Long Range Wakefield Suppression at X-band Frequencies

Transcription

1 NLC - The Next Linear Collider Project Y (norm.) Y (norm.) Y (norm.) Y (norm.) Long Range Wakefield Suppression at X-band Frequencies Snowmass July 1-20, 2001 R.M. Jones, July 16, 2001 R.M. Jones, Feb. 2, 2000

2 Outline Why damp long range wakefields? BBU and Emittance dilution issues Analysis of DDS (Damped and Detuned Structures). Modal and spectral function analysis. ASSET results: theory vs. experimental results (DDS1 through RDDS1) Stack measurements of RDDS1 (Rounded Damped and Detuned Structures) Built-in diagnostic properties of (R)DDS. Influence of fabrication errors on BBU and emittance dilution. New high phase advance structures (TW 5π/6 and SW π). Traveling wave and standing wave designs -interleaving of frequencies, limited local damping and manifold damping.

3 Features of Wakefields Short-range wakefields a -3.8 : sets a lower limit on aperture: a 0.18λ Long-range wakefields: disrupt the trailing bunches (190 in the present design), dilute the beam emittance and can give rise to an instability known as BBU (Beam Break Up) The driving bunch excites an EM field in the cavity which persists long after the original bunch has left the cavity. The transverse force exerted on the trailing particles has a complicated dependence r on position within the structure. However, the integral along the axis of the transverse force (F ) is much simpler and this defines W(s), the wakefield: Fdz=qqrLW(s) 1 21 φ s q 2 Long range wakes are suppressed by careful detuning and damping. Both local and remote (manifold) damping schemes are investigated. q 1 r 1 v =ce z z z

4 Damping Dipole Modes B Field E Field In phase quadrature

5 Manifold Features of (R)DDS The manifold is a single mode TE 10 and it is cut off to the accelerating mode (thus there is little impact on the accelerating mode) Each manifold is tapered to maintain good coupling RDDS has circular manifolds (superior pumping compared to rectangular guide). From mechanical considerations it is required to decouple 4 cells from either end of the structure. Frequency (GHz) ψ=π 14.5 f s 14 f x ψ= Cell Number Detuned structure modes are localised standing waves with a spectrum of phase velocities. Both beam coupling and manifold coupling as functions of frequency are localised around particular cells.

6 Multi-Functional Properties of Manifolds The long range wake is reduced by coupling out the field into four collinear manifolds. The radiation coupled out provides information regarding the alignment of cells Additionally the manifolds provide pumping

7 Circuit Model of RDDS1 Brillioun Diagram for RDDS1 Cell Upper dipole band Frequency GHz Avoided crossing Phase Deg. Three cells in the chain are illustrated. TM modes couple to the beam. Both TM and TE modes and excited and the coupling to the manifold is via TE modes. The manifold is modeled as a transmission line periodically loaded with L-C elements.

8 Circuit Model Equations Coupling Between Manifold-Cell n= n n+ n κ n n n V j I /C i / C c / ω n= n n+ n κ n n n v j i /c I / C c / ω Matrix Elements R = 2cos φ, R = 1 nn n nn± φ n= φ 0n α n π n n φ 0n cos cos L/c F / F f sinc φ 0n = 2 π L/c f F 2 2 cn nn = n +Γn αn n H /f / / F f H = η / 2f f nn± 1 n ± 1/ 2 n 1 n± H / f fˆ nn± =± ηx, n ± / n n± 1 H ˆ ˆ ˆ ˆ ˆˆ 2 nn = 1/f n, Hnn± 1 = ηn ± / / f nfn± nn =Γn π n n 2 φ 0n G L/c F / F f sinc Network Equations in Matrix Form: RA=Ga ˆ 2-1 H-1/f a+hxa=ga =GR Ga H-1/f ˆ 2 a+h ˆ t 2 x=b/f In the 9-parameter model each parameter is determined from MAFIA or Omega3 simulations to produce Brillioun diagrams for a limited number of fiducial cells. The remaining cells are obtained by interpolation and non-linear error function (Erf) fits.

9 Determination of Parameters There are nine parameters to be determined (5 associated with the cells and 4 with the manifold). These are determined by specializing to uniform structures as we did in the case of a DS. The dispersion curves for the three lowest modes are matched to those determined from simulations using MAFIA and Omega3: $"""""" TE Cell "%""""""""& Mode $""""%""""& TM Cell Mode TE-TM Coupling $""%""& Manifold 1 cos 2 $"""%"""& Mode 1 1 ˆ cos 1 2 sin2 +η ψ cos cos + Γ η ψ f ˆ2 2 2ˆ2 0 F f f η ψ ψ φ f f 0 f0f α F L 1 ˆ cos 1 π η ψ sin φ F f c ˆ f 0 f 0 """"""""" " """""""""""! Coupling to Manifold Setting Γ = 0 it is evident that the above breaks up into 3 equations: cosψ =cosφ, the manifold equation, and a two band dispersion relation =

10 We require 9 points on the curves. Three dispersion are used and the 6 ψ = 0, π points. This guarantees that the mode curves given by the circuit match the end points. The remaining 3 points are taken near the avoided crossing. They guarantee that the curves cross at the correct phase and that the shape of the avoided crossing is well represented (the coupling strength is determined in this region). Thus we form: D(p) 1... =0... D (p) 9 where D n (p) represents the dispersion relation obtained from the nth ψ point and p represents 9 parameters. These 9 coupled non-linear equations are solved for the 9 parameters and the 3 dispersion curves are obtained. This procedure was followed for 11 cells of DDS1 and intermediate cells are obtained by interpolation and error function fitting procedures.

11 Determination of Wake Function Firstly, consider the eigensystem without the driving beam: the source-less eigensystem: p Ha =λ a p H = Complete matrix of system where the coupling to the manifold is included (symmetric non-hermitian) 2 p p q p λ =1/f a = i / c a.a = δ pq p p To obtain the solution in the presence of the driving current, the beam, we make an expansion over the source-less eigenfunctions. This enables the envelope of the wake function to be obtained as: where the kick factors are given by: K p p p p Ŵ(s) = K exp k sj 1/( 2Q ) p p p a n n m m n Ksfsexpjn φ p a s s p m K f exp jm φ = n m p p p Nf p( 1 df p/df)a a and φ p = ω pl/c (L= length of structure), f n s is the nth synchronous frequency and Kn s is the nth cell uncoupled kick factor. p

12 Also, the double band kick factor is obtained in a similar manner as: K p p a n n n m m m n ε Ksfsexp jn φ p a s s p m ε K f exp jm φ = n m p p p Nf p( 1 df p/df )a a However, the determination of the eigenfunctions becomes a complex process in which 612 eigenvectors are required to be carefully sorted. A straightforward perturbation solution does not yield sufficiently accurate results as in shown in the next transparency. Thus, in order to be able to routinely design (R)DDS it is necessary to utilize a new method, the Spectral Function

13 Perturbation Analysis 1400 Quality Factor The exact modal method reveals the Q and frequency shift to oscillate about the perturbed values (characteristic of an overcoupled system) Mode Q via Perturbation formula Exact Q Coupling to the manifold changes the modal Qs and frequencies of the linac. Frequency (GHz) 0.01 Perturbation Frequency Frequency Shift (GHz) Exact Frequency Shift Mode Freq (GHz)

14 Spectral Function Method The sum over damped modes, which provides the main contribution to the transverse wake of the (R)DDS, is replaced by a Fourier-like integral of a spectral function over the propagation band of the manifolds. Recall the circuit equations for the TE and TM cell amplitudes in matrix form: Hˆ Ht x ˆ 1 ˆ a a 1 B - = -1 a f2 a f2 H H-GR G 0 x Here, H x is a tridiagonal matrix with vanishing diagonal elements which describes the TE-TM cross coupling, R which describes propagation in the manifold is also tridiagonal and G, which describes the TE coupling of the cells to the manifold is diagonal. The elements of the column vectors are themselves N element vectors. The above readily lends itself to a further condensed form: Ha f 2a = f 2B The drive beam represented by the N component vector B couples to the TM wave only: n s s B = 4 π f n/cknlexp j2 π L/cn

15 where L is the periodicity of the cell. The transverse wake function (transverse potential per unit length) for a particle traveling behind a velocity c particle (per unit drive charge per unit witness charge) is written: ( ) 2 Ws= Z(f j ε )exp π js/c f j ε df where ε is an infinitesimal quantity (required to ensure the integral is performed away from the real axis) and the wake impedance is given by: with the 2N x 2N matrix given by: ( ) 1 2 N n m s s π nm n,m Z(f ) = π K K exp jl/c f n m H H = H f H Because W(s) is real we require Z(f) =Z * (-f*) for f in the lower half plane. Due to the presence of the manifold we find that Z is discontinuous across the real axis and thus cuts are introduced to ensure that Z remains single valued on the physical sheet. It also an even function of f in the complex plane Z(f) = Z(-f).

16 Wake Function Regimes 1. Zero Damping (Pure Detuning) W(s) = θ(s) 2Kpsin kps Here s is the distance behind the bunch, θ(s) is the unit step function and 2πf k p p = c 2. Weak Damping k W(s) = θ(s) 2Kpsinkpsexp 2 p s Q 3. Strong and Moderate Damping cut p W(s) = θ(s) S p(f)sinkpsdf The spectral function technique covers a broad class of regimes: 1 and 2 are special cases of 3.

17 Application of Spectral Function Method Spectral Function for DDS1 with Matched Terminations in HOM Coupled Loads and Realistic Reflection Coefficient (Right) Wake Field Computed for DDS1 Compared with Experimental (ASSET) Results. Also shown is the pure damping due to pure copper losses and a DT (Detuned Structure)

18 Conspectus of DDS Shown uppermost is twice the kick factor weighted density function for DT (left) and the corresponding wakefield (right). The succeeding curves show DDS1 (exp. Points from ASSET data) and DDS2. The effect of the reflection from the HOM coupler is included in the modeling. The long range wake is improved for DDS1 by an order of magnitude over DT and DDS2 by a factor of 3 over DDS1.

19 Conspectus of DDS (Cont.) DDS3 uses similar HOM and loads as DDS2 but the cells frequencies have been redistributed using a mapping function method such that Kdn/df is now Gaussian (rather than dn/df) The signifcant improvement in the matching of the HOM loads for DDS2 reduces the amplitude of the oscillations of the spectral function.

20 Fundamental Mode Coupler Loading The standard output coupler for the DDS structures has been fitted with a pair of WR90 rectangular output waveguides polarized in the electric field direction emerging on opposite sides of the coupler cell in the horizontal direction. In order to provide an output for both dipole mode polarizations these output waveguides have been replaced by four WR62 waveguides oriented so as to form an "X" with respect to the horizontal and vertical axes. The coupler was matched for the accelerator mode by varying the cell radius and wavguide iris width using the mesh shown above.

21 Fundamental Mode Coupler Loading (Cont.) In fabricating RDDS1 mechanical design considerations require 4 cells to be decoupled from the manifold at both the input and output ends of the structure. This gives a number of high Q modes at the high frequency end of the spectral function (where modes with large kick factors are located) In the ASSET measurement of the wakefield in DDS3 it was noticed that one plane of the wakefield was significantly better than the other and this was discoved to be due to the fundamental coupler loading down the wakefield. In RDDS1 we explicitly designed the coupler so that it loads down the dipole modes at the high frequency end with a Q ~36 (designed with the MAFIA code)

22 Fundamental Mode Coupler Loading (Cont.) Decoupling 4 cells at the output end significantly impacts the wakefield. High Q modes are seen to dominate. Loading down the final cell with the fundamental mode coupler (Q~36) improves the wakefield. Little emittance dilution results due to this wake.

23 RDDS1 ASSET Measurement ASSET Measurement Circuit Model No Errors Frequency Error MHz Cell # After diffusion bonding the 206 cells of RDDS1 an error was discovered in the first few and middle 6 cells due to the structure adhering to the supports. The flaring gives rise to frequency errors of the order of 30MHz in the central cells. This effect has been cured by choosing a support with similar expansion coefficients as the structure itself.

24 RDDS1 ASSET Measurement (Cont.) The dipole wakefield is particularly sensitive to absolute frequency errors. The separation of modes in the central region (~15.2GHz) is approx 7MHz and any errors in the cell frequencies of this order will impact the wakefield severly. A relative error which is smooth from cell to cell has little impact on the spectral function and therefore on the wakefield. The spectral function provides a useful tool to predict the wakefield under various regimes, including errors in the synchronous frequency that result from fabricational errors The present structure may at some point be re-measured in an ASSET test after squeezing the middle cells back to their design values

25 Equivalent Circuit for Short Stacks of RDDS1 Disks V 1 V L 1 L 2 Manifold C 1 C 2 L 1 2 C 1 C 2 L 2 2 L 1 2 L 2 2 TE Frequency (GHz) C 1 1/2 L 1 2 L /2 L 2 2 TM L C 3 21/ Phase (Deg.) 12 Circuit diagram and Brillouin diagram corresponding to RDDS1 cell stack 98 to 103 (average cell 100.5). The points are obtained from an experimental measurement and the lines are obtained from the circuit model in which the original design was prescribed prior to the experiment ψ f exp f mode f av

26 Manifold Radiation and a Structure Diagnostic 1. Use of Manifold Output to Minimise W(s) for small s Monitoring a small fraction of the wakefield radiated into the manfolds enables the wakefield to be minimised by moving beam towards the electrical center of the structure. The procedure adopted is to pick two representative frequencies and minimise the manifold output power Measure W(s) at the same time. For a proper (for example in DDS1 the procedure is to choose frequency points close to the structure misalignments) choice of frequency pair, this minimises W(s) at the source adjusted drive beam position. This allows the beam position to be determined to 10µm or better. 2. Application of Manifold Radiation to Probing Structure Imperfections Once the accelerating structure has been finally bonded there will be errors in the cell-to-cell alignment and a bookshelving effect. The cell-to-cell misalignments are measured from the radiation picked up from the manifolds.

27 A circuit theory method is used to compute the cell misalignments from the manifold power. The circuit theory allows the power picked up from the manifiolds to be calculated The relation between the minimised power spectrum to cell offset is linear at a given frequency. The modes are localised and this is revealed through the circuit model

28 Circuit Model of Power Radiated to Manifold Coupling to manifold (TE) H GR G Hx a a f U = f Ht x Hˆ aˆ ˆ a B = 4 n n B πfsksl n exp[ 2πjLn/c] c Also, the manifold wave amplitude is related to the field in the structure by: A= R 1Ga A R G a R â A = = Power coupled to the manifold: faa sinφ 8π ( 1+ R) 2 where the manifold mode has the dispersion relation: cos φ = cos φ α ( π L/c) F /(F f )sinc φ n n n n 0n

29 Mode Localisation and Cell Offsets Power spectrum emitted from the downstream DDS manifold illustrating the narrow frequency bands emitted by localized offsets. The pairs of numbers above each peak indicate the central cell and the span specifier of the localized offsets. A single cell is excited and we successively add adjacent cells until the power picked up from the manifold reaches that of the power calculated when all cells are allowed to interact. This process is carried out for a number of fiducial cells and the frequency location of the maximum in the power allows a frequency mapping function to be obtained, together with the associated degree of localization

30 Illustration of the deviation of the synchronous frequency from the uncoupled one due to cell-to-cell detuning. The short horizontal lines indicate the extent to which cell offsets may be localized by frequency Comparison of the CMM (Coordinate Measuring Machine) data set versus the ASSET power minimization position data remapped from frequency to cell number.

31 Structure Breakdown and Group Velocity Recent experiments have indicated that breakdown occurs in (R)DDS. Pitting of the irises was observed from boroscope pictures. Most of the damage was concentrated towards one end of the structure the high group velocity end An R-L-C circuit model was utilised to calculated the power absorbed during a breakdown event. This confirms the group velocity dependence. Circuit model of power absorbed in breakdown P abs R A vg sin φ (grad) (R/Q) 2 ω/c φ sin φ + 2v cos φ g Circuit model for fundamental mode in accelerating Structure R A resistance of breakdown, v g is the group velocity, φ is the synchronous phase advance (=2π/3 for DDS1), grad the accelerating gradient, and R and Q are the shunt impedance and quality factor evaluated at the synchronous phase DDS1 Group Velocity Profile

32 Low V g Accelerating Structures A vigorous experimental program is now in progress at SLAC influence of structure length, group velocity are being studied for TW structures. Standing wave π structure are also being investigated. (see Adolphsen, Tuesday T3 session). Initial structures focused entirely on understanding (and curing) breakdown. Recent structures will also incorporate detuning of cell frequencies. The fundamental mode s phase advance has been raised to 5 π /6 (from 2π/3) to reduce the group velocity whilst maintaining a/λ = 0.18 These structures, with appropriate loading, will also be candidates to suppress transverse wakefields: eg, H60VG3 (90cm long, v g =0.03c) H90VG5 (60cm long, v g =0.05c) We investigate the the wakefields for these shorter, low-group velocity structures moderate loading and interleaving of cell frequencies, together with moderate loading and a manifold damped version are considered in particular.

33 High Phase Advance Accelerating Structures c n-1 c n c n+1 i n-1 i i n n+1 R n-1 R n-1 i n c n R n R n+1 R n R n+1 i n-1 i n+1 c n-1 I n-1 I n c n+1 I n+1 TE TM Circuit model for locally damped structure illustrating 3 cells of an n-cell chain Wake VpCmmm Wake VpCmmm H90VG s m H90VG5 with 2-Fold Interleaving s m Wakefield for single high phase (5π/6) advance structure: H90VG5 with moderate loading (Q ~ 1000). This structure is 90cm long. The initial group velocity is c. Wakefield from a single structure is clearly insufficiently damped. The frequencies of adjacent structures are interleaved

34 High Phase Advance Accelerating Structures (Cont.) Shown is the H90VG5 structure. Varying the Q- loading each individual cell indicates that a Q of 500 or lower is required to adequately damp the wakefield. Unless interleaving of structures utilized.

35 High Phase Advance Accelerating Structures (Cont.) Spec. Fnc VpCmmmGHz 100 Spec. Fnc VpCmmmGHz H60VG Frequency GHz. H60VG3 3-Fold Interleavi ng Frequency GHz. Wake VpCmmm Wake VpCmmm H60VG s m H60VG3 with 3-Fold Interleaving s m Spectral function and wakefield for single high phase (5π/6) advance structure: H60VG3 with moderate loading (Q ~ 1000). This structure is 60cm long. The initial group velocity is 0.03c. A single structure is clearly insufficiently damped. The frequencies of adjacent structures are interleaved 3-fold and the wake is well suppressed.

36 High Phase Advance Accelerating Structures (Cont.) Shown is the spectral function (upper) and wakefield (lower) for the H60VG3 structure. Varying the Q-loading each individual cell indicates that a Q of 500 or lower is required to adequately damp the wakefield.

37 Beam Dynamics SΣ s b Standard deviation of sum wake function from the mean value for H90VG5 with two-fold interleaving (shown dashed in red) compared to H60VG3 with threefold interleaving (solid black) The issue of BBU is addressed by offsetting the beam by σ/4 and tracking it through to the end of the linac (10km). Y norm Y norm Y norm Y norm. Emittance % BPM Position km Phase space and emittance growth for H90VG5 with 2-fold interleaving of structures Emittance % BPM Position km Phase space and emittance growth for H60VG3 with 3-fold interleaving of structures

38 RDDS1 Fabrication Tolerances Frequency Deviation [MHz] del_sf00 del_sf0pi del_sf1pi del_sf Single-disk RF-QC Disk number Small dimensional errors, generated when fabricating the irises and cavities of an accelerator structure, give rise to errors in the synchronous frequencies. Presently, it is possible to machine the cells to an accuracy of better than 1 µm However, when fabricating several thousand such structures, looser tolerances may reduce the fabrication costs The linac will consist of 4720, nominally identical structures, each of which contains 206 slightly different cells. Shown here is a measurement of the cell dimensions performed at KEK

39 An error type which is repeated in every cell of a structure but differs in every structure is referred to as: a systematic-random error. An error that is repeated in every structure, but varies from cell-to-cell, we refer to as a random-systematic error. We also consider random-random and systematic-systematic (potentially the most damaging) error types making a total of 4 error types. RMS Dev. of Sum Wake % Increase in Spacing RMS sum wakefield for 3MHz RMS errors An indicator for the onset of BBU is provided by the wakefield at a particular bunch which is formed by summing all wakefields left behind by earlier bunches which is denoted as the sum wakefield. BBU will likely arise when the RMS of the sum wake is the order of 1 V/pC/mm/m or larger. When not in the BBU regime, the sum wakefield also provides an accurate method of calculating the multi-bunch emittance dilution

40 Percentage Emittance Growth Identical structure-tostructure errors Random structure-tostructure errors BPM Position (km) Emittance growth due to 3MHz RMS errors that are (a) reproduced in every structure and (b) random from structure-to-structure Y (norm.) Y (norm.) Phase Space (3MHz RMS error). Phase space for a linac composed of 4720 structures assumed to have identical random errors in each structure Y (norm.) Y (norm.) Phase space for a linac composed of structures with a different random error in the synchronous frequency (non-identical structures).

41 Manifold Damped 5π/6 Structure 20 Light Line 20 Light Line Frequency GHz Upper Dipole Lower Dipole TE TM Frequency GHz Upper Band Dipole Mode Coupled Dipole Modes Avoided Crossing Spec. Func. VpCmmm 12 Uncoupled Manifold Phase Deg. Dispersion curve for cell 42 of purely detuned accelerator (prior to manifold loading) Uncoupled Kdndf LDS DDS dndf Integral Frequency GHz. Spectral function for locally damped(q~1000) and manfifold damped 83 cell structure Wake VpCmmm Phase Deg. Dispersion curve of a manifold damped version of cell 42 LDS DDS s m DDS wake compared LDS

42 Standing Wave π Structures 20 Light Line Kdndf,K,fsyn Frequency GHz Upper Dipole Mode Lower Dipole Mode TM TE Avoided Crossing Phase Deg. 1000K f syn Kdndf Frequency GHz Consider an amalgamated structure consisting of 6 separate structures (each structure consisting of 23 cells). The first cell in each structure is damped heavily (Q ~ 10). Dipole dispersion curve for cell 69 of an amalgamated SW structure Uncoupled kick Factors (K), synchronous frequency distribution (f syn ) and kick factor weighted density function (Kdn/df) for standing wave structure. 10% bandwidth and 8σ. The kick factors do not increase with increasing frequency in contrast to those in the traveling wave structures

43 Spec. Func. VpCmmm Frequency GHz. Spectral function for the lower band frequencies of a standing wave structure consisting of 6 separate structures (each structure containing 23 cells). A limited number of modes in the center of the distribution Q ~ 300 Wake VpCmmm Envelope of wake function for a 138-cell standing wave amalgamated structure s m

44 Summary The circuit model combined with the spectral function method has proved a useful design tool in predicting the envelope of transverse wakefields in NLC structures (covering the regimes of zero through to strong and moderate damping) Decoupling the last few cells results in several high Q modes which are readily damped with the fundamental mode coupler. Microwave measurements and modal predictions of the dipole mode frequencies for short (6-cell) stacks are in good agreement. Manifold radiation serves as an excellent structure diagnostic BPM and structure straightness monitor. RDDS1 meets the emittance budget for the NLC (provided that flaring in the cells is not repeated results obtained from KEK indicate that this effect has been cured!). As built, the measured RMS error is approx 0.5 MHz. Considerably looser tolerances are allowable provided errors in the cell-tocell frequencies are not repeatable from structure-to-structure (randomsystematic and systematic-random errors are serious errors)

45 Summary (cont.) Errors that are random from structure-to-structure result in an effective interleaving of frequencies and allows even large wakefields to be tolerated. This may require sorting or binning of structures As BNS damping is utilized to minimize emittance dilution that results from short range wakefields, a similar technique may be employed to reduce the dilution resulting from long range wakefields. Recent designs for low-group velocity, short structures indicate that manifold damped verions provide adequate transverse wakefield suppression. Interleaving of the dipole frequencies of structures also provides excellent wakefield damping. Studies on SWπ structures reveals the lower dipole mode kick factors to fall off at the high frequency end of the band (in contrast to TW structures where the kick factors continue to increase linearly with frequency). Heavy (Q ~10), but limited to a sparse number of cells, results in excellent damping. Upper dipole bands are under active consideration (the second dipole band for example, has kick factors of a similar order of magnitude to the lower band)

46 ASSET Test of DDS1 Wake Function (V/pC/mm/m) DT DDS s (m)

47 Design Procedure Wake Function (V/pC/mm/m) DDS s (m)

48 Design Considerations Suppressing wakefields Short-range wakefields a -3.8 : set lower limit on aperture: a 0.18λ Long-range wakefields: require specific detuning, proper mode dispersion properties, and damping To achieve reasonable cell-to-cell and structure-to-structure alignment tolerances Reducing breakdown damage - low group velocity Cost effective high efficiency, optimal T fill Minimizing surface field and RF pulse heating Mechanical constraints

49 Lossy Circuit For Local Damping (or SW)

50 Summary We have an established procedure for structure design We have tools to optimize structure parameters and wakefields Advanced numerical tools and massive parallel computer resources made possible to model complicated structures accurately. to reduce number of prototypes - shortening development cycle

51 Summary (cont.) High phase advance is advantageous for low v g design X-band using 5π/6-mode for low v g design to cope with highpower breakdown damage Can maintain large aperture: a/λ=0.18 Can achieve lower group velocity: <5%c Acceptable bandwidth Good RF efficiency: τ~0.5; t f ~100-ns Reasonable length: 0.9-m or 0.6-m X-band detuned standing wave structure being studied Structures yet to be tested at high power X-band Wakefields yet to be studied