Further Developments For A Particle-In-Cell
Code For Efficiently Modeling Wakefield
Acceleration Schemes: QuickPIC
James H. Cooley1 and Thomas M. Antonsen Jr.
Institute for Research in Electronics and Applied Physics
University of Maryland, College Park, MD 20742
Chengkun Huang, Viktor Decyk, Shuoqin Wang, Evan S. Dodd, Chuang
Ren, Warren B. Mori
University of California Los Angeles, Los Angeles, CA 90095
Tom Katsouleas
University of Southern California, Los Angeles, CA 90089
Abstract We describe the present status of the development of a three-dimensional fully parallel
Particle-In-Cell (PIC) code, called QuickPIC. QuickPIC embeds a two-dimensional plasma
simulation code in a three-dimensional beam and/or laser simulation code utilizing the quasi-static
approximation [1,2]. In this paper, we will provide a brief review of the quasi-static approximation
and the current algorithm in QuickPIC, as well as the future plans to improve the code. Preliminary
results will be presented, which include comparing QuickPIC Plasma wakefield simulations with
OSIRIS [3] simulations, and which include laser propagation simulations.
1 [email protected]
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
232
INTRODUCTION
The challenge of modeling any plasma physics problem is to self-consistently
evolve the electromagnetic fields through Maxwell's equations and the trajectories,
currents, and charge densities of the particles through the relativistic equations of
motion. The technique that uses the fewest physics approximations is the fully explicit
particle-in-cell (PIC) technique in which the full set of Maxwell's equations is solved
together with the individual trajectories of each plasma particle. While PIC codes
make the fewest approximations of any plasma model, they are also the most
computationally demanding. Therefore, to use such a code on even some simple
problems requires that it be implemented using parallel algorithms. The group at
UCLA has been developing parallel PIC algorithms since 1987 and currently has a
variety of fully parallelized PIC codes. One of the electromagnetic codes, called
OSIRIS, has been used extensively to model advanced accelerator concepts
[4,5,6,7,8,9,10]. Unfortunately, even utilizing the largest supercomputers, it is still not
practical to use a parallel PIC code such as OSIRIS to model many interesting
problems in plasma based acceleration.
To model plasma wave excitation due to an intense particle or laser beam
traversing a plasma requires the use of a particle code since trajectories are nonlinear
and cross each other. The acceleration gradient in these wakes may approach 100
GeV/m. It turns out that to model a beam propagating through a few meters of
1016cm3 plasma, i.e., a 100 GeV stage, using OSIRIS will take -100,000 hours of
CPU time. Therefore, from a practical standpoint we need to develop a different type
of code to model this problem.
Fortunately, the beam evolves slowly while it moves through the meter of plasma.
We can therefore develop an approximate code, which exploits the disparity in time
scale between plasma wave excitation and beam evolution, while still keeping a high
degree of accuracy [2]. We have developed a code, QuickPIC, which is capable of
simulating many of the previously impractical problems using part of the quasi-static
approximation. Rapid development of QuickPIC was possible since it exploits the
existing numerical routines and expertise at UCLA, by constructing a framework for
parallel algorithms [11].
During the first stage of the development of QuickPIC [11], a modified set [12] of
the quasi-static equations were included. In this paper, we describe our plans for
implementing the full set of quasi-static equations. We start by reviewing the quasistatic approximation and the method we propose to use to solve them. Then we
describe the equation, which governs the evolution of the laser vector potential and
discuss the method used to solve this equation and also how the laser is coupled to the
plasma. Last we will review some preliminary results and some comparisons between
OSIRIS and QuickPIC [11].
233
PLASMA AND PLASMA BEAM MODEL
QuickPIC is based on the quasi-static approximation. We start from Maxwell's
equations in the Lorenz gauge for the wake fields generated in response to the moving
beam or laser pulse,
(1)
(—2 —2 - V 2 ) A = — j ,
c dt
(2)
c
where p and j are the charge and current densities.
The quasi-static equations are obtained by first transforming to coordinates (x, y,
s , %) where s = ct is the distance the beam has propagated, and % = ct - z measures
time in the speed of light frame. With this coordinate transformation the quasi-static
approximation consists of neglecting derivatives with respect to s, i.e., ds ~ 0 . Then
the full quasi-static field equations can be written as,
V2±A±=-4n>Jc,
(4)
' A "~~^~,
(5)
where \j/ = (l)-Az. Note that we have dropped Ampere's law for the axial component
of the vector potential using instead Eqs. (3) and (4) along with the gauge condition
(5).
Two different forms of the equations of motion are used depending on whether one
is considering plasma particles that pass quickly through the wake, or one is
considering beam or energetic particles that travel with the wake. In the former case
trajectories are advanced in £,
dfa
1
|
..^
, Ta
^^^
i/ a
_j ,
|z |
/s-\
d% tl-VT/e///c/
while in the latter case trajectories are advanced in
where V, P are velocity and momentum and the subscript b, e denote beam electron
and plasma electron. In Eqs. (6) and (7) we have introduced the laser electric and
magnetic fields, which are generated from the laser vector potential Aiaser. Also, the
plasma particles are assumed to respond to the ponderomotive (wave period-averaged
force) force of the laser and
f = l+—^ |lf+2kJ .
(8)
me |_
\c \
A simplified version of the equations (originally due to D. H. Whittum [12]) is
obtained in the case of beam driven wakes by noting that the longitudinal beam current
dominates both the transverse current of the beam and that of the plasma. Therefore,
one can neglect the transverse current j ± , and hence the transverse component of the
234
wake vector potential A ± . The longitudinal plasma current can also be neglected
compared to the longitudinal beam current. To lowest order, the above quasi-static
equations can then be reduced to:
V^0 = -4np = -4n(pb + pe + pion ) ,
p,,J,
(9)
(10)
V ± <P,
(ID
The above equations only involve the two dimensions perpendicular to the beam
propagation direction, so Eqs. (9) and (10) can be solved in 2D space. Once the
potentials are calculated, the momentum and position of the particles can be updated
using Eqs. (11) and (12). For plasma electrons, this is done for every time step A£,
which needs to resolve the plasma frequency. While for beam electrons, the time step
is As , which only needs to resolve the betatron frequency or the hosing growth rate.
The above algorithm was adopted in the first phase of QuickPIC development.
Eqs. (9) and (10) are solved in Fourier space by a parallel Poisson solver, either with
periodic or conducting boundary conditions. We also added three major corrections.
These corrections include: (a) a relativistic plasma electron pusher; (b) a correction to
the plasma charge density due to the longitudinal motion of plasma electrons by using
the following deposition scheme:
p -Y^^>
Pe =
v^i-ve//i/c
and (c) an addition of the parallel plasma current,
l Z ^_ f .
e
"
(13)
(14)
V~l-Ve//i/c
These three corrections can be explicitly turned on or off from the input.
Currently, QuickPIC can achieve results reasonably similar to those from more
computational intensive full electromagnetic code such as OSIRIS for a medium
beam/plasma density ratio. We find that in this parameter regime QuickPIC gives
results that are within a few tens of percent of those from OSIRIS but at a saving of
1000 times in computing time. As the beam density increases, transverse currents
begin to play an important role, and the accuracy of the above, reduced quasi-static
description deteriorates. We are currently implementing the full quasi-static equations
described in Eqs. (3)-(7) in QuickPIC. One difficulty that arises in the full quasi-static
description is that the momentum depends on the wake vector potential through E in
Eq. (6), and the wake vector potential depends on the momentum through j in equation
(4). This requires the implementation of an iteration scheme on the short time scale
A£ . This correction is also being implemented with code debugging underway. With
the full equations implemented we expect to get much more accurate results, based on
other full quasi-static implementations [2], at only modest decrement to the overall
time performance.
235
LASER EVOLUTION
In this section we focus on the numerical solution of the laser evolution equation.
We assume an envelope approximation in which the laser vector potential is expressed
as a slowly varying complex amplitude modulating a plane wave,
YYIC
(15)
In the speed of light frame the wave equation for the laser becomes,
where, the right hand side represents the response of the plasma electrons and the
coefficients fa and fa account for the group velocity and dispersion of any background
gas that may be present. We note that in the absence of background gas fa=fa=Q, but
nevertheless, Eq. (16) still accounts for plasma group velocity and dispersion due to
the mixed s-% derivative.
We solve equation (16) using a split-step algorithm where we separate the
evolution operator into two separate operators, one involving transverse derivatives
and one involving axial derivatives,
a
1
(17a)
(17b)
Solution of Eq. (16) is then obtained by alternately solving Eqs. (17a) and (17b) for a
single time step As. The method is robust because each of the Eqs. (17) can be solved
implicitly. The axial equation (17b) is solved using finite difference methods on a grid
in £ The transverse equation (17a) is solved in Fourier space. This requires two 2D
FFTs at each grid point in % and at each time s. Due to the mixed derivative appearing
in (17b) there is a stability requirement !%&%> 1. In addition, the presence of the
plasma causes a frequency shift. This places a condition on the time step for accuracy,
namely As < kQ / k2p .
Time centering of the split step algorithm and communication with the 2D PIC
portion of QuickPIC is illustrated in Fig. 1. Communication between the laser
evolution solver and the plasma particle solver occurs on the grid in time s. Thus, we
center Eq. (17b) at this time, and advance the transverse operator (17a) between time
grids. This then necessitates a half time step evaluation of Eq. (17b) to provide the
ponderomotive potential Vp = rac2y|a| to the plasma particle solver. This provides
second order accuracy in ds for the algorithm.
The Laser evolution algorithms have been tested for vacuum propagation and for
propagation in a nonlinear medium characterized by a second order modification of
the dielectric constant. The results for these tests all agree with analytic results, where
236
Second Order Accurate Split-Step Scheme
time-s
s+ds
FIGURE 1. Diagram of laser split-step scheme coupled to plasma. Note the half time
step axial push, this push is necessary in order to time center the laser with the plasma
evolution.
available, e.g., beam spreading due to group velocity dispersion and diffraction in
vacuum. The results for self-focusing are qualitatively correct and we are currently
benchmarking it against WAKE [2] to quantify these results. We are also in the
processes of coupling the vector potential to the plasma response as well as putting the
laser evolution process in the full parallel QuickPIC framework as described in [11]
PRELIMINARY RESULTS FOR PLASMA
We have used QuickPIC to conduct some benchmarking runs to simulate E157/
afterburner beam and plasma parameters.
For the simulations of El 57 experiments we
used a plasma density of 2.1xl014 cm"3 and a beam charge of l.SxlO10 electrons with a
gaussian distribution, ar=0.07mm and c^=0.63mm and with a beam energy of SOGev
(y=60,000). The dimensions of the simulation domain were 2.9x2.9x6.4mm, divided
into 256x256x256 cells with 128x128x320 particles in the beam and 512x512 particles
in the plasma. Figure 2 shows a comparison of the results from QuickPIC and OSIRIS
for the wake field for the El57 experiment.
237
For the Afterburner experiment we use a plasma density of 2.1xl016 cm"3 and a beam
size of ar=0.007mm and c^=0.045mm. The domain size is 0.59x0.59x0.59mm with
256x256x256 cells; other parameters are identical to E157. Figure 3. Shows a
comparison between QuickPIC and OSIRIS for the afterburner parameters. QuickPIC
has also been used to examine the hosing instability associated with the beam centroid
oscillation as seen in Figure 4.
CONCLUSION AND FUTURE DIRECTION
We have described a new program, which is capable of efficiently simulating
interactions in both plasma and laser wakefield accelerator schemes over extended
interaction times. The assumptions made in the approximations are valid for a wide
range of accelerator schemes and the accuracy of the model is adequate both for
experimental design and experimental predictions and simulations.
Future efforts will focus both on improving the accuracy of the plasma particle
evolution and on implementing a fluid plasma simulation for examining weakly
nonlinear interactions. Including non-plasma generated external fields e.g., magnetic
focusing fields, is under development for advanced accelerator designs with multistage plasma wakefield accelerator. Additionally, including ionization for both the
particle and fluid models will be implemented.
This work was supported by DOE under contract numbers DE-FC02-01ER41179,
DE-FG03-92ER40727 and DE-FG02-97ER41039 and by NSF grants Phy-0078508,
Phy-0078437, and ECS-9617089, and by LLNL under grant # W-07405-ENG48.
'5>
-200
-
FIGURE 2.: Longitudinal accelerating field (El57 parameters) from QuickPIC(dot-dash, conducting
boundary condition; dash, periodic boundary condition) and OSIRIS (solid)
238
FIGURE 3. Longitudinal accelerating field (Afterburner parameters) from QuickPIC(solid and dash,
plotting accelerating field at different time steps) and OSIRIS (dot-dash), the Y-axis is accelerating field
in Gev/m.
FIGURE 4. beam centroid oscillation (Afterburner parameters)
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