Theory for Wake Fields and Bunch Stability in Planar
Dielectric Structures
S. Y. Park,1'2 Changbiao Wang,3 and J. L. Hirshfield2'3
1
POSTECH, Pohung, Korea
Omega-P, Inc., 199 Whitney Ave., New Haven, CT06511 USA
3
Beam Physics Laboratory, Yale University, 272 Whitney Ave., New Haven, CT 06511
2
Abstract. Wake field forces that are induced within the vacuum gap of a planar dielectric-loaded
waveguide along which a matched planar beam is propagating have been analyzed analytically.
When femtosecond, pico-Coulomb bunches from a LACARA-driven optically-chopped beam are
used, it is shown that accelerating wake fields in the range of 100's of MV/m can be produced.
Stability issues for this arrangement are examined based on field expansion in symmetric and antisymmetric LSM and LSE modes.
INTRODUCTION
Wake field accelerators [1-9] are generally attractive because no external source of
radiation is needed to generate the accelerating fields. In such an accelerator, an electron
bunch (drive bunch) passes through a waveguide partially filled with dielectric where the
Cerenkov condition is satisfied, and hence excites wake radiation fields, which are used
to accelerate a second electron bunch (test bunch) at a suitable distance behind the drive
bunch; in other words, the drive bunch imparts its energy to the test bunch through
Cerenkov and inverse Cerenkov effects. In a dielectric-loaded waveguide, the Cerenkov
radiation spectrum is discrete, since waves radiated from charged particles and those
reflected from boundaries interfere with one another [10].
Although all the
electromagnetic modes excited by the drive bunch have different frequencies, they have
an identical phase velocity, equal to the velocity of the drive bunch [1].
Gai and his coworkers have demonstrated experimentally wake field effects in a
single-mode dielectric structure and found that experimental results agree reasonably well
with theoretical predictions for a single drive bunch [1]. Different approaches have been
developed to examine wake fields in a dielectric-loaded cylindrical wavegiude [2,3,6]
and in a planar dielectric structure [4,5,9]. A fair degree of attention was aimed at
transverse stability in the wake field acceleration scheme. It was shown that transverse
wake fields do not vanish even in the ultrarelativistic limit [2,3], but are proportionately
smaller than in an iris-loaded metallic structure [3]. Results for bunch dynamics from an
analytic theory [6] indicate that use of the high wake field acceleration gradients that can
be generated in a cylindrical dielectric-loaded waveguide will involve serious issues of
transverse stability, since strong destabilizing transverse forces automatically accompany
strong longitudinal acceleration gradients [8].
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
527
The idea of coherent superposition of Cerenkov radiation in a wake field accelerator
was proposed [5], in which a succession of drive bunches are positioned in such a way
that they all lie in the decelerating wake fields of the preceding drive bunches; these drive
bunches lose additional energy to produce a stronger accelerating field for a trailing test
bunch. Since all the modes excited by the drive bunch have the same phase velocity,
multimode constructive interference to produce a sharp peaked wake field is also
possible. An experimental demonstration of a multimode wake field caused by a train of
drive bunches has been carried out at ANL, where it was confirmed that the wake field
from a train of bunches can be superimposed [7]. A further supporting experiment has
been proposed [11].
The planar dielectric-lined rectangular waveguide in a wake field accelerator has
some advantages over the cylindrical dielectric-loaded waveguide. The planar structure
can be easily tuned to correct eigenmode frequency errors, and can store more energy for
a given bunch charge [9]; it can pass more beam charge and has better stability to
transverse beam deflection than a cylindrical structure of comparable dimension [4],
particularly if the ratio of the beam height to width is > 10. Prior investigations on the
planar structure have been made based on a two-dimensional (2D) model [4,5]. Although
a 2D-analysis can simplify calculation and enable one to obtain useful conceptual
information, it will obviate knowledge of important details about field transverse
distributions, especially when transverse wake field effects are important. Recently, a
full three-dimensional analysis has been published [9]. The authors employed
longitudinal section electric (LSE) modes and longitudinal section magnetic (LSM)
modes [12] to match boundary conditions and obtained dispersion relations for
symmetric modes, in which the axial electric field is symmetric about the central plane
parallel to the two dielectric slabs loaded in the waveguide. Symmetric LSE and LSM
modes are sufficient to expand wake fields produced by ideal symmetric drive bunches.
However, anti-symmetric LSE and LSM modes (see Appendix I) should be included
when any asymmetry of drive bunches is involved or stability issues need to be
examined. Thus, the use of only symmetric LSE and LSM modes to describe wake fields
with neglect of anti-symmetric LSE and LSM modes will not allow a full analysis of
bunch stability.
A complete theory for excitation of wake fields in 3D planar dielectric-lined
waveguides, similar in structure to a prior theory for cylindrical waveguides [6], has been
formulated. In addition to finding analytic solutions for the wake fields themselves,
analytic formulas have been derived for the forces on a test particle moving in the wake
of other particles. These latter solutions are needed to explore questions of stability.
This paper presents calculation results with this theory for femtosecond, pico-Coulomb
bunches from a LAser Cyclotron AutoResonance Accelerator (LACARA) driven
optically-chopped beam [13]. The schematic of LACARA-chopped slab bunch in the
planar dielectric structure of the waveguide (not drawn) is shown in Fig. 1. For the
examples to be discussed in this paper, the bunch is taken to be 3.5 fsec in duration (~1
jim) and 10 jim x 150 jim in transverse dimensions, containing ~1 pC and having energy
-500 MeV.
528
a
b
FIGURE 1. Schematic of slab bunch within a planar dielectric structure.
WAKE FIELDS AND FORCES
A summary is given below of the analytic forms for the wake fields, and of the
corresponding longitudinal and transverse forces, for a rectangular dielectric-lined
waveguide structure as shown in Fig. 1 with height d, outer width 2b, and inner (vacuum)
width 2a. The source function for the wake fields is a point charge qQ located at (x0,y0)
that moves parallel to the z-axis with velocity v0 = c f i z . For finite-extent bunches, fields
and forces are obtained by summing over point charges.
(ezn(x)coskns
v*>,
H,
d
l,n=\
with s = z — v0t. When / is odd,
if > (>;) = cos A;
and when / is even,
with k =—
529
The ortho-normalization constant is
-6
V
*2
_\b
^x2
(2)
The transverse field components can be derived from the longitudinal components, as
follows:
(3)
e—
c
and
e—
c
ft)
vk
where
(4)
j
k]_ = e/ii —z- — k2. In the dielectric regions, the axial field components are
c
given by
(
hz(x)~\Bcosk2(b-x)/cosk2(b-a)
(5)
or
B cosh k2 (b-x)/ cosh k2 (b - a)
, if-k22=k22=k2±l-k2<0.
(6)
While in the vacuum region, where - k2 = kxl =k2u-k2y < 0, one has a symmetric
solution,
hz(x)
(7)
.Ssi
and an anti-symmetric solution,
530
z (x) ^ _
(A sinh k^x I sinh k^a
z (*) I
I
/
(8)
V
The eigenvalue k and the ratio B/A are given by the condition that hy and ey be
continuous at the dielectric-vacuum interface, leading to
Mn
Af12
=0
M12 M22
(9)
From this one obtains the dispersion relation MUM22 -M,22 = 0, and the amplitude ratio
= -Mn/Mn=-Mn/M22.
The three components of the force on a test charge q are given simply by
(e)
(-s).
(10)
l,n=\
These components can easily be shown to satisfy the Panofsky-Wenzel theorem [14]
—F
F -V VF
~
'
A strict general proof of the above theorem is given in Appendix II.
One can also derive the forces from a pseudo-potential, i.e., F = -VO , with
(11)
7,71 = 1
This psuedo-potential O is expected to be useful in analyzing and visualizing regions of
stability for wake fields. For beams of finite extent, the above results for a point charge
source can be generalized by integrating over the extended source, employing the
relationship
(12)
For a rectangular distribution of width 2A in x, for example, this gives rise to a form
factor
531
-A<x<A
otherwise
V.
and to the replacement
I
1=-
I sinfcc I
I sinfcc
Similar considerations apply for finite charge distributions in y. As a consequence, one
can see that in the limit as the structure height d —> °o, if the beam height equals the
structure height, one has for the form factor sin(/7r/2)/(/7z;/2). This is zero for / even, and
decreases in magnitude as l~l for / odd, limiting the number of modes that make a
significant contribution.
NUMERICAL EXAMPLES
Some examples of the calculated longitudinal force Fz = -qEz on a test electron at x =
y = 0 are presented in Ref. [13] for one to ten 1-pC, 500 MeV symmetric sheet bunches
each with width, height and length of 10, 150, and 1 jim injected symmetrically into the
planar dielectric structure described Fig. 1 with 2a = 15 jim, 2b = 18.8 jim, and dielectric
constant s =3.0. The bunch spacing is chosen to be equal to the fundamental wake field
period of 20 jim, so that cumulative build-up of the field from successive bunches can
occur. It has been shown in Ref. [13] that the calculated peak wake field amplitude of 40
MV/m from a single bunch is in good agreement with the value found in KARAT
simulations. Uniform build-up of the maximum wake field to nearly 600 MV/m from ten
bunches can be reached. These results affirm the expectation that injection of a train of
pC sheet bunches into a micron-scale dielectric wake field structure could lead to
accelerating wake fields that approach 1 GV/m.
Here the same parameters of beam bunch and dielectric structure mentioned above
are used to examine the issue of stability with the analytic formulas (Eq. 10, etc.) for the
forces behind a moving bunch. In Figs. 2 and 3, examples are given for Fz and Fx, the
longitudinal and transverse forces at locations behind a drive bunch where one would
position either a test bunch for acceleration, or a second drive bunch for building up the
wakefield. The locations chosen are for a 1 pC drive bunch that has advanced to z = 800
jim. For a symmetrically-injected 10-|im wide drive bunch, at z = 788 jim the on-axis
accelerating force is 43.05 MeV/m, while at z = 779 jim the on-axis decelerating force is
-41.9 MeV/m. Figs. 2a and 2b show the longitudinal and transverse force profiles at
these two locations. For reference, the drive bunch profile is superimposed on each
figure (although it is located ahead at z = 800 jim). One sees a gentle transverse variation
in longitudinal force, which will give rise to spreads in either acceleration (Fig. 2a) or
deceleration (Fig. 2b). One sees a transverse force profile that passes through zero on the
structure axis; it is stabilizing (i.e., focusing) at the acceleration location (Fig. 2a), but is
destabilizing (i.e., defocusing) at the deceleration location (Fig. 2b).
532
FIGURE 2a. Longitudinal ( F z ) and transverse ( F x ) force profiles on an electron in the planey = 0, z =
788 Jim, the location for maximum acceleration for a test bunch at z = 800 Jim. Note that the transverse
force is stabilizing (i.e., focusing) and symmetric about x = 0. The projection on this plane of the
symmetric 10-|im wide drive bunch is also shown.
FIGURE 2b. Longitudinal ( F z ) and transverse ( F x ) force profiles on an electron in the plane y = 0, z =
779 Jim, the location for maximum deceleration for a drive bunch at z = 800 Jim. Note that the transverse
force is de-stabilizing (i.e., defocusing) and symmetric about x = 0.
In a second example, the 1-pC drive bunch is taken as being injected parallel to, but
displaced 1.5 jim from, the structure axis. Fig. 3 shows the force profiles, also at the
accelerating location z = 788 jim (Fig. 3a) and at the decelerating location z = 779 jim
(Fig. 3b). Again, the transverse drive bunch profile is superimposed on the figures. The
results show the longitudinal force profiles to be non-symmetric, with variations that are
about a factor-or-two greater than for the symmetric bunch case of Fig. 2. Likewise, the
transverse force profiles are non-symmetric: stabilizing at z = 788 jim, but with respect
to the point x = -1.3 jim rather than the axis; and destabilizing at z = 779 jim.
533
E
>
0
4
x (microns)
FIGURE 3a. Same as Fig. 2a, at the z-plane of maximum acceleration, but for a drive bunch displaced
+1.5 Jim transversely from the structure axis. Note that the non-symmetric transverse focusing is with
respect to the point x = -1.3 Jim.
E
>
<D
0.0
FIGURE 3b. Same as Fig. 2b, at the z-plane of maximum deceleration, but for a drive bunch displaced
+1.5 |im transversely from the structure axis. Note that the destabilizing transverse force is positive except
in the interval z < -5 Jim, where charge would not be intentionally placed. In this case, the entire bunch
would be accelerated towards the right-hand dielectric wall.
Determination of the exact consequence of a finite transverse force Fx as shown in
Figs. 2 and 3 requires a dynamical calculation of transverse motion of test particles in the
given non-uniform field. However, a simple estimate can be made for the case of a
constant force Fx . An electron with no initial transverse velocity will follow a trajectory
(Fx/2moy)(t-t0)2 , where (x0,t0) are initial values.
Since we have
(t-t0)=(z-z0 )/c = Az/c , then the distance Az that an electron can travel before being
x =x
displaced by a distance Ax is Az = ^2ym0c2Ax/\Fx\ . For \FX\ = 0.5 MeV/m, Ax= 2.5
534
|im,
above, ∆
Axx == 2.5
2.5
µm, and
and yγ == 1000,
1000, one
one finds
finds Az=
∆z = 7.1
7.1 cm.
cm. Since,
Since, in
in the
the examples
examples taken
taken above,
jim
would
cause
the
bunch
edge
to
scrape
the
dielectric
wall
(for
a
symmetric
10-|im
µm would cause the bunch edge to scrape the dielectric wall (for a symmetric 10-µm
wide
from this
this estimate
estimate that
that the
the accelerating
accelerating module
module should
should be
be
wide bunch),
bunch), one
one can
can conclude
conclude from
shorter
than
7.1
cm.
The
first
prototype
of
a
precision
planar
dielectric
structure
with
an
shorter than 7.1 cm. The first prototype of a precision planar dielectric structure with an
internal
gap
of
15
jim
would,
as
a
practical
matter,
be
no
more
than
a
few
cm
in
length
so
internal gap of 15 µm would, as a practical matter, be no more than a few cm in length so
that—based
beam motions
motions without
without wall
wall interception
interception
that—based on
on the
the above
above estimate—transverse
estimate—transverse beam
should
be
possible
in
the
first
experiments.
Of
course,
this
estimate
should
be
refined by
by
should be possible in the first experiments. Of course, this estimate should be refined
use
of
an
exact
dynamical
calculation.
use of an exact dynamical calculation.
ItIt is
stability with
with respect
respect to
to motions
motions along
along y,
y, the
the long
long
is also
also instructive
instructive to
to examine
examine stability
vertical
dimension
within
the
dielectric
structure.
Figs.
4a
and
4b
show
F
and
F
as
z
y
vertical dimension within the dielectric structure. Figs. 4a and 4b show Fz and Fy as
functions
10-|im wide
bunch with
with height
height of
of 150
150 µm.
jim. The
The
functions of
of yy at
at xx =
= 00 for
for aa symmetric
symmetric 10-µm
wide bunch
forces
are
shown
in
the
planes
z
=
788
jim
(Fig.
4a)
and
z
=
779
jim
(Fig.
4b),
where
forces are shown in the planes z = 788 µm (Fig. 4a) and z = 779 µm (Fig. 4b), where
maximum
are found.
found. The
The F
Fzz -profiles
-profiles are
are seen
seen to
to be
be
maximum acceleration
acceleration and
and deceleration
deceleration are
reasonably
of each
each conducting
conducting wall,
wall, but
but thereafter
thereafter fall
fall (or
(or rise)
rise)
reasonably uniform
uniform to
to within
within 25
25 jim
µm of
rapidly
to
zero
at
the
wall
(where
E
is
of
course
zero).
The
strong
^-gradient
in
F
as
z
rapidly to zero at the wall (where E z is of course zero). The strong y-gradient in Fzz as
one
approaches
the
conducting
walls
could
give
rise
to
a
large
spread
in
accelerated
beam
one approaches the conducting walls could give rise to a large spread in accelerated beam
energy.
figures that
that focusing
focusing forces
forces act
act to
to accelerate
accelerate most
most of
ofthe
the
energy. ItIt is
is also
also is
is seen
seen in
in both
both figures
beam
edge
towards
the
structure
axis
at
y
=
0.
But
in
Fig.
4b,
a
narrow
10-|im-thick
beam edge towards
y=
Fig.
10-µm-thick
defocusing
outer edge
edge of
of the
the beam
beam
defocusing layer
layer is also seen, which would tend to accelerate the outer
into
the
wall.
A
full
dynamical
study
of
bunch
motions
under
the
action
of
the
vertical
into the wall. A full
under the action of the vertical
force
the formation
formation of
of wake
wake fields,
fields,
force FFyy is
is also
also clearly needed to accurately characterize the
and
the
subsequent
acceleration
of
test
particles
in
planar
structures.
and the subsequent
planar structures.
50
z
0
0
F = 0
-5
F
( x == 00,, zz == 7788j^m
8 8 µ m ))
z
25
F (M e V /m )
F
5
y
F (M eV /m ) a n d F (M e V /m )
10
y
x
x
bu
n ch pprofile
ro file
bunch
-2 5
-1 0
-7 5
-75
-5 0
-50
-2 5
-25
0
25
25
50
50
7
5
75
y (m
ic ro n s)
(microns)
FIGURE 4a.
4a. Longitudinal
Longitudinal ( FFzz)) and transverse (( F
FIGURE
F yy )) force
force profiles
profiles on
on an
an electron
electronininthe
theplane
planexx==0,0,zz==
788 Jim,
µm, the
the location
location for maximum
maximum acceleration for a drive bunch at z = 800
788
800 µm.
Jim. Note
Note that
that the
the transverse
transverse
forces near
near the
the walls
walls are
are stabilizing (i.e., focusing), tending to accelerate electrons
y << 75
forces
electrons in
in 58
58 << \y\
75 µm
Jim
away from
from the
the walls
walls where
where FFzz is falling
falling rapidly.
away
535
bunch
b u nc h profile
p rofile
0 , z = 779|o,m
779 µ m )
(( xx = 0,
y
0
z
0
F (M e V /m )
F
F = 0
-2 5
x
-5
x
E
>
25
5
y
F (M e V /m ) a n d F (M e V /m )
10
F
-5 0
z
-1 0
-7 5
-5 0
-2 5
0
25
50
75
(m ic ro n s)
yy (microns)
FIGURE4b.
4b. Longitudinal
Longitudinal ((FFz z) ) and
and transverse
transverse (( FFyy)) force
FIGURE
force profiles
profiles on
on an
an electron
electronininthe
theplane
planezz==779
779µm,
Jim,
thelocation
locationfor
formaximum
maximumwake
wake field
field deceleration
deceleration for
for aa drive
drive bunch
bunch at
at zz =
= 800
the
800 µm.
|im. Note
Note that
that electrons
electronsinin
65|im
µmare
are accelerated
accelerated towards
towards the
the axis,
axis, while
while electrons
electrons in
y << 75
4545<< |>>|y <<65
in 65
65 << \y\
75 µm
|im are
are accelerated
accelerated
away
from
the
axis.
away from the axis.
REMARKS
REMARKS
Thesepreliminary
preliminaryresults
results presented
presented in
in the
the paper
paper lead
These
lead to
to several
several tentative
tentative conclusions.
conclusions.
First,
the
expectation
that
3D
elongated
planar
dielectric
structures
First, the expectation that 3D elongated planar dielectric structures with
with an
an aspect
aspect ratio
ratio
d/b
=10
will
be
essentially
stable
is
not
quite
fulfilled;
however,
we
find
d/b =10 will be essentially stable is not quite fulfilled; however, we find that
that both
both
horizontal and
and vertical
vertical stability
stability improve
improve as
as this
this ratio
ratio increases.
horizontal
increases. Second,
Second, dynamical
dynamical 3D
3D
orbit calculations are clearly required to study bunch distortions due to transverse forces
orbit
calculations are clearly required to study bunch distortions due to transverse forces
at the locations where it is planned to inject either a test bunch for acceleration, or further
at the locations where it is planned to inject either a test bunch for acceleration, or further
drive bunches to build up the accelerating field. Third, the transverse forces F and F
drive bunches to build up the accelerating field. Third, the transverse forces Fxx and Fyy
are not simply focusing and defocusing, as in a quadrupole magnet. Fourth, the
are not simply focusing and defocusing, as in a quadrupole magnet. Fourth, the
transverse focusing and defocusing properties of wakefields may offer possibilities for
transverse focusing and defocusing properties of wakefields may offer possibilities for
stable transverse bunch transport and bunch compression not previously appreciated.
stable transverse bunch transport and bunch compression not previously appreciated.
These conclusions indicate that an extensive program of study in the future should be
These
conclusions indicate that an extensive program of study in the future should be
directed at analytical studies of bunch dynamics in dielectric-lined planar structures.
directed
analytical
of bunch
dynamics
planar structures.
But at
even
beyond studies
the results
discussed
above,ina dielectric-lined
further major theoretical
issue remains
But
even
beyond
the
results
discussed
above,
a
further
major
theoretical
issue
to be resolved that was heretofore not discussed widely. This issue affects (a)
theremains
design
toofbewake
resolved
that
was
heretofore
not
discussed
widely.
This
issue
affects
(a)
the design
field structures; (b) the comparisons between theory and PIC-code (KARAT)
ofsimulations;
wake field and
structures;
the comparisons
between theory
and PIC-code
(c) the (b)
interpretation
of experiments.
It should
be stressed(KARAT)
that no
simulations;
and
(c)
the
interpretation
of
experiments.
It
should
be stressed
no
realizable wake field structure exists without axial boundaries, and
injected that
charge
realizable
wake
field
structure
exists
without
axial
boundaries,
and
injected
charge
bunches must traverse these boundaries in entering and exiting the structure. Indeed, the
bunches
traverse
thesethat
boundaries
entering
and exiting
structure.
the
KARATmust
results
[13] show
the wakeinfield
is weaker
and lessthe
localized
nearIndeed,
the entry
KARAT
results
[13]
show
that
the
wake
field
is
weaker
and
less
localized
near
the
entry
boundary than wake field calculations which do not include effects at the entry boundary
boundary
than wake
field calculations
which
do not
include effects atstructures
the entry has
boundary
would indicate.
Recent
work on wake
fields
in axially-bounded
been
would
indicate.
Recent
work
on
wake
fields
in
axially-bounded
structures
been
carried out by Onishchenko et al [15] (for symmetric fields in a dielectric-filled has
cylinder
carried
out
by
Onishchenko
et
al
[15]
(for
symmetric
fields
in
a
dielectric-filled
cylinder
with one axial boundary), and by Park [16] (for fields of arbitrary symmetry in a
with
one axial cylindrical
boundary), waveguide
and by Park
fields
arbitrary symmetry
in a
dielectric-lined
with[16]
one (for
or two
axialofboundaries).
To illustrate,
dielectric-lined cylindrical waveguide with one or two axial boundaries). To illustrate,
536
we cite Park's equation for the axial component of the wake field in the vacuum channel
induced when a charge q0 travels parallel to the axis of a cylindrical dielectric-lined
bounded waveguide of length L, but at a displaced radius r0, namely
o7icq0 \^
=
co2 -co;
-^T±
m,n=l
-(a>0sma>t—^~
r.
(13)
Here a>0 = kmv, k\0 = k^ (ft2 -l), £^— = k^+kl, and km = mn/L . The second term
in Eq. 13 (containing sma>0t) describes Cerenkov radiation, and is identical to that
obtained for an axially-unbounded cylinder [6,8]. The first term (containing sinfttf) is
due to the presence of the boundaries, and describes transition and precursor radiation. It
can be seen that, for each mode (m,ri) an interference between the two terms results. An
important manifestation of this interference is the apparent decay from phase mixing of
the wake field at relatively short distances from the boundary, after summing over all the
excited modes. The KARAT simulations give undeniable evidence for this interference
phenomenon. However, only as the analytical tools develop further, and as the theory is
applied to rectangular structures of the type discussed in this paper, will a full
understanding of this complex effect be understood. As stated above, a full
understanding is crucial for the design and interpretation of wake field accelerator
experiments.
ACKNOWLEDGMENT
This work was supported by the DoE, High Energy Physics Division.
APPENDIX I
In this Appendix, a derivation of dispersion relations in a planar dielectric-loaded
waveguide is presented. Two planar dielectric slabs are symmetrically disposed on
opposing walls of the rectangular waveguide, with e the permittivity and JJL the
permeability, as shown in Fig. AI-1.
sxxxxxxx;
xxxxVxx*
|("-L);
V)
pn
N$S§S
XxX^xX^
x$x\$x§
s 2v 2\
^Cv^C^Cx^
-b
-a
(0,
Nn
8
1
1
N
^g
X
N
2^ 2s
\xxx\xx"
-d
a
b
FIGURE AI-1. Planar dielectric-loaded rectangular waveguide.
537
In an empty uniform waveguide with perfect conducting boundaries, TE and TM
modes are eigenmodes and they can exist independently. In a dielectric-loaded
waveguide, a single TE or TM mode usually is not an eigenmode and thus cannot exist
independently; they may couple to one another through boundary conditions to constitute
eigenmodes. Here we use TE and TM mode fields to match boundary conditions and
derive dispersion relations including symmetric and anti-symmetric LSE and LSM
modes.
In region-I, the field solutions with respect to the variable x need only satisfy
continuity relations at the dielectric-vacuum boundary, unlike in region-II where the
fields are specified (i.e. vanishing of tangential E and normal H). Therefore, in region-I,
four possible combinations for functions with respect to x can be possible. These are
given by
1. {£z ~ cos(£ljcx), Hz - sm(klxx)} ,
2. {Ez - sin(£ljcx), Hz - cos(klxx)} ,
3. {Ez - cos(klxx\ Hz - cos(£ljcx)} ,
4. {Ez - sin(£ljcx), Hz
where klx is the wave number in the x-direction. For symmetric LSE and LSM modes,
the axial electric field is symmetric about x and the axial magnetic field is anti-symmetric
and vice versa. Therefore, combination- 1 corresponds to symmetric LSE and LSM
modes, combination-2 corresponds to anti-symmetric LSE and LSM modes. However, it
can be shown that combinations-3 and -4 cannot consistently satisfy boundary conditions
at x = a and x = -a.
Symmetric LSE and LSM Modes
In the region-I ( - a < x < a ), the axial field components are given by
where kly = —— , and k2x + k2y + k2 = /Ltlel a*2 , and the propagation factor el(0)t~kz^ in the
above is suppressed. All the transverse components can be obtained from the axial
components. For the right region-II (a < x < b) , the axial field components are given by
538
where k2x + k2y + k] = /J,2£2 co2 .
When these fields are used to match boundary conditions at x = a, we obtain the
symmetric LSM and LSE mode dispersion equation, given by
(symmetric LSM mode factor)
(symmetric LSE mode factor)
These are the modes considered by by Xiao, Gai, and Sun [9].
Anti-symmetric LSE and LSM Modes
In the region-I ( - a < x < a ), the axial field components are given by
v
bc"v
The dispersion equation for anti-symmetric LSM and LSE modes is given by
(anti-symmetric LSM mode factor)
•lx cos(klxa) k2x cos[k2x (a-b)]}=Q
sin(£ljca) ^2 sm[k2x(a-b)]
(anti-symmetric LSE mode factor)
(AI_g)
APPENDIX II
The original formula in Ref. [14] was derived for the increment in transverse
momentum imparted to a fast charged particle passing through an rf cavity along its axis,
with boundary conditions at the cavity ends taken into account. Strictly speaking, this
formula cannot be used to obtain the local relationship between transverse and axial
forces. Tremaine and Rosenzweig generalized Panofsky and Wenzel's formula with the
assumption that the differential operator V can be taken out from the integrand within a
path integral [4]. The same assumption is taken in Ref. [17]. Usually that procedure is
not justifiable. Here we present a strict proof of the Panofsky-Wenzel theorem for a
translationally-invariant system, such as a dielectric-loaded wake field accelerator.
Suppose that in a longitudinally uniform waveguide a test particle with a charge of q
moves following a drive particle, both with the same constant velocity v = zv with z the
longitudinal unit vector. The relation between the transverse and axial forces
experienced by the test particle will here be shown to be given locally and
instantaneously by
539
dz
F^VJV
(Aii-i)
Since the test particle is at rest relative to the drive particle, the fields produced by the
drive particle and experienced by the test particle are not changing in time. Thus the
fields seen by the test particle obey the equations
Using the above and V x E = - 9B/8/ , we have
z.
(AIM)
From the Lorentz force equation, with Eq. (AII-3) we have
~\
dz
~\
±
dz
dz
(vxB) ± = V±(qE,) = V±F,. QED
The last step used the fact that (v x B)Z = 0 .
Similarly, we have VxF = qVxE + qVx(\xB)=-q(dB/dt + v - V B ) = 0 , from
which F can be obtained from a scalar potential, as indicated in Eq. (1 1).
Obviously, the above proof holds for particles moving through a longitudinally
uniform dielectric-loaded waveguide system when the test particle is synchronous with
all the wake fields produced by the drive particle. However, when a test particle moves
through an rf cavity or waveguide which supports fast wave fields, Eq. (AII-1) does not
hold because the test particle cannot remain synchronous with the fast wave, resulting in
breakdown of Eq. (AII-2). In such a case, the relation between transverse and axial
forces experienced by the test particle becomes more complicated [18].
REFERENCES
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2. Ng, K.-Y., Phys. Rev. D 42, 1819-1828 (1990).
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540
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