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The role of plasma in advanced accelerators*
Jonathan S. Wurtele+Plasma Fusion Center and Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachussets 02139
(Received 18 December 1992; accepted 18 February 1993)
The role of plasma in advanced accelerators is reviewed with emphasis on three significant areas
of research: plasma guiding of beams in accelerators, plasma focusing of beams in high-energy
linear colliders, and plasma acceleration of beams,
I. INTRODUCTION
The exponential increase in beam energy delivered by
accelerators over the last seventy years is due in part to
improvements and enlargements of fixed technology, and
in part to the introduction of new acceleration techniques
to replace those that had reached the outer bounds of their
performance. The present technology of synchrotrons and
linear accelerators is clearly near the end of the road. A
machine significantly larger than the Superconducting Su-per Collider’ will almost surely require more resources
than are available to build it. Electron machines, such as
the Stanford Linear Collider’ (SLC), have perhaps one
more iteration: Increasing their energy by a factor of ten
might be accomplished using conventional techniques.
There is an intensive effort underway to develop new ac-
celeration techniques which can replace the old and allow
for a new generation of particle accelerators. The develop-
ment and status of advanced accelerator concepts, plasma
based, as well as others, can be found in a series of work-
shop proceedings.3-7
Transverse fields can be created by a beam propagating
through a plasma and used for transport and focusing. This
plasma guiding of beams, especially in the “ion focused”
regime,**’ accelerators,8-‘5 and radiation sources,‘6-23 such
as free-electron lasers, has proved effective in numerous
experiments conducted over the last decade.1°-15
The need to maximize the event rate in high-energy
colliders requires beams focused down to submicron di-
mensions. Plasma lenses, which, in principle, can reach
focusing strengths much greater than conventional mag-
netic focusing, have been actively studied in this
regard.2”3’ Plasma lenses and transport in the ion focused
regime do not have substantial plasma return current flow-
ing through the beam. The return current in a plasmawould flow through and partially current neutralize the
beam if the beam radius is of order a plasma skin depth.
Current neutralization has been proposed3’ as a method to
reduce the detrimental effects of beamstrahlung at the in-
teraction region in a linear collider.
The use of plasma as an accelerating medium requires
the generation of an intense longitudinal plasma oscillation
with a phase velocity equal to the speed of light. Three
methods for exciting a plasma oscillation have been inves-
*Paper 5RV1, Bull Am. Phys. Sot. 37, 1468 (1992).
‘Invited speaker.
tigated. Two methods use laser beams to excite the plasma:
the beat-wave accelerator,3345 and the laser wake field
accelerator.33’46-52The accelerator application requires in-
teraction lengths much longer than the natural diffraction
length, so that diffraction must be overcome,46*53-56and
instabilities57-59 avoided. The third scheme uses a relativ-
istic beam, and is known as the particle beam wake field
accelerator.6~67 While most studies have concentrated on
using plasma to accelerate charged particles, photon accel-
eration is another area of active experimental and theoret-
ical interest.68-72
This paper will first examine the most immediate role
of plasma in accelerators, as a replacement for, and sup-
plement to, conventional transport and focusing magnets.
It will then review the experimental status and the critical
physics issues of plasma accelerator concepts.
II. PLASMA GUIDING OF BEAMS
A. Plasma guiding of beams in accelerators
There has been intense experimental and theoreticalresearch on plasma guiding of beams in accelerators. This
body of work has been reviewed recently” by Swanekamp
ef al., and will not be discussed in detail. The underl ying
physical mechanism for plasma guiding is the expulsion of
plasma electrons by the self-electric field of an electronbeam. The radially outward force on beam electrons from
the beam self-electric field is balanced, to order
[ 1 - (v/c) ‘1, where u is the beam velocity and c the speed of
light, by the inward pinch force from the self-magnetic
field. Plasma electrons do not experience a radial force
from the beam magnetic field, but do feel an radially out-
ward force from the beam electric field.
When the plasma is underdense ( np < nb), the beam
expels all plasma electrons from the path of the beam,
creating a rarefaction region extending out to the charge
neutralization radius (the radius r, such that n&=n&).
The resulting focusing force is independent of the beam
density (except at the head of the beam, which is not fo-
cused since plasma electrons have not had time to exit
from the beam path). With a uniform ion density the fo-
cusing is linear, with a betatron wave number
kp=wpc=wpc/(2y)“2, where w;=(4re2ndm) with -e
the electron charge and m the electron mass, and
fl= l/[l- (v/c)‘] is th e b earn energy in units of mc’. The
betatron wavelength obtainable with pl asma of even mod-
2363 Phys. Fluids B 5 (7), July 1993 0899-8221/93/5(7)/2363/8/$6.00 @ 1993 American Institute of Physics 2363
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est density ( 1014/cm3) is higher than can be reached with
conventional magnets.
When the plasma is overdense ( np > nb), enough
plasma electrons will be expelled so that the beam is space-
charge neutralized. The beam will also generate a return
current in the plasma. If the plasma skin depth, c/w,, is
large compared to the beam radius, the return current will
flow primarily outside the beam and current neutralization
will not be significant. The beam will then be focused by its
self-magnetic field, and the focusing force will vary with
the beam density.
As will be seen in Sets. II and III, the fields of a beam
with a properly tailored density can be used to accelerate
and focus trailing particles, and the head of a beam can
generate fields which will focus the bulk.
For plasma guiding to be effective, the electron-beam
pulse should be long compared to l/up, so that the plasma
has time to respond to the beam, and short compared to
ion time scales, so as to avoid undesirable instabilities. The
regime most suited for plasma use is the ion focused re-
gime, where the density is low, the focusing can be linear
and the return current effects are small.
For relativistic beams, the main radial outward forceon the beam envelope is a pressure from the beam emit-
tance (i.e., the area in transverse phase space occupied by
the beam). Propagating beams by using plasmas requires
that they be sufficiently intense so that the combined beam
and plasma fields provide enough focusing to overcome the
tendency of the beam to expand. In the original electron
beam ionization experiments, the guiding was provided by
a plasma which was, itself, created by the ionization of a
background gas by the beam. There, the gas density needed
to be sufficiently high so that the beam ionization rates
would create a plasma of the requisite density. This had the
undesirable effect of causing a strong longitudinal depen-dence in the focusing strength (which scales with the
plasma density, increasing back from the head of the
beam).
A significant advance was made when the concept of
laser guiding1’-‘5 was introduced in the mid-80s. In laser
guiding, a laser ionizes the plasma so that the plasma den-
sity and radius become independent of the beam. The gas is
chosen so as to have a low ionization potential and the
plasma density is fixed by the laser intensity and gas pres-
sure. The beam electrons then experience a focusing force
which does not depend on their longitudinal position. Theplasma radius can then be made narrow, of order the mean
beam radius. This will create a nonlinear focusing force, sothat particles in the beam will have a spread in betatron
frequency. Experiments have shown that the nonlinearity
thereby introduced can effectively damp coherent transport
instabilities.
Electron pulse propagation in an underdense plasma is
subject to a beam breakup (BBU) instability in both radial
and slab geometries.16 For the radial geometry, in the long
pulse limit, (~65) > (kpz), and the growth rate is
0.8(0ps)~‘~(k& .’3 This scaling is typical of BBU insta-
bilities, where the head of the beam drives the tail of the
beam through interactions with its electromagnetic envi-
ronment. In conventional accelerators the coupling is
through cavities and other sources of impedance, while
here the plasma provides the coupling needed to generate
an instability.
B. Plasma in radiation generation devices
More recently, plasma has been used” to guide elec-
tron beams through radiation generation devices such as
the free electron laser (FEL). This use of plasma couldlead to a simplification of FEL design since transverse fo-
cusing, by either axial magnetic fields or by carefully con-
structed wiggler fields, may be rendered unnecessary in
systems with pulse lengths between the electron and ion
time scales.
There are other important reasons for guiding beams
in FEL’s with a plasma.18 The focusing which can be pro-
vided by conventional magnets is inadequate for optimal
FEL performance. The FEL coupling is proportional to
the beam plasma frequency and inversely proportional to
the beam energy. Optimization assuming pure wiggler fo-
cusing often yields poor performance at high frequency. A
beam with a smaller radial spot size than can be reached
with wiggler focusing will have stronger coupling to the
optical pulse and generate a higher growth rate. In addi-
tion, due to optical guiding in the FEL,19 the higher
growth rate can lead to an increase in the total number of
e-foldings. Plasma guiding of the beam may be important
in microwiggler systems where the FEL focusing is negli-
gible (less than a betatron wavelength over the wiggler)
and the use of external quadrupoles is precluded by the
permanent magnets in the wiggler. Plasma has also been
employed” in microwave sources, such as backward wave
oscillators, which are powered by low-energy beams.
Beams propagating through an ion channel can, undercertain conditions, have a resonant electromagnetic
instability. ‘l-z3 This instability occurs when the plasma is
wide compared to the beam radius and the force experi-enced by the electrons in the channel is linear. The physics
of the instability, the ion channel laser (ICL), is akin to
that of the cyclotron resonance maser, where bunching is
in transverse momentum space (rather than in physical
space, as in an FEL). The ICL instability has a different
resonance condition from the FEL instability-it requires
synchronism between the oscillations of electrons in the ion
channel and the electromagnetic wave, w - kv=wp. Since
the betatron frequency scales as y-i”, the net frequencyupshift in the ICL will scale as y3”, weaker than the y
upshift in the FEL.
The transverse oscillations in the ion channel provide
the coupling between the electrons and the electromagnetic
wave. Detailed theoretical studies of the instability based
on techniques similar to those used for the FEL have
yielded calculations of growth rates for various initial beam
distributions. These are comparable in magnitude to those
of an FEL. Since no external focusing is required, a system
based entirely on plasma could be designed, where the
plasma focuses the electrons in the accelerator and the ICL
mechanism is used to extract the beam power and convert
2364 Phys. Fluids B, Vol. 5, No. 7, July 1993 Jonathan S. Wurtele 2364
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it into coherent radiation. Preliminary experimental
studies” have verified the ICL mechanism, but further ex-
perimental work, on areas such as saturation efficiency and
sensitivity of the growth rate to temperature and emit-
tance, is required before any conclusions about its potential
use as a radiation source can be drawn.
III. PLASMA FOCUSING OF BEAMS
The transverse fields generated by a beam propagating
through a plasma can be made extremely strong-much
stronger than can be obtained from conventional quadru-
pole or solenoidal magnets. The strong fields of the plasma
make this an attractive lens for high-energy linear collid-
ers, where the interaction rate is proportional to the inverse
of the spot size of the beam at the collision point.The plasma lens has been vigorously investigated2”29
for use in future linear colliders. Numerical studies28,29 n-
dicate that potential luminosity enhancements of an order
of magnitude can be reached by the SLC with a plasma
lens at the final focus.The design of pl asma lenses uses the envelope equation
along with simulations to confirm the results. Here, b is the
amplitude of the beam envelope, related to the beam radius
by rb= (DE) I”, where the emittance is E. As with plasma
guiding, when in the underdense regime, the focusing force
is due to the (almost) stationary ions, almost linear and
independent of the beam. In this case, the focusing param-
eter is K=2mp,,/y, where r, is the classical electron ra-
dius. In the overdense regime, the beam is space-charge
neutralized, and if the system is not too overdense, one stillhas rb <c/w,, the return current flows mainly outside the
beam, and K-22nrgzb(s,z)/y. Here, the dependence of the
density on both distance from the head of the bunch and
propagation length in the lens will change the focusing of a
given slice of the beam. In the underdense regime, since
beams are typically Gaussian in shape, the head of the
beam will propagate in the overdense region and there will
be longitudinal variations of the focusing strength until the
beam density equals that of the plasma.
Nonlinear effects arise in the overdense regime from
the transverse variation of the beam density-the central
core of the beam is focused differently than the wings. This
lens aberration has been estimatedz4 to degrade focusing by30%. Numerous focusing schemes based on tailoring the
plasma density so as to minimize the spot size at the final
focus have been considered,24’28-30 with varying levels ofsophistication.
Experiments at Argonne65 and Tokyo University I3
studied plasma lenses for electron beams in the overdense
regime. These proof-of-principle experiments are consis-
tent with theoretical predictions. They used beams with
densities of order 10”/cm3, which is about six to seven
orders of magnitude less than the beam densities required
for high-energy physics experiments. Thus further experi-
ments and detailed parameter studies with more realistic
beams are evidently warranted. Experiments on plasma fo-
cusing with heavy ion beams have also reported some suc-
cess.
All focusing lenses give particles acceleration which, in
turn, leads to synchrotron emission. The adiabatic plasma
lens was proposed3’ as a method for avoiding the Oide
limit,31 a fundamental limit on beam size due to both the
unavoidable synchrotron radiation and to the inherent
quantum nature of the emission process. Particles which
pass through a lens off-axis have some probability of emit-ting a synchrotron radiated photon, and changing their
energy. The energy change leads to different focal lengths
for particles, and, hence, averaging over the beam, the spot
size is larger.
Note that if all particles entering the lens at a given
radius emitted exactly the same synchrotron radiation,
then this effect could be compensated for, in principle, by
modifying the lens design. It is the quantum nature of the
emission that creates the limit, not the classical synchro-
tron emission. The adiabatic lens simply focuses particles
more and more intensely, so that the emittance growth
from synchrotron radiation is compensated for by in-creased focusing strength. Calculations have shown3’ that
the Oide limit does not apply, but, rather, a new, much less
severe, limitation is placed on the spot size (if the initial
emittance is sufficiently small). In this scheme, the plasma
is present at the interaction point, which is undesirable.
In a collider, both positron and electron beams need to
be focused. Positrons behave in a fundamentally different
way than electrons. The positron beam sucks in electrons
in the surrounding plasmas and these electrons oscillate.
The resulting focusing field is rather nonuniform, with re-
gions of higher intensity than that of a corresponding elec-
tron lens. There have been no lens experiments with posi-trons.
A significant difficulty with all plasma lens schemes is
the background increase in the detectors from collisions
between electrons or positrons and plasma ions. These are
not desirable from a luminosity and data analysis stand-point, and careful consideration to background must be
made in the design of plasma lenses in the final focus re-
gion. Other, more straightforward, concerns are engineer-
ing issues such as shot-to-shot variation in plasma param-
eters.
A. Plasma compensation of beamstrahlungA major problem for designers of the next generation
of linear colliders is the synchrotron radiation, or beam-
strahlung, of a particle in the collective magnetic field of
the opposing bunch. Energy loss can be substantial, greater
than 10% for some designs. In addition to energy loss, an
energy spread is induced in the beam, which reduces the
event rate for narrow resonances. For colliders with TeV
energies, the beamstrahlung photons interact with the
beams to produce low energy electron-positron pairs, in-
creasing the background in the detectors.
The severity of the problem has led to consideration of
colliding very small bunches with widths of from 10 to 100
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times greater than their height (typical widths are of order
tens of angstroms). While these designs avoid many of the
difficulties from beamstrahlung, they introduce severe tol-
erance and emittance constraints on the accelerator.
A second, plasma-based, approach to eliminatingbeamstrahlung has been proposed3* which is essentially an
extremely overdense lens with a skin depth of order the
beam radius. It has been shown in Ref. 32 that beamstrahl-
ung can be substantially alleviated simply by introducing a
plasma into the interaction region. When the plasma skin
depth is comparable to the beam size, the return current
will propagate within the beam and partially current neu-
tralize it. The beam self-magnetic field will then be reduced
by the magnetic field of the return current, as will the
beamstrahlung. Potential difficulties with this idea are an
increased background level in the detectors due to the col-
lisions of the beam with the plasma ions and the rather
high plasma densities that are required3* ( 1022-1023/cm3).
IV. PLASMA ACCELERATION OF BEAMS
The previous sections of this paper have examined the
role of plasma as a replacement for and supplement to
conventional transport and focusing magnets. This has re-
quired that beams be long compared to the plasma period,
so that longitudinal plasma oscillations are not excited.
The plasma accelerators operate in exactly the opposite
regime, exciting a longitudinal oscillation and then using it
to accelerate an electron or positron bunch.Plasma-based accelerators have been an area of active
research for over a decade.3347 The accelerating fields in
the plasma can be extremely intense compared to thoseobtainable by conventional technology. Conventional
radio-frequency (rf) accelerators have field gradients that
are limited to around 100 MV/m. The SLC operates at 20MV/m, and extending the technology to higher frequency
is not expected to yield gradients of more than 200 MV/m.
An accelerator with a 10”/cm3 plasma could, in principle,
achieve gradients of order 10-100 GV/m.
There are some obvious limits on the final energy that
can be reached using plasma acceleration. First, if the dif-
ference in phase velocity between the plasma wave and the
accelerated bunch leads to a phase shift of about ninety
degrees, the acceleration mechanism will cease. Second, the
interaction length is limited by the natural tendency of the
light to diffract. This diffraction length of a laser pulse is
known as a Rayleigh range, Zr=rrw2/jl, where w is thelaser waist at a focus and A is the laser wavelength.
There are two competing constraints important in de-
termining the ideal laser spot size. The need to excite a
large wave and, hence, to obtain a high gradient, demands
an intense laser field and, therefore, a small spot size. The
total acceleration will be reduced, however, as the spot size
is made smaller, because of the reduction in total interac-
tion length from increased diffraction. With present laser
technology, l-10 TW of beam power at 1 ,um, and requir-
ing u,,,/c > -0.3, diffraction limits the interaction to be of
order 1 mm, while dephasing takes much longer, about 1 m
for a 101’/cm3. A third limit comes from pump depletion.
As a plasma wave is generated, the drivers must lose en-
ergy, which will eventually be depleted.
A. The beat-wave accelerator
The most mature plasma acceleration scheme is the
beat-wave accelerator, first proposed33 by Tajima and
Dawson. The basic idea is simple: Two co-propagating la-
ser beams with frequencies separated by the plasma fre-
quency are injected into a plasma. The laser pulses beat
together satisfying the Manley-Rowe relations, so that thedifference in wave numbers imposes the wavelength of the
plasma wave. Simple analysis then shows that when the
wave frequencies are much greater than the plasma fre-
quency, the group velocity of either electromagnetic wave
is equal to the phase velocity of the plasma wave. Since the
group velocity is nearly the speed of light, the plasma os-
cillation can be used for acceleration.
Experimental work on beat-wave acceleration has been
conducted3H2 at a number of laboratories worldwide. Ini-
tial experiments concentrated on generating and diagnos-
ing large amplitude plasma waves. Recent measurementsN
from UCLA show that electrons injected at 2.1 MeV areaccelerated to 9.1 MeV, the detection limit of the diagnos-
tics. Accelerating gradients of order 1 GV/m are consis-
tent with a measured 10% density perturbation. Relativis-
tic quiver velocities are of order 0.3~. The interaction
length, and hence the final energy is limited by laser dif-
fraction, not by plasma inhomogeneity or dephasing. The
UCLA experiments use CO2 lasers with lines at 10.59 and
10.29 pm. Results4’ from Ecole Polytechnique using a
YAG laser with lines at 1.05 and 1.06 pm in a 10”/cm3
plasma have the same accelerating gradient. The saturationof the beat wave can occur through various mechanisms:
relativistic detuning,45 mode coupling,43 plasma wave
(modulational) instability,‘@ or laser-plasmainstability.57*58
6. The laser wake field accelerator
The beat-wave scheme requires two lasers and a
plasma frequency precisely tuned to the difference fre-
quency between them. The laser pulses are long compared
to the electron plasma period and must have a beat fre-
quency near the plasma frequency. This imposes a unifor-
mity constraint on the plasma density. An alternative idea
is the plasma wake field accelerator,33 in which an intense
laser pulse is injected into a plasma and leaves behind it awake, in the form of a plasma oscillation. The physics of
laser-pulse propagation in underdense plasmas has recently
been reviewed;48 what follows are the basic ideas and some
recent results. The plasma wake will travel at the group
velocity of the laser pulse. The laser pulse must be ex-
tremely intense for a substantial wake to be generated; the
intensity can become sufficiently great so that electron mo-
tion in the pulse is relativistic. Estimates,46 based on a cold
plasma relativistic fluid theory, have been made for the
maximum obtainable field in the beat-wave configuration.
It is potentially higher than that of the beat-wave scheme if
the motion is relativistic.
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In order to study the physics of pump depletion, dif-
fraction and plasma wave generation, a coupled self-
consistent system of equations is required. The simplest
coupled equations which exhibit much of the interesting
physics are
(2i$ &+Vf )a=kir-a-a*),,
a* Sn
( )+k: y=V*(a*a*).
(1)
(2)
In these equations, a=eA/mc is the dimensionless laser
field amplitude, the relativistic motion is assumed to be
small but not negligible (so that the relativistic factor y can
be expanded to quadratic order in perpendicular momen-
tum), the unperturbed laser frequency is wo, and the den-
sity perturbation is given by &r/n. The first term on the
right-hand side of Eq. ( 1) is from the density perturbation
and the second is from the mildly relativistic motion in the
wave field. For short pulses, the density perturbation is
quadratic in the wave amplitude and the two terms tend to
cancel each other out.
A related concept, the plasma fiber accelerator wasproposed5’ several years ago. In it, the laser propagates
down a hollow overdense plasma. The acceleration is pro-
vided by the axial component of the laser field which is
present when the laser is guided in the plasma fiber. There
is resonance absorption in the overdense plasma, and con-
comitant pulse degradation. In the underdense channel,
there are no resonant losses and the wake field is used to
accelerate.
Two concepts for overcoming diffraction have been in-vestigated: relativistic guiding and plasma channel guiding.
In relativistic guiding, the relativistic motion of the elec-
trons reduces the plasma frequency where the wave is
large. Since a plasma has a dielectric constant less than
unity, it thereby increases the dielectric constant at the
center of the pulse. This results in a system similar to an
optical fiber. Detailed calculations show that a total power
greater than 16.2( w/w,)*GW is required for relativistic
guiding.
For short pulses, which are designed to generate large
wakes, the plasma motion must be included in anyanalysis.48 Relativistic guiding is nearly canceled by the
tendency of the ponderomotive force to push the plasma
forward ahead of the pulse, generating an increased plasma
density. Thus4* relativistic guiding cannot be used for
pulses shorter than a plasma period. Furthermore, the
sharp rise of a pulse can lead to a plasma oscillation and,
therefore, to a modulation of the plasma density. This, in
turn, modulates the index of refraction of the plasma, and
the pulse” tends to break into smaller subpulses. Oneproposed5’ method for circumventing this problem is the
use of tapered pulses which are wide at the head (and
therefore diffract slowly), and narrow further back where
relativistic guiding works.
The propagation of laser pulses in underdense plasma
has long been known to be unstable due to Raman and
Brillouin scattering. Of interest here is the short-pulse na-
ture of the interaction and the resulting requirement thatany analysis must necessarily include both relativistic ef-
fects and the plasma density perturbations. Relativistically
guided pulses have been examined57’58 in recent work and
found to be subject to Raman instabilities which limit the
propagation of pulses longer than a plasma wavelength to
a few Rayleigh lengths. The analysis is two dimensional
and shows that the electron dynamics must be included
even for pulses longer than a plasma period. The electron-
density perturbation provides the feedback between the
head and the tail of the pulse which leads to pulse erosion.
There is an interesting analogy with conventional beam-
breakup instabilities in accelerators. A given longitudinalslice of the laser pulse distorts the plasma which then cou-
ples to longitudinal slices which follow it. If the feedback is
unstable, the focusing properties of the distorted plasma
are changed in such a way that they further degrade the
beam. Analytical and numerical results5’ indicate that long
pulses (longer than a plasma period) will be unstable with
e-folding distances of roughly a few diffraction lengths.
A suitably tailored plasma channel can also provide
optical guiding of the laser pulse. Here, the plasma itselfhas a radial density profile which is minimum at the center
of the laser pulse. This works independently of the power
in the pulse. In fact, an intense pulse will itself modify the
channel; thus the theory is clearest for a pulse with
vosc/c < 1.
The plasma channel suffers from the laser hose
instability.5g If the axis of the channel is not aligned with
the axis of the laser pulse, the laser pulse will make a dipole
perturbation to the channel which, in turn, will tend to
move the rear of the pulse further off axis. This instability
has an analogy in the electron hose instability,16 where an
electron beam slightly displaced from the axis of an ion
channel perturbs the channel, which then moves the rear of
the pulse further off axis. The laser hose instability imposes
a constraint on the misalignment tolerance of the opticalpulse axis and the channel axis. Growth rates scale as
(ops)1’3(z/z,)*‘3 and, for typical parameters, are of order afew diffraction lengths.
A hollow plasma channel has been shown5’ to have The laser hose instability differs from the electron hosetwo useful properties for plasma acceleration. First, a laser instability due to the different response of the particle beampulse propagating in a hollow channel will be optically and the light (Lorentz equations versus paraxial waveguided and will not diffract. Second, the pulse generates a equation). The growth rates are similar and have the char-
surface mode on the inside of the channel. The fringe fields acteristic of all beam-breakup calculations for traveling
of this mode extend into the center of the channel and are
well suited for acceleration. The surface mode produces an
accelerating field inside the channel which is uniform to
order (w/w) *. The period of oscillation is predicted to be
reduced by about 30%, and is seen in the simulations. This
has been analyzed theoretically and confirmed by simula-
tions. Using a hollow channel reduces, by a factor of 4, the
amplitude of the plasma oscillation, when compared with a
uniform plasma.
2367 Phys. Fluids B, Vol. 5, No. 7, July 1993 Jonathan S. Wurtele 2367
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waves: The dependence of the growth rate on interaction
length and distance behind the head of the pulse is com-
bined into the exponent with fractional powers that depend
on the details of the parameter regime under study. The
head of the beam sees no wake and experiences no insta-
bility, while the tail sees a large wake and has a short
e-folding distance.
The fact that a particular pulse shape may be unstable
does not preclude interacting with the plasma, and gener-
ating useful wakes, over many diffraction lengths. Pulsesmay evolve into more complicated (more structured)
pulses which are relatively stable and can propagate a longdistance. For the accelerator application, the important
factor is the shape and phase velocity of the plasma wake,
not the laser pulse itself.
C. The electron beam wake field accelerator
Electron beams can also be used to generate wakes in
plasmas.60-67The advantage of using an electron beam is
that a high-energy beam can be rather stiff, and will not
degrade as quickly as a laser pulse while it propagates in aplasma. The disadvantage is that a high-energy beam is
needed to drive the plasma wave, and may not be readily
available without access to large accelerators.
In the overdense plasma limit, where the perturbed
plasma density can be assumed small compared to the un-
perturbed density, the response of the plasma to the driving
beam is reasonably easy to calculate analytically and quite
different from the response of the plasma to the laser pulse.
The plasma wave in the overdense limit satisfies
where Sn is the perturbed plasma density. Note that theleft-hand side of this equation is almost the same as that for
the laser driver, but the dependence on the shape of thedriving term is quite different. For the electron-beam wake,
the driving term is proportional to the beam density, while
for the laser wake it is proportional to the second deriva-
tive of the longitudinal profile (the ponderomotive force).
Since, ideally, one would want to decelerate an intense
beam at low energy while accelerating a more dilute beam
to very high energy, the ratio, R, between the accelerating
field and the decelerating field at the drive bunch should be
as large as possible. It can be shown that, when the drive
bunch is a point charge and the analysis is one dimen-sional, R has a limit of less than 2.0. To enhance this ratio,
the drive bunch must have a rise time that is long com-
pared to a plasma period and must fall off fast on the
plasma time scale. If the rise is over N-plasma periods, the
ratio R is limited to 27rAr or a triangular pulse profile. The
experiments65 at Argonne demonstrated that a plasma
wave can be excited by a relativistic beam.
Unfortunately, a drive beam which is well suited toproduce wakes with properties desirable for acceleration is
not necessarily well suited to survive its journey throughthe plasma.66 For the long bunches needed to excite large
oscillations, the transverse forces on the driver are nonlin-
ear and functions of radius. This is detrimental to the sta-
ble propagation of the drive bunch, and suggests operation
in the underdense regime.
The drive beam will generate a plasma perturbation
within itself, and this will result in decelerat ing fields
which are functions of radius and distance back from the
head of the bunch. These forces will disrupt the smooth
propagation of the drive beam. One way this can be alle-
viated is the use of an underdense plasma.
Recent work@ has studied wake excitation in an un-derdense plasma. The drive beam expels all electrons in its
path. After it passes a given point, the expelled pl asma
electrons swing back into the plasma, generating a largeaxial electric field in front of them. After the first oscilla-
tion behind the pulse, the particle motion becomes non-
laminar, and the wave is no longer suitable for accelera-
tion. This regime of operation has several attractive
features: The accelerating field is uniform radially out tothe channel radius (not the drive beam radius), the trans-
verse force is linear in radius if the background ion density
is uniform, and the plasma density can be orders of mag-
nitude smaller than in the overdense regime. The trans-
former ratio R must still be kept large by tapering the drivebunch to have a slow raise and fast falloff.
D. Plasma acceleration of photon beams
It was proposed6* that a photon bunch could be accel-
erated, as well as an electron bunch. The physical mecha-
nism behind this is the variation of the index of refraction
with time. This can be achieved in passive media with
moving plasma waves,68moving ionization fronts, or flash
bulk ionization. Moreover, the medium need not be pas-
sive. Temporal variation in an FEL has been shown
analytically” and measured experimentally’t to generatefrequency shifts. The temporal variation in the Raman
growth rate as an intense laser pulse hits a dense plasma
has also been seen to generate frequency shifts.‘*
V. CONCLUSIONS
The quest for ever higher energy particle beams has led
to significant advances in our understanding of beam-
plasma and laser-plasma interaction. An accelerator for
high-energy physics must produce not only the desired en-
ergy but also satisfy a luminosity requirement. It seems
clear that plasmas can reach extremely high gradients.
Further experimental work will study not only the energybut also the energy spread and the emittance of the accel-
erated bunch. The generation of a monoenergetic, well-collimated beam may impose limits on the efficiency of
acceleration and wave generation. Many of these questions
should be answered by experiments based on the high
power, table-top laser systems, and high brightness elec-
tron beams now becoming available.
ACKNOWLEDGMENTS
The author would like to thank T. Katsouleas and G.
Shvets for useful conversations.
2368 Phys. Fluids B, Vol. 5, No. 7, July 1993 Jonathan S. Wurtele 2368
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This work was supported by the U.S. Department of
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