Jonathan S. Wurtele- The role of plasma in advanced accelerators

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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




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





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.





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.





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.





This work was supported by the U.S. Department of

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