A DAMPED DETUNED STRUCTURE FOR THE MAIN LINACS OF THE
COMPACT LINEAR COLLIDER
V. F. Khan , R.M. Jones and I. Shinton; The Cockcroft Institute, Daresbury, WA4 4AD, UK;
The University of Manchester, Manchester M13 9PL, UK.
Abstract
Here we present initial results on an alternate design
for the CLIC main accelerating linacs which is a
moderately damped and detuned structure. The detuning
provides a spread in the higher order mode frequencies
which results in a rapid decay of the wakefields;
recoherence of the wakefields due to a finite bandwidth
of the modes is suppressed by means of the damping
waveguide like structures referred to as manifolds. The
manifolds are non-propagating at the acceleration mode
frequency and provides a moderate damping of the higher
order modes (Q~ 500).
INTRODUCTION
At present the combination of X-band frequency and
normal conducting cavities is the only viable option to
achieve higher accelerating gradients, e.g. NLC [1] for
colliding electrons and positrons at a centre of mass
energy of the order of few TeV. The disadvantage of
choosing the X-band operating frequency is the
wakefields left behind the accelerated bunches, it has a
potential to kick the bunches off axis which will lead to
beam dilution or in the worst case beam breakup; hence
these wakefields need to be suppressed adequately.
The NLC was a proposed normal conducting linear
collider, designed to operate at 11.424 GHz with an
accelerating gradient of 65 MV/m [1] to achieve a centre
of mass energy of 1 TeV. The wakefields in the
accelerating structures were forced to decay rapidly by
spreading the dipole frequencies in a Gaussian fashion
and its recoherence was suppressed adequately by means
of manifolds with Q~500 [2]. The compact linear collider
(CLIC) is aiming to achieve a 3 TeV centre of mass
energy and is designed to operate at 11.9942 GHz. The
goal is to achieve an average accelerating gradient of
100MV/m to reduce the overall collider length to ~ 48 km
[3]. In order to achieve a higher gradient with nearly same
operating frequency, the iris dimensions for CLIC are
smaller compared to that of NLC [3],[4]. As the wakefield
(intrabunch) is inversely proportional to the fourth power
of iris radius [5], the wakefileds in CLIC are much more
severe than in NLC. This leads to accelerating less
number of particles per bunch, so as to maintain overall
efficiency of the collider the current is subsequently
increased by reducing the bunch spacing. Keeping a
balance between all these constraints the CLIC main linac
structure is designed with a strong damping Q<10 [6].
The strong damping scheme may have its own
disadvantage such as perturbing the fundamental mode.
We are looking into a possible damped and detuned
scheme for CLIC structure. The RF and beam dynamics
constraints force to keep the dipole frequency spread to
~1GHz. The detuning scheme for CLIC is explained in
detail in [7] and [8]. Here, we present the initial results on
the structure presented in [8] with the implementation of
manifold geometry. We make use of a circuit model [9] to
calculate various coupled parameters and use the spectral
function method to calculate the coupled frequencies and
wake-function [10]. In the present structure, the lowest
dipole Q is in the neighbourhood of 500, which is clearly
not sufficient to damp the wake-field envelope for first
few bunches. But in reality, the wakefield experienced by
a bunch is the wake amplitude and not the envelope
which allows the possibility of using a zero crossing
scheme instead.
CELL GEOMETRY
The cell parameters and the fundamental mode
properties are very similar to the initial design presented
in [8] with the exception of a rise in pulse temperature
heating due to the inclusion of manifolds. Manifolds are
nothing but circular TE10 waveguides running parallel to
the accelerating structures and are coupled to main
accelerating structures through coupling slots. The
dimensions of the manifolds and the coupling slots are
chosen carefully so as not to perturb the fundamental
mode. Thus, every accelerating cell is coupled to four
manifolds and every structure (consisting of 24 cells) will
be connected to a higher order mode coupler. The higher
order mode energy propagating through manifolds will be
coupled out to loads and this allows dielectric materials to
be located remotely. This is the biggest advantage of
using the manifold-scheme over the wave guide damping
scheme as the threat of dielectric breakdown is avoided.
The disadvantage is, it is difficult to get strong damping
(Q~10) with this method due to mechanical and
fabrication complexities. One-fourth of a CLIC_DDS cell
is illustrated in Fig.1, Fig.1a) displays the manifold mode
and 1b) displays lowest dipole mode.
Manifold
Coupling slot
a)
b)
Figure 1: One-fourth of a single accelerating cell. a) A
manifold mode, b) dipole mode.
line.
CIRCUIT MODEL
The simulation of an entire structure using a finite
difference or a finite element computer code is a time and
memory consuming operation. Instead, we perform
simulations for selected cells to calculate: dispersion
curves, synchronous phases, synchronous frequencies and
kick factors. Intermediate cell parameters are obtained
using interpolation and spline fits. All these parameters
are then used as an input in a manifold implemented
circuit model [9]. A dipole by its nature is neither TE nor
TM but both (hybrid). This model is a double chain LC
circuit model, it represents TE and TM components of the
dipole fields, each component magnetically interacts with
both the components of the neighbouring cells. The
manifold is capacitively coupled to the accelerating cells
and it propagates the TE10 mode. As we go down the
structure towards upper energy end the iris dimensions
reduces, hence the capacitive coupling of the manifold to
the cell modes also reduces. In order to achieve a nearly
constant dipole Q, we need to increase the coupling
between the manifold and the cell modes, this is achieved
by inserting the coupling slots deeper in the cavity i.e
manifold insertion increases down the structure.
We use the finite element code HFSSv11 to determine
the eigen frequencies of the three lowest modes (two
dipole and one manifold mode) for a periodic single cell
using specified phase advance boundary conditions. The
circuit model prediction of the frequency-phase
dispersion relation is shown in Eq.1) [9]:
1  η cos ψ/ f
1  η̂ cos ψ
2
0
/ f̂ 02
cos ψ - cos φ 




/ f f̂ sin ψ

  2 / α / F 2  f 2  f -2
f
-2

 η 2x

2
0 0
1)

 2 F 2 / F 2  f 2 πL/c 2 1  η̂ cos ψ f̂ 0-2  f̂ -2 sincφ
where: f 0 , f̂ 0 , η, η̂ are resonating frequencies and
coupling coefficients of the corresponding TE and TM
mode respectively; η x is the cross coupling between TE
and TM modes; φ corresponds to the local phase advance
per section of the manifold; ψ to the phase advance per
cell; Γ to the manifold to TE coupling; α to the shunt
capacitance; F to the resonant frequency of the series
capacitance-inductance shunt and L to the section length.
The above mentioned parameters are determined by
fitting the circuit model prediction to the simulation
results, the three predicted curves should pass through the
selected frequency phase pairs. The resulting brilloun
diagram for the first cell is illustrated in Fig.2). The
region near a phase advance of 100 deg. is the avoided
crossing region, which represents the coupling to the
manifold. For no coupling to the manifold, the dispersion
curve of the lowest dipole and manifold mode will cross,
as it is shown by the dashed curves. The red dots are used
to predict circuit model curves and the black dots are
additional points to verify how good the prediction is. The
dashed curves represents dispersion curves before
coupling to a manifold and the dashed line is the light
Avoided crossing
Figure 2: Brillouin diagram for the first cell of
CLIC_DDS. The points represent simulation and curves
represent the circuit model result.
THE SPECTRAL FUNCTION METHOD
For no damping or weak damping the wake function is
calculated using a modal sum method [11]. But, to
maintain a moderate damping the manifold has to be
strongly coupled to the cells resulting in a strong
perturbation of the modes. In this case, it is difficult to
calculate the wake function using the modal summation
method. For a strong coupling of the manifold, the
transverse wake function for a particle trailing by a
distance “s” behind a velocity “c” drive bunch can be
calculated as follows [11]:
W s  
 Sf  sin2 π f /c df θs
2)
f 0
In Eq. 2) θs  is the unit step function, the wake in this
case consist of a continuum of modes and this is the
spectral function method. The spectral function S(f) is
defined as [12]
4
3a)
Sf   ImZf  jε 
π
where Z(f) is defined by
N
~
Z f    K n K m f n f m exp 2π j L/c f n  m  H
s s s s
nm
3b)
n,m
Here f’s and K’s are the synchronous frequencies and
kick factors respectively, ε is an infinitesimal quantity, L
~
is the cell period and H is defined by Eq. 6) of [10].
nm
The envelope of the wake function is given in terms of the
absolute value of the Fourier transform of S(f):[10],[11]

Wc s   θs  Sf exp 2 π f s /c df

4)
0
The spectral function of a 24 cell CLIC_DDS structure is
illustrated in Fig. 3. The peaks in the spectral function are
the resonating modes, the Q of the modes are calculated
from these peaks as is shown inset of Fig.3. Due to a
small bandwidth the mode separation is likewise small
and hence the peaks are overlapping, resulting in a less
number of peaks than should be seen.
maintain the zero crossing for first few bunches where
envelope wakefield is well above the acceptable limit. As
there are only 24 modes to sample the Gaussian
distribution the sampling is not as good, interleaving the
neighbouring structure frequencies will help in bettering
the sampling. We study a four-fold interleaved structure
to enhance wake suppression. Fig. 6 illustrates the
resulting wake function of a four-fold interleaved
structure.
Bunch location
Figure 3: Spectral function of a 24 cell structure, the
resulting Q shown inset.
The wake function for this structure is calculated using
Eq. 3) and 4) and is displayed in Fig.4).
Bunch location
Figure 6: Envelope wake-field of a four-fold interleaved
structure.
DISCUSSION
Figure 4: Envelope wake-field of a 24 cell structure
As can be seen from Fig. 4) that wakefield is not
adequately suppressed for the first few bunches, instead
we look into the actual wake experienced by these
bunches through the zero crossing scheme, illustrated in
Fig.5).
Envelope wake
Actual wake at bunch
The present geometry with the inclusion of manifolds has
damped the higher order mode Q up to 500, there is
further room to improve upon this. Interleaving the
neighbouring structure frequencies will further enhance
the wakefield suppression. Optimisation of the manifold
geometry for bettering wakefield suppression is an aspect
of our future research.
ACKNOWLEDGMENTS
The authors are thankful to The Cockcroft Institute for
supporting this work. These results were first presented at
the weekly Manchester Electrodynamics and Wakefields
group meeting (MEW) at the Cockcroft institute and we
thank all the members of the group for many useful
discussions. Useful discussions have been conducted with
W. Wuensch, A. Grudiev, J. Wang and T. Higo regarding
X-band structures.
REFERENCES
Bunch separation
Figure 5: Amplitude wakefield of a 24 cell structure.
For the structure being discussed here, it requires a
systematic shift of ~80 MHz in the mode frequencies to
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