The RF power necessary to accelerate the main beam of the Compact Linear Collider (CLIC) is produced by decelerating a high-current drive beam in Power Extraction and Transfer Structures (PETS). The reference structure is not cylindrically symmetric but has longitudinal waveguides carved into the inner surface. This gives rise to a transverse component of the main longitudinal mode which can not be damped, in contrast to the transverse dipole wake- field. The field is non-linear and couples the motion of the particles in the two planes. Limits of the stability of the decelerated beam are investigated for different structures

Millich, Antonio

Riche, A J

Schulte, Daniel

CERN Document Server

BEAM STABILITY IN THE DRIVE-BEAM DECELERATOR OF CLICUSING STRUCTURES OF HIGH-ORDER SYMMETRYA. Millich, A. Riche, D. Schulte, CERN, 1211 Geneva, SwitzerlandAbstractThe RF power necessary to accelerate the main beam ofthe Compact Linear Collider (CLIC) is produced by decel-erating a high-current drive beam in Power Extraction andTransfer Structures (PETS). The reference structure is notcylindrically symmetric but has longitudinal waveguidescarved into the inner surface. This gives rise to a trans-verse component of the main longitudinal mode which cannot be damped, in contrast to the transverse dipole wake-field. The field is non-linear and couples the motion of theparticles in the two planes. Limits of the stability of thedecelerated beam are investigated for different structures.1 INTRODUCTIONThe CLIC study investigates a possible future linear col-lider based on acceleration with a frequency of 30 GHz [1].The necessary power is produced by decelerating a high-current, low-energy beam which runs parallel to the mainbeam. This is done in 20 consecutive drive beam decel-erators per main linac each of which is fed with a newbeam [2]. During the deceleration the beam develops alarge energy spread of a factor ten and is subject to trans-verse deflections due to wakefields. For reliable operationthe beam stability in the decelerator is critical since thebeam contains a large energy per pulse. In addition, thedown-times of the 40 decelerators add to the total down-time.2 DECELERATOR MODELThe decelerators are on average 767 m long and consistof 550 PETS [3]. The layout of the drive beam deceler-ator is strongly coupled to that of the main beam accel-erator. It consists of a simple FODO lattice with a con-stant quadrupole spacing of 1:115 m. Between each pairof consecutive quadrupoles a power extraction structure isplaced. It feeds two accelerating structures in the mainlinac. The gradients of the quadrupoles are varied alongthe beam line to achieve a constant phase advance of about88  per FODO cell for the lowest energy particles. The en-velopes of the particles with higher energies will then au-tomatically be smaller than those with lower energies [4].3 STRUCTURE MODELThe active length of the power extraction structures is0.8 m; longer structures would lower the main linac fill fac-tor. The inner bore of the structure is cylindrically sym-metric except for the longitudinal waveguides that are cutinto the surface. Here four structures are considered withfour, six and eight waveguides, respectively, and a circu-larly symmetric one. Their main properties are listed inTable 1. The longitudinal and transverse wakefields can bewell represented by a single mode each, with a frequencyof 30 GHz for the longitudinal and 24 MHz less for thetransverse mode. The group velocities of the wakefieldsare close to the speed of light, e.g. ?  k = 0:441 forthe four waveguide structure.The power the structure extracts from the beam has to be512 MW. This is simply given by the product of averagedeceleration of the beam particles times the current. Thelongitudinal wakefield Wk is a function of the structure ra-dius a, Wk / a−3 for the four-waveguide structure. So onecan in principle choose a high-energy, low-current beam ina small aperture structure or vice versa. Simulations of thetransverse wakefield have shown that the best beam stabil-ity in a given decelerator using four waveguide structuresis achieved by choosing the highest beam current possi-ble. The injector complex is expected to allow for bunchcharges q of up to q = 20 nC.The high group velocity leads to a concentration of lon-gitudinal and transverse wakefields at the end of the struc-ture. In the simulation, the passage of a particle throughthe structure is therefore simulated in a number of steps.During each step the longitudinal and transverse field is as-sumed to be constant. But from step to step, the fields varyaccording to the field profiles in the structure. The trans-verse wakefields not only drain out of the structure but aredamped with a quality factor Q  50 [5].Since the structure is not cylindrically symmetric, thelongitudinal field varies with the transverse position.Ek(r; ) =1∑i=02kirmsiamsicos(2s=) cos(msi):For the number of waveguides ms 6= 0 this corresponds toa transverse field which can be written asE?(r; ) =1∑i=02kimsirmsi−1amsi2sin(2s=)[−~er cos(msi) + ~eφ sin(msi)]: (1)Here,  = 0 lies in a symmetry plane in the centre of awaveguide. In contrast to the transverse wakefield, this ef-fect also plays a role if the beam is perfectly centred in thestructures.0-7803-5573-3/99/$10.00@1999 IEEE. 1863Proceedings of the 1999 Particle Accelerator Conference, New York, 199900.511.522.530 2 4 6 8 10 12 14E/E 0r [mm]Φ=0fitΦ=pi/2fitFigure 1: Fit of the field non-uniformity in the six wave-guide structure.Table 1: Parameters of the different power extraction struc-tures used in the simulations. R0=Q is given in linac ohms.The last structure is circularly symmetricms 4 6 8 1a [mm] 13 15 16 12R0=Q [Ω=m] 41 50.8 49.0 292.6W^? [V=pC=m2] 225 250 260 3170k 0.441 0.59 0.64 0.78qb [nC] 16.1 16.7 17.7 8.0E0 [GeV] 1.34 1.29 1.22 2.76G^=G 4.3 3.0 2.5 1.0k1=k0 1.8 1.6 1.4 —k2=k0 1.0 0.65 0.21 —For the six and eight waveguide structures the coeffi-cients can be determined easily. Figure 1 shows the resultof the fit in the two planes  = 0 and  = =2 using eightcoefficients. Only the first two are important, the others arequite small. In the case of the four waveguide structure, thefit is less reliable since the curves are not smooth due to themesh used in the computations with MAFIA. Also in thiscase the first two coefficients are taken, even so the follow-ing ones are not much smaller. The most important effectis however expected from the lowest orders.4 BUNCH TRAINThe bunch train used to produce the RF power consists offour parts. In the main part, the flat top, the charge does notvary from bunch to bunch. It is used to produce the powerduring the passage of the main beam. It is preceded by anumber of bunches in which the charge is increasing frombunch to bunch. This ramp is necessary to compensate themain linac beam loading by providing a field profile as ifthe first bunch of the main linac pulse had been precededby others. An additional ramp of charge preceding the onedescribed may be necessary if the drive beam injector can-not achieve the sudden change in charge. It is neglectedin the following since its length and shape still have to beinvestigated. After the flat top, another ramp with decreas-ing charges will follow which is needed for beam loadingcompensation in the drive-beam accelerator, see [6].The bunch length is assumed to be z = 400 m, thetransverse emittances γx = γy = 150 m. During thedeceleration, particles can lose up to the 90 % of their initialenergy, while the first bunches will be hardly decelerated.5 ADDITIONAL ENERGY LOSSDue to the non-uniformity, off-axis particles can be decel-erated more or less than on-axis. If the deceleration is toolarge these particles will be lost since they are over-focusedby the quadrupoles. For a four-waveguide structure, the ad-ditional energy loss E can be estimated asE = G38c0∫ L0hr4(z)ia4dz:Here, L is the linac length, r^(z) the maximum amplitudeat position z and the factor 3=8 is due to the integrationover the betatron motion. For the lowest energy particlesthe beta-function is constant, so the integration leads toEE0 − Ef =38c0r^5 − r55(r^ − r)a4E0Ef:Here, r^ and r are the minimal and maximal amplitudes inthe first FODO cell. If E=Ef  0:3 the lattice will beover-focusing. Assuming k1=k0 = 3:3 in the four wave-guide structure (attributing the full amplitude of the non-uniformity to this value) a particle can be lost if it has aninitial offset of about 8. Simulations confirm this value.If consecutive structures are rotated by =ms the lowestorder term of the non-uniformity will about cancel. Thiswill allow the particles to have even larger initial ampli-tudes without being lost.6 TRANSVERSE EFFECTSIn order to stabilise the beam, it is useful to rotate everysecond structure by  = =ms [3]. Since the beta-functionis significantly longer than the structures, the kick due tothe lowest order in (1) averages out over a short distance.In Fig. 2, the maximum envelopes along the deceleratorare shown for all three structures. A damping with Q = 50and Q = 200 is assumed, the beam is offset by one sigmainitially and has a size of 4x and 4y . The envelope isnormalised to the half-aperture. Even with Q = 200 thebeam is sufficiently stable in all structures. The ones withsix and eight waveguides are better than the one with fourdue to the larger aperture and the smaller non-uniformity.7 AZIMUTHAL DEPENDENCEThe kick a particle experiences due the field non-uniformity is purely radial in a symmetry plane, but else-where also has an azimuthal component. To investigatethe different effects, a four sigma beam is used, contain-ing test particles with Courant-Snyder Invariants Ax =1864Proceedings of the 1999 Particle Accelerator Conference, New York, 199900.10.20.30.40.50.60.7envelope/aperture4 WG,Q=506 WG,Q=508 WG,Q=5000.10.20.30.40.50.60.70 100 200 300 400 500envelope/aperturequadrupole no4 WG,Q=2006 WG,Q=2008 WG,Q=200Figure 2: The normalised envelopes for the different struc-tures with waveguides, the upper graph is for the expecteddamping Q = 50, the lower for Q = 200.-0.6-0.4-0.200.20.40.6-0.6 -0.4 -0.2 0 0.2 0.4 0.6y/ax/a4 WG6 WG8 WGFigure 3: The envelopes in the different angles 0.16 cos2(0)x; Ay = 16 sin2(0)y . The beam has aninitial offset in the direction of this plane of one sigma:x = cos(0)x; y = cos(0)y . Figure 3 shows themaximal excursion of any particle for the different angles.The variation is essentially due to the optics only.8 CIRCULARLY SYMMETRICSTRUCTUREThe circularly symmetric structure has no field inhomo-geneities. Its transverse wakefield as well as its impedance4567891011120 50 100 150 200 250 300 350envelope [mm]Q∆x=1.5σx∆x=1.0σx∆x=0.7σx∆x=0.3σxFigure 4: The beam envelopes for the circularly symmetricstructure for damping with different Q.are high. While it therefore seems natural to use a struc-ture with larger diameter, it is very difficult to achieve alow group velocity in this case. Already at r = 12 mm, itis very high: g = 0:78. The initial energy of the beamis very high and its current significantly lower than in theformer cases, see Table 1. In this case, it is more difficult todamp the transverse modes than in the former ones. Firstmeasurements indicate Q  150 [7]. Figure 4 shows themaximal amplitudes of a four-sigma beam with differentinitial offsets for different values of Q. While the struc-ture is clearly not excluded, the wakefield effect seems tobe significantly stronger than in the previous ones.9 CONCLUSIONCalculations of the beam stability in the drive beam decel-erator have been performed for new transfer structures. Thelongitudinal and transverse effect of the non-uniformityof the longitudinal field have been included together withthe usual wakefield effects. The stability improves com-pared to the previous case. Especially with the six andeight waveguide structures almost no effect of the non-uniformity is visible.10 REFERENCES[1] J.-P. Delahaye and 30 co-authors, “CLIC a 0.5 to 5 TeVCompact Linear Collider.” EPAC 1998 and CERN-PS-98-009-LP.[2] H. Braun and 14 authors, “A new method of RF powergeneration for two-beam linear colliders.” EPAC 1998 andCERN-PS-98-014-LP.[3] A. Riche and D. Schulte. “The drive beam decelerator ofCLIC”. LINAC 98 and CERN-PS-98-043-LP[4] A. Riche. “Maximum energy transfer efficiency in CLICdrive beam”. CLIC-Note 266 (1994).[5] L. Thorndahl. Private Communication.[6] D. Schulte. “The drive beam accelerator of CLIC”. LINAC1998 and CERN-PS-98-042-LP.[7] G. Carron and L. Thorndahl. Private communication.1865Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Beam Stability in the Drive-Beam Decelerator of CLIC Using Structures of High-Order Symmetry

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