1
WAKEFIELD AND SURFACE ELECTROMAGNETIC FIELD
1
OPTIMISATION OF MANIFOLD DAMPED
ACCELERATING
2
STRUCTURE
S
FOR CLIC
3
4
V. F. Khan
†*
, A. D’Elia
†*‡
, R. M. Jones
†*
,
5
A. Grudiev
‡
, W. Wuensch
‡
,
G. Riddone
‡
, V. Soldatov
‡
§
6
†
School of Physics and Astronomy, The University of Manchester, Manchester, U.K.
7
*
The Cockcroft Institute of Accelerator Science and Technology, Daresbury, U.K.
8
‡
CERN, Geneva, Switzerland.
9
§
JINR, Dubna, Russia
10
Email: Vasim.Khan@hep.manchester.ac.uk
11
Abstract
12
The main travelling wave linacs of the compact linear collider
(CLIC) operate at a frequency of 12 GHz with a
13
phase advance per cell of 2π/3. In order to minimise the overall footprint of the accelerator, large accelerating
14
gradient
s are
sought
. The present baseline design for the main linac
s
of CLIC demands an average electric field
15
of 100 MV/m. To achieve this in practical cavities entails the dual challenges of minimising the potential for
16
electrical breakdown and ensuring the beam excited wa
kefield is sufficiently suppressed. We present a design to
17
meet both of these conditions,
together with a description of
the
structure
,
CLIC_DDS_A
,
expressively designed
18
to experimentally test the ability of the structure to cope with high powers.
19
Key wor
ds
20
Beam dynamics
,
Breakdown
,
CLIC
,
CLIC_DDS
,
CLIC_G
,
DDS
,
HOMs
,
Linear collider
,
Manifold damped
,
21
NLC
,
Wakefields
.
22
PACS: 29.20.Ej
–
Linear accelerator
23
29.27.

a
–
Beams, charged particles

in accelerators
24
77.22.Jp
–
Breakdown electrical, dielect
r
ic
s
25
1.
Introduction
26
The aim of the CLIC project is to collide multiple bunches of electrons and positrons
at a 3 TeV centre of
27
mass energy. In order to achieve high accelerating gradients within the cavities, normal conducting (NC) linacs
28
are employed [1

3].
The CLIC baseline design aims at an accelerating gradient of 100 MV/m [4

5] with an X

29
band frequency of
12 GHz. This frequency resulted from a detailed optimisation procedure based on various
30
simulations [4

5].
The curves representing the optimisation parameters are relatively flat in the vicinity of 12 to
31
15 GHz. The 12 GHz frequency was chosen as it is
close to the frequency used in the next linear collider
32
(NLC) [6] programme and hence the wealth of knowledge developed over two decades can be capitalised upon.
33
2
There are two phenomena which must be taken into account when designing these accelerating
structures:
34
electrical breakdown and beam

excited wakefields. The former can be addressed both by carefully designing the
35
structure such that the surface e.m. fields are minimised and by paying attention to the surface morphology.
As
36
for the wakefields, t
hey have both short

range (along the bunch) and long

range (along the bunch train)
37
components. The short

range wakefield is a strong function of the iris aperture and is not the focus of this study.
38
Here we present a design for suppressing the long

range w
akefield, whilst minimising the surface e.m. fields on
39
the walls of the accelerating structures. The method for damping the wakefields entails detuning each of the cell
40
frequencies, by tapering down the irises along the structure, and providing moderate (
Q
~1000) coupling to four
41
attached manifolds [6]. This method is similar to that adopted in the NLC, however, with stronger constraints
42
imposed due to the larger gradients required. This resulted in
markedly
different outer cavity wall design.
The
43
design pre
sented here is an alternative
to
the baseline design for the main linacs of CLIC, which relies
on
heavy
44
damping (
Q
~10 [5]) through strongly coupled waveguides attached to each cell. However, we maintain the same
45
number of cells in
the
CLIC_DDS_A design.
Other wakefield suppression strategies are possible [
7
].
46
In the
CLIC
damped and detuned
structures (DDS)
,
the focus of this work
, both the fundamental
47
(accelerating) mode and the higher order dipole modes
(HOMs)
are calculated. In both cases, the
48
elec
tromagnetic (e.m.) fields in single cells is calculated using codes which rely on
representing the geometry
49
with a finite element based mesh. For a sufficiently fine mesh, an accurate representation of the e.m. fields is
50
obtained. The beam loaded accelerat
ing field is calculated from an integral [
8
] representation of the energy flux
51
within the overall cavity, based on the field in
individual
cells.
The transverse dipole field is calculated from a
52
circuit model [
6,
9

11
], designed to represent dipole mode sl
ot

coupled
to waveguide like manifolds. This
53
circuit model and its spectral function [
6,
11
] generalisation, provides a design tool to allow the influence of
54
geometrical modifications to be rapidly accounted for in the wakefield calculations. This unique t
ool has been
55
validated on several
previous
NLC structures and has proved to be an accurate prediction of the wakefield [6
,
56
11
].
57
The CLIC_DDS design
is based upon the knowledge gain
ed
and documented from the NLC studies. However,
58
the geometrical change implemented in CLIC_DDS
to minimise the pulsed temperature rise
is
based on
the
59
baseline design of the CLIC main linacs which is waveguide damped structure
(
WDS
known as
CLIC_G
[4

5]
).
60
W
e report on various stages involved in evolving CLIC_DDS so as to satisfy the stringent constraints imposed
61
by the rf breakdown and beam dynamics criteria. In order to rapidly realise the bandwidth necessary for the
62
wakefield suppression,
detuned
structure
s
(DS)
have been studied
first.
In all cases, we prescribe a Gaussian
63
distribution for suppression of the wakefields.
We
learned that a bandwidth of ~3.3 GHz is necessary to
64
suppress the wakefield to satisfy the beam dynamics criterion for an inter

bunch s
pacing of 6 rf cycles.
65
The paper is structure
d
such that the next section presents an overview of early designs for CLIC which fail to
66
subsequently satisfy both the electrical breakdown and beam dynamics constraints. This is followed by a design
67
to
overcome these limitations. The final section provides some concluding remarks together with future
68
prospects for a design which will be tested for high power.
69
70
71
2.
Means To Independently Satisfy RF Breakdown And Beam
72
Dynamics Constraints
73
In all cases, w
e utilise
a
finite element code HFSS [
1
2
] to model accelerating structure
s
, calculate
t
he fields
74
and eigen modes within the sructure.
T
he si
n
gle infinitely periodic cell is tuned
by varying cavity radius (
b
)
to
75
the accelerating frequency of 11.994 GHz for
a given iris raius (
a
) and iris thickness
(
t
)
.
Dispersion curves are
76
obtained from the circuit
model
[
1
3
]
. For a cell subjected to infinite periodic condition the dispersion relation
77
between frequency ω/2π and phase advance per cell ψ
is:
78
3
cos
ψ
η
1
ω
ω
r
(
1
)
79
The resonating frequency (
ω
r
) and coupling co
efficient
(η)
of the neighbouring
cells are obtained from the 0 and
80
π mode (simulation results).
81
2
π
2
0
2
π
2
0
r
ω
ω
ω
2
ω
ω
(
2
)
82
2
π
2
0
2
0
2
π
ω
ω
ω
ω
η
(
3
)
83
The calculation of group velocity
(derivative of eq. 1
[14]
)
is fol
l
owed by the
c
alculation
of the dipole mode
84
synchronous frequency.
We model seven single cells to
represent
a structure of 25 cells.
As the power absorbed
85
in the breakdown is
strongly dependent on both surface e.m. fields and
the
fundamental mode group velocity
86
[
1
5
]
, we
maintain
a
low group velocity by changing the iris
thickness from 5.7
mm
(cell 1)
to 0.5
mm
(cell 25)
.
87
The taper
ed
iris radii
and tickness
es
in
this
structure resu
l
t
in
a
group velocity variation from 1.93 %c to 1.0
%
c
88
Parameters of this large bandwidth DS are
presented in Table 1.
The ratio of average iri
s radius (<a>) to
89
accelerating wavelength (
λ
) for this structure is 0.142.
It is important to minimise this ratio so as to reduce the
90
breakdown possibility
[6]
.
91
An
optimal design, in terms of rapid damping of the dipole wakefield results in an iris radius taper down from
92
4.95
mm to 2.15
mm results in a dipole bandwidth of ~3.3 GHz
[
16
]
.
The iris radius follows an erf distribution
93
with cell number.
The bandwidth (
∆
f
) in terms of the standard
deviation
of a Gaussian distribution
(
σ
) is:
∆
f
=
94
3.6
σ
. The detuning spread
in this structure
is 20% of the central frequency.
Once the synchronous
frequencies
95
and kick
factors are calculated using
computational
tool
(uncoupled m
ode)
,
we calculate
the
coupled mode
96
frequencies and kicks so as to account for the cell

to

cell interaction
s
using a
double band circuit model [1
3
]
. In
97
this structure, we observed
an approximately
200 MHz shift in the coupled mode
frequencies with respect to the
98
uncoupled mode
due to the interaction
s
of the
fields coupled through
irise
s.
The representation of a Gaussian
99
distribution needs better sampling, 25 cells are clearly not
sufficient
for this purpose. In this case, wakefield
100
decay in a 25 cell structure is not
adequate
to meet the beam dynamics criterion. Hence
,
we interleave
a number
101
of structures to satisfy the beam dynamics criteria.
An
8

fold interleaving provides the necessary suppression of
102
the wakefield.
The transverse
long

range wakefield is calculated using the modal sum method as follows [
17
]
103
)
(
2Q
i
1
t
i
ω
Exp
K
2
(t)
W
p
p
N
1
p
p
T
t
(
4
)
104
where
ω
p
is the synchronous frequency,
Q
p
is the quality factor of the synchronous mode
and
θ
(
t
)
is the
105
Heaviside step function
.
A comparison of the uncoupled and coupled mode frequencies is illustrated in Fig.
1.
106
Similarly
, a
comparison
of the designed
uncoupled and coupled mode
kick factor weighted density function
107
Kdn/df
is presented in Fig. 2. In this case
,
a non

smooth behaviour of
Kdn/df
is observed due to non

smooth kick
108
factors of t
h
e coupled mode.
The e
nvelope of the wakefield for
an entire train of
3
12 bunches
is illustrated in
109
Fig. 3
.
I
n this case
,
various
damping
Q
s are artificially imposed
.
The wakefield in a DS with
losses
due to finite
110
conductivity
(
Q
cu
~6000
)
is also shown
.
W
akefield suppression
well beyond
the
beam dynamics requirements
i
s
111
obtained
.
For
this geometry, the surface e.m. field on the copper walls, on the other hand is too large. The
112
electrical breakdown constraints are not met.
113
The motivation behind investigating a reduced bandwidth structure is
also
to
enable
the rf breakdown
114
constraints
to be satisfied
.
This leads us to match the end cell iris dimensions to the CLIC_G structure, with a
115
tapering of Gaussian function. In this case, the ratio of
<a>/λ
reduces
to
0.1 and
the
average group velocity also
116
reduces by ~20%
, [
18,19
]
.
The structure
now satisfies rf breakdown constraints. However, t
he
structure
117
4
bandwidth reduces significantly
to ~0.9 GHz resulting in severe wakefields on the
first two trailing bunches
and
118
is illustrated by the
blue curve
in Fig. 4
.
For a moderate damping with reduced bandwidth,
a
possible way to
119
s
a
tisfy both the constraints is to increase the bunch spacing by a factor of 3 i.e. to 18 cycles (1.5 ns).
In this case
,
120
the rf

to

beam efficiency reduces
down
to
an unacceptable value of
8 t
o
10%.
The other
possible option is to rely
121
upon zero crossing scheme. When wakefield is calculated, an excursion of the envelope is cal
culated.
H
owever
,
122
the wake
experienced
by the bunch may be small. In this case, the iris dimensions of the structure
(
s
)
a
re tuned in
123
such a way that the bunches see almost zero
amplitude
of the wakefield (and not the envelope). Wakefield in a
124
structure implemented with zero crossing is illustrated in Fig. 4, here the location of
the
dots represent
bunches.
125
The envelope of wakefield for this structure in presented in Fig. 5 with
several
damping
Q
s.
126
The
CLIC
project requires more than
140
,
000
[
7
]
accelerating structures
.
In
practic
e,
it will be
difficult to
127
maintain the zero crossing schem
e for
all
structure
s
. M
eeting the mechanical
tolerances
to build these structure
s
128
is also challenging.
A
possible solution for
a
moderately damped DDS is to relax the bunch spacing
,
whilst
129
loosing a few percentage in overall efficiency and to choose a structure wi
th a moderate dipole bandwidth
.
In
130
this manner a
trade off between the bandwidth and efficiency is
investigat
ed in the next section.
131
3.
A structure Satisfying RF Breakdown And Beam Dynamics
132
Constraints
133
After realising the necessary bandwidth range for
satisfying the beam dynamics constraint by studying DS, a
134
conventional circular cell incor
porated with manifold geometry wa
s studied.
The p
rofile of a typical DDS cell is
135
presented in Fig. 6, where
R
M
is the radius of the manifold and
R
c
is the radial distance of the manifold coupling
136
slot from the electrical centre of the cavity.
This structure
consists
of 24 accelerating cells
and
is known as
137
CLIC_DDS_C. The taper in the iris radius ranges from 4
mm to 2.3 mm to provide a bandwidth of ~
2.3 GHz
.
138
The ratio of
<
a>
/
λ
for this structure is 0.126.
DDS_C incorporate
s
manifolds,
slot

coupled to the accelerating
139
cells
. Th
e
s
e
coupling slot
s
perturb the cell wall,
and cause
more current to flow in the vicinity of the slot
,
140
leading to excessive surface magnetic field
s
(H

field).
The average peak power requirement of
an
8

fold
141
interleaved DDS_C i
s ~73 MW to maintain an average accelerating
gradient
of 100 MV/m
.
The
bunch
142
population
in this case is
chosen to be
4.2 x 10
9
. Howe
ver, for this structure
bunches can be populated
up to
5.0
143
x 10
9
. I
n
this case, the input power requirement will increase to ~76 MW.
The average rf

to

beam efficiency is
144
~23%
. The enhancement of the H

field in the coupling slots results in a pulsed
surface
temperature rise of 72° K
145
for an rf pulse length of
~
250 ns
.
The pulsed
surface
temperature rise along the structure length in each of the 8
146
structures of DDS_C is illustrated in Fig. 7. As can be seen, the structure observes nearly 30%
(
the tolera
ble
147
limit is 56° K [4,
5])
temperature rise towards the downstream end
and
fails to meet the rf
breakdown constraint
.
148
An accurate determination of the dipole properties of this structure is facilitated by the circuit model [6
,
9] and
149
spectral function
method [
6, 11
].
This is necessary in order to accurately predict the wakefield for a multi

cell
150
structure
,
slot

coupled to wave guide like manifolds.
The
l
owest dipole bandwidth
in
this structure
is:
∆
f
= 3.6
σ
151
= 2.33 GHz
and
t
he detuning spread is 13.7%
of the central frequency
.
The
dispersion relation for a manifold
152
damped single infinitely periodic cell is defined as [6
,
9
]
153
0n
0n
2
2
0
2
0
0
2
2
2
0
2
2
0
ψ
sin
ψ
f
f
ˆ
cos
η
ˆ
1
c
πP
Γ
cos
ψ
cos
sin
f
ˆ
f
η
f
f
ˆ
cos
η
ˆ
1
f
f
cos
η
1
(
1
)
154
where f
0
and η are the resonating frequency and
coupling coefficient of the
TE mode
respectively
and
̂
and
̂
155
of the TM mode
, Γ coupling of the manifold with cell,
φ
phase advance per c
ell, ψ local phase advance per
156
waveguide section and P is the period of the cell. Here
,
the cross coupling term between TE and TM mode
s
for a
157
th
in iris [
2
0
], can be approximated
as
ˆ
.
The d
ispersion curves of the first three dipole modes in a typica
l
158
DDS_C cell are
illustrated in Fig. 8.
We utilise
the s
pectral function method to calculate the impedance of the
159
structure
[6
, 1
1]
160
S(ω)= 4 Im {Z(ω+j
ε
)}
(
2)
161
5
where
ε
is
an
infinitesimal displacememnt and Z(ω)
is the
impedance of the structu
re and
is
define as [
6, 11]
162
nm
N
m
n,
m
s
n
s
m
s
n
s
2
H
~
m
n
P]
c
j
ω
exp[
ω
ω
K
K
2
π
1
ω
Z
(
3)
163
Here N is the number of cells in a structure, K’s and ω’s are the synchronous kicks and frequencies respectively.
164
The matrix
nm
H
~
contains various circuit parameters involved and is defined in [
6, 11
].
165
The
wakefield is calculated by taking in
verse Fourier transform of the s
pectral function.
The s
pectral function of
166
an
8

fold interleave
d DDS_C is illustrated in Fig. 9
and
the corresponding
wakefield in Fig.
10
.
As the dipole
167
bandwidth in this case is moderat
e,
the
wakefield decay should be rapid enough to meet the beam dynamics
168
criterion.
In order to meet the beam dynamics constraint, t
he
inter

bunch spacing
in this case
is
relaxed to
8 rf
169
cycles (0.67 ns)
from the base

line 6 rf cycles
.
It is inevitable to
relax the bunch
spac
ing
in a structure with
170
moderate bandwidth
.
I
n this way
,
the beam dynamics criterion is satisfied at the cost of few percentage loss in
171
the efficiency.
The wakefield in this case is damped beyond the beam dynamics limit which is shown b
y dashed
172
line in Fig.
10
.
Though
the
wakefield suppression in this structure is
adequate
, DDS_C
need
s
further
173
optimis
ation
to meet the rf breakdown criteria, this is discussed in the next section.
174
The
H

field in a
standard
circular cell is uniformly
distributed along the surface of the cell. When the surface
175
is perturbed, to inc
orporate for manifold coupling,
the
field in this region gets enhanced.
For a circular
un

176
d
a
mped
cell of iris radius 4
mm, the
normalised
H

field
(
with respect to the
accelerating field
)
on the cell wall is
177
~3.8 mA/V. When the cell
wall
is perturbed by a coupling slo
t
of width
3 mm, the enhancement in the
H

field
178
peaks up
to 6 mA/V i.e. nearly 60% enhancement.
The p
ulsed temperature rise is proportional to the square o
f
179
the H

field [
21
]. Reducing the iris radius also reduces the H

field, however, it also
affects
the dipole bandwidth.
180
In this case it is necessary to re

distribute the H

field on the cavity wall, and insert the manifold coupling slot at
181
a location where
the
field is minimum. This re

distribution reduces
the
field enhancement.
In order to study the
182
field distribution in the absence of manifold slots
i.e. an un
damped cell
, a range of cells with modified walls
183
have been studied and are illustrated in Fig 1
1
.
The modified cavity
shape
is
defined in terms of an ellipse
ε
with
184
A
and
B
as semi

major and semi

m
i
nor axis respec
t
ively
[
22
]
.
For
B
= 0,
ε
=
∞
and the cell wall is rectangular
185
and a circular wall corresponds to
ε
= 1
.
The variation of
the
normalised
H

field along the contour
of an
186
undamped cell is presented in Fig. 12
.
The
dashed line in this figure represents an approximate location where
187
manifold slot will be introduced.
A
manifold of slot width
2.5
mm was introduced
.
T
he field enhancement
for
188
selected
shapes
is illustrated in Fig. 1
3
.
As can be seen, for an elliptical cell of
ε
=
1.38, the field enhancement
189
is
a
minim
um
.
T
here is no field enhancement
in the vicinity of the coupling slots
compared to the peak field
190
within this cell.
The peak normalised H

field on the cell contour is now ~4
.4
mA/V. However, there is still some
191
field enhancement towards the tip of the manifold slots which
is
~5
mA/m.
192
The
iris thickness was also
optimised
to minimis
e
the surface electric field
(E

fi
eld)
.
The new structure
193
incorporating an elliptical outer wall and modified iris thickness is known as DDS_E.
A change in iris thickness
194
primarily affects
four
rf parameters:
1
)
the
surface
E

field,
2
)
the
fundamental mode group velocity
(
v
g
)
,
3
)
shunt
195
impedance
(R)
, which affects
the
input power requirement and hence efficiency of acceleration (
rf

to

beam
196
efficiency
)
and
4
)
dipole bandwidth.
Several structures with a range of iris thickness
es
were studied by
197
comparing the
ir rf
properties such as surface E

field, input power requirement, rf

to

beam efficiency and dipole
198
bandwidth.
In this process, the rf properties of the structures
(
DDS_E
)
were compared with
a
reference
199
structure
.
The cavity wall of the reference structure
is
elliptical with
iris
radii and
thickness
es
retained from
200
DDS_C
.
A comparison of the rf properties of the reference structure with various other structures is
shown in
201
Fig. 1
4
.
In this optimisation we realised that b
eyond
the
average iris thickness of
2.65
mm, surface
E

field
202
remains almost
invaria
nt
.
The
input power
is reduced and
hence the efficiency of acceleration increases
.
203
However,
the
dipole bandwidth
is reduced
.
Considering the trade

off between the efficiency and dipole
204
bandwidth, an average iris t
hickness of 2.65
mm
was optimised which gives a taper in the iris from 4
mm to 1.3
205
mm.
This demands an
average input power for the 8

fold interleaved DDS_E o
f
69.5 MW. The
overall
average
206
rf

to

beam efficiency in this case is 24%.
The maximum
surface E

field is 251 MV/m and the pulsed
207
6
temperature rise is 52° K, which is reduced by
~28
%
compared to DDS_C
.
As the fields on the cell wall
s
were
208
reduce
d
due to modifications in the geometry, the coupling
of the dipole modes also reduced
. This affects
the
209
wakefield suppression adversely. However, the wakefield is still suppressed beyond the beam dynamics limit. A
210
comparison of
the
wakefield in DDS_C a
nd DDS_E is presented in Fig. 15
.
A test structure, which is
the
first
211
out of
eight

fold interleaved st
ructures of DDS_E is being
fabricated.
The properties of the test structure
are
212
discussed in the next section.
213
4.
S
tructure
Optimised For High Power Testing
214
In order to test the high power performance of DDS_E, a test structure known as DDS_A has been
designed.
215
HOM couplers are omitted as the purpose of this structure is to evaluate the ability of the accelerator to sustain
216
high powers.
The first structure
(out of eight
DDS_E
) is
used
b
ecause
it has the largest
aperture
compared to
217
other
interleaved
structures
(and is also the reason
why it
ne
eds relatively more input power
)
.
Therefore
, the
218
breakdown rates in this structure are expected to be severe compared to the remaining
interleaved
structures.
In
219
order to make the design of the structure
easy
as
far as
the
mechanical and cost point of view is concerned,
the
220
manifold dimension
s
are kept constant
throughout
the structure. The consequence of which is poor coupling of
221
the dipole modes to the manifold, hence
the
wakefield is non

optimal in this case.
As the primary aim of this
222
non

interleaved structure is to test the high power performance, we do not expect wakefield to be damped
223
adequately.
Detailed geometric parameters of DDS_A are presented in Table 2.
In [
2
3
]
,
a
new local
quantity
(
S
c
)
224
is defined, and is termed as
modified Poynting vector,
to calculate the
complex
power flow from the structures.
225
It
provides
the limit on
the
rf
gradient in presence of
electrical
breakdown.
The m
axima
in
the E, H and S
c
fields
226
in DDS_A ce
lls are presented
in Fig. 16 [
22
].
The fundamental mode rf
parameters of the s
ingle cells are
227
illustrated in Fig. 17 and overall structure properties both in beam loaded and unloaded conditions are presented
228
in Fig. 18.
229
The spectral function of DDS_A is illustrated in F
ig. 19. The
Q
of the dip
o
le m
odes is calculated by fitting a
230
Lorentzian [
17
] to the peaks in
s
pectral function.
The average dipole
Q
in this structure is ~1650
and is
231
illustrated in Fig. 20.
The w
akefield in DDS_A is illustrated in Fig. 21.
232
The calculations involved in optimising the structure for fundamental as well as dipole mode properties are
233
based on single infinitely
long
periodic cells. However, wakefield calculations do involve circuit parameters to
234
account for the coupled mode intera
ctions. In order to build a realistic structure, we need to design matching
235
cells at the either ends of the structure (regular cells) to match the impedance of the structure to minimise the
236
reflection.
Instead of a conventional rf power coupler,
CLIC_
DDS_A
will be powered using a mode launcher
237
[
24, 25
]
. In order to minimise the overall reflection in the structure, we design matching cells, at either ends of
238
the structure.
The matching procedure
begins with designing the cells
as indicated in Fig. 22. Here,
the geometry
239
in the middle is the first (or last) regular cell provided with matching cells at the either ends and beam pipes at
240
the extreme ends. This, in principle, is similar to a constant impedance structure. The matching parameters such
241
as matching ir
is
a
, matching cavity radius
b
and matching gap length
L
are varied to minimise the reflection
242
(S
11
) at the operating frequency.
In this way, both the end cells are designed. However, the real geometry is not
243
the constant impedance but constant gradient type, hence the matching parameters (
a
,
b
and
L
) need to be fine
244
tuned for a real tapered structure. After defining a complete 3D
structure of 24 regular cells + 2 matching cells
245
in
simulation software (HFSS [
1
2
]), we fine tune the matching parameters for the whole structure using Kroll
246
method [
26
]. This time we minimise the standing wave ratio (SWR) in the structure. I
n
this way
,
th
e complete
247
structure is tuned including the matching cells. The
accelerating
field in the fully tuned structure is illustrated
in
248
Fig. 23. The extreme peaks in this plot correspond to the matching cell accelerating field
s
. These peaks are
249
dissimilar to the
regular cell peaks due to the fact that
the
matching cell lengths are different compared to the
250
regular cell
lengths. The erf tapering of the regular cells is evident in the full
y
tuned structure accelerating field.
251
The accelerating field phase advance pe
r cell is also illustrated in Fig. 23. Here, a nearly triangular shape profile
252
reflects the 120° phase advance. The maximum deviation in the phase advance per cell is no more than 6°. The
253
discrepancy
from a perfect triangular shape can be understood i
n
the following ways: i) difference in the
254
extreme (regular) irises due to error function tapering, ii) use of only 9 cells to represent full structure of 24
255
cells, iii)
difference in the cell length
s
of the matching cells
compared to regular cell length
.
T
he S parameters
256
7
of th
e
fully tuned structure are presented in Fig. 24.
In this case, the simulation results of S
11
=

54 dB (2.24 x
257
10

6
) has been achieved
at the operating frequency
.
The
quality factor
as a function of frequency
is
presented in
258
Fig. 2
5
.
259
The fabrication of the D
D
S_A cells is in progress and the test cells
(discs)
are shown in Fig. 2
6
. A complete
260
CAD drawing of DDS_A, consisting of 24 regular cells and 2 matching cells is illustrated in Fig. 2
7
.
The overall
261
parameters of the DDS_A are su
mmarised in Table 3.
262
5.
Final Remarks
263
Though the rf breakdown and beam dynamics constraints are stringent in
the
CLIC main linacs,
a design
264
incorporated with
a relaxed bunch spacing, moderate bandwidth and modified
outer
cell wall
meets the design
265
constraints
provided
eight

fold inte
rleaving
of dipole frequencies is employed
.
266
Acknowledgements
267
We acknowledge
illuminating
discussions with J. Wang, Z. Li, T. Higo
, R. Zennaro
and I. Syratchev on
linac
268
structures and beam
dynamics. Research leading to these results has received funding from European
269
commission under the FP7 research infrastructure grant no. 227579.
270
References
271
1.
F. Tecker, 2008, CLIC And CTF3, Journal Of Physics: Conference Series
110
(2008) 112005.
272
2.
J. P. Dela
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273
SuperB Factories, , Journal Of Physics: Conference Series
110
(2008) 012009.
274
3.
E. Jensen, 2005, Normal Conducting CLIC Technology, CLIC

Note

641, Switzerland.
275
4.
H. Braun
,
et
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al
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Note

764, Switzerland.
276
5.
A. Grudiev and W. Wuensch, 2008, Design Of An Accelerating Structure For The CLIC Main Linac,
277
LINAC’08, Canada.
278
6.
R. M. Jones,
et
.
al
., 2006, Wakefield Suppression In A Pair Of X

Band Linear C
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279
STAB,
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, 102001.
280
7.
R. M. Jones, 2009, Wakefield Suppression In High Gradient Linacs For Lepton Colliders
,
Phys. Rev.
281
STAB,
12
, 104801.
282
8.
W. Wuensch and I. Wilson, 1999, Beam Loading Voltage Profile Of
An Accelerating Section With A
283
Linearly Varying Group velocity, CLIC Note 399, Switzerland.
284
9.
R. Jones,
et
.
al
, 1996, Equivalent Circuit Analysis Of The SLAC Damped Detuned Structure,
285
Proceedings Of The European Particle Accelerator Conference, EPAC’96, SLA
C

PUB

7187, Spain.
286
10.
R. Jones,
et
.
al
, 1996, A Spectral Function Method Applied To The Calculation Of The Wake Function
287
For The NLCTA, Proceedings Of The Linac Conference, LINAC’96, SLAC

PUB

7287,
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288
11.
R. M. Jones,
et
.
al
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tion Errors On Wake function Suppression In NC X

289
Band Accelerating Structures For Lepton Colliders, new Journal Of Physics,
11
(2009) 033013.
290
12.
www.ansoft.com
291
13.
K.L.F. Bane and R. Gluckstern, 1992, The Transverse Wakefield Of A Detuned X

Band Accelerator
292
Struc
ture, SLAC

PUB

5783, USA.
293
14.
R. M. Jones, 200
7
,
Fundamentals Of Collective Effects, Wakefields And Impedances, Contribute To
294
Cockcroft Institute Accelerator Course, U.K.
295
15.
R. M. Jones,
et
.
al
., 2006, Dipole Wakefield Suppression In High Phase Advance Detuned
Linear
296
Accelerators For The JLC/NLC Designed To Minimise Electrical Breakdown And Cumulative BBU,
297
Proceedings Of The Particle Accelerator Conference, PAC’01, SLAC

PUB

8887, USA.
298
16.
V. F. Khan and R. M. Jones, 2008, Wakefield Suppression In The CLIC Main Linac
s, Proceedings Of
299
The European Particle Accelerator Conference, EPAC’08, Italy.
300
17.
R. Jones, 2004, A Study Of Higher Band Dipole Wakefields In X

Band Accelerating Structure For The
301
NLC/GLC, Proceedings Of The
Linac Conference
,
LINAC
’
04, SLAC

PUB

10682, German
y.
302
8
18.
V. F. Khan and R. M. Jones, 2008,
B
eam Dynamics And Wakefield Simulations For The CLIC Main
303
Linacs, Proceedings Of The Linear Accelerator Conference, LINAC’08, Canada.
304
19.
V. F. Khan and R. M. Jones, 2008, An Alternate Design For The CLIC Main Linac
Wakefield
305
Suppression, Proceedings Of The X

Band And beam Dynamics Workshop, XB’08, U.K.
306
20.
R. M. Jones, 2005, Fundamentals Of Wakefields and Impedances: From Physical

Mathematica
307
Analysis To Practical Applications, Contributed to the U.S. Particle Accelerato
r School, USA.
308
21.
I. Wilson, 1987, Surface Heating Of The CLIC Main Linac Structure, CLIC

Note

57, Switzerland.
309
22.
V. F. Khan
,
et
.
al
, 20
1
0, Recent Progress On A Manifold Damped And Detuned Structure For CLIC,
310
Proceedings Of The International Particle
Accelerator Conference, IPAC’10, Japan.
311
23.
A. Grudiev,
et
.
al
,
2009, New Local Field Quantity Describing The High Gradient Limit Of
312
Accelerating Structures, Phy. Rev. STAB.
12
,102001(2009).
313
24.
I. Syratchev,
2002
, Mode Launcher As An Alternative Approach To The
Cavity
–
Based RF Coupler
314
Of Periodic Structures, CLIC Note 503, Switzerland
.
315
25.
C.
D.
Nantista,
et. al
,
2004,
Low Field Accelerator Structure Couplers And Design Techniques,
Phys.
316
Rev. STAB
,
7
,
072001
.
317
26.
N. M. Kroll,
et. al
,
Application Of Time Domain
Simulation To Coupler design For Periodic
318
Structures
,
LINAC00, 2000.
319
27.
A. Grudiev, 2008, Updates On Structure Optimisation, Procedure, Input And Results, CLIC Reference
320
Structure, Talk Presented In The Second CLIC Advisory Committee, CLIC

ACE, Switzerland.
321
322
Fig. 1
323
324
Fig. 2
325
326
9
Fig. 3
327
328
329
330
Fig. 4
331
332
Fig. 5
333
334
10
335
336
Fig. 6
337
338
Fig. 7
339
340
Fig. 8
341
11
342
343
Fig. 9
344
345
346
Fig. 10
347
348
12
349
Fig. 11
350
351
352
353
Fig. 12
354
355
13
356
Fig. 13
357
358
359
360
361
Fig. 14
362
363
14
364
Fig. 15
365
366
Fig. 16
367
368
369
370
371
372
373
374
375
15
376
Fig. 17
377
378
Fig. 18
379
380
381
382
16
383
Fig. 1
9
384
385
386
Fig.
20
387
388
389
390
391
392
393
394
17
395
396
Fig. 2
1
397
398
399
Fig. 2
2
400
401
402
403
404
405
406
407
18
408
409
Fig. 2
3
410
411
Fig. 2
4
412
413
414
415
416
Fig. 2
5
417
19
418
419
Fig. 2
6
420
421
422
423
F
ig. 2
7
424
20
425
Figure captions
426
Fig. 1: A
comparison of uncoupled and coupled mode frequencies
427
Fig. 2: A comparison of uncoupled and coupled mode kick factor weighted density function
428
Fig. 3: A comparison of
uncoupled and coupled mode frequencies. Dashed line represents tolerable limit on
429
wake.
430
Fig. 4: Amplitude of wake in a reduced bandwidth structure. Dots reprsent the location of the bunches.
431
Fig. 5: Envelope of wake in a reduced bandwidth structure. Dashed
line represents tolerable limit on wake.
432
Fig. 6: Quarter symmetry cross section view of a DDS_C cell
433
Fig. 7: Pulsed temperature rise in each of the structures of DDS_C.
434
Fig. 8: Dispersion curves of first three dipole modes in an infinitely periodic single
cell of DDS_C. Solid curves
435
represent circuit model prediction and the dots HFSS simulation results. Red dots are used to predict the curve
436
and the black dots additional points to show how good the prediction is. Dashed curves indicate the dipole
437
modes in
absence of manifold coupling. Dashed line indicates the light line.
438
Fig. 9: Spectral function of 8

fold interleaved DDS_C structure.
439
Fig. 10: Envelope of wakefield in 8

fold interleaved DDS_C structure.
440
Fig. 11: Various contours to study H

field in an un

damped cell.
441
Fig. 12: A comparison of normalised H

field in various geometries of an un

damped cell.
442
Fig. 13: Filed enhancement in various geometries due to manifold slot.
443
Fig. 14: A comparison various rf properties as function of iris thickness. The rf p
roperties of DDS_E with iris
444
thickness of DDS_C were attributed to 100% to compare the effect of iris thickness variation.
445
21
Fig. 15: A comparison of wakefield suppression in DDS_C and DDS_E.
446
Fig. 16: Maxima of fields in single cells (1/8
th
symmetry) of DDS_A.
447
Fig. 17: RF parameters of DDS_A.
448
Fig. 18: Overall rf properties of DDS_A. Lower and upper black dashed lines indicate allowable temperature
449
rise and E

field respectively. The black line in the middle represents the average beam loaded
accelerating
450
gradient.
451
Fig. 19: Spectral function of DDS_A.
452
Fig. 20: Dipole Q of DDS_A
453
Fig. 21: A Envelope of wakefield of DDS_A
454
Fig. 22:
M
atching cell de
sign geometry
455
Fig. 23:
RF properties of fully tuned structure. Left: Accelerating field, Right: Phase advance per cell
456
Fig. 24: Final S parameters
457
Fig. 2
5
:
Quality factor as a function of frequency
458
Fig. 2
6
: DDS_A discs.
459
Fig. 2
7
: DDS_A: Full structure of 24 regular cells + 2 mat
ching cells.
460
Tables
461
Table 1: Single cell parameters of the large bandwidth structure
462
Cell
a
b
t
v
g
/c
f
syn
Number
mm
mm
mm
mm
GHz
1
4.95
11.23
5.72
1.93
15.00
5
4.53
10.79
4.83
1.86
15.56
9
4.23
10.53
4.19
1.73
15.97
13
3.95
10.34
3.65
1.62
16.35
17
3.65
10.16
3.24
1.47
16.75
21
3.26
9.99
2.4
1.3
17.25
25
2.15
9.69
0.5
1.03
18.37
463
Table 2: Single cell parameters of DDS_A
464
Cell
a
b
t
v
g
/c
Q
R’/Q
f
syn
K
syn
Number
mm
mm
mm
mm

kΩ/m
GHz
V/pC/mm/m
1
4.00
11.05
4.0
2.07
5020
10.18
15.91
46.66
2
3.85
10.95
3.88
1.85
5091
10.65
16.07
50.22
5
3.61
10.78
3.55
1.62
5325
11.72
16.38
57.23
9
3.39
10.64
3.13
1.51
5604
12.90
16.67
63.86
13
3.21
10.52
2.76
1.42
5838
13.95
16.93
69.58
17
3.02
10.41
2.39
1.34
6061
15.05
17.18
74.88
21
2.8
10.29
1.94
1.22
6307
16.42
17.50
81.11
22
23
2.63
10.21
1.65
1.11
6451
17.41
17.73
85.41
24
2.50
10.16
1.47
1.00
6534
18.13
17.89
87.95
465
Table 3: Summary of
DDS_A parameters
466
Parameters
Units
CLIC_DDS_A
Accelerating mode properties
<
a
>/λ

0.13
First, last iris radius
(
a
)
mm
4.0
,
2.5
First, last iris thickness
(
t
)
mm
4.0
,
1.47
First, last (
Q
)

5020, 6534
First, last
(
v
g
/c
)
%
2.01
,
1.0
First, last shunt impedance
(
R’
)
MΩ/m
51
,
118
Filling
(
t
f
), rise (
t
r
)
time
n
s
45.4
, 23
Pulse length (t
p
c
)
ns
251
No. of bunches
(N
b
)

312
Bunch population
(
n
b
)
10
9
4.2
Peak input power
(
P
in
)
MW
70.8
Maximum loaded, unloaded
E
acc
MV/m
105, 132
Maximum
E
sur
MV/m
220
Max
imum
∆T
sur
°K
51
Maximum S
c
MW/
μ
m
2
6.75
RF

beam

efficiency
(
η
)
%
23.5
P
in
(
t
p
p
)
1/3
/C
in
[
27
]
MWns
1/3
/mm
16.93
Luminosity
per bunch crossing
[
27
]
10
34
(m

2
)
1.36
Figure of merit [
27
]
a
rb.
uni.
7.6
Lowest dipole mode properties
Dipole bandwidth
(
∆
f
)
GHz
2.0
Standard deviation of Gaussian (σ)

∆
f/3.48
Detuning spread (∆f/f
c
)
%
11.8
467
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