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					 1   WAKEFIELD AND SURFACE ELECTROMAGNETIC FIELD
 2   OPTIMISATION OF MANIFOLD DAMPED ACCELERATING
 3                STRUCTURES FOR CLIC
 4
 5                                V. F. Khan†*, A. D’Elia†*‡, R. M. Jones†*,

 6                        A. Grudiev‡, W. Wuensch‡, G. Riddone‡, V. Soldatov‡§
              †
 7                School of Physics and Astronomy, The University of Manchester, Manchester, U.K.
                  *
 8                    The Cockcroft Institute of Accelerator Science and Technology, Daresbury, U.K.
                                             ‡
 9                                               CERN, Geneva, Switzerland.
                                                   §
10                                                  JINR, Dubna, Russia

11                                     Email: Vasim.Khan@hep.manchester.ac.uk

12                                                      Abstract
13     The main travelling wave linacs of the compact linear collider (CLIC) operate at a frequency of 12 GHz with a
14   phase advance per cell of 2π/3. In order to minimise the overall footprint of the accelerator, large accelerating
15   gradients are sought. The present baseline design for the main linacs of CLIC demands an average electric field
16   of 100 MV/m. To achieve this in practical cavities entails the dual challenges of minimising the potential for
17   electrical breakdown and ensuring the beam excited wakefield is sufficiently suppressed. We present a design to
18   meet both of these conditions, together with a description of the structure, CLIC_DDS_A, expressively designed
19   to experimentally test the ability of the structure to cope with high powers.

20                                                     Key words
21   Beam dynamics, Breakdown, CLIC, CLIC_DDS, CLIC_G, DDS, HOMs, Linear collider, Manifold damped,
22   NLC, Wakefields.

23   PACS: 29.20.Ej – Linear accelerator

24          29.27.-a – Beams, charged particles- in accelerators

25          77.22.Jp – Breakdown electrical, dielectrics

26                                                     1. Introduction
27     The aim of the CLIC project is to collide multiple bunches of electrons and positrons at a 3 TeV centre of
28   mass energy. In order to achieve high accelerating gradients within the cavities, normal conducting (NC) linacs
29   are employed [1-3]. The CLIC baseline design aims at an accelerating gradient of 100 MV/m [4-5] with an X-
30   band frequency of 12 GHz. This frequency resulted from a detailed optimisation procedure based on various
31   simulations [4-5]. The curves representing the optimisation parameters are relatively flat in the vicinity of 12 to
32   15 GHz. The 12 GHz frequency was chosen as it is close to the frequency used in the next linear collider (NLC)
33   [6] programme and hence the wealth of knowledge developed over two decades can be capitalised upon.



                                                             1
34     There are two phenomena which must be taken into account when designing these accelerating structures:
35   electrical breakdown and beam-excited wakefields. The former can be addressed both by carefully designing the
36   structure such that the surface e.m. fields are minimised and by paying attention to the surface morphology. As
37   for the wakefields, they have both short-range (along the bunch) and long-range (along the bunch train)
38   components. The short-range wakefield is a strong function of the iris aperture and is not the focus of this study.
39   Here we present a design for suppressing the long-range wakefield, whilst minimising the surface e.m. fields on
40   the walls of the accelerating structures. The method for damping the wakefields entails detuning each of the cell
41   frequencies, by tapering down the irises along the structure, and providing moderate (Q~1000) coupling to four
42   attached manifolds [6]. This method is similar to that adopted in the NLC, however, with stronger constraints
43   imposed due to the larger gradients required. This resulted in markedly different outer cavity wall design. The
44   design presented here is an alternative to the baseline design for the main linacs of CLIC, which relies on heavy
45   damping (Q~10 [5]) through strongly coupled waveguides attached to each cell. However, we maintain the same
46   number of cells in the CLIC_DDS_A design. Other wakefield suppression strategies are possible [7].

47     In the CLIC damped and detuned structures (DDS), the focus of this work, both the fundamental (accelerating)
48   mode and the higher order dipole modes (HOMs) are calculated. In both cases, the electromagnetic (e.m.) fields
49   in single cells are calculated using codes which rely on representing the geometry with a finite element based
50   mesh. For a sufficiently fine mesh, an accurate representation of the e.m. fields is obtained. The beam loaded
51   accelerating field is calculated from an integral [8] representation of the energy flux within the overall cavity,
52   based on the field in individual cells. The transverse dipole field is calculated from a circuit model [6, 9-11],
53   designed to represent dipole mode slot-coupled to waveguide like manifolds. This circuit model and its spectral
54   function [6, 11] generalisation, provides a design tool to allow the influence of geometrical modifications to be
55   rapidly accounted for in the wakefield calculations. This unique tool has been validated on several previous
56   NLC structures and has proved to be an accurate prediction of the wakefield [6, 11].

57     The CLIC_DDS design is based upon the knowledge gained 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 are waveguide damped (known as CLIC_G [4-5]). We report on
60   various stages involved in evolving CLIC_DDS so as to satisfy the stringent constraints imposed by the rf
61   breakdown and beam dynamics criteria. In order to rapidly realise the bandwidth necessary for the wakefield
62   suppression, detuned structures (DS) have been studied first. In all cases, we prescribe a Gaussian distribution
63   for suppression of the wakefields. We learned that a bandwidth of ~3.3 GHz is necessary to suppress the
64   wakefield to satisfy the beam dynamics criterion for an inter-bunch spacing of 6 rf cycles.

65     The paper is structured 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 sections provide details on a structure which will be high power tested
68   and some concluding remarks.

69

70

71           2. Means To Independently Satisfy RF Breakdown And Beam
72                              Dynamics Constraints
73     In all cases, we utilise a finite element code HFSS [12] to model accelerating structures, calculate the fields
74   and eigenmodes within the sructure. The single 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 [13]. For a cell subjected to infinite periodic condition the dispersion relation
77   between frequency ω/2π and phase advance per cell ψ is:




                                                             2
                                                                   ωr
 78                                                    ω                                                            (1)
                                                                1  η cosψ

 79   The resonating frequency (ωr) and coupling coefficient (η) of the neighbouring cells are obtained from the 0 and
 80   π mode (simulation results).

                                                                   2
                                                                2ω0 ω2π
 81                                                ωr                                                                (2)
                                                                ω0  ω 2
                                                                 2
                                                                       π


                                                                ω 2  ω0
                                                                  π
                                                                       2
 82                                                        η                                                        (3)
                                                                ω0  ω 2
                                                                  2
                                                                       π


 83   The calculation of group velocity (derivative of eq. 1 [14]) is followed by the calculation 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   [15], 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 tapered iris radii and ticknesses in this structure result 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 iris 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   3.6σ. 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 mode), we calculate the coupled mode
 96   frequencies and kicks so as to account for the cell-to-cell interactions using a double band circuit model [13]. 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 interactions of the fields coupled through irises. 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]

                                                       N                    
                                                                                  i 
103                                       WT (t)  2   K     p Expiω p t
                                                                   
                                                                             1 
                                                                             
                                                                             
                                                                                         (t )
                                                                                  2Q p 
                                                                                       
                                                       p 1                                                         (4)

104   where ωp is the synchronous frequency, Qp 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
108   kick factors of the coupled mode. The envelope of the wakefield for an entire train of 312 bunches is illustrated
109   in Fig. 3. In this case, various damping Qs are artificially imposed. The wakefield in a DS with losses due to
110   finite conductivity (Qcu ~6000) is also shown. Wakefield suppression well beyond the beam dynamics
111   requirements is obtained. For this geometry, the surface e.m. field on the copper walls, on the other hand is too
112   large. The 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, the structure


                                                                    3
117   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   satisfy 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 to10%. The other possible option is to rely
121   upon zero crossing scheme. When wakefield is calculated, an excursion of the envelope is calculated. However,
122   the wake experienced by the bunch may be small. In this case, the iris dimensions of the structure(s) are 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 Qs.

126     The CLIC project requires more than 140,000 [7] accelerating structures. In practice, it will be difficult to
127   maintain the zero crossing scheme for all structures. Meeting the mechanical tolerances to build these structures
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 with a moderate dipole bandwidth. In
130   this manner a trade off between the bandwidth and efficiency is investigated 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 incorporated with manifold geometry was studied. The profile of a typical DDS cell is
135   presented in Fig. 6, where RM is the radius of the manifold and Rc 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 incorporates manifolds, slot-coupled to the accelerating
139   cells. These coupling slots perturb the cell wall, and cause more current to flow in the vicinity of the slot,
140   leading to excessive surface magnetic fields (H-field). The average peak power requirement of an 8-fold
141   interleaved DDS_C is ~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 109. However, for this structure bunches can be populated up to 5.0
143   x 109. In 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 tolerable
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 lowest dipole bandwidth in this structure is: ∆f = 3.6σ
151   = 2.33 GHz and the 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
           1  η cos  1  η cos  η2
          
           2
          
                        
                      2  ˆ 2
                               ˆ
                                   2 
                                                  
                                         sin  cos  cosψ  Γ 2 πP 1  η cos  f 0  f 2 sinψ0n
               f 0  f  f 0  f  f 0f 0ˆ       
                                                                       c
                                                                              ˆ        ˆ2         ψ0n
                                                                                                                      (5)
                                                

154   where f0 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 cell, 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 modes for a
157   thin iris [20], can be approximated as    . The dispersion curves of the first three dipole modes in a typical
                                                    ˆ

158   DDS_C cell are illustrated in Fig. 8. We utilise the spectral function method to calculate the impedance of the
159   structure [6, 11]

160                                              S(ω)= 4 Im {Z(ω+jε)}                                                    (6)

                                                                 4
161   where ε is an infinitesimal displacement and Z(ω) is the impedance of the structure and is define as [6, 11]

                                                  N
                                 Zω                                          P]n  m H nm
                                           1                                  jω           ~
162                                                     K s K s ωs ωs exp[
                                                          n m n m
                                                                                                                      (7)
                                          2π 2   n, m
                                                                               c

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 H nm contains various circuit parameters involved and is defined in [6, 11].

165   The wakefield is calculated by taking inverse Fourier transform of the spectral function. The spectral function of
166   an 8-fold interleaved DDS_C is illustrated in Fig. 9 and the corresponding wakefield in Fig. 10. As the dipole
167   bandwidth in this case is moderate, the wakefield decay should be rapid enough to meet the beam dynamics
168   criterion. In order to meet the beam dynamics constraint, the 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 spacing in a structure with
170   moderate bandwidth. In 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 by dashed
172   line in Fig. 10. Though the wakefield suppression in this structure is adequate, DDS_C needs further
173   optimisation 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 incorporate for manifold coupling, the field in this region gets enhanced. For a circular un-
176   damped 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 slot of width 3 mm, the enhancement in the H-field
178   peaks up to 6 mA/V i.e. nearly 60% enhancement. The pulsed temperature rise is proportional to the square of
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 undamped cell, a range of cells with modified walls
183   have been studied and are illustrated in Fig 11. The modified cavity shape is defined in terms of an ellipse ε with
184   A and B as semi-major and semi-minor axis respectively [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. The field enhancement for
188   selected shapes is illustrated in Fig. 13. As can be seen, for an elliptical cell of ε = 1.38, the field enhancement
189   is a minimum. There 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 minimise the surface electric field (E-field). 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 (vg), 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 thicknesses were studied by
197   comparing their 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 thicknesses retained from
200   DDS_C. A comparison of the rf properties of the reference structure with various other structures is shown in
201   Fig. 14. In this optimisation we realised that beyond the average iris thickness of 2.65 mm, surface E-field
202   remains almost invariant. 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 thickness 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 of 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


                                                                    5
207   temperature rise is 52° K, which is reduced by ~28% compared to DDS_C. As the fields on the cell walls were
208   reduced 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 and DDS_E is presented in Fig. 15. A test structure, which is the first
211   out of eight-fold interleaved structures of DDS_E is being fabricated. The properties of the test structure are
212   discussed in the next section.

213                        4. Structure 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 because it has the largest aperture compared to
217   other interleaved structures (and is also the reason why it needs 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 dimensions 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 [23], a new local quantity (Sc)
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 maxima in the E, H and Sc fields
226   in DDS_A cells are presented in Fig. 16 [22]. The fundamental mode rf parameters of the single 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 Fig. 19. The Q of the dipole modes is calculated by fitting a
230   Lorentzian [17] to the peaks in spectral function. The average dipole Q in this structure is ~1650 and is
231   illustrated in Fig. 20. The wakefield 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 interactions. 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 iris a, matching cavity radius b and matching gap length L are varied to minimise the reflection
242   (S11) 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 [12]), we fine tune the matching parameters for the whole structure using the
246   Kroll method [26]. This time we minimise the standing wave ratio (SWR) in the structure. In this way, the
247   complete structure is tuned including the matching cells. The accelerating field in the fully tuned structure is
248   illustrated in Fig. 23. The extreme peaks in this plot correspond to the matching cell accelerating fields. These
249   peaks are dissimilar to the regular cell peaks due to the fact that the matching cell lengths are different compared
250   to the regular cell lengths. The erf tapering of the regular cells is evident in the fully tuned structure accelerating
251   field. The accelerating field phase advance per cell is also illustrated in Fig. 23. Here, a nearly triangular shape
252   profile reflects the 120° phase advance. The maximum deviation in the phase advance per cell is no more than
253   6°. The discrepancy from a perfect triangular shape can be understood in 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 lengths of the matching cells compared to regular cell length. The S parameters

                                                                6
256   of the fully tuned structure are presented in Fig. 24. In this case, the simulation results of S11 = -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. 25.

259     The fabrication of the DDS_A cells is in progress and the test cells (discs) are shown in Fig. 26. A complete
260   CAD drawing of DDS_A, consisting of 24 regular cells and 2 matching cells is illustrated in Fig. 27. The overall
261   parameters of the DDS_A are summarised 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 interleaving 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.


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291       13.   K.L.F. Bane and R. Gluckstern, 1992, The Transverse Wakefield Of A Detuned X-Band Accelerator
292             Structure, SLAC-PUB-5783, USA.
293       14.   R. M. Jones, 2007, 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 Linacs, 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, Germany.

                                                              7
302   18. V. F. Khan and R. M. Jones, 2008, Beam 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 Accelerator 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, 2010, 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

                                                      8
326   Fig. 3
327




328

329

330   Fig. 4




331

332   Fig. 5




333


        9
334

335

336   Fig. 6




337

338   Fig. 7




339

340   Fig. 8




        10
341

342

343   Fig. 9




344

345

346   Fig. 10




347


        11
348

349   Fig. 11




350

351


352

353   Fig. 12




354

        12
355

356   Fig. 13




357

358


359


360

361   Fig. 14




362


        13
363

364   Fig. 15




365

366   Fig. 16




367

368

369

370

371

372

373

374


        14
375

376   Fig. 17




377

378   Fig. 18




379

380

381


        15
382

383   Fig. 19




384

385

386   Fig. 20




387

388

389

390

391

392

393


        16
394

395

396   Fig. 21




397

398

399   Fig. 22




400

401

402

403

404

405

406


        17
407

408

409   Fig. 23




410

411   Fig. 24




412

413

414

415

416

417


        18
418   Fig. 25




419

420

421   Fig. 26
422




423

424

425

426

427



        19
428

429                                                      Fig. 27




430

431                                                Figure captions
432   Fig. 1: A comparison of uncoupled and coupled mode frequencies

433   Fig. 2: A comparison of uncoupled and coupled mode kick factor weighted density function

434   Fig. 3: Envelope of wakefield with several artificially imposed damping Qs

435   Fig. 4: Amplitude of wake in a reduced bandwidth structure. Dots represent the location of the bunches.

436   Fig. 5: Envelope of wake in a reduced bandwidth structure. Dashed line represents tolerable limit on wake.

437   Fig. 6: Quarter symmetry cross section view of a DDS_C cell

438   Fig. 7: Pulsed temperature rise in each of the structures of DDS_C.

439   Fig. 8: Dispersion curves of first three dipole modes in an infinitely periodic single cell of DDS_C. Solid curves
440   represent circuit model prediction and the dots HFSS simulation results. Red dots are used to predict the curve
441   and the black dots additional points to show how good the prediction is. Dashed curves indicate the dipole
442   modes in absence of manifold coupling. Dashed line indicates the light line.

443   Fig. 9: Spectral function of an 8-fold interleaved DDS_C structure.

444   Fig. 10: Envelope of wakefield in an 8-fold interleaved DDS_C structure.

445   Fig. 11: Various contours to study H-field in an undamped cell.

446   Fig. 12: A comparison of normalised H-field in various geometries of an undamped cell.



                                                             20
447   Fig. 13: Filed enhancement in various geometries due to manifold slot.

448   Fig. 14: A comparison various rf properties as function of iris thickness. The rf properties of DDS_E with iris
449   thickness of DDS_C were attributed to 100% to compare the effect of iris thickness variation.

450   Fig. 15: A comparison of wakefield suppression in DDS_C and DDS_E.

451   Fig. 16: Maxima of fields in single cells (1/8th symmetry) of DDS_A.

452   Fig. 17: RF parameters of DDS_A.

453   Fig. 18: Overall rf properties of DDS_A. Lower and upper black dashed lines indicate allowable temperature
454   rise and E-field respectively. The black line in the middle represents the average beam loaded accelerating
455   gradient.

456   Fig. 19: Spectral function of DDS_A.

457   Fig. 20: Dipole Q of DDS_A

458   Fig. 21: A Envelope of wakefield of DDS_A

459   Fig. 22: Matching cell design geometry

460   Fig. 23: RF properties of fully tuned structure. Left: Accelerating field, Right: Phase advance per cell

461   Fig. 24: Final S parameters

462   Fig. 25: Quality factor as a function of frequency

463   Fig. 26: DDS_A discs.

464   Fig. 27: DDS_A: Full structure of 24 regular cells + 2 matching cells.


465                                                                Tables
466                     Table 1: Single cell parameters of the large bandwidth structure

          Cell                   a                   b                   t                vg/c                    fsyn
         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
467
468                                            Table 2: Single cell parameters of DDS_A

        Cell            a             b          t          vg/c        Q        R’/Q          fsyn           Ksyn
       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

                                                              21
       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
      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
469
470                                        Table 3: Summary of DDS_A parameters
                      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 (vg/c)                        %                      2.01, 1.0
            First, last shunt impedance (R’)                MΩ/m                       51, 118
                Filling (tf), rise (tr) time                   ns                     45.4, 23
                     Pulse length (tc) p                       ns                        251
                   No. of bunches (Nb)                          -                        312
                 Bunch population (nb)                        109                         4.2
                 Peak input power (Pin)                      MW                          70.8
            Maximum loaded, unloaded Eacc                   MV/m                      105, 132
                      Maximum Esur                          MV/m                         220
                     Maximum ∆Tsur                            °K                          51
                       Maximum Sc                          MW/μm2                        6.75
                RF-beam-efficiency (η)                         %                         23.5
                   Pin (tp )1/3/Cin [27]
                         p                             MWns1/3/mm                       16.93
           Luminosity per bunch crossing [27]          1034 (m-2)                        1.36
                  Figure of merit [27]                 arb. uni.                          7.6
                                        Lowest dipole mode properties
                Dipole bandwidth (∆f)                    GHz                            2.0
           Standard deviation of Gaussian (σ)              --                         ∆f/3.48
                Detuning spread (∆f/fc)                   %                            11.8
471




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