PLASMA ACCELERATORSCHARACTERIZED BY RF LINAC TERMINOLOGY
A. Ogata, AdSM, Hiroshima University, Hiroshima, JapanJ. B. Rosenzweig, UCLA, Los Angeles, CA, USA
M. Ferrario, LNF dell’INFN, Frascati, Roma, Italy
Abstract
Plasma accelerators are described by using RF linac terminology such as shunt impedance, filling time, transit time
factor, quality factor etc.. Some notions are hard to import from RF linacs into plasma accelerators. If we dareto evaluate the plasma accelerators using such parameters
as Q, Zs , etc., however, we have to admit that the plasma
accelerators are not so efficient as the RF linacs.

1INTRODUCTION

Recent effort have shown that plasma accelerators can havean acceleration gradient exceeding a GeV/m[1]. Design ofa linear collider based on these acceleration mechanisms
has been seriously discussed[2]. It is the time to examine the possible performance of the plasma accelerators inreal applications using understandable terms. One methodof examination is to compare the plasma accelerators withRF linacs, which have a history of more than 50 years
and their design procedures and performance estimation
are well established[3].
The next section of this paper describes differences between the plasma accelerators and the RF linacs. The following sections aim at applying the design criteria and theterminology of the RF linacs to the plasma accelerators.
Dependence of the parameters on the plasma frequency is
discussed in a separate section. The description is limited
to electron acceleration. This paper is an expansion of theprevious work[4].

2

PLASMA ACCELERATORSAND RF LINAC

We can regard plasma accelerators as a type of travelingwave, constant-impedance linacs. There exist, however,some definite differences between plasma accelerators andRF linacs.
First is in that the plasma accelerators have their driverinside the accelerating structure, while electromagnetic
waves are fed from outside of the structure in RF linacs.
In the RF linacs, it is possible to discuss wave generatorssuch as klystrons separately from the accelerator structure,but it is impossible in plasma accelerators. Pump depletionof the drivers is one of their characteristic problems. Thedrivers are either lasers in the laser wakefield accelerator(LWA) and in the beat wave accelerator (BWA), or particle
beams in the plasma wakefield accelerator (PWA), which

drive plasma waves sacrificing their energy. The pump depletion length in the LWA is given as[5]µ¶2µ ¶2EL
ωLLW ALpump = `L= `L,(1)Ez
ωpwhere `L is the laser pulse width. Assuming a set of typical LWA parameters, n = 1018 cm−3 , laser wavelength
λL = 1µm, and `L ∼ λp , we have Lpump = 3.6cm. Inthe PWA, energy balance arguments give a depletion lengthstraightforwardly. According to the fundamental theoremof beam loading, the peak deceleration in a particle in thedriver is ∼ eEz /2. We then haveWALPpump =2(γ − 1)me c2,eEz0(2)where the numerator is energy of a particle in the driver. If
a 100MeV drive particle attains 100MeV/m initial acceleration gradient, the depletion length is 2m.
Next, the RF linacs supply the power to beams as pure
electromagnetic waves, while the plasma wave is an electrostatic wave. We can study RF linacs analyzing the power
flow of the RF with finite group velocity. To the contrary,group velocity of the plasma wave is zero if we neglect
the plasma temperature, though its phase velocity is almost
equal to the speed of light.
The RF linacs use linear waves. Some plasma accelerators, however, use nonlinear plasma waves, because
they attain higher acceleration gradient. They are the selfmodulated LWA and the PWA in the ’blow-out regime’.
In these nonlinear plasma waves, the wave-breaking phenomenon occurs. The present paper mainly describes linear plasma accelerators, because the previous paper refers
to nonlinear plasma accelerators[4].
Lastly, there exists one complication that the plasma accelerators do not have but the RF linac does posses. It is theexistence of higher spatial harmonics, or nonsynchronous
phase velocity waves, as well as higher-order longitudinalmodes in the RF linac.
Processes involved in the RF linac and two typicalplasma accelerators, LWA and PWA, are shown in Fig. 1.
In the LWA (Fig. 1(a)), the pump laser has problems. It
is diffracted in a plasma and the diffracting photons canexcite plasma only weakly. We do not worry about thediffraction seriously in this paper, hoping that the opticalguiding will solve this problem in the near future. Thelaser excites a plasma wave sacrificing its power. Pump
depletion is thus caused. Additionally, if the pulse lengthFigure 1: Energy flow in an LWA(a), in PWA(b) and in an RF linac(c).
is much longer than the plasma period, the excited plasma
wave in turn modulates the laser pulse waveform in selfmodulation instability. The self-modulated LWA positively
utilizes this mechanism. After the wave excitation, we haveto consider processes such as beam loading, wave decay,
and wave breaking in some cases. Moreover, the accelerated driven beams overrun the plasma wave. This phaseslippage dictates the acceleration length in some cases.
Fig. 2 shows the linear plasma wave in the LWA before
and after propagation of certain distance. The amplitudeof the wavefront decreases, because of the pump depletion(”a” in the figure) due to ionization and wave-excitation.
The amplitude of the wave following the wavefront also
decays which is indicated as ”b” in the figure. This wave
decay is discussed in the next section. If the particle isaccelerated to a speed greater than the phase velocity of theplasma wave, it slips forward from the optimum phase. The
effective field for acceleration they obtain thus decreases,by an amount indicated as ”c” in the figure.
Processes in the PWA are simpler than in the LWA. Neither diffraction nor phase slippage is involved. However,
we have to prepare plasmas in the PWA, though tunneling ionization using drive beams is possible under certain
conditions[6].
The plasma accelerators are usually classified by theexciting principles of plasma waves into PWA, LWA andBWA. From the viewpoint to find their similarity with RFlinacs, we can divide them into two; whether they excite plasma wave pulsively or continuously. Whatever theplasma acceleration scheme is, the evolution of the electricfield associated with the linear plasma wave is expressedby the second order differential equation:
dEz
d2 Ez
+ Ez ωp2 = f (t),+ γw
(3)
2dt
dt
where ωp is the electron plasma frequency and γw is thesum of all of the wave decay constants. The drivingterm f (t) generally describes the wave excitation mechanism. The standard LWA and PWA excite plasma waves
pulsively, while the BWA, the self-modulated LWA, andthe wakefield accelerators with pulse-train drivers[7] excite them continuously. In the latter case, the driving termf (t) has a component of the plasma frequency to excites
the plasma wave resonantly. Though this is not explicitin the self-modulated LWA, we can understand that its
plasma wave is excited by the beating between the pumpand forward-Raman scattered radiations, so that it is a kindof BWA. The following description shows that those belonging to the latter with continuous drivers are more closeto RF linacs.

33.1

ANALOGIES TO RF LINACPARAMETERS

Quality Factor

In the absence of the driving term, solution of Eq. (1) is aexponential with a decay constant, which defines the quality factor:
ωp.
(4)
Qw =2γw
We can identify at least three wave-decay processes assources of γw ; collisional damping, modulational instability and Landau damping. In order to discuss the collisional

Figure 2: A Plasma wave amplitude before (left) and after (right) propagation of certain distance. Amplitude of thewave decreases during the propagation due to the pump depletion (”a” in the figure). Because of the plasma wave decay,
the amplitude is maximum at the wavefront and decreases (”b” in the figure). Because the accelerated particle gets thevelocity greater than the phase velocity of the plasma wave, it slips forward from the optimum phase. The effective field
which the driven particle accepts decreases (”c” in the figure).
damping, we distinguish here the colliding species by lowercase letters. We then approximate the collision frequencies asωpe log Λc,(5)νei ∼ νee ∼Λcand νii ∼ (me /mi )1/2 νei , νie ∼ (me /mi )νei , wherelog Λc is the Coulomb logarithm[8] with Λc = 12πnλ3D ,λD = (²0 kTe /ne2 )1/2 is the Debye’s shielding length, nis the plasma density and Te is the electron temperature.
The decay constant due to the modulational instability isµ ¶1/4µ¶1/23ωpe vte
ωpi,γmod =2
ωpi vL(6)

where ωpi is the ion plasma frequency, ωpe = ωp , vte =(kB Te /me )1/2 and vL = eEz /(me ωpe )[9]. These two decay constants are depicted in Fig. 3 as functions of plasma
density and electron temperature, in which an assumptionvL = 0.2c is made. In a low-density region, the collisions
dominate the wave decay, while in a high-density region,
the modulational instability dominates.
At Te =10eV and ne =1018 cm−3 as a typical set of parameters of the linear LWA, we have γmod ∼ νei + νee andQw ∼ 20 aside from the Landau damping. Though thisvalue may have a large error, it is far smaller than the Q
value of a standard RF linac, which usually exceeds 1000.
The quantity attenuation constant of the RF linac given
by α = ωRF /2Qvg is the attenuation per unit length of
the field of the acceleration mode due to the loss to theaccelerator structure. It resembles to decay constant of theplasma wave. We cannot forget, however, that the groupvelocity vg of the plasma wave is nearly zero so that thedirect correspondence does not stand.
The Landau damping occurs only when the plasma electrons have drift velocities near the plasma wave velocity.
This can be serious when the plasma electrons are trapped
and accelerated. One is the case in the experiments using
lasers with powers exceeding 20TW[10], and the other isthe case of laser-injection laser acceleration (LILAC)[11].

Figure 3: Decay constant of plasma wave due to the modulational instability (solid lines) and collisions (νei + νee ,dashed lines) as a function of plasma density and electrontemperature.

3.2

Filling TimeIn the RF linac, the filling time is the time necessary for
drive waves to fill the accelerating structure, which is theproduct of the group velocity of the RF wave and the lengthof the acceleration structure. It is not directly applicableto plasma accelerators, in which the group velocity of theplasma wave is almost zero. If we interpret it as the time forplasma wave amplitude to reach saturation, the BWA, theself-modulated LWA and the wakefield accelerators withmultiple drivers (or driver train) have the filling time.
If the equilibrium is attained by balance between plasma
wave decay and excitation by drivers in these schemes, thefilling time should be identical to the decay time. In practice, the quiver motion of the plasma electrons cause saturated equilibrium in the BWA. If the wave amplitude makes
the electron motion ultrarelativistic, it lowers the plasma
frequency to break the resonant condition against the beatfrequency in the BWA[12]. The same phenomenon could
occur in the wakefield accelerators with multiple drivers.
The standard wakefield accelerators, however, cannot

have the notion of the filling time, in which the drive waves
are excited in a shorter time than the wave period.

3.3

Transformer Ratio

Problems caused by the wake in the plasma accelerators are
not so difficult as in the RF linacs. Because of the plasma,
the driven beams do not see the surrounding structure.
Moreover, the plasma wave itself has no higher harmonics. However, the notion of transformer ratio, peculiar to
wakefield accelerators, is important in the PWA. It is themaximum accelerating field for electrons behind the drivebeam, divided by the maximum decelerating field in thedrive beam. This ratio cannot exceed 2, if the bunch lengths
are smaller enough than the plasma wavelength[13]. If westretch this definition to the LWA, definition of its transformer should have the maximum decelerating gradient for
the laser photons as the denominator, because the energy
transfer from the laser to the plasma is predominantly inthe form of frequency shift of the laser[14]. It has beenshown that the transformer ratio of the LWA is nearly equal
to 2, in the limit where the laser pulse length is much longer
than the plasma wavelength, using the laser red-shift as ameasure of laser deceleration[15].

3.4Shunt Impedance

The shunt impedance is one of the measures to evaluate
the accelerating tube characteristics. What if we apply thisnotion to the plasma tube with a cross section S? The energy loss in a unit length of the accelerating tube −dP/dz
is proportional to the square of the accelerating field Ez .
The shunt impedance is the coefficient between these quantities,(7)
Zs = Ez2 /(−dP/dz).
Incidentally, the physical meaning of the quality factor isQw = ωp W/(−dP/dz),(8)where W is the energy of the field in a unit length:·¸
12z
1, (9)
W (z) = ²0 Ez (z)2 S = ²0 E02 S exp −22Lpump
in the plasma accelerator, with Lpump being the pump depletion length[4]. Deleting −dP/dz from these two relations, we have Zs = Qw Ez2 /ωp W . Assuming Lpump islong enough and using Eq.(4), we haveZs =1.
2²0 γw S(10)

In order to make a rough estimate, let us assume S = π/kp2 ,where kp = ωp /c is the plasma wavenumber. We then have
Zs =ωp2.
2π 2 ²0 c2 γw(11)For example, the parameters n = 1018 cm−1 and γw =1012 s−1 give Zs ∼ 200kΩm−1 . It is much lower than thatof a tube of the RF linac, >MΩm−1 .

Because the impedance is defined as the transfer function from current to voltage, and because the PWA converts drive beam current to electric field or voltage per unit
length, its wake amplitude can define the impedance. Approach from this viewpoint is described in the previouspaper[4]. To the contrary, laser as a driver is not well described by the impedance concept.

3.5

Beam Loading

The driven beams also excite wakefield in a plasma whose
phase is always into the direction to decrease the existing
field. Existence of the driven beams thus weakens the fieldof the plasma wave. We can understand that this is thebeam loading in the plasma accelerators. The linear theory predicts that amplitude of the decelerating field causedby the beam loading of N particles should be equal to theplasma wakefields induced by the N particles in its absolute value in any plasma accelerators[17]. The peak fieldthen becomes¸·2eN
4+2K(kr),(12)1−Ezload =2 p b
π²0 rb2
kp2 rb2
where rb is the beam radius. This equation ignores thefunctional dependence of the beam loading field on the radial position, an effect which is strong especially in linearregime.
If the acceleration is stationary as in the case of RF linacs
with continuous RF filling, it is possible to write the electric field as Ezef f = Ez − IZb , defining the driven-beam
impedance Zb . Acceleration in the PWA and the LWA,however, cannot be stationary. Though some continuouslydriven plasma accelerators such as the BWA have the possibility of the stationary operation, no such trial has been
ever made. It is more practical to write the effective field in
the multiple driven bunch operation as
XkEzload(t + 2πk/ωp ). (13)Ezef f (t) = Ez (t) −k

If a train of driven bunches are spaced at integer multiple ofthe plasma wavelength (we could define the beam currentas I = eN ωp /2π in this case), the resultant energy gainand energy spread would differ from bunch to bunch. It ispossible, however, to equalize these quantities by choosing
the bunch spacing slightly different from this.
In RF linacs, the beam loading factor, the ratio of thepower for acceleration to the total power stored in the accelerating structure, is designed around 10%. It is relevant
to follow this criterion in designing the charges of driven
beams of plasma accelerators.
As was mentioned previously, the Landau damping occurs when the plasma electrons have velocities near theplasma wave velocity. If we change our point of view, wecan consider that the plasma wave decay due to the beamloading is due to the fast electrons. We can thus regard
the beam loading phenomenon as a generalized Landau
damping[16].

3.6

Phase Slippage and Transit Time Factor

The phase velocity of the plasma wave in laser accelerators
is equal to the group velocity of the laser in a plasma, whichis smaller than the speed of the light in a vacuum. Thebunches with the speed of light satisfying z = ct surfs onthe phaseµ ¶2¶µ1πz11 λL−
ωp z = −,ωp t − kp z =c vp2 λpλpwhere ωp /kp = z/t = vp . Effective electric field thedriven beam feels is"#µ ¶2πz1 λL,(14)
Ezef f ∼ Ez0 cos −2 λpλpwhere we have neglected the pump depletion. If we define£¤RL
Ez0 cos −(λL /λp )2 (πz/2λp ) dz
0,(15)T =RL
Ez0 dz0
it is the ratio of the energy gained in the course of phaseslippage in a length L to that in a dc field Ez0 . This
has a similar form of the transit time factor in the Alvarez RF linac. Equating L = (λp /2)(λp /λL )2 in eq.(15),which is called as ”dephasing length”[4], we have T =2/π ∼ 0.636. Typical LWA parameters, λL = 1µm andn = 1018 cm−3 gives the dephasing length to be 18.6mm.
The transit time factor of a synchronous particle in theAlvarez linac is π/4 ∼ 0.785.

4

PLASMA-FREQUENCY DEPENDENCEOF PARAMETERS

The acceleration gradient in existing RF linac structures islimited by breakdown and the maximum gradient scales as
7/8∼ ωRF . As the frequency is increased, the breakdown isless of a problem, but there appear other limits such as surface melting of the tube wall. The plasma accelerators arefree from such limits, so the large acceleration gradient is
its greatest merit. It is proportional to ωp . We examine here
plasma frequency dependence of other parameters such as
the quality factor.
−1/2
We know Q ∼ ωRF in the RF linac[3]. In the linear plasma accelerators, the frequency dependence depends
mainly on the plasma density and partly on the plasma temperature. Eq.(5) stands in a low density region where themodulational instability dominates the plasma wave decay.
It tells us ωp /νei ∼ Λc . Because Λc ∼ n−1/2 = 1/ωp , wehave Qw ∼ ωp−1 . Collisions dominates in a higher density
region. In this region Eq.(6) stands, in which ωpi is proportional to ωpe , and vL is independent of ωp , because Ez
in the numerator of its definition is proportional to ωp . We
then conclude that Qw is independent of ωp in this region.
Consequently, the shunt impedance Zs is independent of
ωp in the collisional region and ∼ ωp in the modulational
1/2
region, while we have Zs ∼ ωRF in conventional linacs.

5DISCUSSION AND CONCLUSIONS

Though it was not discussed in this paper, the efficiency
from wall plug electric power to either laser pulses (inBWA and LWA) or drive beams (PWA) has to be taken into
consideration. The conversion efficiency of klystrons usedin standard RF linacs is ∼ 50%. To the contrary, the wall
plug efficiency of T3 lasers presently used in LWA experiments is less than 10−4 . This is small enough to discourage any accelerator design including linear colliders. We
hope that development of semiconductor-laser techniques
will improve this value significantly. The PWA could havemuch higher efficiency at the present, because of the possibility of obtaining wall-plug efficiency in the 40% range
using a heavy beam-loaded multibunch drive RF linac.
In conclusion, we have tried to image plasma accelerator
from the viewpoint of the RF linac. Some notions are hard
to import from RF linacs into plasma accelerators. If wedare to evaluate the plasma accelerators using such parameters as Q, Zs , etc., however, we have to admit that thatplasma accelerators are not so efficient as the RF linacs.

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