The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

PIC simulation of the Proton Acceleration in Plasma Waves Producedby Backward Raman Scattering
Masahiro Adachi, Shinichi Masuda\ Tom Katsouleasb, and Atsushi Ogata
ADSM, Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, JAPAN
8

Advanced Photon Research Center, Kansai Research Establishment, JAERI, Kyoto 619-0215, JAPAN~niversity of Southern California, Los Angeles, CA 90089-0484 USAABSTRACT
Plasma waves which have phase velocity same with thatof the incident beam can accelerate the beam. The phasevelocity must be far smaller than the speed oflight if theincident beam is of protons. Moreover, the phasevelocity of the wave has to increase gradually as theproton test beam is accelerated. Plasma waves producedby laser backward Raman scattering can meet theserequirement, because their wave phase velocity can becontroled by changing the plasma density distribution. In
order to verify this acceleration scheme, . a
one-and-two-halves PIC simulation code was developed.
In the simulations, we can give the plasma density
distribution, the laser power and frequency, can initial
velocity of the test beam, etc ..

has to increase in accordance with the proton
acceleration.
An idea to solve the above problem is proposed in thispaper [8], which uses backward Raman scattering
stimulated by laser produces a slow plasma wave via
parametric instability. It excites plasma waves whose
phase velocity is controlled by plasma density variation
along the laser axis.
In the next section, this new proton acceleration
scheme is explained. Section 3, 4, and 5 describes a
particle-in-cell (PIC) simulation code which wasdeveloped to study the scheme. Section 6 contains
summary and discussion.

2. Proton acceleration in Plasma wave
Produced by Backward Raman Scattering

1. Introduction
A longitudinal electrostatic wave (plasma wave) canaccelerate charged particles, if the phase velocity of thewave is synchronized with the particle. Numerous
experiments of the laser- or electron-beam-driven
plasma accelerators based on this principle have been
demonstrated the accelerations of the relativistic electron
beams [ 1]. However, no experimental studies ofproton/ion acceleration using plasmas have been madefor the last twenty years.
Study of proton/ion acceleration using plasmas has·longer history than that of electron acceleration. The"electron ring accelerator" had been studied extensively
until early 1970's in USSR, USA, Germany and Japan,
in which a small amount of ions in an electron ringshould have been forced to acquire the speed of theelectrons. However, the studies have terminated without
remarkable success [2]. Recently, similar ideas havebeen rekindled by Russian scientists [3]. Some otherideas of plasma-based ion accelerators are found inreferences [4-6].
The same technique as electron acceleration using therelativistic plasma waves cannot be applied to theacceleration of protons with energy below -1 Ge V,
because velocity of protons is not constant in this energy
region. To trap the protons in the plasma wave, the phasevelocity of the plasma wave has to match with that ofthe
injected protons, firstly. Secondly, this phase velocity

Dispersion relations of waves in a plasma arem2 =m~ +c2e

(1)li12 = li1~

(2)
for the electromagnetic and electrostatic waves,respectively, where aJ and k are frequency and wave
number of eigen mode, mP is a plasma frequency and cis the speed of light. Eq.(2) tells us that the electrostatic
mode is fixed at mP but can have any wave number.
The plasma wave with larger k propagates with smaller
phase velocity.
The backward Raman scattering is a three-wave
parametric instability [9-11]. An incident electromagnetic wave (pump laser) decays into an electromagnetic wave (Raman-scattered wave) and a longitudinal electrostatic wave (plasma wave). The phasematching conditions in three-wave interaction:kL =kR+kp,
liJL =li1R+li1p,(3)must be satisfied, where suffixes L and R mean thepump laser and the scattered radiation, respectively.
Eqs.(l) and (3) give the wave number of the scatteredradiation as a function of li1 L and li1 P ,-289-(4)The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

where, and henceforward, + and - signs correspond toforward and backward scattering, respectively. Substituting Eq.(4) in Eq.(3), the wave number of the excitedplasma wave is obtained as4. Trapping of Protons by a Plasma Wave
The protons in the slow plasma wave is simulated asshown in Fig.3 and 4. The protons with velocity 0.3c are
injected, as a test beam into a uniform plasma under thesame conditions with those in Fig.l. Depending on thephase of the plasma wave at the injection, each proton
becomes accelerated or decelerated. The proton
distribution is highly distorted to form a sinusoidal
velocity distribution (Fig4.(a)), and it becomes wider as
the test particles propagate with the plasma wave (Fig. 3and 4.(b)).
Inserting the initial proton velocity 0.3c, into Eq. (6),we get UJ P I UJ L =0.405 . If the laser wavelength is 1 J-l m,
the electric field at the wave breaking limt,
C 1u11 = meUJ Pcf3P, becomes 390GeV/m.
Fig.4(a) shows that some test particles are accelerated
from 0.3c to 0.306c~ 98.7(GeV) in the distance ~60/kp
or 23.61J.!m. The acceleration gradient of this simulation,
Gsim , becomes 83.7GeV/m. As the result, the ratio
GsiniGtull becomes 0.214.

_uhg;;Pk =kL +- 1-2cp=lih-clihl

1--2 +1-2- .
[nf~-g;;Pfih(iJr(5)

Finally, the phase velocity of the plasma wave iscalculated asf3

1= (jJP = (jJP

P ckP

UJL(6)/1-UJ21UJ2 + /1-2UJ IUJ'.}

PL'.}

pL

This equation shows that the plasma wave producedby backward Raman scattering has slow phase velocity
which increases with the plasma density.
There still remain two differences between the protonand electron accelerations. First, the acceleration gradient is much smaller in ·the proton acceleration. Theacceleration gradient at the wave breaking limit of theelectron acceleration is given by meUJ Pc , while it isin the proton acceleration.
Second, the Landau damping suppresses the Ramanscattering. It is not serious in a low-temperature plasma
as long as the pump laser exists. However, once thepump laser vanishes, the plasma wave also fades out
because of this damping mechanism. Though a shortlaser pulse leaves the wake behind in the laser wakefield acceleration of electrons, this scheme does notwork in the proton acceleration, because the wakecannot survive until a slow proton test beam arrives. In
other words, the plasma wave has to be excitedcontinuously by a de-like laser.
Calculation given in Ref.[7] shows that a proton beam
can be accelerated from ~2 to 50MeV in ~12 ps in aplasma9 in which the density is changed from~3x10 1 cm·3 to ~3xl0 20 cm· 3 in ~0.5mm. The requiredlaser energy for this acceleration is less than 500 mJ, if
we use a laser with wavelength of 800 nm.

0.300000010

me(jJ p v p"' 0.300000000~test proton beam

0t1t

a-0.300000005

0.299999995

1.0x1o·'"Ci!

0.51;;;~:~ i:';

E 0.299999990

0.00-0.5"'

plasma particle :. i < -1.00.299999985"tjtw,=40
0.299999980-120-100-80-40-60-200

posision : xkx

Fig. I. Distribution of the plasma particles and theinjected protons.

j__../·

50··········Ex.-.---- Backward~ scattering peak

40~

30

3.PIC Simulation of Backward Raman
Scattering-:EzPlasma wave peak//II·= Laser peakH'l2010We have developed a one-and-two-halves PIC
simulation code to study the proton acceleration. As inFig.1, we distribute the uniform plasma from x=O tox=100/k, and test protons from x=--130/kp to -80/kp
with vefocity 0.3c (energy 98.5 MeV). Fig.2 shows
spectrum of the electric field at t = 400 I UJ P • A laser is
injected into a uniform plasma at x=O, whosefrequency is 2.47UJ P , and amplitude grows linearly10kxc/wPFig.2. Spectra of the plasma wave and the laser.
0.30150.10

0.3010"'~0.30050.05

0.30000.00 ~E0from t=O to 11 UJ P and then becomes constant. Its
normalized vector potential is 0.1. Fig.2 shows that aslow plasma wave by the backward Raman scattering is
produced at kx ""l.08UJ P I c , whose phase velocity isapproximately 0.3c. Fig.3 shows that the electric field ofthe plasma wave is -0.14meUJpcle =0.47meUJpvp le.

~~c.

~
~~

~
....__.3
00.2995-0.05

0.2990-0.10
0.2985
020

40

60

80

100xk,Fig.3.Test proton beam (dot) and electric field of theplasma wave (line).

-290-The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

6. Summary and Discussion

0.306
0.10
0.304

A method is proposed to accelerate protons by aslow plasma wave produced by the backward Raman
scattering. It adjusts the plasma density so that the phasevelocity of the plasma wave harmonized with the groupvelocity of the accelerated protons. Using a uniform
plasma, we succeed in proton acceleration in thesimulation. About synchronizing the phase velocity withthe proton velocity in the inhomogeneous plasma, more
simulation study is needed.

0.05.!"

0.302

0.00

E':~

0.300

-0.05

0.298

-0.10

20

40

60

10080

(a)

xk,
0.20
0.306

.
~

<:Q.

..

~

0.10.r;,;:/iit;m@mli~i0.302

il_0.300

0.298;;

6080

References

0.15

....

0.304

0.05
0.00

E':~

-0.05
-0.10

i100

120

160

140

(b)

xk,

Fig.4.Phase spce plot of the proton beam injected intothe plasma. (a) t = 600 I UJ P . (b) t = 800 I UJ P .

S.Growth Process of the Plasma Wave
Figs.5-6 summarize some results of simulations.
Fig.5 shows the time evolution of electric field of theplasma wave at various UJ P I UJ L ratio. We find that, in aplasma with a higher density, the waves grow more
quickly, and are saturated more quickly. Fig.6 shows theelectric field at the saturation as a function of theUJPIUJL.

>.
0~-~

[1] E. Esarey et al., IEEE Trans. Plasma Sci. 24 (1996)
252.
[2] M. Resiser, Seminar on New Trends in Particle
Acceleration Techniques, Capri, June 1982, Servizio
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·[4} E. Comet et al., Phys. Rev. Lett. 46 (1981) 181.
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[6] B. Rau, T. Tajima, Phys. Plasmas 5 (1998) 3575.
[7] A. Ogata, T. Katsouleas, Proceedings of the 1998
Workshop on Advanced Accelerator Concepts, AlP
Conference Proceedings 472 (1999) 551-562.
[8] T. Katsouleas et al., Proceedings of Laser
Acceleration of Particles, AlP Conf. Proc. 130 (1985)
63-90.
[9] C.S. Liu, P.K. Kaw, in: P.K. Kaw et al. (Eds.),
Advances in Plasma Physics, Vol. 6, Wiley, New York,
1976, pp. 83-119, 121-177[10] J.M. Manley, H. E. Rowe, Proc. IRE 44 (1956) 904.
[11] D.W. Forslund, J.M. Kindel, E.L. Lindman, Phys.
Fluids 18 (1975) 1002.

0.1

o~~"'"''"'>
0:;;..c:o~0.01

c"'.:;iif~:J><
w-0.001~"'..:9- 0.0001
100200

300

400

500

600

700

800!/(plasma frequency)

Fig.5. Growth of Ex at each ratio, UJ P IUJ L.
I'"""*-maximl..lll value at each ratio

g o.,! 0.01 . ....
~

•L

IX;

0.1

0.2'"

.

0.001

i
0.0001

0.3

0.4

0.5

rat io(pl asmafrquency/laser f reqquency)

Fig.6. electric field at the saturation as a function of theUJPIUJL.

-291-