PIC Simulations of Plasma Beat-Wave
Acceleration Experiments at UCLA
R. Narang, C.E. Clayton, C.V. Filip, S.Ya. Tochitsky, D.F. Gordon*,
C. Joshi, W.B. Mori
University of California, Los Angeles, Los Angeles, CA 90095
*Naval Research Laboratory, Washington, D.C. 20375
Abstract. The plasma beat-wave accelerator (PBWA) in the Neptune Laboratory at UCLA
utilizes a ~1 terawatt two-wavelength laser pulse to tunnel ionize hydrogen gas at conditions of
resonance for driving relativistic plasma waves. This plasma wave is used as an accelerating
structure for an externally injected ~11 MeV electron beam from the Neptune Photo-injector.
Simulations in 2-D have been done to model this experiment for laser ionized plasmas with
mobile ions for two focusing geometries, f/3 and f/18. Simulations have shown that ion motions
in the transverse direction for small spot size cases (f/3 case) cannot be neglected, and that the
acceleration of electrons is therefore limited by shortening of the effective interaction length due
to deviations from the resonant density. In the f/18 case, while ion motions are not as severe as
in the f/3 case, ionization induced refraction begins to limit the peak intensity of the laser. In
addition, injection of the electron beam into the plasma wave is modeled to determine what
acceleration is to be expected in experiments.
INTRODUCTION
Current efforts to model the PBWA in the Neptune Laboratory [1] utilize the PIC
code turbo WAVE [2]. This code allows modeling of the experimental parameters
present in the lab. Simulations are done for laser-plasma acceleration experiments in
which the plasmas are produced by focusing a two wavelength (10.27 |um and
10.59 |iim), -120 ps laser pulse with f/3 or f/18 optics with intensities of ~1015 W/cm2
and ~2 x 1014 W/cm2, respectively [3]. In these experiments the separation of the laser
frequencies is equal to the plasma frequency (A(jo=oop). From the simulations we are
able to determine the normalized relativistic plasma wave amplitude, Ex, which allows
the estimation of the field gradient present in the plasma. Determination of the wave
amplitude allows the estimation of the magnitude of acceleration expected for
electrons that are externally injected into the plasma wave. In general for 1-D theory
the longitudinal field gradient is ~eno172 V/cm, where no=1016 cm~3 is the resonant
density and 8 is the density perturbation. The maximum energy gain is given by:
Wmax «E X L[4]. In an actual experiment other factors, such as finite interaction
length, will limit the maximum energy gain of accelerated electrons.
The propagation of relativistic electron beams inside of a relativistic plasma wave is
also simulated for laboratory conditions. The Neptune Photo-injector [5] produces an
electron bunch with an energy of 11.4 MeV with approximately 50 pC reaching the
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
213
interaction point (IP). This beam is focused at the IP to a transverse spot size of
Orms=125 |iim and has a bunch length corresponding to ox=1.8 mm (6 ps). Simulations
are done with these conditions taken into consideration.
A laser rise-time and fall-time of 275COP"1 (-50 ps) is used for all simulations and
the frequency ratios are taken to be oo1/(jop=3 1 (10.59 um) and co2/cOp=32 (10.27 um).
Table 1 gives the relevant parameters for the simulations presented in this paper,
where each run corresponds to a total time of 90 ps (c/cop ~ 53 |um and ai,2 is
normalized to ——^). The laser propagates in the x-direction with y as the
eE
transverse coordinate. The simulation window in the f/18 case is twice the length of
the f/3 box, therefore the focus is reached on a later time step (the focus is placed at
the center of each simulation window). The Rayleigh range, 2ZR, is -1.5 mm for f/3
focusing with a simulation box that is ~5 mm, while for the f/18 case 2ZR~2.5 cm.
Table 1. Beat-wave Simulations for Neptune Parameters_________________________
Run
x-cells
y-cells
Steps
ai=a2
WQ(C/COP) dx(c/cop) dy(c/cop)
1024
f/3
1.0
0.1
0.1
0.05
128
10,000 0.3
f/18
4.0
0.05
0.1
0.025
4096
256
20,000 0.1
2-D SIMULATION RESULTS
Images of Ex normalized to the cold wave-breaking amplitude, mccop/e, and the
charge density, p (where negative and positive values on the color-table mean an
excess of electrons or ions respectively), are shown in Figure 1 for the f/3 focusing
geometry at t=50.4 ps.
80
100
c/up
FIGURE 1. Simulation off/3 focusing, where (a) is snapshot of Ex and (b) is of p, both at t=50.4 ps.
214
After 50.4 ps the laser has blown-out plasma at the focus to less than 50% of the
resonant density, the wave is disrupted, and the amplitude of the wave is reduced near
focus. At the end of the f/3 simulation (t=90 ps) the focal region has nearly all of the
plasma blown-out, with an excess of ions at the edge of the plasma. The peak wake
amplitude achieved in the f/3 focusing is -0.1 mccop/e near focus, while it is
-0.3 mccop/e away from focus. From this amplitude an average accelerating gradient
of 2 GeV/m (calculated from the expression for the longitudinal field gradient) over a
distance of approximately 3 mm can be expected, which implies a net acceleration of
-6 MeV that could be achieved.
Figure 2 illustrates that blow-out of plasma is not significant for f/18 focusing,
where WQ is -200 um. In the f/18 simulation, where the spot size is now 4.0 c/cop, ion
motions do not affect the plasma wave structure and the results are similar when
immobile ions are used. The peak wave amplitude achieved is -0.1 mccop/e which
corresponds to an accelerating gradient of 1 GeV/m (for a 1 cm interaction length we
should expect -10 MeV acceleration). Each laser line has an amplitude of vosc/c=0.1
which corresponds to an intensity of -2 x 1014 W/cm2, and since the threshold for
ionizing hydrogen is 1.47 x 1014 W/cm2 [6], the peak wave amplitude may be limited
by the laser field amplitude. The position of the best focus appears to move -20 c/cop
upstream as the laser beam ionizes the hydrogen, which is indicative of the onset of
ionization induced refraction and this may also limit the peak wave amplitude.
50
100
150
200
50
100
c/up
150
200
20
10
FIGURE 2. Simulation results for f/18 focusing where (a) is an image EX and (b) is an image of p, both
at t=70.2 ps.
The plasma wave structures are quite different for these two simulations and can be
attributed to the differences in the transverse fields. Figure 3 shows that the amplitude
of Ey is 10 times larger for f/3 than for f/18 focusing. For the f/3 case, since Ey is
215
-0.3 mcoop/e, the blow-out of electrons can be attributed to the transverse
ponderomotive force produced by this field. For the f/3 case the parameter kpR is ~1
(where R is the transverse dimension) while it is approximately 4 in the f/18 case
which, according to Fedele et al.[7], implies that the ratio of Ey/Ex would be
approximately 4 times larger for the f/3 case. This is consistent with the f/18
simulation, which gives the electron plasma wave structure for Ey that has been
observed in previous 2-D simulations [8].
FIGURE 3. (a) is an image and lineout of Ey for the f/3 simulation at t=50.4 and (b) is for the f/18 run
at t=70.2 ps.
ACCELERATED ELECTRONS
To get an estimate of the electron acceleration an electron beam is injected on axis
into each of the accelerating structures with y=22.3 (11.4 MeV), neglecting the
emittance of the beam. This injected electron bunch has a transverse spot size of
oy=1.25 c/GOp and a length corresponding to ox=36 c/cop (6 ps). The transverse size of
the beam is taken to be smaller than the experimental value due to limitations in the
transverse size of the f/3 simulation box, and to limit the interaction region to the
center of the plasma wave structure. The electrons are injected at the same time for
each simulation (t^36 ps) such that the bunch interacts with the wave before and after
it reaches its maximum amplitude. The maximum wave amplitude is reached at the
center of the simulation window at t~50 ps for the f/3 case and t~70 ps for the f/18
run.
In Figure 4 snap shots of the relativistic electron phase space are shown for the f/3
simulation (for y>10). At the time when the beam exits the simulation window y is
approximately 34 (~17 MeV) for the most energetic electrons. This is in reasonable
agreement with the estimate of ~6 MeV calculated from the normalized longitudinal
wave amplitude obtained from the simulation, which corresponds to a field gradient of
~2 GeV/m.
216
(a)
(a)
(b)
(b)
c/ω
c/ω
c/co
c/ωpp
c/ωnpp
FIGURE 4.
4. (a)
(a) is
is the
the relativistic
relativistic electron
electron phase
phase space
space for
for the f/3 simulation
simulationproduced
produced att=36.0
t=36.0 ps(b)
(b)
FIGURE
FIGURE
4. (a)
is the
relativistic electron
phase space
for the f/3
f/3
produced at
at t=36.0 ps
ps (b)
is at t=72.0
t=72.0 ps.
ps.
isisatatt=72.0
ps.
InFigure
Figure 555 the
the electron
electron energies
energies produced
produced for
for the
the f/18
f/18 simulation
simulation are
are shown
shown (for
(for
In
Figure
the
electron
energies
In
f/18
simulation
are
shown
(for
γ>10).
This
simulation
gives
a
maximum
energy
of
approximately
18
MeV,
which
is
y>10). This
This simulation
simulation gives
gives aa maximum
maximum energy of approximately 18 MeV, which
γ>10).
which is
is
in
close
agreement
with
the
estimate
obtained
for
a
uniform
wave
that
is
1
cm
long.
close agreement
agreement with
with the
the estimate
estimate obtained
obtained for a uniform
inin close
uniform wave that is
is 1 cm
cm long.
long.
Although the
the simulation
simulation box
box is
is only
only 111 cm
cm in
in length,
length, for
for aaa Rayleigh
Rayleigh range
range of
of
Although
the
simulation
box
is
only
cm
in
length,
Although
for
Rayleigh
range
of
2Z
≈2.5
cm,
the
main
contribution
to
energy
gain
is
from
the
central
1
cm
region
with
r
2Zr r≈2.5
~2.5cm,
cm, the
the main
main contribution
contribution to
to energy
energy gain
gain is
is from
from the
2Z
the central
central 11 cm
cm region
region with
with
eitheredge
edgeof
ofthe
theplasma
plasmacontributing
contributing minimally
minimally to
to the
the energy
energy gain.
gain. For
Forthe
thef/18
f/18run
run
either
edge
of
the
plasma
contributing
minimally
to
either
the
energy
gain.
For
the
f/18
run
the
final
energy
is
not
significantly
different
than
for
the
f/3
case
(see
Figures
4b
and
the final
final energy
energy isis not
not significantly
significantly different
different than
than for
for the
f/3 case
the
the f/3
case (see
(see Figures
Figures 4b
4b and
and
5b), which
which may
may not
not be
be the
the case
case for
for larger
larger aσyy since
since the transverse
transverse fieldswould
would affect
5b),
5b),
which may
not be
the case
for larger
σy since the
the transverse fields
fields would affect
affect
injected electrons.
electrons. For
For the
the f/18
f/18 run
run a longer
longer interaction
interaction length
length would
would produceaalarger
larger
injected
injected
electrons. For
the f/18
run aa longer
interaction length
would produce
produce a larger
energy
gain
given
that
the
wave
amplitude
is
maintained
for
that
distance.
energygain
gain given
given that
that the
the wave
wave amplitude
amplitude is
is maintained
maintained for
for that
energy
that distance.
distance.
(a)
(a)
(b)
(b)
200
226
c/ωp
c/ωp
c/co
c/ωp
c/ωpn
relativistic electron
electron phase
phase space
space for
for the
the f/18
f/18 simulation
simulation atatt=36.0
t=36.0psps(b)
(b)isisatat
FIGURE 5. (a) is the relativistic
FIGURE
t=81.0 ps. 5. (a) is the relativistic electron phase space for the f/18 simulation at t=36.0 ps (b) is at
t=81.0ps.
t=81.0 ps.
CONCLUSIONS
CONCLUSIONS
CONCLUSIONS
the PBWA
PBWA in
in the
the Neptune
Neptune Laboratory
Laboratory atat UCLA
UCLA has
hasbeen
beendone
doneusing
using
Modeling of the
Modeling of the PBWA in the Neptune Laboratory at UCLA has been done using
PIC code
code turboWAVE.
turboWAVE. Both
Both f/3
f/3 and
and f/18
f/18 focusing
focusinggeometries
geometriesthat
thatare
areused
usedininthe
the
the PIC
the PIC code turboWAVE.
Both
f/3f/3
andcase
f/18the
focusing
geometries
that are used
in the
simulated. For
For the
the
transverse
fieldsblow-out
blow-out
theplasma
plasma
experiment are simulated.
f/3 case
the transverse
fields
the
experiment are simulated. For the f/3 case the transverse fields blow-out the plasma
217
which has been shown to be one of the limiting factors in the production of large
amplitude plasma waves. For f/18 focusing the intensity of the laser and ionization
induced refraction of the laser beam may be the limiting factors in producing large
amplitude waves for electron acceleration experiments. The relativistic plasma wave
amplitudes are consistent with the phase space plots for accelerated electrons produced
by injecting an electron bunch. For experimental parameters an accelerating gradient
of ~2 GeV/m and ~1 GeV/m could be achieved for f/3 and f/18 configurations,
respectively. Clearly the f/18 focusing geometry produces a more uniform
accelerating structure for the PBWA than f/3 focusing.
ACKNOWLEDGMENTS
Simulations were done at NERSC on the IBM SP RS/6000. This work is supported
by U.S. DOE Grant number DE-FG03-92ER40727.
REFERENCES
1. Clayton, C.E., et al., Nuclear Instruments & Methods in Physics Research A, Amsterdam: Elsevier
Science, 1998, pp. 378-387.
2. Gordon, D.F., et al., IEEE Transactions on Plasma Science, 28, No. 4, 1135-1143 (2000).
3. Tochitsky, S.Ya., et al., Optics Letters, 26, No. 11, 813-815 (2001).
4. Tajima, T. and Dawson, J.M., Physical Review Letters, 43, No. 4, 267-270 (1979).
5. Anderson, S.G., et al., Advanced Accelerator Concepts, AIP Conference Proceedings 569, New
York: American Institute of Physics, 2000, pp. 487-499.
6. Augst, S., et al., Physical Review Letters, 63, No. 20, 2212-2215 (1989).
7. Fedele, R., et al., Physical Review A, 33, No. 6, 4412-4414 (1986).
8. Decker, C.D., et al., IEEE Transactions on Plasma Science, 24, No. 2, 379-392 (1996).
218