WAKEFIELD AND SURFACE ELECTROMAGNETIC FIELD OPTIMISATION OF MANIFOLD DAMPED ACCELERATING STRUCTURES FOR CLIC

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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
haye, 2008, Lepton Colliders at The Energy And Luminosity Frontiers: Linear Colliders And
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
.
al
. 2008, CLIC 2008 parameters, CLIC
-
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
olliders, Phys. Rev.
279

STAB,
9
, 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,

Switzerland.

288

11.

R. M. Jones,
et
.
al
., 2009, Influence Of Fabrica
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