Wakefields in Dielectric-Loaded Rectangular
Waveguide Accelerating Structures
Liling Xiao*, Chunguang Jing*, Wei Gai* and Thomas Wong^
*High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439
Illinois Institute of Technology, Chicago, IL 60616
Abstract. Employing modal analysis, an accurate formulation to represent the electromagnetic
wakefields generated by an electron charge bunch traveling along an off-center axis in a
dielectric loaded rectangular waveguide accelerating structure is presented. The resulting xdipole and y-dipole modes are described in detail. Specific analytical results were obtained for
the dispersion characteristics, the transverse force, and the longitudinal wakefields.
I. INTRODUCTION
Following the successful wakefield analysis of center-beam excited dielectric-loaded
rectangular waveguides [1], an effort has been made to investigate the case of offcenter excitation. Based on Fourier theorem, the wakefield resulted from traveling
electron bunch can be decomposed into an infinite series of normal modes, each
satisfying the boundary conditions required by the waveguiding structure [2]. Such a
representation is more accurate than the single monopole mode approach, and is a
necessity to account for the asymmetry in the total field caused by an off-center charge
beam. At the same time, the analysis and computation become inevitably more
involved. From a computation standpoint, the summation over the normal modes must
be truncated at some point. Under the assumption of a very small off-center
displacement for the charge beam, it can be expected that the monopole and the dipole
modes should be adequate to provide an accurate account of the wakefields for
practical considerations.
It is known [3] that the modes in a dielectric-loaded rectangular waveguide can be
classified as LSM (longitudinal section magnetic) and LSE (longitudinal section
electric) modes that have no H or E components normal to the interface respectively.
This corresponds to assuming the transverse direction to the interface normal vector to
be the direction of propagation. Applied to our case, when a short bunch moves
through dielectric-loaded rectangular waveguide at transverse position (x0, yo) instead
of on-axis where the coordinate is (0,0), the modes contributing to the longitudinal
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
542
wake field should consist of monopole modes LSMin plus LSEin [1] and the dipole
mode in x-direction, (which we denote as x-dipole) and the dipole mode in y-direction
(y-dipole). Here, we neglect the other higher order modes because we assume that the
bunch trajectory does not deviate far from the axis, and keeping terms to this order is
sufficient to ensure the accuracy of the final results. We have verified this by
comparing our analytical results with a numerical simulation using a commercial EM
code.
The organization of this paper is as follows. In section II, we obtain the analytical
expressions for the EM field components of multipole modes through potential
functions, and then we calculate R over Q [R/Q], a very important structure parameter,
for each related mode. Because R/Q has a very simple relation to the loss factor ki of
beam loading [4], we can obtain the longitudinal component of the wakefield
amplitude excited by a charged particle traveling beam easily, rather than computation
of the wakefield using the Panofsky-Wenzel theorem as [5].
II. EIGENMODES
The field components of some special accelerating modes in a dielectric-loaded
rectangular waveguide have been analyzed using a mode matching method in our
previous work [1]. The cross-section of a dielectric-loaded rectangular waveguide is
shown in Figure 1. The general structure considered here is limited to the case of two
//-plane slabs placed symmetrically. In [1], we only considered those modes with
nonvanishing longitudinal electric components at the central point since any other
modes will not couple to an on-axis beam. This implies that the central plane in the yz view is an open plane. Using symmetries, the remaining modes with vanishing
longitudinal electric field components on axis can be obtained by considering the
central plane in the y-z view as a short plane.
Fig.l Dielectric-loaded rectangular guide
The transverse equivalent circuit corresponding to open and short planes for this
symmetrical H-plane waveguide can be established as shown in Fig.2 (a) and (b)
respectively. In this paper, we will use the superscript (open) and (short) to represent
these modes with and without the longitudinal electric components nonvanishing at the
543
central point. To clarify our procedure, we summarize all fundamental electromagnetic
field physics in this kind of H-plane dielectric slab loaded rectangular waveguide. First
of all, we obtain dispersion relations of the open and short central plane cases
respectively by using transverse resonance method [6].
From [6], we can deduce the dispersion relations as follows:
ZL=0
ZL=0
(0)
kymn
aa
K
ymn
ZL=0
ZL=°°
(a)
(b)
Fig.2 a) open plane and its equivalent circuit b) short plane and its equivalent circuit
(open)
- Z <!» cot( *£> a} + Z <1 tan
(short)
tan(
a) + Z <& tan[ k^n (b - a)] = 0
(0)
where Z™ =
COS
(b - a)] = 0
ry (1)
> ^ Qmn
_
(la)
(Ib)
l)
k (ymn
~
are the values of the characteristic impedances of each mode, and the transverse
propagation constants are expressed in terms of the longitudinal propagation constant
pmn using the following conditions:
_ ft 2
,
and
W
Pimn
(2)
544
Here k= 2nf/c is the propagation constant in free space and the notation of
superscript (0) or (1) indicates the vacuum or dielectric region of the waveguide
respectively.
Transcendental equation (1) is a complex function of pmn and f, which gives the field
components and dispersion relations of all the eigenmodes for a dielectric-loaded
rectangular waveguide. For the inhomogeneous guide considered here, the dispersion
relation must be solved at each frequency. An infinite number of discrete solutions
exist, but we are interested only in those consisting of monopole and dipole modes.
The field components of LSM(open), LSE(open), LSM(short) and LSE(sWt) modes in a
dielectric-loaded rectangular waveguide can be derived from the solution of vector
potential wave equations as the following formulations [6].
LSM mode:
Ve=di i/se(x,y,z)
. i ay.
=
cojue dxdy
(3a)
'dy2
. i ay.
7
"- dydz
LSE mode:
€
H
dz
=- i
(OfJ£
O
E„ =
E, = - —
dxdy
n
•> ^
(3b)
H z = -j
e dx
where \|/e and \|/h must satisfy the scalar wave equations of
V2ys (x,y,z) + f}2\// (x,y,z) = 0
q = e,h. (4)
Applying the boundary conditions at the perfectly conducting guide walls (x=±w/2,
y=b) and the boundary condition at the magnetic or electric wall(y=0), the potential
function x|/e for open or short modes are
(5a)
a<y<b7
i——(
w
545
win ,
w
w
(
(56)
a<y<b
2
In each region, the fields for the LSE(opeQ)mn or LSE(short)mn derive from an magnetictype potential function. Using the boundary conditions at the perfectly conducting
guide walls (x=±w/2, y=b) and the boundary condition at the magnetic or electric wall
(y=0), the potential function \|/h is
,
w
w
(60)
2
cos
win
w
w
0<y<a
,
a<y<b
,
_
0<y<a
(6b)
,
2
a<y<b
The corresponding field components derived from equation (3) are reviewed briefly
and listed separately as the Appendix of this paper.
III. TRANSVERSE WAKEFIELDS
The monopole and dipole modes are of greatest interest in determining beam
stability. We have shown that the LSM(open)n mode is the lowest luminal mode in our
the structures under consideration, and its longitudinal electric field is distributed
symmetrically in both x and y. In [1], the general acceleration properties of the
LSM^open\i mode had been calculated, such as the ratio of the peak surface electric
field to the axial acceleration field ES/E0, the group velocity vg, the attenuation constant
a, and R/Q which measures the efficiency of acceleration in term of the given stored
energy, etc. We also have analyzed the longitudinal wake fields of monopole modes
LSM(open)in+LSE(open)in. Some results are presented and compared with numerical
results from the MAFIA code and are found to be in good agreement.
In this paper, we will consider the transverse wake fields of dipole modes first by
using the same method as used in [1] for the longitudinal wake fields of the monopole
modes, then together with result from monopole analysis, we can obtain accurate wake
field numerical simulations.
546
The longitudinal electric field components of LSM^U, LSE^U, LSM(short)
and LSE^^mn modes have the following form in the vacuum region:
(7a)
——si* £*-(* + -^) cos * <«i, y
w
w
2
(7b)
)* <?L sin ^f(* + y) sin * <»1 7
sin ^L(* + f ) sin *<«! y
>v
Obviously,
LSM(open)2n/LSE(open)2n
2
modes
are
the
x-dipole
(7c)
(7d)
modes
and
We assume a Gaussian longitudinal beam shape (with bunch length az and charge e).
The transverse forces can be directly calculated from Ez by using the Panofsky-Wenzel
theorem [2]
(8)
dz
For synchronous x-dipole modes (k(0)y=-j(27t/w)), the longitudinal electric field
component and transverse forces can be expressed as
2
27T
= ^£0/ (% ,0) sin —x cosk^y cosfiz
W
where E0i(x0, 0) is the off-axis (x0, 0) accelerating/ decelerating gradient of the ith
mode. Thus,
Qi
cos(—
For synchronous y-dipole modes (k(0)y=-j(7r/w)), the longitudinal electric field
component and transverse forces can be expressed as
oi(0,y0)sm
-(x + ^)si
w
2
n
—xsin
547
where E0i(0, yo) is the off-axis (0, yo) accelerating/ decelerating gradient of the ith
mode. Then,
F(Q,y0,z) = e^=l—— fY £w*<°> cos(*<°Vo)cos #(z-z';
(12)
However, since we have already obtained the field distribution, and the normalized
shunt impedance R/Q of each mode can be calculated using mode analysis [1], the
longitudinal component of the wakefield amplitude excited by a charged particle beam
traveling on axis can be easily obtained as follows [2].
03)
where
R
( —) =
'Q
coP
Ez(x0, yo) is the value of the longitudinal electrical field component at point (x0, yo), P
is the power flow in cross-section of waveguide /> = i HEtxfftdxdy> an(^ vg *s me grouP
velocity.
We assume a Gaussian longitudinal beam shape (with bunch length az and charge
q). The longitudinal wake Wz(z) at distance z behind the drive electron beam is then
1
given by Wz (z) = ,
Z
oo
/2
E
j^ u
cos
z
z/ ex
A ( ~ ) P( ~ ~^^dz'
(14)-
Together with the monopole mode [1], the whole wakefield generated by the electron
bunch traveling at the off-center position (x0, yo) can be obtained accurately.
IV. CALCULATED RESULTS
This section we will be concerned with presenting the data by applying the theory we
introduced above to a specific case. Here, the dimensions of the X-Band waveguide
are a =3 mm, b =5mm and w =23mm, and the dielectric constant =10. Fig.3 shows the
dispersion characteristics of x-dipole and y-dipole modes, and we found the
corresponding synchronous accelerating parameter for each mode, i.e. the intersection
points between the dispersion curve of each mode and the light speed line.
548
3500
3000
3000
LSM21
LSM11
LSM12
LSM22
3000
LSE22
2500
E2500
LSE23
light
2000
12000
LSE11
Propagation constant(rad/m)
LSE21
propagation constant(rad/m)
2500
2500
LSM23
LSE12
T)
2000
E2000
Light
= 1500
1500
c
1500
f 1000
1000
a
1000
11000
i
i 500
500
500
500
0
5
0
11
15
20
25
25
30
30
35
35
40
40
45
50
9
14
1919
24
24
29
29
34
39
Frequency(GHz)
Frequency(GHz)
frequency(GHz)
frequency(GHz)
Fig.3 a)
Fig.3
a) dispersion
dispersion characteristics
characteristics of
of x-dipole
x-dipole mode
mode b)
b) dispersion
dispersion characteristics
characteristics of
of y-dipole
y-dipole mode
mode
Table
results of
x-dipole modes
Table (1)
(1) shows
shows the
the calculated
calculated results
of x-dipole
modes synchronous
synchronous with
with and
and
(open)
(open) , LSE(open)
(open) ,
acting
upon an
ultra-relativistic electron
(β=2πf/c),
in
which
LSM
21
21
acting(open)
upon
an ultra-relativistic
electron
(p=27if/c),
in
which
LSM
i,
LSE
2
2i,
(open)
(open)
(open)
r
pen)
(open)
(open)
(open)
LSM
,
LSM
,
LSM
,
and
LSE
are
considered.
23
24
22
LSM ° 22
,
LSM
,
LSM
,
and
LSE
are
considered.
22
23
24
22
TABLE
TABLE 1.
1. Parameters
Parameters of
of x-dipole
x-dipole synchronous
synchronous accelerating
accelerating modes
modes
Mode
Freq.
βPi(rad/m)
i(rad/m)
Mode
Freq. (GHz)
(GHz)
(open)
11.95
250.413
LSM (open)21
11.95
250.413
21
LSM
(open)
(open)
14.77
309.617
LSE
14.77
309.617
21
LSE
21
(open)
33.92
710.998
LSM (open)22
33.92
710.998
22
LSM
(open)
38.33
803.381
LSE (op%
38.33
803.381
22
LSE
(open)
56.88
1192
(open)
LSM
1192
56.88
23
LSM
23
(open)
(openJ
80.44
1686
LSM
80.44
1686
24
LSM
24
(R/Q)
(R/Q)ii
313.224
313.224
268.956
268.956
171.053
171.053
27.304
27.304
70.91
70.91
32.785
32.785
In Figure
Figure 4,
In
4, the
the transverse
transverse wake
wake fields
fields obtained
obtained using
using equations
equations (10)
(10) is
is shown.
shown. For
For
this
example,
a
q
=1nC
bunch
with
length
σ
=2mm
located
at
x
=1mm
and
z
0
this example, a q =lnC bunch with length az =2mm located at x0 =lmm and yyo0 =0
=0 is
is
moving along
moving
along the
the axis
axis with
with aa speed
speed of
of light
light in
in dielectric-loaded
dielectric-loaded waveguide.
waveguide. The
The first
first
six
six x-dipole
x-dipole modes
modes are
are used
used in
in the
the sum
sum over
over modes;
modes; higher
higher order
order modes
modes do
do not
not
contribute
contribute significantly.
significantly.
549
A
A
-40
-32
-24
0
zfrtun)
Fig.
4.
Calculated
transverse
wakefield
Fig.
in
an
X-Band
structure of
of x-dipole
x-dipole mode
mode (a=3mm,
(a=3mm,b=5mm,
b=5mm,
Fig. 4.
4. Calculated
Calculated transverse
transverse wakefield
wakefield in
in an
an X-Band
X-Band structure
structure
of
x-dipole
mode
(a=3mm,
b=5mm,
w=23mm,
w=23mm,
=10,
and
azzz=2
=2
mm,
q=lnc)
w=23mm, εeεrrr=10,
=10, and
and σσ
=2 mm,
mm, q=1nc)
q=1nc)
The
corresponding results
results for
y-dipole
The
for y-dipole
modes are
are shown
shown in
in table
table (2),
(2),and
The corresponding
y-dipole modes
(2),
and here
here only
only
four
modes
are
considered.
four
four modes
modes are
are considered.
TABLE
2.
Parameters
TABLE
of y-dipole
y-dipole synchronous
synchronous accelerating
accelerating modes
modes
TABLE 2.
2. Parameters
Parameters of
Mode
Freq.
ββ
i(rad/m)
Mode
Freq. (GHz)
(GHz)
Pi(rad/m)
Mode
(rad/m)
i
(short)
(snort)
7.319
153.392
(short)
LSM
7.319
LSM
153.392
n
LSM(short) 11
11
(shon)
15.44
323.512
(short)
LSE
323.512
LSE
15.44
LSE (short)n11
11
27.45
575.244
(short)
(short)
LSM
575.244
27.45
12
LSM
12
LSM(short) 12
(snort)
38.68
810.657
LSE
i2
(short)12
LSE
38.68
810.657
LSE
12
(R/Q)
(R/Q)iii
(R/Q)
461.556
461.556
227.808
227.808
24.98
24.98
20.542
20.542
The
transverse
wake
The
fields
obtained using
using equations
equations (12)
(12) are
are shown
shown in
in Figure
Figure 5.5. The
The
The transverse
transverse wake
wake fields
fields obtained
bunch
is
located
at
x
=0
and
y
=1mm.
The
first
four
y-dipole
modes
are
bunch
in the
the
bunch is
is located at x000=0 and yo=lmm.
y00=1mm. The first four y-dipole modes are used
used in
in
the
sum
over
modes, and
and again
sum
again higher
higher order
order modes
modes do
sum over
over modes,
do not
not contribute
contribute significantly.
significantly.
5
f
1.8
/
0.6
J
'\\
\
-0.6
V
s^\
-1.8
40
-32
-24
- 1 6 - 8
0
8
16
\l
24
I
32
4
a (mm}
Fig. 5. Calculated transverse wakefield in an X-Band structure of y-dipole mode
Fig.
Fig. 5.
5. Calculated transverse wakefield in an X-Band structure of y-dipole mode
550
In Fig.6,
Fig. 6, analytical
analytical simulation
simulation results
results for
for the
the longitudinal
longitudinal wakefields
wake fields for
for x and ydipole modes obtained by using Eq.14 are shown respectively. It is evidently that the
x-dipole has a larger contribution to the bunch wakefield than the y-dipole.
1.5-10*
1J-1D*
-40
-32
-1J-1D1
-24
(a)
(b)
Fig.6 Longitudinal wakefield analysis (a) x-dipole case (b) y-dipole case
The modes that we choose to compute the whole wakefield are with respect to the
(R/Q)ii for each corresponding mode which includes both monopole
contribution of (R/Q)
and dipole. The table (3) is the list of mode and their parameters we choose. Fig.7 is
final behavior of the wakefield including all contributions from the dipole modes.
Modes used
used for
for calculating wakefield
TABLE 3. Modes
(GHz)
Mode
Freq. (GHz)
βPi(rad/m)
i(rad/m)
(mono)
(mono)
11.17
LSM 11
11.17
234
n
LSM 12(mono)
(mono)
LSE 11
n(mono)
LSM 13(mono)
(monoJ
LSM 14(mono)
di 1
(y-dipole)
LSM 111/yp° ^
LSM
x(x-dipole)
dl ole
LSM 221/ - P J
LSM
LSM 15(mono)
LSE 221/x(x-dipole)
-dlP°leJ
(y-dipole)
LSE 111/y-dip°le'>
LSE
dl ole
(x-dipole)
^P ^
LSM 22
22
33.28
13.19
13.19
56.24
79.84
7.319
11.95
11.95
103.9
103.9
14.77
14.77
15.44
15.44
33.92
697
276
1179
1179
1673
1673
153
153
250
2177
310
324
711
551
(R/Q)
(R/Q)ii
12610
12610
3743
3009
1334
1334
574
462
313
288
269
228
171
171
3.5-10^
1.5-10*
2.1-10'
«
g
MO*
-7-10*
V
-IHQ'
-40
-32
-24
-16-8
0
8
16
24
32
40
-1.5-10*
-40
-32
-24
-16-8
0
iM
8
16
24
32
40
Fig.7 (a)analytical
(a)analytical result
result of
of full
full wakefield
wakefield (b)
Fig.7
(b) contribution
contribution of
of dipole
dipolemodes
modestotowakefield
wakefield
V. CONCLUSIONS
CONCLUSIONS
V.
exact solution
solution for
for wakefields
wakefields generated
generated by
by aa bunch
AAexact
bunch traversing
traversing aa dielectric
dielectricloaded
loaded
rectangularwaveguide
waveguide accelerating
accelerating structure
structure off
rectangular
off axis
axis is
is developed.
developed. ItItisisan
anextension
extensionofof
theprevious
previousapproach
approach for
for the
the on
on axis
axis beam
beam case.
case. The
the
The idea
idea behind
behind this
this effective
effective and
and
accurate
analytical
method
to
solve
the
complicated
problem
of
wakefields
accurate analytical method to solve the complicated problem of wakefields isisbased
basedon
on
thefact
factthat
thatthere
there exists
exists an
an orthogonal
orthogonal EM
the
EM field
field modal
modal decomposition
decompositionand
andthe
the
dependence of
of wakefield
wakefield on
on the
the linear
linear combination
dependence
combination of
of accelerating
acceleratingstructure
structuredimension
dimension
factors
R/Q.
factors R/Q.
ACKNOWLEDGEMENTS
ACKNOWLEDGEMENTS
This work is supported by Department of Energy, High Energy Physics Division,
This work is supported by Department of Energy, High Energy Physics Division,
under the contract No. W-31-109-ENG-38
under the contract No. W-31-109-ENG-38
552
APPENDIX
The formulas of EM components for inhomogeneous H-plane dielectric loaded
rectangular waveguide are listed here. The z dependence exp(-jpmnz) has been omitted
in these expressions for simplicity.
LSM(open)
^Lk^ cos ^L(x
w
w
Amn
Exmn =
mn
Bmn
B
?)>
fl < 7 < b
0 < 7 < a
Ln sin -^-(* + —)cos kyn (b - y),
~w
2
a <y <b
Amn (~jpmn )*£ sin ^-(jc + y)cos *£ ;,,
0 < y <a
B
(- JPM )*£i sin ^-(* + y) sin *£ (ft - y\
mn
xmn
x
W sin
*2iy (*
w ( + 2)
k
mn
H
cos mn
w *£>
o < y <a
Amnk]mn sin —— (* + — )sin *£> 7,
w
2
Eymn = •
Ezmn = •
2L) cos * (o)y,
2
+
w
= •
- B
a<y<b'
2
sin —— (jc + — )cos ky(l) (b - y),
w
2
H^ = 0
cue Q£ ft
(1)
a <y <b
/ .
. mn
mn .
w . . ,m
^ ^ ( y ^ o ) - ^ c o s —— (x + y)sm k^y,
0< y <a
H zmn =
B
mn
where AAmn
i
_
LSE
Amn
1
Bmn
(9pen)
(J®£ Q£r)— — cos —— (x + y)cos k^ (b - y\a < y < b
Kt C nsmk^n(b-a)
ym
erCosk^n(b-a)
'"; ^K
i^ c o s ^ a
sin^>«
UJ
'
nmmod es:
C m n (- jPmn ) cos ^-(^ + ^) cos A:^ ^,
0 < y <a
Exmn =
Dmn (~ JPmn ) cos ^^(x + ^) sin A: <ii (6 - y\
w
2
E^ = 0
C.^-sin^-U + ^ c o s t ^ j - ,
Ezmn =
^
mn
mn ,
0<y<a
w,
, m ,,
Z) mM ——— sm ——— (x + — )sm fr£l
(6 - ^),
w
w
2
*
^
k ^ mn . mn ,
w. . , ( 0 }
jcoju
H
.
w
0
w
a <y <b
<ar < ^ < 6
2
(3)
xmn
k(y
-)
™
jd)ju
1
Cmn
Hymn
m
"sm
w
0
k mn
'
jcoju
mK
w
mH
cos
w
0
W
)cos £ ( 1 ) (b
2
(x+
y),
(x + W ) cos k(y^n y,
2
a<y<b
0 < y <a
=
Z)
c
*«"
jcoju
Pmn^n
ftjt/
Hzmn =•
r.
£™
cos
^
0
w
(jc+
m n
^
W
2
+
)sin k(l) (b
W
w
0
P mn * ymn
coju
m;r
0
cos
mn
w
y),
a< y < b
^ ^
0
< ^ < «
2
(^ +
W
2
i (1) / i
)cos k(;mn (b
^Cma k®,<xxk®,(b-a)
"^
C^*>
Wh
553
\
y\
L
a <y <b
C
sinO*-«)
A.
cosO '
M
,
(1)
LSM(short)
modes.
•- -\1
m n (Q)
mn
w .
m
-^L« —— *iJ cos —— (* + —)sin £)J.y,
w
w
2
m ft r (D
mn ,
w, . ,
,.
,
B mn —— &J^
cos —— O + —) sin k(rn
mn (b - y),
w
w
2
Amnk2cmn sin ^(;t + —)cos k™y,
w
2
E^ = \
B.kL sin ^(* + ^) cos *« (b-y),
W
2
[
w
a <y <b
0 < y <a
2
E'- (-#« )*iii «n ^L(» + |.) sin * <Ji (6 - ,),
xm,
=•
„
-B
„
coe Qe B
. mn ,
w,
»m^r
\
sin —— ( ;c + —) cos k ;^; (6 - y ),
w
2
"3- =0
. mn
mn ,
w,
,r(n
A
mn ( 7 ^ o ) ————— COS ————— (* + —— ) C O S
"
.
-I5T ^ X(jcoe
1
w
w
2
a <y <b
^ ^ J,
a <y <b
0 < y <a
. mn
mn ,
w,
,mx,
—— cos —— ( ;c + —) cos A; ^ (b - y), a < y < b
w
w
2
in
a)
t £.\
_d „„ _ - C s C (* - «) or Am _ er cos C (ft -——
,
(o)
ww
where
Qer)
B
*2sin*2«
LSE(short)mn modes
'
5™
cos*™«
C mB (-7^^ )cos ^^(X + —)sin /t<^ y,
Exmn =
w
2
Dm (~jpm) cos ^^(^ + ^)sin *^ (6 - j-X
w
2
E^n
0 < y <a
a <y <b
=0
_ mn . mn ,
w . , ^
C.« ——— sin ——— O + —)N sin £ r<°o >,,
0 < y <a
_
w
w
2
Ezmn ^ mn . mn ,
WN. ,mx,
a <y <b
^.« ——— sin ——— (x + — )sm A:^i (6 - y),
w
w
2
k Z^_(_
, mn
r o( ^) y,
———.) S.m mn
———,(x + w
—, )cos ,kymn
(:mn —vmn
0 < y <a
jcoju o
w
w
2
Hxmn = •
^
k vmn mn . mn ,
w,
,mx,
£) mn —^^
———— sin ——— (x + — )cos kJ( ) (b - y\
a <y <b
jcoju 0 w
w
2
""
^
^i»
mn i, v 1 w .i °in
. Ir, r^ o ^' T
f
cmn
pr»°
0 < y < a
C
" 7^0
wU + 2)Sm^^'
1
H ymn
=
D MW ^ cos m" (x+ W)smk%n(b
jcoju 0
w
2
cc
m«
H
zmn
(5)
0 < y <a
- Am (OS Opm sin ^-(x + y) cos fc <!> y,
H
a <y <b
0 < y <a
Am jpm *£ sin H^-(X + ^) sin k™ y,
E ™, = '
0 < y <a
= «
D
^WH
y\
- ^^
^^r
w
cos
{x + ;cos K(0) y,
coju 0
w
2
Pmnkymn
coju
0
cos
m ft
w
^A -i-
W
2
j cos A.
(1)
where C^ _-^co<(fe- fl )
554
{D
a < y <b
0 < y <a
y ),
a < y <b
C w w _sin^>-a)
"
sin£(0)a '
(«\
(7)
REFERENCES
[1] L. Xiao, Wei Gai and Xiang Sun, Phys. Rev. E, 65, 016505 (2001)
[2] Magdy Z.Mohamed and John MJarem, IEEE MTT vol.40., No. 3, Mar. 1992
pp.482-494
[3] Robert E. Collin Field Theory of Guided Waves McGraw-Hill, Inc. 1960 pp.224232
[4] H.H.Braun et al., CERN-CLIC-Note 364, 1998 (unpublished)
[5] W.K.H.Panofsky and W.A.Wenzel, Review of Scientific Instruments, Vol. 27, 967
(1956)
[6] Constantine A.Balanis Advanced Engineering Electromagnetics John Wiley
&Son, Inc. 1989 ISBN 0-471-62194-3 pp410-414
[7] Marc E. Hill, Phys. Rev. Lett. 9, 094801 (2001)
[8] A. Tremaine, et al Phys. Rev. E. 56, 7204-7216 (1997 )
[9] T-B. Zhang, J. L. Hirshfield, et al. Phys. Rev. E, 56, pp.4647-4655 (1997)
[10] S.Y.Park, et al. Proc. AAC2002
555