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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 Expiω 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 Zω 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. 270 References 271 1. F. Tecker, 2008, CLIC And CTF3, Journal Of Physics: Conference Series 110 (2008) 112005. 272 2. J. P. Delahaye, 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 Colliders, 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, SLAC-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 Fabrication 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 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 22

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