Recent Advances and Some Results in PlasmaBased Accelerator Modeling
W.B. Mori
Departments of Physics and Astronomy and of Electrical Engineering
University of California, Los Angeles, Los Angeles, CA 90095
Abstract. Simulation, using particle-in-cell (PIC) methods, has played a critical role in the
evolution of the field of plasma-based acceleration. Early on, simulations allowed the testing of
new ideas using so-called cartoon parameters. These simulations were done in either one or twodimensions using single processor supercomputers. Through the development of new algorithms
and parallel computing, today, we can now use PIC simulations to model the full-scale of
ongoing experiments in three-dimensions. These experiments are attempting to accelerate
electrons to ~1 GeV. In this article, I will present recent results in which simulation results are
compared to experiment and I will discuss the future challenges in advanced accelerator
modeling. Principally, these are 1.) to be able to model a 100+ on 100+ GeV collider in threedimensions and, 2.) to develop more efficient, yet still accurate, algorithms so that simulation can
be used for real-time feedback with experiment.
INTRODUCTION
Computer simulation is now considered on equal footing with experiment and theory
as the third discipline in science. The evolution of computer simulations over half a
century, from an intellectual curiosity to an essential tool of scientific discovery, has
been the result of clever algorithm development, clever software development,
physical insight, and tremendous advances in computer hardware. In fact, computer
simulation is rapidly moving forward as a means of validating the design of expensive
machines before large capital expenditure is spent to build them.
The above basically holds true for all areas of science. In this article, I will show
how computer simulation is impacting plasma-based acceleration. In particular, I will
quickly review how far we have come, assess the current status of plasma-accelerator
modeling, and describe what challenges lie ahead while offering possible paths toward
successfully meeting these challenges.
To begin, it is important to understand the problem that is being modeled. There are
four basic concepts being studied in plasma-based acceleration. These are the Laser
Wakefield accelerator (LWFA [1]), the Self-Modulated Laser Wakefield Accelerator
(SMLWFA)[2], the Plasma Beatwave Accelerator (PBWA)[3], and the Plasma
Wakefield Accelerator (PWFA)[4] These ideas are summarized pictorially in figure 1.
It is worth noting that the late John Dawson was the inventor of each of these schemes.
In each of these schemes a driver, either a laser (LWFA, SMLWFA, PBWA) or a
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
11
particle
The phase
particle beam
beam (PWFA),
(PWFA), traverses
traverses a plasma creating a plasma wave wake. The
velocity
velocity of
of the
the wake
wake is
is roughly
roughly equal to the velocity of the driver in complete analogy
with
wake
with the
the water
water wave
wave wake
wake left
left behind a motor boat in a lake. The plasma wave wake
has
has aa longitudinal
longitudinal electric
electric field
field so it can efficiently
efficiently accelerate a trailing bunch of
charged
charged particles.
particles. Since
Since the
the drivers are usually
usually moving at nearly the speed of light (if
one
one wanted
wanted to
to accelerate
accelerate protons for example then the driver might have a lower phase
velocity) then
then the
the phase
phase velocity
velocity of the driver moves nearly at the speed
velocity)
speed of
of light,
light,
making itit ideal
ideal to
to accelerate particles to relativistic energies. Plasma-based
making
accelerators are
are of interest because the accelerating fields in the
accelerators
the plasma
plasma wave
wave
structures can
can in
in principle
principle be
be many
many orders
orders of
of magnitude
magnitude above
structures
above current
current RF
RF technology.
technology.
A nice
nice review
review of
of plasma-based
plasma-based acceleration
acceleration prior
prior to
to 1996
A
1996 can
can be
be found
found in
in ref.
ref. [5]
[5] by
by
Esarey et
et al.
al.
Esarey
Plasma Wake
Wake Field
Accelerator(PWFA)
•l Plasma
Field Accelerator(PWFA)
A/V
A high
high energy
energy electron
electron bunch
bunch
A
•l
Laser Wake
Wake Field
Accelerator(LWFA)
Laser
Field Accelerator(LWFA)
A single
single short-pulse
short-pulse of
of photons
photons
A
Plasma Beat
Beat Wave
Wave Accelerator(PBWA)
Accelerator(PBWA)
•l Plasma
Two-frequencies, i.e.,
i.e., aa train
train of
of pulses
pulses
Two-frequencies,
Self Modulated
Modulated Laser
Wake Field
•l Self
Laser Wake
Field Accelerator(SMLWFA)
Accelerator(SMLWFA)
Raman forward
forward scattering
scattering instabili
instability
Raman
evolves
evolves to
to
FIGURE 1.
1. Simple
Simple schematics
schematics of
of the
the four
four basic
basic plasma-based
plasma-based accelerator
FIGURE
accelerator schemes.
schemes.
Based on
on the
the above
above description,
description, to
to model
model the
the full
full scale
scale of
of aa plasma-based
plasma-based
Based
accelerator,
one
needs
a
code
(or
codes)
that
can
model
the
evolution
accelerator, one needs a code (or codes) that can model the evolution of
of the
the driver,
driver, the
the
generation and
and evolution
evolution of
of the
the wake,
wake, and
and the
the acceleration
acceleration of
generation
of the
the trailing
trailing bunch
bunch of
of
particles. ItIt turns
turns out,
out, perhaps
perhaps not
not surprisingly,
surprisingly, that
that in
in most
most cases
particles.
cases to
to do
do this
this properly
properly
one needs
needs particle
particle based
based models.
models. That
That is,
is, one
one
one needs
needs to
to follow
follow self-consistently
self-consistently the
the
trajectories
of
particles
in
their
self-consistent
fields.
The
reasons
trajectories of particles in their self-consistent fields. The reasons for
for this
this are
are that
that in
in
many cases
cases the
the wake
wake excitation
excitation process
process is
is highly
highly nonlinear
nonlinear with
many
with particle
particle trajectories
trajectories
passing through
through each
each other
other and
and that
that any
any reasonable
reasonable beam
beam loading
passing
loading scenario
scenario will
will require
require
very
tight
spot
sizes
which
cannot
be
modeled
using
fluid
descriptions.
very tight spot sizes which cannot be modeled using fluid descriptions. Fluid
Fluid
12
descriptions [2,6] are useful for modeling certain pieces of the problem, e.g.,
wakefield excitation in a channel by weak to moderate intensity pulses; however,
reduced description particle techniques are now incredibly fast and accurate as well.
Simulations are also an indispensable bridge between theory and experiment. The
wakefield generation process can be very nonlinear making exact analytic expressions
impossible. Furthermore, experimental observables are limited. In most cases the
diagnostics involve particles or photons hitting a detector, therefore, the exact
acceleration mechanism and the fields in the plasma need to be inferred. It is worth
noting that just as there has been an explosion of computing power, there has also
been an explosion of experimental diagnostic techniques. This could be the topic of an
analogous paper.
This article is therefore the story of the evolution of particle-in-cell tools used to
model advanced accelerators, their successes, and the possibilities for the future. I will
discuss the advantages and disadvantages with fully explicit PIC, ponderomotive
guiding center PIC, and quasi-static PIC. I will also show a few sample results from
each of these approaches.
A BRIEF REVIEW AND A LITTLE HISTORY OF THE PIC
METHOD
Fully explicit PIC simulations [7,8]follow the self-consistent trajectories of a group of
charged particles due to forces from the electromagnetic fields calculated from the
charge and current densities due to these particles. It is also important to use the
appropriate boundary and initial conditions. The fields are calculated from the full set
of Maxwell's equations. Therefore, to the extent that the time steps and cell sizes do
not eliminate physics, such codes make "no" approximations. The particles are
weighted to the grid so they have an effective size on the order of the grid. This
modifies large angle Coulomb scattering allowing one to model a plasma problem
using fewer particles than in an actual experiment. However, this alteration of the
physics is well understood and the statistical mechanics of such a system of "finite
size" particles can be studied analytically[8]. The drawback to fully explicit PIC
simulations is that they are also the most computationally intensive.
Since the seminal paper on the subject by Tajima and Dawson in 1979[1], fully
explicit PIC simulations have played an important role in the development of plasmabased accelerators. Their initial one-dimensional simulations followed 5000 particles
on a 512 cell grid and were run for 500 to 1000 time steps. Today such a run would
take less than 10 seconds on a cheap desktop computer. By the early 80's, twodimensional simulations on a 600x128 grid with 106 particles were being run for
10,000 or so time steps[9]. These simulations were done on Cray XMP vector
processors using the code WAVE[10]. Today's desktop computers have more
computing power than the Cray XMP supercomputers. These simulations were useful
for testing out new ideas and modeling experiments using so-called cartoon
parameters. For example, the ratio of the laser frequency to the plasma frequency, i.e.,
the frequency ratio, might be 30 in the experiment and the ratio would be chosen to be
5 in the simulation.
13
In these early simulations the laser pulse length was much longer or comparable to
the plasma length. In the late 80 's with the advent of the PWFA concept, simulations
in which the drive beam propagated much longer distances than the beam's length
were of interest. Since only the region around the driver is of interest, it was very
advantageous to develop an algorithm that followed the driver. One solution, put forth
by Joyce et al.[l 1], was to write a code which solved the field equations after making
a mathematical transformation into the frame moving at the speed of light. This code
was specifically written for studying particle beam drivers. Another solution, put forth
by J.J.Su and co-workers [12]was to use a moving window in which cells were taken
from the back of the box and added to the front. Fresh particles were added to the front
and the fields were initialized to zero. This cyclic mesh algorithm was implemented
into an r-z 2D cylindrical code called ISIS[13]. A Cartesian geometry version of the
moving window was developed by JJ. Su, C.D.Decker et al., in the early 90's for
studying the LWFA and SMLFWA schemes[14].
Another breakthrough occurred in the early to mid 90's when K-C. Tzeng, JJ. Su
and co-workers set out to write a parallelized version of the moving window
algorithm. The code ISIS (a serial code) was stripped (a major endeavor), but the basic
particle push, current deposit, and field solve routines were kept. The new code,
called PEGASUS[15], was written in Cartesian geometry and included the ability to
launch laser pulses. This code was used extensively to model many of the initial shortpulse laser plasma experiments and studied the generation of self-trapped electrons in
the SMLWFA concept[16]. During this time a typical simulation followed 25 million
particles on a 8192x512 grid for 10,000's of time steps. These simulations also
utilized the moving window. The original simulations were done on an SP 16 node
computer at UCLA.
The next breakthrough was the successful development of a code that used a quasistatic PIC description. The simplifying idea is that the driver evolves on a much
slower time scale (propagation distance) than the frequency (wavelength) of the wake.
In the moving window, the driver and hence the wake directly behind evolve on
distances of the order of the Rayleigh length (laser pulse) or betatron wavelength
(particle beam). Both of these distances are very large compared to the pulse length of
the driver.
Interestingly, there were two independent, yet simultaneous developments in the mid
90's. Whittum [17]developed a quasi-static (frozen field) approach for particle beam
drivers, while Mora and Antonsen[18] developed a code for laser drivers. The quasistatic assumption was used in both codes, but the implementation and level of
approximation were very different. Whittum's code was 3D (he used it to study
particle beam hosing which is inherently a 3D process), while Mora and Antonsen's
code (WAKE) was 2D (Cartesian or cylindrical). Furthermore, Whittum was
interested in the so-called blowout regime in which the driver is much narrower than a
skin depth (c/cop). As a result, he assumed that the plasma electrons only moved
transversely and not along the parallel direction, i.e., the direction of the driver (there
were other approximations as well). On the other hand, in WAKE the full dynamics of
the plasma electrons (and ions) were kept, but the high-frequency motion in the laser
field was averaged out. Therefore, as long as the plasma electrons are not self-trapped
then the model uses the proper ponderomotive force. It is sometimes referred to as the
14
ponderomotive guiding center motion of the particles. The algorithm in WAKE was
fully relativistic, but it breaks down if there are self-trapped electrons or if the axial
velocity of a single particle in the laser's field approaches the speed of light. If the
normalized laser amplitude exceeds ~3 then the algorithm typically has problems. The
basic quasi-static equations will be briefly given in the next section.
In the late 90's there were several further significant developments to plasma-based
accelerator modeling. First, several groups set out to develop parallelized 3D PIC
codes. The most well known are OSIRIS (Hemker, Tsung, Fonseca, Lee et al.)
[19]and VLPL. (Puhkov)[20]. These codes are written in modern object programming
languages (OSIRIS in Fortran95 and VLPL in C++). These codes allowed one to
model the full-scale of actual experiments in 3D without any fitting parameters. For
example, OSIRIS has been used extensively to model the E-157 and E-162
experiments at SLAC [21]. To model these experiments, requires following 10 to 64
million particles on a 3 to 10 million cell grid for 200,000 to 300,000 time steps.
These simulations require -10,000 node hours on the SP machine at NERSC. It
should also be noted that the OSIRIS effort also resulted in the first parallel 2D
cylindrically symmetric PIC code and this was critical during the early stages of the E157 collaboration [22].
Second, Gordon et al., implemented a ponderomotive guiding center algorithm into a
standard fully explicit PIC code [23]. Such a code need not resolve the laser
frequency, but it still must resolve the plasma frequency. This code called
turboWAVE, was originally 2D. It used the WAVE algorithm and was parallelized. I
will summarize this algorithm in the next section.
These advances have also coincided with a tremendous increase in computer power
so that today simulations can follow a billion particles for 10,000 time steps. These
advances have also led to a tremendous increase in the amount of data that is
generated. Therefore, to use these codes successfully one needs sophisticated
diagnostic and visualization packages. One such package is OSIRIS_analysis which
was originally developed by F.A. Fonseca[19].
Today there are five major 3D parallel PIC codes being used by members of the
plasma- based accelerator community. These are OSIRIS [19], VLPL [20],
turboWAVE [23], VORPAL (J.Cary et al.) [24], and a code developed by H.Ruhl
[25]. It is worth noting that while there are similarities, there are also differences
between each of these codes. For example, turboWAVE solves for the potentials (|)
and A, and has a Poisson solve to maintain charge conservation (the algorithm is
identical to that in the old WAVE code); and as mentioned earlier it has a
ponderomotive guiding center option. On the other hand, the other codes all solve for
the E and B fields directly and avoid the need for a Poisson solve by using a rigorous
charge conserving current deposit solve (some codes have options for what is called
the Langdon improvement to the Marder's scheme[26]). There is not a unique particle
weighting scheme, nor a unique way to calculate the current. Some codes have
several options for depositing the current, e.g., in OSIRIS one can choose between the
current deposit schemes used in ISIS [13]or TRISTAN [27]. Some of these codes have
field and impact ionization (the most detailed version of these is in a code called
OOPIC [28]which is a parallelized, 2D code), as well as various filtering or
smoothing options. The boundary conditions and initialization routines are also not the
15
same. I make these points because each of these choices can be a tradeoff between
speed, accuracy, simplicity, and noise. For these reasons it is useful to have several
codes so that results can be checked and rechecked. This is similar to how scientific
knowledge is advanced through laboratory experiments. If one lab's results cannot be
reproduced in another lab then the results are useless. In complete analogy, if the
results from one code cannot be reproduced by another code, then the results are of
little value.
The most recent breakthrough in plasma-based accelerator modeling has been the
development of a parallelized 3D quasi-static PIC code. This code, called
quickPIC[29], embeds a 2D parallel electrostatic PIC code inside a 3D parallel
electrostatic PIC code. In its current version it uses Whittum's approximations and is
therefore limited to studying PWFA in the blow-out regime. However, it has already
been useful in studying physics for longer propagation distances than can be modeled
using OSIRIS, and in giving insight into the physics in the E-157 and E-162
experiment. In the next section I will describe the algorithm in a bit more detail and
describe how we plan on adding the full quasi-static description into it. More detail
can be found in ref. [29], and in the paper by Cooley et al. in this proceeding. It is also
noteworthy that this code has been applied to the study of the electron-cloud
instability. This instability is a topic of importance in circular accelerators. More
detail can be found in the paper by Ghalam et al., in this proceedings.
As one can see, there is now a tremendous effort in PIC modeling of plasma-based
acceleration. This is very gratifying to those of us who have been in this business
almost from the beginning, and it made the late John Dawson chuckle with delight. In
fact, this community has produced more 3D parallel fully explicit PIC codes than any
other. In the 70's when the concepts behind multi-dimensional PIC codes were being
developed, researchers were driven by the need to understand the types of laserplasma interactions that occur in inertial confinement fusion. Today, many of the ideas
and visualization advances for 3D parallel PIC codes come from the plasma-based
accelerator community.
COMPUTER REQUIREMENTS AND BASIC ALGORITHMS
In this section I will give a simplified review of the various PIC algorithms, i.e.,
fully explicit PIC, ponderomotive guiding center PIC, and quasi-static PIC. I will give
simplified estimates on how much computing savings is obtained from each level of
approximation and give personal opinions on the applicability of the various
approximations.
Computing requirements:
One can estimate the computer time and memory requirements to model a stage that
provides a specific amount of energy gain. For example, in this section, I discuss the
requirements for a IGeV stage. First, the cell size must be chosen to resolve the
smallest spatial scale of interest. In the PWFA this is the wavelength of the wake (the
16
collisionless
), while
in the
the LWFA
LWFA (or
(or SMWLFA
SMWLFA or
or PBWA)
PBWA) this
this isisthe
the
collisionless skin
skin depth,
depth, c/co
c/wpp),
while in
laser
wavelength
(to
model
some
SMLWFA
stages
one
also
needs
to
resolve
the
laser wavelength (to model some SMLWFA stages one also needs to resolve the
Raman
backscatter
plasma
wave
wavelength).
In
the
transverse
direction,
one
needs
to
Raman backscatter plasma wave wavelength). In the transverse direction, one needs to
resolve
and/or the
the drive
drive bunches
bunches spot
spot size.
size. IfIf one
one uses
uses the
the
resolve the
the collisionless
collisionless skin
skin depth
depth and/or
moving
an axial
axial box
box length
length of
of aa few
few bunch
bunch lengths
lengths or
or wake
wake
moving window
window then
then one
one needs
needs an
wavelengths
(~
20
c/co
)
and
a
transverse
dimension
of
a
few
collisionless
skin
wavelengths (~ 20 c/wpp) and a transverse dimension of a few collisionless skin
depths
depend on
on the
the strength
strength of
of the
the driver
driver and
and are
are
depths (~2jtc/eo
(~2pc/wpp).). These
These values
values depend
sometimes
simulations parameter
parameter study.
study.
sometimes determined
determined empirically
empirically through
through aa simulations
In
amplitude that
that is
is still
still below
below the
the wavebreaking
wavebreaking
In addition,
addition, one
one wants
wants aa large
large wake
wake amplitude
[30]
assume that
that eE
eEzz=.
=. 5mcw
5mccopp.. IfIf one
one wants
wants to
to gain
gain an
an
[30] amplitude,
amplitude, ~mceo
~mcwpp/e,
/e, here
here we
we assume
energy
acceleration length
length of
of ~2000mc
~2000mc22/./. 5mcw
5mccopp
energy of
of aa GeV
GeV then
then one
one needs
needs to
to have
have the
the acceleration
=4000
=4000 c/eop.
c/wp.
When
smaller than
than the
the smallest
smallest
When using
using a fully
fully explicit PIC code the time step must be smaller
cell
assume Dt=.02).
At=.02). In
In the
the PWFA
PWFA the
the
cell size
size as
as required
required by the Courant condition (we assume
cell
size
is
typically
is
.05
c/co
and
in
the
LWFA
it
is
roughly
the
frequency
ratio
p
cell size is typically
c/w
LWFA it is roughly the frequency ratio
smaller.
~4 particles
particles per
per cell.
cell. Putting
Putting all
allof
ofthis
this
smaller. In
In addition, in a 3D run, one can use ~4
13
13
together,
the
number
of
particle
pushes
is
=.5
x
10
.
Assuming
a
speed
of-7.5
together, the number
10 . Assuming a speed of ~7.5
jis/particle/step
speed of
of OSIRIS
OSIRIS on
on the
the SP
SP
ms/particle/step for the
the entire loop in 3D (this is the current speed
at
NERSC)
gives
an
estimate
of-10,000
node
hours
on
an
SP.
We
usually
run
several
at NERSC)
of ~10,000
on an SP. We usually run several
short
(number of
of cells)
cells) until
until the
the
short runs
runs where
where we reduce the transverse dimension (number
physics
simulation over
over the
the full
full stage.
stage.
physics changes.
changes. Then we use these parameters for a simulation
The
(the 2D
2D algorithms
algorithms are
are typically
typicallyaa
The above
above arguments are summarized in figure 22 (the
factor
factor two
two faster).
faster).
Beam-driven wake*
Dz
Az
Dy, Ax
Dx
Ay,
Dt
At
grids in
## grids
inzz
#
grids
in
x, yy
# grids inx,
steps
## steps
Nparticles
N
particles
Fully
Fully Explicit
Explicit
£< .05
.05 c/w
c/wp
p
£< .05
.05 c/w
c/Wp
p
£< .02
.02 c/w
c/wp
p
≥350
>350
≥150
>150
≥2
x 105b
>2x10
~.25
x 108 (3D)
-.25x10°
(3D)
6
~1
x
10
(2D)
~1x10 6 (2D)
13
~.5
x 10l o (3D)->10,OOQhrs
(3D) - ≥ 10,000 hrs
~.5x10
11
1
~1
x 10 1(2D)
- ≥ 75 hrs
~1x10
(2D)->75hrs
Particles xx steps
steps
Particles
22
*Laser-driven GeV
GeV stage
o/wp )p)=1000
*l_aser-driven
stage requires
requires on
on the
the order
order of
of (w
(o)o/w
=1000 xxlonger,
longer,
however,
the
the
resolution
can
usually
be
relaxed.
however, the the resolution can usually be relaxed.
FIGURE 2. Computer time necessary to model a IGeV stage.
17
To simulate, a lOOGeV stage will require 100 times more computing time (but not
more memory). To simulate a IGeV LWFA stage will require finer spatial resolution
(and hence more memory) since the laser wavelength needs to be resolved. Assuming
that the resolution increases by co0/cOp in the propagation direction and the number of
particles per cell remains fixed, then a LWFA run will require ((jo0/cop)2 more computer
time (number of particles and number of time steps both go up). In reality, the
resolution does not have to increase by this much. One should use -20 cells per laser
wavelength, Ax=.3c/eo0, which corresponds to .05/.3 (co0/cOp) finer resolution. Using
the ponderomotive guiding center approach makes modeling a PWFA and a LWFA
stage comparable. However, depositing the susceptibility and solving the envelope
equation slows down the code.
Fully explicit PIC:
The fully explicit PIC algorithm is straightforward. Basically, a chosen number of
particles are loaded onto a grid. The charge and current densities can then be
calculated by "depositing" the particles onto the grid; p = Vq and j = ^qv. These
particles
particles
current and charge densities are used to advance the fields via Maxwell's equations,
_ V x £ = I^
c ft
and V x B = ^j + I^
c
c dt
The updated fields are used to advance the particles to new positions and velocities via
the relativistic equation of motion,
dP_
dt
(-
vxB
As alluded to earlier, although simple in concept, there are many subtle issues with
solving these equations on a computer, e.g., the deposited p and j may not satisfy the
continuity equation. However, there are numerous papers and books on these issues. In
my opinion the best is Birdsall and Langdon [8].
Ponderomotive guiding center PIC:
When using a fully explicit PIC code the time step must be smaller than the smallest
cell size. Therefore, when modeling a laser-plasma accelerator scheme the smallest
cell size must be ~.3c/eo0 in order to resolve the laser wavelength (-20 cells per X). In
the ponderomotive guiding center scheme one separates out the fields into plasma
18
fields and laser fields. These equations were clearly derived by Antonsen and Mora
[18].
The plasma fields are solved explicitly and the time averaged laser field is solved
using an envelope, i.e, a paraxial wave type, equation,
The plasma particles are pushed using the plasma fields and the ponderomotive force
from the laser's envelope as follows
dP
(- vxB\ 1 q \ ,2 ,
— = q\E + — — — - — — V a where r7 =
dt \
c ) 47 ' '
The particles are weighted onto the grid in the standard way to evolve the plasma
fields and are used to calculate a susceptibility term for the laser's envelope equation
via, x = -^! ~~ • The code turbo WAVE has the option of using this scheme.
The turboWAVE algorithm is based largely on the well tested code WAVE. Although
the explicit portions of turboWAVE use similar algorithms as WAVE, it is an entirely
new code written in C++ and it makes use of object oriented techniques.
When using the ponderomotive guiding center approximation the smallest spatial
scale length is now the wavelength of the wake. In addition, satisfying the Courant
condition requires that the time step resolve the plasma frequency. Therefore, one
might think that using such a code would lead to a savings of (co0/cOp)2; however, the
savings is not quite this much it is found that one still needs to resolve spatial scales of
the harmonics of the wake since it typically gets nonlinear. It is also worth noting that
this approach also fails when self-trapped electrons are produced since the expression
for the ponderomotive force does not apply for this group of particles. In addition, the
envelope approach cannot model Raman backscatter which can also be an important
process in producing self-trapped electrons.
Quasi-static PIC:
Another level of approximation is to use the quasi-static or frozen field
approximation. There are various ways of attempting to implement such an
approximation, e.g., the choice of gauge. One such implementation is that in
QuickPIC, which starts from the Maxwell equations in Lorenz gauge (the code WAKE
uses the transverse Coulomb gauge which is more appropriate for large spot sizes),
(~T-V 2 )4> = 4,rp
(1)
(ll-v>)A=^j
(2)
19
In (x, y, s, £) coordinates, where s = z (z is the direction in which the beam is
moving),%=t-z/c, then the quasi-static approximation amounts to assuming ds « 0.
Then a set of full quasi-static equations can be written as,
V*0 = -4jtp
ft)
*
and the equations of motion are,
(5)
ds
,
(6)
In Eq. (5) and (6), *P = 0 - 4/ > where A/7 is the longitudinal component of vector
potential, q is the charge of particle, and p is the charge density. V, P are velocity
and momentum respectively. The subscript b, p denote beam, plasma respectively.
Notice that in the blow-out regime of the PWFA, the longitudinal beam current
dominates the transverse currents by the beam and the plasma. For this situation one
can neglect the transverse current j± , and hence the transverse component of the
vector potential A ± . The longitudinal current by the plasma can also be neglected.
The above quasi-static equations then reduce to:
V* 0 = -4jtp = -4jt(pb + pe + pion ),
P;J,
(8)
(9)
(10)
The equations above only involve two dimensions, which are perpendicular to the
beam propagation direction, so Eq. (8) and (9) can be solved in 2D space. Once the
potentials are calculated, the velocity and position of particles can be updated using
(10) and (1 1). 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 resolve the betatron frequency or the hosing growth.
The above algorithm is identical to that of D. H. Whittum [17] and was adopted in
the first phase of QuickPIC development. Eq. (8) and (9) are solved in Fourier space
by a parallel Poisson solver, either with periodic or conducting boundary conditions.
There are also three additional effects that can already be included. These corrections
include: (a) a relativistic plasma electron pusher; (b) a correction to the plasma charge
20
density due to the longitudinal motion of plasma electrons by using the following
deposition scheme,
p
'
_Iy_£*_,
(12)
V^l-Vejc
and (c) the inclusion of the parallel current of plasma ,
.
1 y ft.K/// z£
le
"~vli-ve/lllc
(13)
The second phase in the development of QuickPIC is still ongoing, and the goal is to
implement the full quasi-static equations as described in Eq. (3)-(7), as well as the
ponderomotive guiding center approximation. The details of this effort is reviewed in
the paper by J.Cooley et al, in these proceedings.
The quasi-static approximation can reduce the computational costs substantially. For
example, consider the IGeV PWFA stage discussed above. Using a quasi-static code
the grid size remains the same, but the time 3D time step is reduced from 300,000 to a
few hundred (one needs only to resolve a betatron oscillation and there were three in
the E-157 experiment). To estimate the number of particle pushes one recognizes that
within a 3D time step the 2D code goes through Nz (the number of 3D cells in the
beam propagation direction) time steps. So if the number of particles per cell is similar
between that in the full PIC code and that in the 2D part of the quasi-static code, then
the savings is roughly the reduction in the number of 3D time steps. As noted above
this savings is -three orders of magnitude. However, the full quasi-static code will
require iteration loops that might slow the code down by an order of magnitude.
The limitations of the quasi-static model are that it is limited to the blow-out regime
and beam drivers with moderate charge, that the full quasi-static code is not working
yet, and that it cannot straightforwardly include self-trapped electrons. However, I
believe such a code will become the workhorse for the plasma-based community
within the next few years.
SAMPLE RESULTS
In this section I will briefly give a few results from each of the PIC type codes. I will
show some 3D OSIRIS result [31] on beam-plasma and laser-plasma interactions,
some 2D turboWAVE simulation results on PBWA, and some quickPIC results on
electron beam hosing. My goal is not to show a great amount of detail, but to show
that these codes are being used to study real problems and are providing a new
paradigm of research in which full-scale 3D modeling is now closely coupled to
experiments. I am also only showing results from work that I am closely involved
with. More details about these results, as well as results from others, can be found in
these proceedings.
A powerful example of the usefulness of full PIC is its role in understanding what is
now referred to as particle beam refraction [31]. When a tightly focused electron beam
traverses a plasma it blows out the background plasma electrons leaving behind an ion
column. If the blowout is symmetric then from Gauss' Law the force on the electron
beam comes only from the space charge of the ion column inside the beam. When the
beam crosses a gas/plasma boundary then there can be no ions on the vacuum or gas
side. As a result, the electron beam will be attracted, i.e., bent, back towards the
21
plasma. Due to its analogy with how a pencil appears bent at the water/air interface
plasma.
Dueintoa cup
its analogy
pencilparticle
appears
bentrefraction.
at the water/air interface
when
it sits
of water,with
thishow
was acalled
beam
when
sitsobtain
in a cup
water, this
wasLaw
called
beam
One itcan
an of
effective
Snell's
forparticle
this effect
byrefraction.
calculating the impulse the
Onegets
can obtain
an asymmetric
effective Snell’s
Law for(and
this effect
by calculating
the impulse
the
beam
from the
ion column
the blown
out electrons).
This effect
beam
gets
from
the
asymmetric
ion
column
(and
the
blown
out
electrons).
This
effect
was studied as part of the E-157 experiment at SLAC and it was also modeled one-to7
was without
studied as
part
of theparameters.
E-157 experiment
at SLAC and
it was 2
also
modeled
one-toone
any
fitting
The simulations
followed
x 10
7 particles on a
one
without
any
fitting
parameters.
The
simulations
followed
2
x
10
particles
on a
160x120x88 grid.
160
x
120
x
88
grid.
A comparison between the experimental and simulation results is shown in figure 3.
A comparison
the images
experimental
andexperiment
simulation results
is shown
figureThese
3.
Time
integrated between
Cherenkov
from the
are plotted
in topinrow.
Time
integrated
Cherenkov
images
from
the
experiment
are
plotted
in
top
row.
These
images can be thought of as the line density of the beam, i.e., a time integrated image
images can be thought of as the line density of the beam, i.e., a time integrated image
of the beam's transverse profile. On the left is an image of the beam without an
of the beam’s transverse profile. On the left is an image of the beam without an
ionizing laser, i.e., there is no gas/plasma interface. On the right is the beam after it
ionizing laser, i.e., there is no gas/plasma interface. On the right is the beam after it
crosses an interface. This image can be understood if one looks at the simulation
crosses an interface. This image can be understood if one looks at the simulation
results on the bottom row. On the left is a 3D isosurface plot of the beam and plasma
results on the bottom row. On the left is a 3D isosurface plot of the beam and plasma
electron density. The beam is moving from left to right. The back of the beam is seen
electron density. The beam is moving from left to right. The back of the beam is seen
to
be attracted back towards the interface when compared to the front of the beam. The
to be attracted back towards the interface when compared to the front of the beam. The
front
is not
not effected
effected because
because itit takes
takestime
timefor
forthe
theplasma
plasmaelectrons
electronstotobebe
front of
of the
the beam
beam is
expelled.
On
the
bottom
right,
a
time
integrated
image
of
the
beam
is
shown
fromthe
the
expelled. On the bottom right, a time integrated image of the beam is shown from
simulation.
From
the
simulation
results,
it
is
clear
that
the
middle
part
of
the
image
simulation. From the simulation results, it is clear that the middle part of the image isis
due
of the
the beam
beam while
while the
the bright
bright spot
spot isisdue
duetotothe
theback
backofofthe
thebeam.
beam.The
The
due to
to the
the front
front of
same
features
are
seen
in
the
experimental
plot.
same features are seen in the experimental plot.
Experiment
(Cherenkov
images)
Laser off
Laser on
3-D
3-D OSIRIS
PIC Simulation
FIGURE 3.
3. Refraction
Refraction of
FIGURE
of aa particle
particle beam:
beam: Comparison
Comparison between
betweenexperiment
experimentand
andfull-scale
full-scale3D
3DPIC
PIC
simulation.
simulation.
22
In
in interpret
interpret aa new
new effect.
effect. As
As part
part of
ofthe
theEEIn the
the above
above example,
example, the
the goal
goal was
was to
to help
help in
157
experiment,
simulations
were
also
used
to
help
design
the
experiment.
As
an
157 experiment, simulations were also used to help design the experiment. As an
example
contours from
from 3D
3D PIC
PIC simulations
simulations using
using
example of
of this,
this, in
in figure
figure 4,
4, isosurface
isosurface contours
OSIRIS
from aa run
run with
with aa symmetric
symmetric drive
drive beam
beamwhile
while
OSIRIS are
are shown.
shown. In
In the
the left
left are
are plots
plots from
on
the
right
are
plots
when
the
aspect
ratio
was
4
to
1.
The
simulations
showed
that
the
on the right are plots when the aspect ratio was 4 to 1. The simulations showed that the
imperfect
beam
reduced
the
phase
space
volume
for
maximum
acceleration
gradient.
imperfect beam reduced the phase space volume for maximum acceleration gradient.
These
350 xx 200
200 xx 200
200 grid
grid and
and used
used ~64
-64million
millionparticles.
particles.
These simulations
simulations were
were done
done on
on aa 350
Symmetric
1 ))
Symmetric Beam
Beam (( aspect
aspect ratio
ratio 11 :: 1
Asymmetric Beam
Beam ((aspect
aspectratio
ratio22:: 1/2
1 / 2))
Asymmetric
FIGURE 4.
4. Plasma
Plasma wave
wave accelerating
accelerating structure
FIGURE
structure for
for aa PWFA
PWFA in
in the
the blowout
blowout regime.
regime.
Another powerful
powerful example
example of
of the
the usefulness
usefulness of
Another
of full
full PIC
PIC isis its
its role
role in
inunderstanding
understanding
the SMLWFA.
SMLWFA. In
In this
this scheme
scheme aa laser
laser pulse
pulse goes
the
goes unstable
unstable to
to forward
forward Raman
Ramanscattering
scattering
types of
of instabilities.
instabilities. In
In typical
typical experiments,
types
experiments, the
the resulting
resulting plasma
plasma wave
wave grows
grows toto
sufficient
amplitude
that
it
self-traps
electrons.
With
today’s
computers
and
sufficient amplitude that it self-traps electrons. With today's computers and 3D
3DPIC
PIC
codes,
such
experiments
can
now
be
modeled
one-to-one.
Obviously,
one
is
codes, such experiments can now be modeled one-to-one. Obviously, one is never
never
exactly sure
sure of
of the
the laser's
laser’s beam
beam quality
quality and
exactly
and the
the plasma
plasma uniformity,
uniformity, so
so one
one does
does not
not
expect
perfect
agreement.
But
in
the
simulation
one
can
directly
measure
the
expect perfect agreement. But in the simulation one can directly measure the wake
wake
amplitude and
and the
the laser
laser fields
fields so
so that
that the
the source
amplitude
source of
of the
the electrons
electrons and
and their
their acceleration
acceleration
mechanism can
can be
be unequivocally
unequivocally understood.
understood.
mechanism
Using OSIRIS,
OSIRIS, simulations
simulations have
have been
been performed
Using
performed with
with parameters
parameters similar
similarto
tothose
thoseinin
ongoing experiments
experiments at
at LOA[32].
LOA[32]. The
simulations
modeled
aa 35
fs
.8mm
, 88TW
laser
ongoing
The
simulations
modeled
35
fs
.8jim,
TW
laser
18
20
-3
propagating through
through aa 1.38
propagating
1.38 x10
xlO 18toto 1.7
1.7x0
xO20cm
cm"3density
densityplasma
plasmafor
for~.6mm.
~.6mm.The
The
simulations follow
follow 200
200 million
million particles
particles on
simulations
on aa 2000x300x300
2000x300x300 cell
cell grid
grid (the
(theplasma
plasmawas
was
only in
in aa 150x150
150x150 region
region in
in the
the transverse
only
transverse plane)
plane) for
for 10000’s
10000's of
of time
time steps.
steps.AAsample
sample
23
result
figure 5.
5. We
We plot
plot the
the electron
electron pps3 vs.
vs. ppi,
pi vs. xs, and pi vs. X2 phase
result is
is shown
shown in
in figure
1, p1 vs. x3, and p1 vs. x2 phase
space
the laser
laser polarized
polarized in
in the
the xxs3 direction
directionand
andpropagating
propagatingininthe
the
space 3D
3D simulation
simulation with
with the
xx direction.
The
simulations
show
that
the
electrons
have
an
asymmetric
transverse
direction. The simulations show that the electrons have an asymmetric transverse
profile
in the
the laser’s
laser's field.
field. The
The simulations
simulationsalso
alsoshowed
showedthat
thatthe
the
profile due
due to
to the
the oscillation
oscillation in
at
higher
densities
there
are
both
a
fewer
number
of
accelerated
electrons
and
a
lower
at higher densities there are both a fewer number of accelerated electrons and a lower
peak
with the
the published
published experimental
experimentalresults[31].
results[31].Details
Detailswill
willbe
be
peak energy,
energy, in
in agreement
agreement with
presented
in
a
forthcoming
publication
by
F.S.
Tsung,
J.C.Adam
et
al.
presented in a forthcoming publication by F.S. Tsung, J.C.Adam et al.
p3 vs.
vs. p1
p1
p3
p1
p1 vs.
vs. x3
x3
p3
p3vs.
vs.x2
x2
FIGURE 5.
5. Self-trapped
FIGURE
Self-trapped electrons
electrons from
from aa 3D
3D PIC
PIC simulation
simulationof
ofaa SMLWFA
SMLWFAusing
usingaa35fs
35fslaser.
laser.
is not
not possible
possible to
to model
model the
ItIt is
the beat-wave
beat-wave experiments
experiments in
inthe
theUCLA
UCLANeptune
NeptuneLab
Lab
using aa full
full PIC
PIC simulation
simulation in
using
in either
either 2D
2D or
or 3D.
3D. The
The frequency
frequency ratio
ratioisis30.
30.InInaddition,
addition,
the laser
laser pulse
pulse length
length is
the
is much
much longer
longer than
than the
the plasma
plasma length
length so
so the
the quasi-static
quasi-static
approximation
is
also
not
useful.
Therefore,
one
must
resort
to
the
approximation is also not useful. Therefore, one must resort to the ponderomotive
ponderomotive
guiding center
center approximation.
approximation. The
guiding
The code
code turboWAVE
turbo WAVE has
has been
been used
usedextensively
extensivelytoto
model
the
UCLA
experiment.
In
this
experiment,
two
lasers
with
wavelengths
model the UCLA experiment. In this experiment, two lasers with wavelengthsofof
10.27mm and
and 10.59
10.27jim
10.59mm
jim are
are focused
focused into
into aa plasma
plasma with
with either
either f/3
f/3or
or f/18
f/18 optics.
optics.The
The
frequency
ratio
is
~30.
We
show
a
sample
result
from
a
simulation
which
modeled
frequency ratio is -30. We show a sample result from a simulation which modeledthe
the
f/3 case.
case. The
The simulations
simulations use
use aa 1024
f/3
1024 xx 256
256 grid
grid with
with 10
10 particles
particles per
per cell.
cell. The
The
simulations ran
ran for
for 10000
timesteps, which
simulations
10000 timesteps,
which corresponded
corresponded to
to 90ps.
90ps. The
Thelaser
laserhad
hadaarise
rise
and
fall
time
of
50ps
and
each
beam
had
a
peak
amplitude
of
v
/c
=
.3.
The
cell
o
and fall time of 50ps and each beam had a peak amplitude of v0/c = .3. The cellsizes
sizes
were .lc/co
.1c/wp in
in both
both directions
were
directions and
and the
the spot
spot size
size was
was 1c/w
lc/copp atatfocus.
focus. The
Thesimulation
simulationbox
box
p
was
5mm
and
the
Rayleigh
length
was
1.5mm.
was 5mm and the Rayleigh length was 1.5mm.
24
In figure 6 we show contour plots of the accelerating electric field and of the
electron density 50.4 ps. The peak value of the electric field was .lmccop/e and the
density is localized to the middle because the plasma was formed via tunnel ionization
of Hydrogen. The initial gas density was 1016 cm"3' It is clear that for this narrow
focus the transverse ponderomotive force is important. If a full PIC simulation was
used then the simulation would have required -15 times more resolution (Ax=.2c/eo0)
and therefore required -225 times more computer time. More detail can be found in
the paper by Narang et al., in this proceedings.
80
100
FIGURE 6. Modeling a PBWA experiment using the ponderomotive guiding center approach in
turboWAVE.
The last example I will give is of electron beam hosing. Hosing is perhaps the major
obstacle towards the successful development of an afterburner stage. We used the code
quickPIC to study a 30 GeV beam with 1.8 x 1010 electrons, with ar=10mm and with
az=.06mm propagating through a plasma of density 2 x 1016 cm-3. The beam had an
initial tilt of .011. In some of the hosing simulations we used a512x512x512 grid
with 16 particles per cell in the 2D grid. The simulations were run on 32 processors at
NERSC. In figure 7, we plot a 3D isosurface plot of the beam density as it enters the
plasma (left) and after it propagated through 2.4 meters of plasma (right). The beam is
moving toward the lower left corner. The beam is seen to hose, although with a lower
growth rate predicted by linear theory [31]. In addition, the current version of
quickPIC has been shown to have limitations for these parameters, i.e., the full quasistatic description is needed. An important area of future research for the afterbruner
concept is to develop a theory for hosing for short pulses, to understand how hosing
might effect beam loading, to understand the transverse two-stream instability for
short pulse positron drivers, and to quantify how much hosing might be expected in
25
simulations, both
both full
full PIC
PIC and
and quasi-static
quasi-static PIC,
PIC,will
will be
be
afterburner designs. PIC simulations,
essential in such a study.
Intial
Intial beam
beam with
with aa tilt
tilt
The
The beam
beam after
after 2.4
2.4 meters
meters
FIGURE
FIGURE 7.
7. Modeling
Modeling afterburner
afterburner parameters
parameters using
using aa 3D
3Dquasi-static
quasi-staticcode.
code.
THE
THE FUTURE
FUTURE
In
In this
this paper,
paper, II have
have discussed
discussed the
the progress
progress in
in plasma-based
plasma-based accelerator
accelerator modeling
modeling
over
past twenty
over the
the past
twenty years.
years. Today
Today on
on large
large parallel
parallel computers,
computers, one
one can
can use
use full
full PIC
PIC to
to
model
model 1GeV
IGeV stages
stages in
in full
full 3D.
3D. This
This makes
makes itit possible
possible to
to use
use PIC
PIC simulations
simulations to
to model
model
the
the full
full scale
scale in
in 3D
3D of
of ongoing
ongoing experiments.
experiments. This
This capability
capability has
has resulted
resulted in
in aa new
new
paradigm of
paradigm
of research
research in
in which
which experimental
experimental and
and simulation
simulation results
results are
are closely
closely coupled
coupled
in
in making
making scientific
scientific discovery.
discovery. However,
However, the
the turn
turn around
around time
time for
for these
these simulations
simulations isis
weeks.
Therefore,
it
is
currently
not
possible
for
real-time
feedback
weeks. Therefore, it is currently not possible for real-time feedback between
between
simulation
simulation and
and experiment.
experiment. The
The development
development of
of quasi-static
quasi-static PIC
PIC codes
codes may
may reduce
reduce
the
the turn
turn around
around time
time from
from weeks
weeks to
to minutes
minutes making
making real-time
real-time feedback
feedback aa reality.
reality. The
The
use of
use
of full
full PIC
PIC is
is still
still required
required as
as aa validation
validation mechanism
mechanism for
for the
the reduced
reduced description
description
codes.
codes.
Another major
GeV
Another
major effort
effort for
for the
the future
future is
is the
the modeling
modeling of
of 100+
100+
GeV stages.
stages. This
This is
is
currently
currently not
not practical
practical using
using full
full PIC
PIC codes.
codes. However,
However, quasi-static
quasi-static PIC
PIC codes
codes will
will also
also
make
make this
this possible.
possible. Full
Full PIC
PIC will
will also
also be
be required
required as
as aa method
method of
of validation.
validation. II am
am
optimistic
optimistic that
that itit will
will be
be possible
possible to
to model
model key
key aspects
aspects of
of the
the afterburner
afterburner concept
concept [33]
[33]
(i.e.,
(i.e., the
the self-consistent
self-consistent acceleration
acceleration of
of electron
electron and
and positron
positron bunches
bunches to
to 100
100 ++ GeV
GeV
26
behind electron and positron drivers, and the focusing of these bunches in plasma lens
elements), in 3D within the next 5 years using codes such as QuickPIC.
In addition to the above, the vast array of existing codes will undoubtedly lead to
new scientific discovery and perhaps new acceleration concepts. Full PIC codes are
already being used to study novel all-optical injection schemes, ion acceleration, and
IGeV LWFA stages using plasma channels. Some work on these topics can be found
in these proceedings. The future for plasma-based accelerator modeling is bright.
ACKNOWLEDGMENTS
I would like to acknowledge my interactions with the late John M. Dawson during
the past 20 years. He was an inspiration, a role model, and a guiding light. I would
also like to thank my many collaborators and co-authors, which are too many in
number to thank individually. These include the UCLA plasma simulation group, the
USC plasma simulation group, the 1ST Portugal plasma simulation group, the E-162
collaboration and the SciDAC team, as well as J.C.Adam who spent a summer
sabbatical at UCLA in 2001. The work reported here was supported by U.S. DOE
Grant numbers DE-FG03-92ER40727 and DE-FC02-01ER41179, NSF contract
number Phy-0078508, and LLNL contract number W-07405-ENG-48.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Tajima, T. and Dawson, J.M., Physical Review Letters, 43, No. 4, 267-270 (1979).
Joshi, C., et al., Physical Review Letters 47, 1285-1288 (1981); Antonsen Jr, T.A. and Mora, P.
Physical Review Letters. 69, 2204 (1992); Krall, J., et al., Phys. Rev. E. 48 2157 (1993); Andreev,
N.E., et al., Pis 'ma Zh. Eksp. Teor. Fiz., 55 551 (1992).
Joshi, C., et al, Nature 311, 525 (1984).
Chen, P., Physical Review Letters 54, 693-696 (1985).
Esarey, E., et al., IEEE Transactions on Plasma Science, 24, 252-288 (1996).
Shadwick, B.A., et al., American Institute of Physics Conference Proceedings, no. 569, 154-162,
(2001); Andreev, N.E., et al. Phys. Plasmas 2, 2573 (1996).
Dawson, J.M, Rev. Mod. Phys. 55,403 (1983).
"Plasma Physics via Computer Simulation" by C.K. Birdsall and A.B. Langdon, McGraw-Hill
Book Company.
Forslund, D.W., et al., Physical Review Letters. 54, 558 (1985); Mori, W.B., et al., Physical Review
Letters. 60, 1298 (1988).
Morse, R.L. and Nielson, C.W., Phys. Fluids 14, 830 (1971), see Appendix B; Forslund, D.|W.,
Space Science Reviews 42, 3(1985).
Joyce, G., et al., Laser & Particle Beams, 12 273-282 (1994); Krall, J. and Joyce, G., Phys.
Plasmas 2 1326-1231 (1995).
Su, J.J,. 1990, PhD Dissertation, UCLA.
Morse, R.L. and Nielson, C.W., Phys. Fluids 14, 830 (1971), see Appendix A; Gisler, G., et al., in
Proc. 11th Int. Conf. Numerical Simulations of Plasmas (Montreal, Que., Canada), 1985, Paper 1B-01.
Decker, C.D.,et al., Phys. Rev. E. 50, R3338 (1994); Decker, C.D.,et al, Phys. Plasmas 3, 2047
(1996) Decker, C.D., et al., IEEE Transactions on Plasma Science, 24, No. 2, 379-392 (1996).
K-C.Tzeng, 1998, Phd Dissertation, UCLA.
27
16. Coverdale, C.A., et al., Physical Review Letters 74, 4659 (1995).; Tzeng, K-C., et al., Phys. Rev,
Lett, 76, 3332 (1996); Tzeng, K-C., Physical Review Letters 79 5258 (1997); Gordon, D., et al.,
Phys. Rev, Lett, 80, 2133 (1998); Tzeng, K-C., Physical Review Letters 81 104 (1998); Tzeng, KC., et al., Phys. Plasmas 6, 2105-2116 (1999); Duda, B.J., et al., Physical Review Letters 83, 1978
(1999).
17. Whittum, D.H., Phys. Plasmas 4, 1154 (1997).
18. Mora, P. and Antonsen Jr., T.A.,Phys. Plasmas 4, 217 (1997).
19. Hemker, R.G., et al., Proceedings of the 1999 Particle Accelerator Conference, New York, NY, 5,
3672 (1999); Hemker, R.G., 2000, PhD dissertation, UCLA; Fonseca, F.A, et al, "Lecture notes in
computer science", 2329, III-342 (Heidelberg: Springer-Verlag).
20. Pukhov, A., Journal of Plasma Physics 61 425-433 (1999) and references therein.
21. Hogan, M.J., et al., Phys. Plasmas 7, 2241-2248 (2000); Clayton, C.E., et al., Physical Review
Letters 88, 154801/1-4 (2002).
22. Lee, S., et al, Phys. Rev. E. 61, 7014 (2000); Hemker, R.G, et al., Phys. Rev, Spec, Top.-Acc. &
Beam 3,061301(2000).
23. Gordon, D.F., et al., IEEE Transactions on Plasma Science, 28, No. 4, 1135-1143 (2000).
24. http://www-beams.colorado.edu/~vorpal/vorpaldocs/: VORPAL also has an option for a fluid
description.
25. Borghesi, M., et al., Physical Review Letters 88, 135002/1-4 (2002) and references therein;
Sentoku, Y., et al., Physics of Plasmas 6, 2855-2861 (1999).
26. Marder, B., J, Comput, Phys. 68, 48(1987).
27. Villasenor, J. and Buneman, O., Computer Physics Communication, 69, 306 (1992).
28. Bruhwiler, D.L., et al., Phys. Rev, Spec, Top.-Acc. & Beam 4, no. 10 (2001).
29. Huang, C.K., et al., Proceedings of the 2001 Particle Accelerator Conference, 5 4005-4007 (2001);
Huang, C.K., et al., Proceedings of ltfh annual review of progress in applied computational
electromagnetics, Monterey, California, March 18-22, 2002.
30. Mori, W.B. and Katsouleas, T., Physica Scripta T30, 127 (1990) and references therein.
31. Muggli, P., et al, Nature, 411 32 (2001); Ren, C., et al, Physical Review Letters 85, 2124 (2000)
Lee, S., et al., Phys. Rev. E 64, 045501(R) (2001) Dodd, E.S., et al., Physical Review Letters
88,125001/1-4 (2002) Tsung, F.S., et al., 99, 29-32 (2002)
32. Malka,V., Qtal,Phys. Plasmas*, 2605 (2001).
33. Lee, S., et al., Phys. Rev. Spec. Top.-Acc. & Beam 5, 011001 (2002).
28