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

Plasmon Linac
Naoko Saito, Yoshio Wada and Atsushi Ogata
Department of Quantum Matter, Graduate School of Advanced Science of Matter,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan
Abstract
This paper proposes a new accelerator using plasmons.
Size of the accelerator structure is reduced to the order ofthe laser wavelength. The structure is made from material
whose plasma frequency is around the laser frequency. Alaser pulse excites plasmons on the surface of the accelerator tube, and the electric field of the plasmon accelerates charged particle beams. This acceleration scheme isclassified into a wakefield acceleration, in which the electric field of the plasmon can survive after the laser pulse
passes through the structure. It does not, however, requirea high power laser as other laser-plasma accelerations, because this acceleration is based not on the ponderomotive
force but on a lower-order force.

1 INTRODUCTIOIN
This paper proposes a new linac using the plasmons,
which can accelerate charged particles with the speed oflight. It has been reported that the laser wakefield acceleration using electric field of an ionized gas successfully
accelerated electron beams[1]. This technique, however,requires a special laser with energy above 1TW and pulse
which nearly equal to the plasma wavelength.
In this proposal, structure of the accelerator has periodicity similar to that of the RF linac. Material with a solidplasma frequency around the laser frequency is used in thestructure. The laser propagates through the structure as TM
waves to excite plasmons on the wall surface. We can regard the electric field of the plasmon as a wakefield, and thetechnique of the proposal as a kind of wakefield acceleration. This acceleration is, however, based not on the ponderomotive force as the laser wakefield acceleration, but
on a lower-order force. Consequently, it does not require ahigh power laser as other laser-plasma acceleration methods.
Three dispersion relations, that of the TM mode of thelaser, that of the plasmon and that of the test particles, mustsatisfy certain relation to realize the acceleration successfully. Section 2 discusses this problem. Section 3 calculates the acceleration gradient of this linac. Section 4 discusses this linac from practical viewpoints.

Figure 1: Disk-loaded structure.

have the point where the phase velocity equals c. Let thispoint be A(w1 , kl). The laser then must have a TM mode,
whose dispersion curve has the point A on it. The diskloaded structure shown in Fig.1 can support the TM modes,
whose dispersion curves cross with that of the light, w =ck. Fig.2 shows it, which has bandgaps between ™oo andTM01 mode, etc ..
The TM modes invokes charges at the intersection ofthe accelerator wall and the line of electric force, that areplasmons. Oscillation of the plasmons accelerate particles,
if the dispersion curve of the plasmon shares the point A
with the curve of speed of light and with the curve of theTM mode. It is certainly necessary to derive the dispersion of the plasmon in the disk-loaded structure. In thispaper, however, we do not dare to carry out the numericalcalculations. Instead, we deduce the characteristics of thedispersion curve of the periodic structure from the curve ofthe plasmon in a inner wall of a straight cylindrical holedug in the homogeneous solid.
The dispersion curve of the plasmon excited on the innersurface of such a hole is found in [2]. Using Bessel functions Im(x),Km(x), with m = ±1,±2,±3, ... , we canwrite this dispersion function as follows:

I:r,(kb)Km(kb)E(w) = Im(kb)Kin(kb)'(1)w

2 DISPERSION RELATIONS OF TMWAVES AND PLASMON
In order that the plasmons can accelerate particles withthe speed of light c, dispersion curve of the plasmon must

Figure 2: Dispersion curves of TM modes in the diskloaded structure.

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

w/w.

W/Wp
1~-----.-------------------,(a)lrr------------,---------1I

(b)

IIII

f w=ck

0.9

0.9

m=OI
I

0.85

III

0 . 8 plasmon

0.8

I--

TMo1

II

0.75

Figure 4: Three dispersion curves.

Figure 3: The dispersion curve of the plasmon.
where b is the hole radius andaI;,(x) = dlm(x)jdx,K;,(x) = dKm(x)jdx.
We approximate the dielectric function· around the plasma
frequencybyt:(w) = 1-w;fw(w+i'y) ""l-w;jw 2 . We
then havew! = w;kblm(kb) I K;,(kb) I.

(2)

Eq. (2) is depicted in Fig. 3(a) for the case m = 0.
In general, we can obtain a dispersion curve-of the periodic structure by folding the dispersion curve of the homogeneous structure at the Brillouin boundary. Let Fig. 3(a)be the dispersion of the homogeneous structure in thepresent case. If the hole has period 27rb in the axial direction, its dispersion curve has the Brillouin boundary at
kb = 1. Folding the dispersion curve of the straight holeat this boundary, we can conceptually obtain the dispersion
curve of the plasmon in a periodic hole, which is shownby the solid line in Fig.3(b ). The original and folded lines
repel each other to generate band gaps near the Brillouinboundary (in the region encircled in Fig. 3(b)). In following calculations, however, we neglect the existence of thebandgaps.
Now we are going to find the frequency w., at which
three dispersion curves intersects. In the following, weconsider only the lowest mode of the plasmons; i.e., m=O.
We fix the axial period l as b/2. To find thew.,, we havemade the following iterations. 1) We first assume the holeradius b. 2) We then draw the dispersion line of w = ck andfind the intersection A of the dispersion curves of the plasmon and the straight line. 3) We draw the dispersion curveof the electromagnetic wave, and, ifthe dispersion curve of
the electromagnetic wave fails to pass the intersection A,
we change the value of b.
The result is shown in Fig. 4. If we inject a laser withthe frequency w.,, it transmits the tube in the ™o 1 modeand excites the plasmons. Because the phase velocity ofthe plasmons is c, they can accelerate the charged particles
with the speed of light.
In the calculation of the dispersion relation of the laser,
we used the approximate equations of the ™o 1 mode dispersions of the disk-loaded structure[3]:w =WcV1+~~;{1-cos¢exp(-ah)},--0 · 7 '-=o-----=-1-----::-2---'---::------;4---;;-5----;:6 kbo. 7Lo-------!-1------::2------::-3------;-'4 kb .....______~(3)= 2.405a/\; = 37rJ[(2.405)b2 l'(4)(5)

where we put a = l = b/2, h = b/8 as the structure constants.
In the iteration, we started from the b value which gives
We= wpj../2, where we= 2.405c/bis the cutoff frequencyof the ™o 1 mode in a straight hole and Wp / ../2 is the thesurface-plasmon frequency. After a few trials, we have succeeded in depicting Fig. 4. For l = b/2, the Brillouinboundary lies at kb = 27r. Though we have neglected theeffect of a bandgap on the plasmon dispersion, it must not
be serious, because the intersection A exists far from theboundary.
Silver has the smallest plasma frequency in metals [4];wp = 13.2 x 1015 s- 1 , while 1 = 68.9 x 1012 s- 1 [5].
The frequency at the intersection in Fig. 4 then becomes
9.96 x 1015 s- 1 , that is the required laser frequency. The
corresponding wavelength is 189 nm, which is in the ultraviolet region.
If we give the first priority to the laser wavelength in
the design and set it 1 J.lm (w = 1.88 x 10 15 s- 1 ) as anexample, the required plasma frequency of the material becomes 2.49 x 10 15 s- 1 ; in other words, the plasma density becomes 1.91 x 10 21 cm- 3 • No natural material has
a plasma frequency in this range[4], but it is possible to attain this frequency by artificially doping impurities into thewide-gap semiconductors.
At the intersection point of the dispersion curves, thephase velocities certainly coincide, but group velocities aredifferent. In Fig. 4, group velocities v 9 of the electromagnetic wave and plasmon are"' 0.0095c and"" -0.0085c,
respectively. If the cavity length is L, the laser pulse length
has to exceed L/lvul· Assuming L = 1 mm, for example,
and choosing a smaller value of the two group velocities,
we have the minimum pulse length as ,....,390 ps.

3

THE ACCELERATION GRADIENT

Without plasmons, the acceleration electric field vanishes as soon as the laser pulse vanishes. Under the existence of plasmons, we can express the electric field by-302-The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October2001

The result is Ws[J] = 21.3 x 10- 27 E 0 [V/m] 2 , whichmeets with the relation Ws = QPwau/w. The necessary
electric power p = Qp wall then becomesthe equation of the forced oscillation:

P[W] = 40.3 X 10- 12 Eo[V /mt
The right-hand side expresses the laser excitation of plasmons and "( is the relaxation time.
To estimate the acceleration field of the laser in the righthand side of Eq. (6) in the disk-loaded structure, we consider a pill-box cavity with radius band length l. The fieldsof this pill-box the cavity are

E _ Ez-7OJO

2.405r H _ EoJo2.405rb ,0 'b(o(7)

nwhere (o = 376 is the impedance of vacuum.
Following the standard procedure of linac design, we
first derive the cavity loss using the electrical conductivity of the cavity material. In the following calculations, we
assume the laser wavelength to be I J.Lm (w L = 1.88 x 10 15
s- 1 ). We then have b = 2l = 432 nm.
The energy loss in the cavity becomes(8)

whereThis equation tells us that the laser power of 40 MW (4GW) is necessary to obtain the acceleration gradient withGeV/m (10 GeV/m). There is no established theory which
gives the maximum-tolerable acceleration gradient in themetal cavity, though some say it is around 100 GeV/m [6].
Inserting the field of the electromagnetic wave, we havethe electric field of the plasmon as the solution of Eq. (6).
If the right-hand side is transient, the electric field attenuates with the time constant 1h. If the right-hand side isstationary, the amplitude of the electric field is wpEo/2"!
times larger than the case where only the electromagnetic
wave exists.
The small Q means that efficiency of this accelerator isquite poor. This must be one of the reasons why the linac
whose rf is replaced by the laser has never been realized,in spite of the development of the nano-meter techniqueto produce fine structure. In our plasmon linac, the laser
energy is used not only for generating the electromagnetic
field but also for the plasmon excitation. It is necessary toreconsider the efficiency problem from this point of view.

4 DISCUSSION(m = JwJ.Lo/2a,is the surface-resistance. If we adopt silver as the cavity
material, (m = 4.16 nat w = 1.88 X 10 15 s- 1 . The skindepth(9)then becomes 3.52 nm. Modification of the right-hand sideof Eq. (8) yields

EJ{2fob Jr( 2 .~ 5 r 21rrdr)+ 27fblJf(2.405)}.
The present parameters give Pwall [W]10- 15 E 0 (V /m] 2 . The Q value of this cavity

l b
Q=Jb+l'(10)989

X

Is it possible to machine this linac. If we use silver, theradius of the accelerator tube b becomes 81 nm, while if weuse some other material which meets the laser wavelength
1 J.Lm, it is 432 nm. Such sizes can be dealt with the present
lithographic techniques. It is, however, impossible for it
to dig holes deeper than ~ 1 J.Lm. One solution to form alonger tube is to put two surfaces together each has a gutter
on it.
In summary, we have given conception of a new linac
in which plasmons excited by a laser accelerate charged
particles with the light velocity. There remain of coursemany works to be done in order to verify the feasibility ofthis concept, we believe that it is a promising solution toconstruct a miniature high-energy accelerator.

(11)

becomes 40.8. The Q value of standard RF linacs lies between 10 4 and 105 at w= 100 - 1000 MHz. The above Q
value of our pillbox cavity is consistent with the scaling

Q <X w!.

5 REFERENCES[1] A. Ogata and K. Nakajima, Laser Part. Beams 16 (1998)
381.
[2] N.R. Arista and M.A. Fuentes, Phys. Rev. B 63 (2001)
165410.
[3] J. Gao, Nucl. Instr. and Meth. A311 (1992) 437 ;T. P. Wangler, "Principles ofRF Linear Accerelation" Wiley (1998).

We can calculate the accumulated energy(12)[4] H. Raether, "Excitation od Plasmons and Interband Transitions by Electrons", Springer-Vwerlag (1980).
[5] W.-C. Tan et al., Phys. Rev. 59 (1999-1) 12661.

using the relation(13)[6] P. Chen and R.J. Noble, in R.A. Carrigan Jr. and J.A. Ellison,ed., "Relativistic Channeling" (Plenumn Press, New York,
1987) p517.

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