to
TURTLE
to
propagate
the
to the code
particles
particlestotothe
theCherenkov
Cherenkovdiagnostic.
diagnostic. Head
Headof
ofbeam
beamisisleft;
left;higher
higherenergy
energyisis
TURTLE;
particles
to the Cherenkov diagnostic. Head of beam is left; higher energy is
TURTLE; aaa
up.
up.
TURTLE;
up.
correlated
correlated
correlated
energy
added
and
the
beam
propagated
12m
through
imaging
energy chirp
chirp has
has been
beenadded
addedand
andthe
thebeam
beampropagated
propagated12m
12mthrough
through
the
imaging
energy
chirp
has
been
thethe
imaging
spectrometer
fields
to
the
time-resolved
diagnostic
in
E-162.
Fig.
7
shows
spectrometer
fields
to
the
time-resolved
diagnostic
in
E-162.
Fig.
7
shows
the
spectrometer fields to the time-resolved diagnostic in E-162. Fig. 7 shows thethe
corresponding
y-z
values
of
the
beam
particles
there.
corresponding
y-z
values
of
the
beam
particles
there.
corresponding y-z values of the beam particles there.
Fig.
slice-average
energy
change
the
beam
non-ideal
OSIRIS
Fig. 888 shows
shows slice-average
slice-averageenergy
energychange
changeofof
ofthe
thebeam
beaminin
inthethe
the
non-ideal
OSIRIS
Fig.
shows
non-ideal
OSIRIS
simulation
as
well
as
comparisons
simulation
as
well
as
comparisons
to
simulation as well as comparisons to to
the
ideal
case
and
to
the
simulated
the
ideal
case
and
to
the
simulated
the ideal case and to the simulated
3DD
diagnostic.The
The
energy
gain
of
the
diagnostic.
The
energy
gain
diagnostic.
energy
gain
of of
thethe
peak
slice
in
this
case
was
200
MeV
peak
slice
in
this
case
was
200
MeV
peak
slice
in
this
case
was
200
MeV
2DD
over1.4m
1.4moror
or
about
25%
lower
than
over
1.4m
about
25%
lower
than
over
about
25%
lower
than
theidealized
idealizedcase
case
with
no
tilts
and
the
idealized
case
with
tilts
the
with
nono
tilts
andand
1DD
14
14 14
uniformplasma
plasmaofof
of
density
1.5x10
uniform
plasma
density
1.5xl0
uniform
density
1.5x10
3
-3 -3
cm. ..Most
Most
of
the
difference
can
be
cm
ofof
the
difference
cancan
bebe
D
cm"
Most
the
difference
attributed
toto
the
tilttilt
ofof
thethe
beam;
thisthis
attributed
to
the
tilt
of
the
beam;
this
attributed
the
beam;
causes
thethe
z-location
of of
-1DD
-1DD
causesparticles
particlesatat
at
the
z-location
of
causes
particles
z-location
the
field
to to
bebe
offoff
thepeak
peakaccelerating
accelerating
field
to
be
off
the
peak
accelerating
field
-2QQ
-2QQ
axis
thethe
region
of of
peak
axisand
andoutside
outside
the
region
of
peak
axis
and
outside
region
peak
field.
Some
reduction
is
also
due
to
field.
Some
reduction
is
also
due
to
field.
Some
reduction
is
also
due
to
Z(ps)
the
the backward
backward slippage
slippage of of
ofthethe
the
the
backward
slippage
accelerating
peak
inin
thethe
longitudinal
FIGURE
8.8. Average
Average
accelerating
peak
in
the
longitudinal
accelerating
peak
longitudinal
FIGURE8.
Averageenergy
energygain
gainper
perps
slice
FIGURE
energy
gain
per
pspsslice
slice
for
the
non-ideal
simulation
(solid),
for
the
density
gradient.
To
simulate
forthe
thenon-ideal
non-idealsimulation
simulation(solid),
(solid),for
forthe
the
density gradient.
gradient. To
To simulate
simulatethethe
the
for
density
ideal
simulation
experimental
idealsimulation
simulation(dashed).
(dashed).
ideal
(dashed).
experimentaldiagnostic,
diagnostic,wewe
wehave
have
experimental
diagnostic,
have
apertured
the
data
inin
Fig.
7 in
x; x;
this
apertured
the
data
in
Fig.
in
x; this
this
apertured
the
data
Fig.
77 in
is
to
simulate
the
effect
of
the
slit
in
the
streak
camera.
The
result
of
the
beam
simulate the
the effect
effect ofof the
the slit
slit inin the
the streak
streak camera.
camera. The
The result
result of
of the
the beam
beam
isis toto simulate
propagation
and
andand
thethe
propagation and
and aperture
apertureis
filterout
outsome
someofof
ofthe
thetail
tailparticles
particles
reduce
propagation
aperture
isisto
totofilter
filter
out
some
the
tail
particles
andreduce
reduce
the
“measured”
energy
gain
relative
the
“actual”
energy
gain
at at
thethe
plasma
exit.
This
work
“measured”
energy
gain
relative
the
“actual”
energy
gain
plasma
exit.
This
work
"measured" energy gain relative the "actual" energy gain at the plasma exit. This work
is still
in
progress
paper.
stillin
inprogress
progressand
andwill
willbe
bereported
reportedinininaaaforthcoming
forthcoming
paper.
isisstill
and
will
be
reported
forthcoming
paper.
598
ACKNOWLEDGMENTS
Work supported by USDOE #DE-FG03-92ER40745, DE-FC02-01ER41192, DEFG03-98DP00211, DE-FG03-92ER40727, DE-AC03-765F00515, DE-FC0201ER41179, NSF #ECS-9632735, DMS-9722121, and LLNL #W07405-ENG48,
NSF-PHY-0078715, PHY-0078508.
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1.
2.
3.
2.
5.
6.
Lee, S. et al.,. Physical Review £, 61: (6) 7014-7021 Part B JUNE 2000.
Hemker, R. et al., Proc. Particle Accelerator Conf. (PAC), IEEE 5, 3672-4 (1999).
Bruwhiler, D. et al., Physical Review STAccel Beams 4, 101302 (2001)
Dodd, E.S. et al., Phys. Rev. Lett. 88, 125001 (2002).
Muggli, P. et al., Nature 411, 43 May 3 (2001).
Lee, S. et al., Physical Review Special Topics - Accelerators and Beams, 501 (1): 1001-U9 JAN
2002.
7. Hogan, M. et al., these proceedings.
8. Ren, C. et al, in preparation.
9. Rosenzweig, J. et al., Phys. Rev. A 44, 6189 (1991).
599