Photonic Approach to Making a Surface Wave
Accelerator
Gennady Shvets1^
Illinois Institute of Technology, Chicago IL 60302
Fermi National Accelerator Laboratory, Batavia IL 60510
Abstract. Plasma surface wave accelerator is known to have superior accelerating properties. It
also has serious limitations such as collisionless damping of the accelerating field. In this paper I
analyze two novel accelerating structures which share the advantages with, and avoid the limitations
of the hollow plasma accelerator. Both are planar structures with confining walls which are made of
a material with negative dielectric permittivity e. The first material is a thin wire array appropriate
for a radio-frequency accelerator. The second material is a silicon carbide (SiC) which has a negative
c(o)) at the frequency of the CO2 laser (A = 10.6jUm). Prospects for making a miniature dielectric
accelerator using SiC capable of GV/m acceleration are also discussed.
1. INTRODUCTION
Planar hollow plasma channel accelerator [1,2] consists of a vacuum region sandwiched
between two confining layers of plasma. The electric field of the plasma surface wave
reaches into the channel enabling particle acceleration. The appeal of such surface wave
accelerator (SWA) lies in the attractive properties of the accelerating field. First, SWA
supports luminous electromagnetic waves propagating with the speed of light along the
axis of the channel z: (0 = kzc. Second, those waves are predominantly longitudinally
polarized: Ez > |E±|, |#_J, resulting in a high shunt impedance. Third, the accelerating
field Ez is uniform inside the vacuum channel, so that no energy spread is introduced by
the finite beam size. And, fourth, the accelerating mode does not exert any transverse
(deflecting) force on the beam. The third and the fourth properties are derived from the
luminous nature of the accelerating field. The existence of the confined luminous mode
inside the homogeneous (in the z— direction) structure is made possible because the
dielectric permittivity £ is positive in vacuum and negative in the plasma.
Despite these attractions, SWA suffers from a very serious problem identified in
Ref. [2]: the accelerating field is damped via the collisionless damping. The physical
reasons for this damping is that (a) the dielectric constant of the plasma depends on its
density: £p = 1 — co^/co2, where (O2 = 4ne2n/m, n is the plasma density well inside the
confining layers, and (b) the vacuum-plasma interface cannot be made perfectly smooth.
Therefore, the transition layer between £ = 1 (vacuum) and £p < 0 (well inside the
plasma) includes the region of the low density plasma with nr = co2m/4ne2 where e = 0.
Resonant absorption occurring at e = 0 is responsible for the collisionless damping. This
paper identifies two non-plasma materials which have a negative £ and no dissipative
regions of vanishing £.
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
371
2. THIN WIRE ARRAY IMITATES PLASMA
It has long been known [3] (and has been recently re-discovered [4]) that, from the electromagnetic standpoint, a three-dimensional array of conducting wires behaves similarly
to the plasma. The similarities are as follows. (1) There is a well defined effective cutoff
frequency (Op, which we refer to as the plasma frequency because of the obvious analogy
with the plasma, below which there are no propagating waves. (2) Two types of waves
are supported by the three-dimensional wire mesh (3DWM): longitudinally polarized
"plasmons" with frequency Q)(k) « cop, and transversely polarized "photons" with the
dispersion relation co2 w (O2 + c2k2.
A 3DWM is not easily amenable to analytic treatment. Although some supporting
arguments were given in Ref. [4], they are not very convincing. In fact, even the existence
of the cutoff frequency is not very easy to understand. Therefore, the origin of the cutoff
in a much more simple two-dimensional wire array (2DWA) is elucidated first. 2DWA
is a periodic structure in the x — y plane which consists of square elementary cells of
size d (periodicity). Each elementary cell contains a metallic wire infinitely extending
in z—direction. Despite the periodic nature of the 2DWA, it is possible to introduce the
effective dielectric permittivity eeff.
The utility of eeff is that it enables treating the complicated cellular structure as
a continuous medium. Specifically, for small wavenumbers k one can calculate the
dispersion relation (0 v. s. k. The relationship between the cell-averaged electric and
magnetic fields can also be expressed in terms of eeff. And, finally, the Poynting flux can
be also computed in terms of the cell-averaged electromagnetic fields and eeff.
2.1. Two-dimensional array of parallel wires
To understand why the three-dimensional wire mesh is similar to plasma, consider
a square wire lattice of period d consisting of round wires of radius r0 running in z—
direction. For a periodic system it suffices to consider a single square cell as shown
in Fig. 1. The mode periodicity is determined by the phase shifts per cell <t*x and fa
which define the wave vector k± = (fyx/d}ex + (fyy/d}ey. In general, the total wave
vector % = kzez -h& ± . First consider the transverse magnetic (TM) modes which are
characterized by the single scalar quantity 0 which is proportional to the vector potential
Az. The magnetic and electric fields can be expressed in terms of 0:
B = ezxV<p
£z = ^Vi0 £± = *V±<t),
(1)
where the equation for B follows from the definition of the TM mode (Bz = 0) and
V -B = 0 , and the equations for E are obtained from the Ampere's law. Applying
Faraday's law to E yields the Helmholtz equation for 0: V^_0 = (t% - co2/c2)0 = 0.
Therefore, Ez <* (j). For the transverse waves which propagate in x — y plane (kz = 0) the
only non-vanishing fields are Ez andB±.
372
y
θ
x
d
FIGURE 1.
1. Single
Single wire
wire cell,
cell, with
withcell
cell boundaries
boundariesshown
shownasasdashed
dashedlines.
lines.Cutoff
Cutofffrequency
frequencydetermined
determined
FIGURE
from Btt =
= 00 at
at each
each cell
cell boundary,
boundary, and
andEEzz =
= 00atatthe
thewire.
wire.
2 +
2 22 )φ =
frequency ω(Opp isis calculated
calculated by
by the
the (∇
(V^_
+ ω(Qp/c
= 00 for
for φ0 which
which isis
The cutoff frequency
p /c )(j)
⊥
conditions at
at the
the surface
surface of
of the
the wires
wires ((0
0) and
andatatthe
thecell
cell
subject to the boundary conditions
φ == 0)
= 0,
0, where
where ~nn isis the
the normal).
normal). For
For example,
example,atatthe
thexx==d/2
d/2 boundary
boundary
∂~nφ =
boundaries ((dj$
ocB
= 0.
Q.
∂dxQ
By =
xφ ∝
(r =
= rr00) and
and the
the four-fold
four-fold symmetry
symmetryof
ofthe
theproblem,
problem,φ0isisexpressed
expressedasas
Using φ0(r
∞
J00(co
r/c)
r/c)
J4k
Y0(ω p r/c)
YY4k4k(((o
(ω pr/c)
r/c)
(ω ppr/c)
ω ppr/c)
4k(co
cos(4£0),
−
+ ∑ Ak
−
cos(4k
φ=
θ ),
J0(ω p r0 /c) Y0(ω p r0 /c)
J4k (ω p r0 /c) Y4k(ω p r0 /c)
k=i
k=1
(2)
(2)
Bessel functions
functions of
of the
the first
first and
and second
second kind,
kind, respectively.
respectively. The
The
where Jm and YYmm are Bessel
are obtained
obtained by
by enforcing
enforcing the
the boundary
boundary condition
condition ∂dxxφ
(j) ==00
expansion coefficients Akk are
= d/2
d/2 for
for 0 <
< θ9 <
< πn/4.
One can
can obtain
obtain an
an approximate
approximate expression
expression for
forthe
thecutoff
cutoff
at x =
/4. One
’s
and
enforcing
the
boundary
condition
at
a
single
point
frequency by neglecting
neglecting A
A^'s
and
enforcing
the
boundary
condition
at
a
single
point
k
(x
(jc =
= d/2,
d/2,yy =
= 0) between the
the wires:
wires:
J1(ω p d/2c) YY1l((co
ω ppd/2c)
d/2c)
+
=
= 0.0.
J0(ω p r0 /c) Y0(ω p r0 /c)
(3)
(3)
Equation (3)
(3) can be
be further
further simplified
simplified by
by assuming
assuming that
that ω(Op d/c
<1.1.Expanding
ExpandingBessel
Bessel
pd/c <
functions as JJ^otpdfa)
(
ω
d/2c)
≈
ω
d/4c,
J
(
ω
r
/c)
≈
1,
Y
(
ω
r
/c)
≈
(2/
π ) ln (ω pprrQ0/c),
/c),
w (Op pd/4c, /00(coppr00/c) w 1, F00(cop pr00/c) w (2/n)]n(<D
p
1
(ω ppd/2c)
πωppd)
d),
we
obtain
and YY1l(co
d/2c) ≈
w (4c/
(4c/n(O
we
obtain
9
ω p2
ω p2
8
8
√
=− 2
≈
, or
or
2
22
2
2
c
dd2 ln
(
ω
r
/c)
c
d
ln
(d/2
2r0 )
p 0 /c)'
In (oy
c
d
In
(rf/2
v^r
0
(4)
(4)
after even further simplification.
simplification. Similar
Similar expressions
expressions (up
(up to
to aa numerical
numerical factor)
factor) were
were
obtained earlier [4,
5].
Equation
(4)
is
quite
remarkable
in
that
it
predicts
that
[4, 5]. Equation (4) is quite remarkable in that it predicts thatthe
thecutoff
cutoff
2 22
frequency ω
o&pp depends
depends on
on the
the surface
surface fraction
fraction of
of the
the wires
wires rr^/d
onlylogarithmically.
logarithmically.
0 /d only
i2 d2 = 3.56
-0.6
-0.8
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
FIGURE 2. Isocontours of 0 in a single wire cell (d = 2). Magnetic field lines follow the isocontours.
High density of field lines near the wire indicates high magnetic field: B(r = rQ)/B(r = d/2) w 12.
Apart from this logarithmic factor, the cutoff is determined only by the spacing between
wires d.
The Helmholtz equation for (j) with the appropriate boundary conditions was solved
using the commercial FEMLAB finite elements code [6]. The structure of the electric
field in a single cell surrounding an infinitely conducting wire (r^/d = 0.05) is shown
in Fig. (2). The isocontours of 0 correspond to the magnetic field lines. The plasma
frequency is (Op = l.89c/d.
Note that a high density of the magnetic field lines indicates a large magnetic field.
For the example in Fig. 2 the ratio of the magnetic fields at the wire and at the edge
of the cell is 12. The enhanced magnetic field indicates a large current in the wires and
accounts for the high cutoff even for very thin wires.
Next, we investigate the TM modes with non-vanishing phase shifts (f>x and (jty between the adjacent cells. As before, kz = 0 is assumed. The corresponding periodically phase-shifted boundary conditions are: (j)(x = d/2,y) = 0(jc = —d/2,y)el(^x and
0(jc,y = d/2) = <j)(x,y = —d/2}el<^ [5]. The dispersion relation (o(kx^ky) is plotted in
Fig. 3. Two direction of & are investigated: & = (&, 0), and & = (fr/\/2, £/\/2). These two
cases were chosen to demonstrate the isotropy in the (jc — y) plane. As Fig. 3 demonstrates, the anisotropy is barely perceptible even for large k w O.Sn/d. By matching the
dispersion relation co(&,0) to the plasma-like dispersion relation co2 = (O2 + j3^M£2c2,
we find that j3^M = 0.953, i. e. the dispersion is very nearly plasma-like. For finite kz the
dispersion relation is trivially modified: co2 = co2 + k2c2 + ^Mc2k^.
Transverse electric (TE) modes with the vanishing electric field Ez along the wires are
also supported by the 2DWA. These modes do not have a cutoff, and their dispersion
relation is given by co2 = c2(k2 + /^E^!)' where ATE ~ 1 f°r ^n wrres- Our numerical
simulations yielded the following values for r0 = 0.05J and r0 = 0.25J: jSrE(r0/J =
0.05) = 0.992 and f^rjd = 0.25) = 0.798.
TM and TE waves are the only transverse modes ("photons") supported by the 2DWA.
374
FIGURE 3. Dispersion relations c&(k, 0) (dashed line), «(/:/\/2, k/^2) (dash-dotted line), and the light
cone co = \k\ (solid line). Near-coincidence between the dashed and dash-dotted lines indicates isotropy
in the transverse plane. Additional parameters: kz = 0, d = 2, —0,Sn/d <k < 0,Sn/d.
Although the TM mode does have very similar propagation characteristics with the
plasma, the TE mode does not have any analog in the one-component unmagnetized
plasma, where there are no self-sustained electromagnetic disturbances with co < cop.
This is not surprising: 2DWA is a strongly anisotropic material. Also, there are analogs
of the longitudinal electrostatic waves ("plasmons") with co = cop.
The mean-field anisotropic dielectric tensor e- • relates the mean (cell-averaged) electric and magnetic fields according to
(5)
where {} denotes averaging over a single cell of the size d x d. The averaged description of the wire mesh using et- can be very useful in designing various complex structures which contain wire meshes as their constituent elements. Recent attention has been
drawn to photonic bandgap accelerators [7] which enable better mode control than conventional accelerating structures. When a structure consists of various regions (i. e. vacuum opening inside a wire structure) significantly larger than the spacing between wires,
it is convenient to use dielectric tensors et- to describe the wired region. Such description
simplifies field matching at the interfaces.
It turns out that the mean field dielectric tensor can be expressed as eeff = e^ezez +
e^(exex + eyey), where the longitudinal and transverse dielectric permittivities are, correspondingly, £M = 1 — C0p/(co2—k2c2) and e± = 1. The dependence of e,, on kz is crucial.
To see why, consider the Poynting flux of the TM mode with k± = 0 and kz^0 [8]:
p=^
(0
x ^(B,}
- -^——£
-L'
~
375
(6)
The mode in question is purely longitudinal: (E±) = 0 and (#±) = 0. It does, however,
transport energy since the cell-averaged Poynting flux in z— direction Pz = (c/4n) (EL x
B^) does not vanish. Therefore, it is the spatial dispersion of e- • (second term of Eq. (6))
that ensures a non-vanishing power flow.
Hence, the 2DWA differs from the plasma in several respects. First, it is anisotropic:
£
_L ^ £\\- Second, e,, depends on &z, and it is not equal to (1 — co2/co2). The consequence
of that is the absence of the longitudinal waves with co = cop for all k. There are also no
important for particle acceleration luminous waves with co = kc.
2.2. Three-dimensional Wire Mesh (3DWM)
Because the 2DWA does not faithfully mimic the plasma, three dimensional numerical
simulations were carried out in order to check whether the electromagnetic properties
of 3DWM are close to those of the plasma. But first I provide qualitative arguments
which support the claim that 3DWA supports three types of electromagnetic waves: two
mutually orthogonal photons with the dispersion relation co2h w co2 + c2k2 polarized
perpendicularly to the wave vector &, and a plasmon with the dispersion relation a& w
co2 polarized parallel to the wave vector k.
Recall that in a 2DWA two types of EM waves with k = kex exist. The first type is
the TE mode polarized in y— direction. Thin conducting wires along the z— axis do
not significantly disturb this mode (e. g., they do not introduce a cutoff) because they
are orthogonal to the wave polarization. The second type is the TM mode polarized in
z—direction, with the dispersion relation co2 w (O2 + c2k2. The 3DWM is obtained from
the 2DWA by adding to it orthogonal wires which are parallel to x— and y— directions.
It is reasonable to expect that adding these wires does not significantly perturb the TM
mode because they are orthogonal to its polarization. However, the (formerly) TE mode
will be strongly perturbed by the addition of the wires parallel to ey. By symmetry,
this mode will not be different from the TM mode. Therefore, the 3DWM supports
two transverse (E _L k) mutually orthogonal waves (photons) which satisfy the same
dispersion relation co2h w co2 + c2k2.
Let's now demonstrate the emergence of the third mode: the longitudinal plasmon.
For simplicity assume k = kez. In addition, consider the following modification of the
2DWA: conducting (metal) planes are inserted at incremental locations ZN = ±N x d,
where N is an integer. Metal wires are intersecting these planes. Between two adjacent
planes, TM waves are described by Eqs. (1). Boundary condition E^ = 0 must be
satisfied at the conducting planes. Only certain kz satisfy this requirement: kzn = nn/d,
where n = 0,, 1,.... The mode with the lowest frequency corresponds to kz = 0; it is a
purely longitudinal wave with the only non-vanishing electric field component Ez. Next,
require a non-vanishing phase shift 0Z per cell, so that 0(z,jc_j_) = ^(z — d^x^e1^2 is
satisfied for all z.
The plasmon solution corresponds to 0 constant inside each cell and discontinuous
across the conducting planes (which can carry surface charge). Therefore, by requiring
376
FIGURE 4. Unit cell dxdxd consisting of three intersecting wires of length d = 2 and width w = d/5.
The cutoff frequency copd/2nc = 0.42. Plasmon and photons at the edge of the Brillouin zone kz = n/d
were simulated. Electric field Ez is shown in the parallel to (x — y) section plane S which is located at
x = d/4. Detailed vector plots of the electric fields in the S— plane are shown in Fig. 5.
a discontinuous jump in 0 according to 0 (nd + 0) = exp /0Z0 (nd — 0), one constructs a
wave with the appropriate phase shift per cell. Such a wave is purely longitudinal, and
has a frequency (0 = (Op independent of 0Z.
Note that we arrive at the 3DWM by replacing the conducting planes by the planar wire grids consisting of intersecting wires in x and y directions. Of course, a conducting plane and a wire grid are not exactly equivalent. However, for low frequencies
(0 < 2nc/d their electromagnetic properties are very similar, and one can expect the existence of a plasmon in a 3DWM. Numerical simulations confirm this conjecture. Using
FEMLAB, we have simulated the electromagnetic structure consisting of the orthogonally intersecting ideally conducting wires of width w = d/5. The elementary cell which
is periodically translated in space is shown in Fig. 4.
The cutoff (plasma) frequency cop = 0.42 x (2nc/d) was found by setting the periodic
boundary conditions for all fields. The plasmon and the two photons at the edge of the
Brillouin zone (0Z = n) was found by setting anti-periodic boundary conditions at the
z = d/2 and z = — d/2 cell boundaries. The projection of the electric fields (Ey, Ez) of the
plasmon onto the section plane S drawn through x = d/4 (as shown in Fig. 4) is plotted
in Fig. 5(a). Clearly, the field is mostly longitudinal. The frequency of the plasmon
corresponding to kz = n/d is co- = 0.45 x (2nc/d): just barely larger than ODP. On the
other hand, the frequency of the photon is significantly higher: co h = 0.56 x (2nc/d).
In fact, coi w (Op + c2k%. It is also clear from Fig. 5(b) that the photon polarization is
transverse.
The important conclusion of Sec. 2 is that a 3DWM is electromagnetically equivalent
to plasma: it is characterized by the dielectric permittivity £ = 1 — co^/co2, and supports
the same types of electromagnetic waves as the plasma. Therefore, one can envision
developing novel surface wave accelerators which consist of two 3DWM slabs separated
377
(b)
(a)
1
0.5
0.5
N
N
0
-0.5
-0.5
-1
-1
-1
-1
0
y
o
y
FIGURE 5. Vector plots of the Ey, Ez electric fields in the 5- plane (see Fig. 4) for 0Z = n. Dashed
line: conductor, (a) Plasmon, E \\ ez, copld/2nc = 0.45. (b) Photon is polarized in.E\\ ey, cophd/2nc = 0.56.
Second photon with E \\ ey is not shown.
by a narrow gap of width 2b where acceleration takes place. The frequency of the
accelerating surface wave is cosurf = cop/^l 4- (Opb/c. Since cosurf < cop, the mode is
confined between the 3DWA with negative £ and vacuum region with £ = 1. Since
making wires thinner than a millimeter may prove impractical, the SWA based on the
3DWM is unlikely to operate at frequencies higher than 100 GHz.
3. SILICON CARBIDE ACCELERATOR
Making a surface wave accelerator which operates at a much higher frequency (THz or
FIR range) requires a different approach. In this section I describe how surface phonons
supported by a naturally occurring material (silicon carbide, SiC) which has a negative
dielectric permittivity in the FIR range can be utilized for particle acceleration. I assume
that the SiC accelerator will be driven by a CO2 laser.
The schematic of the device is shown in Fig. 6. The two slabs of SiC (referred to as
the confining walls) form a waveguide for the accelerating field. They can be supported
by side walls, or by dielectric posts, or can be attached by the specially designed heatremoving handles attached from the top. The side view in Fig. 6 illustrates the physics of
particle acceleration. Acceleration takes place in the vacuum gap between the SiC plates.
Accelerating field is supported by the surface charges at the SiC/vacuum interface.
Because of the negative dielectric permittivity of SiC, the accelerating wave is the
surface phonon; it decays exponentially inside the confining walls. Unlike the standard
planar dielectric accelerator, the proposed SiC accelerator does not require the outside
metal wall to confine the radiation. The following physical dimensions are envisioned:
length Az = 1cm, height Ax = lOjUm, width Aj = 50jUm.
The 10.6jUm radiation can be coupled into the structure either from the side, or from
378
FIGURE 6. Schematic of the SiC accelerator, side view through the dielectric side walls. Acceleration
takes place inside the vacuum gap between two parallel confining walls made of the silicon carbide. Field
lines extend from the surface of the SiC/vacuum interface into the channel and accelerate particle bunches
(two bunches spaced by one acceleration wavelength are shown).
the top (through the SiC layer). Sideway coupling may be accomplished by connecting
the confining walls by the periodically spaced (in z—direction) dielectric posts. Coupling
from the top of the accelerating structure is enabled by etching a periodic (tooth-like)
structure on top of the SiC layers [9]. The quality factor associated with the leakage
of the accelerating mode should be smaller than the resistive quality factor Qr of the
structure.
3.1. Electromagnetic Properties of the SiC Accelerator
In the SiC accelerator electrons gain energy from the surface mode with the following
properties: its axial (accelerating) field Ez is peaked on axis, and its deflecting electric
and magnetic fields Ex and By vanish on axis. This mode is externally excited in the
structure by the CO2 laser.
Assume that the confining SiC plates are separated by the distance 2b in jc-direction,
and are infinite in the x — y plane. Vacuum-SiC interfaces are assumed at x = ±b. I
consider the propagation in z-direction of the accelerating modes with non-vanishing
field components Ez, Ex, and By. The accelerating field Ez is assumed symmetric while
By and Ex anti-symmetric in jc, respectively. All three field components are assumed to
have the dependence exp/(£zz — cor). Magnetic field By is given by
SUlffyX
By =
for jc <
and By = <
for jcl > 6,
(7)
where Xv = ®2/c2 — k2, Xc = —a>2ec/c2 + k2, and kzc < co is assumed. Electric field
components are expressed through By: Ex = (kzc/co)By and Ez = (ic/£O))dxBy, where
379
k c/co,
FIGURE 7. Dispersion for two separations between SiC plates, coLb/c =1,0 (positive group velocity)
and coLb/c = 0.5 (negative group velocity). SiC optical properties are assumed to be £Q = 9.23, £0 = 6.2,
COL = 182.7 x lO1^-1 (\ = 2nc/coL = 10.3jUm).
£ = 1 in the vacuum gap and e = ec inside the SiC layers.
Accelerating surface waves are supported by the SiC accelerator because its dielectric
permeability is negative in a narrow frequency band co^ < co < COL [10]:
c-c i vj i — c-oo
0)1-CO2- i
,-j
o
•
——————
2
co|, - co -/7co
7
(8)
where COL = 182.7 x lO^s"1 (AL = 2nc/coL = lO^jUm) and % = 149.5 x
(Ay^ = 2nc/a>T = 12.6jUm) are the frequencies of the longitudinal and transverse optical
phonons, respectively. The damping constant 7= 0.9 x 1012s~! is relatively small and
can be neglected (except for the calculations of the g-value of the accelerating structure).
The high-frequency dielectric constant £00 = 6.2 for the 4H Hexagonal SiC crystal.
Requiring the continuity of Ez across the vacuum/cladding interface yields the dispersion relation co v. s kz. Plots of co v. s kz are presented in Fig. 7, where the wave damping
was neglected by setting 7=0. The most valuable for particle acceleration is the luminous mode with co = kc. The frequency of the luminous surface mode sensitively depends on the gap width b. Interestingly, the group velocity of the wave vgr = dco/dkz can
be either positive or negative, depending on the gap width. Wider channels correspond
to positive group velocities. The existence of the waves with negative group velocity
makes the SiC structure suitable as a backward wave oscillator. The decay length of the
surface mode inside the SiC wall is approximately A5 = 0.7c/coL w ljum.
The resistive quality factor of the SiC accelerator can be estimated as Qr ~
[d(co£)/dco]/Im(e). For co w COL, find that Qr w 2coL/7 = 350. This is a fairly
high quality factor for such a short wavelength A w 10.6jUm. One deficiency, however,
is the high energy density in the SiC walls: <9(eco)/<9co « 38. This implies that only
about 10% of the energy resides in the channel and contributes to acceleration.
380
Let's calculate the energy contained in the 5jUm x 50jUm x 1cm device with the
accelerating field Ez = 30 MeV/m. The corresponding energy density in the channel is
0.01//cm~3, in the SiC: 0.4//cm~3. The volume of the involved SiC is (A5) x SOjUm x
1cm contains 5 x 10~7J. Since the decay time of the accelerating field is about j~l = Ips,
the required peak power of a short pulse CO2 laser is 200kW. Such CO2 oscillators (with
a much longer pulse duration of about 100 ns) are fairly standard.
3.2. Why SiC: Description of the Material Properties
Thermal and dielectric properties of SiC are ideally suited for accelerator applications.
As a wide bandgap (V = 3.26 eV) semiconductor, it exhibits a very high DC electrical
breakdown field (300 MV/m). SiC-based electronics and sensors can operate in very hot
environments (600 C= 1112F) where conventional silicon-based electronics (limited
to 350 C) cannot function. Silicon carbide's ability to function in high temperature, high
field, and high radiation conditions are equally important for the field of accelerator
physics. This is because future accelerators must operate at high accelerating gradients
(large electric and magnetic fields), withstand significant heating due to the ohmic
dissipation of the accelerating fields, and tolerate occasional beam loss and radiation
by the accelerating particles executing betatron oscillations.
Equally important is the high thermal conductivity (about 3.8 W/cm/K) which is
higher than most metals. For example, the temperature drop AT = 1000K over a distance
A5/2 = 0.5 across the area A = 50juw x 1cm provides the power outflow of 380 kW
which is twice the CW power inflow required for maintaining a 30MV/m gradient.
For higher accelerating gradients pulsed operation becomes necessary. The fact that
radiation penetrates deep into the SiC (deeper than it would into metal) decreases the
material heating. For example, a 1 MW lOOps long laser pulse heats the SiC by only 50
degrees.
In summary, the SiC accelerator described in this paper is very different from its
closest cousin, dielectric accelerator [11]. Some of these differences result in major
improvements of the accelerating structure when compared to the dielectric accelerator.
These differences and advantages are as follows.
1. The semiconducting material (SiC) with negative dielectric permittivity is employed for mode confinement: £c(%) ~ —1.2 < 0, where co0 is the frequency of
the CO2 laser.
2. SiC accelerator can be powered by a conventional carbon dioxide laser.
3. SiC accelerator does not require outside metallic walls.
4. SiC can withstand extremely high DC electrical breakdown fields (about 300
MV/m). SiC can function at temperatures approaching 1000 C without deterioration. SiC has a very high thermal conductivity (about 3.8 W/cm/K).
5. SiC accelerator structure has a fairly high resistive quality factor Qr w 350
381
ACKNOWLEDGMENTS
This work was supported by the US DOE Division of High-Energy and Nuclear Physics,
the Presidential Early Career Award for Scientists and Engineers (PECASE), the Illinois
Board of Higher Education, and the the Illinois Department of Commerce and Community Affairs.
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