Wuensch, Walter

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wakefield and U is the energy of the bunch.wave number, p is the charge density of the bunch, W is the transverse(from the head back) and s is the distance along the linac, k is the betatronwhere X is the transverse displacement, z is the position along the bunchM(z.SdZ’QX "(wl + E2 X’(z»S) + l<2(w)X(w) = p(z’)Wf(Z · Z’)X(Z’·S)U) U () ·"wakefield is given by [5],The motion of a bunch subject to smooth focusing and a transverseBeam Dynamics Related to BNS Dampingfor CLIC and is not destructive to the beam.measurement is proposed. It is based on the BPM which is being developedIn the second half of the paper a device for an approximate emittancebe found from the analysis as well.number of measurements which are needed along the length of the linac canfeatures of the beam when RFQ strengths are wrong. Some idea of themeasurement scheme can be developed that is sensitive to the essential_:_..____£__.. OCR OutputUJhow the beam degrades when BNS damping is misapplied. In this way a»In the first half of this paper an attempt is made to qualitatively understand \§results [4] Iemittance measurements. Attempts to use five per linac have led to puzzling 2%length of the linac. Most simulation results use a very high density ofsufficient. It is also not clear how many measurements are needed along thethe design of hardware. An approximate emittance measurement may beemittance measurement has not been established in a way in which can guidehow to make the measurement [3]. The necessary performance of themeasurements are less well advanced although there are some ideas aboutAttempts to match beam simulation with hardware for emittancebeam position monitors (BPM)s for CLIC are under development and test [2].of their placement within the linac have been well studied [1] and appropriateThe resolution and precision needed for beam position monitors and detailsset the correct level of BNS damping [1].measurements for adjusting radio frequency quadrupole (RFQ) strengths tobeam position measurements for linac alignment and emittancealgorithms currently used in CLIC main linac beam dynamics simulations:Two types of beam measurements are required for the trajectory correctionIntroductionWalter WuenschMeasurements for Adjusting BNS Damping in CLIC17 August 1994CLIC Note 244X(z0,s) = 1 — i se"“.() OCR Outputas on the first charge , the solution of the motion is,If we add the homogenous solution and impose the same initial conditionsX = aseihThe inhomogeneous solution has the form,X”(z,s) + k2X(s,z) = gg?-e””so has an equation of motion,P0W0Z0X(0»S)The second charge sees a wake,X (0,s) = elinitial displacement of 1 gives the motion,The first bunch sees no wakefield so just executes harmonic motion. Ank(z,s) = k.p<z> = p06(Z> +p06(Z - Z0)charges located at z=O and z=zO and which have the same k ,The first approximation is to consider a bunch made up of two discreteU°=O.No acceleration will be considered,wi (Z — Z') = (Z - Z')W0,be assumed to be linear.compared to the wavelength of the main deflecting mode and wakefields willThroughout the following analysis bunches will be assumed to be shortequation under various approximations [6,7,8].(usually more sophisticated) efforts have been made to solve the aboveconsidered to try to understand the general behaviour of the solutions. OtherInstead of attacking the equation directly, a number of special cases will befrequency, OCR OutputThus the shape of the beam is modulated by an ever increasing spatial_ _ X(z,s) = e"“e `°r<~¤>evolves according to the function,spread. A straight beam which enters the linac with unit displacementmotion with a continuously varying frequency due to the focusing strengthWith no wakefield each point along the bunch will execute harmonicfrom coherent bunch motion - perfect BNS - to too strong RFQS is assumed.is the limiting case of too strong RFQS. A smooth and continuous variationslightly too strong, the motion with wakefields turned off is considered. ThisIn order to gain insight in the evolution of the bunch when RFQs arewith Ak/k=.O5.k(z,s) = k01+ ,{&iJko Z0focusing strength,Through the RFQS the CLIC beam will be subject to a linear variation in`0xr 2 00 r » 1 X (z,s) + k (z,s)X(z,s) = T|(z - z )X(z ,s)dzpWZ ·motion of a bunch with uniform charge density pg from z=O to z=zO isThe motion of a continuous bunch is now considered. The equation of2kUk' = k + po Wozoby a slightly different k for the second bunch.BNS damping cancels this phase shift by introducing a phase shift caused2kUA = _ P0 Wozo (PFor small s the wakefield introduces a phase shiftbunch motion to the give the solution for 3 or more bunches.nonlinearity of the equation of motion prohibits the summation of the twoapplied force were in the direction of the velocity. Unfortunately theis in contrast to the quadratic growth in amplitude one would have if thecharge which is 90° out of phase with the driving term - the first charge. Thisln the limit of large s, there is a linear growth in the amplitude of the secondThis is of course only approximately true as the wakefield alone eventually OCR Outputno wakefields as the two effects compensate each other in BNS damping.energy spread must be essentially the opposite of that for energy spread andThe behaviour of a beam over short distances with wakefields and nonot solvable in closed form, so approximate solutions must be investigated.Unfortunately the author is stuck here. The inverse transform is probablyiwZ X(z,s) = ilel ‘ U emdt.z+·¤ ling-rp0W0Transforming back to the original functions,x(z,t) = e ‘ U .This has the solution,k§— rz U & 1 pOW0 dzzrespect to z,changed to a second order differential equation by differentiating twice withThis integral equation is a homogeneous Volterra equation which can be(kg — t)x(z,t) = w I (z — z')x(z’,t)dz’W 2X(z,s) 4: x(z,t),pairs,Taking the Fourier Transform with respect to the variable s with transformOX11( 2 _ 00 1 1 1 z,s) + k0X(z,s) - —-6-j(z — z )X(z ,s)dzpWZ ‘no focusing strength spread is given by the equation of motion,The opposite limiting case, the equation of motion of a uniform bunch withcycles for CLIC and a full cosine after 20 cycles.beam will evolve to a half of a cosine after k/ (2Ak) betatron cycles - about 10finite length when RFQs are mistuned. For example the displacement of theHere one can draw the first conclusions of the evolution of a bunch withsignal is 90° out of phase with the signal for large beam offsets. OCR Outputwhen it is scanned through the center of the cavity. The minimum voltageThe tilt signal manifests itself as a minimum voltage that the beam produces.17 mm long (1 o) which corresponds to 6° of 30 GHz phase.before the tail arrives. The effect can be used for CLIC because the beam islonger an exact cancellation because the fields excited by the head evolvetail exactly cancel. If the bunch has a finite phase extension then there is nocharge passes through the center of the cavity because the effects of head andwavelength of the TMHO mode does not excite the mode when center ofextension of bunches. A tilted bunch which is very short compared to theresonant cavities. Tilt is measured as natural result of the finite phaseCLIC beam position monitor. CLIC beam position monitors are TMH0The device which will now be considered to measure beam tilt is a standardH Excitation of a resonant cavity with a tilted beamearlier in the linac - the 4th of 5 sectors is being adjusted in this case.the narrower features. The narrower features are thought to originate fromeven with a full emittance measurement mainly tilt is being adjusted and notby eye the least tilt also corresponds to the best emittance. This means thatOne can see that the beam tilts gradually as the RFQs are misadjusted. Alsoquadrupole focusing strength. Corresponding emittances are shown in figurein figure 1. ot is the ratio of the RFQ focusing strength to the staticdisplacements at the end of a sector when RFQ strengths are varied is showntracking simulation results are considered. An example [9] of beamTo verify that the beam tilt is a sensitive measure of RFQ tune beamsimulations and is beyond the scope of this paper.wave. This periodicity constraint can only be determined from beamprobably passed before the beam shape has evolved beyond a quarter sineclosely to limit uncorrectable effects like filamentation and the tolerance isnot included in the smooth focusing model. The bunch must be controlledThe measurement periodicity is however probably determined by effectscycles in a 250 GeV CLIC linac.installed at least once every 10 betatron cycles - there are about 50 betatronuncorrected beyond a half cosine. Thus a tilt measurement should bethe ever increasing spatial frequency the beam would pick up if it evolvedadjusting RFQ strengths. However such a device would not be sensitive tothe level of BNS damping needed for a particular machine set up and thus formeasures the average slope of the beam, the tilt, could be used for determinedwhen BNS damping is not applied correctly in either limit. A device whichFrom all these arguments the beam shape evolves to a fraction of a cosineapproximation) and focusing spread alone involves no amplitude growth.causes a blow up in amplitude (as was shown in the two charge- 5The minimum voltage is,x = 1-OCR OutputThe minimum occurs when the center of charge is on axis,8x= 0-=·& = (2x + d)(1+ cos(A<p))avA minimum occurs when,VMVQN = %(x2 + (x + d)2 + 2x(x + d)cos(A<p)The square of the magnitude is,VW, = -3(x + (x + d)ei°°’)The total voltage is,V, = -£xe*° V2 = §(x + d)e*"""The voltages induced in the cavity by the two charges are,x+d_L °of charge 1 /2, separated by a phase Az and a transverse offset d,This effect can be clearly seen if the bunch is approximated by two charges- 6an unacceptable increase in wakefields. OCR Outputreduced below the current 4 mm but this means of improvement is limited bybeam pipes every cavity must have. The beam pipe diameter could beThe improvement is limited because field lines will always extend into theA shorter cavity would have a lower skew to displacement sensitivity ratio.(less affected by wakefields) as a reference.sec. The tilt monitors could also be aligned using a special low charge beamThis will increase measurement time by roughly a square, so 400 points or 0.3pair of PZTs and nulled iteratively by displacing and changing the angle.align tilt monitors to this precision, the tilt monitors should be mounted on acorresponds to an angle of .1 mrad. Since it is probably not possible to prealigned to about the same value over its active length of a few mm This1 /30 for beam tilt). To keep the resolution to below .1 um the BPM must bewould have a skew to displacement sensitivity ratio of 1/ V2 (as opposed torather than the bunch length. A BPM a quarter wavelength long, 2.5 mm,measurement. The excitation depends on the length of the BPM, 3.332 mm,skew of the BPM results in an out of phase excitation exactly like the tiltis the alignment of the BPM with respect to the trajectory of the bunch. AA more serious limit to the performance of the BPM for a tilt measurementto resolution due to the BPM signal processing system should be minor.deflections limited when making a tilt measurement. Other than this the limitThe dynamic range of the detection electronics must be large and the beamrange or below.the data in figure 1. Clearly the resolution of the BPM must be in the .05 umvoltages induced by the slices . Figure 3 shows the values determined fromreferring displacements to the center of charge, then adding the complexsignal strength is determined by approximating the bunch by 40 slices,estimate the signals that can be expected from a BPM tilt measurement. TheWe can now return to the beam simulation results shown in figure 1 to.01 sec.measurement would be fast as a scan taking 20 steps would only requireresolution must be small but does not need to be calibrated. Themoving the cavity with a piezo electric crystal (PZT) since the movementthe center of the BPM to find the minimum voltage. The scan can be madeA tilt measurement is made by scanning a BPM so the beam passes throughprecision.performance limit for tilt is resolution whereas for beam position it ismass displacement. This may seem to be hopeless but the measurementThis means that the BPM is 1/ 30th as sensitive to tilt as it is to center of30Vm = J`voltage is,Approximating the CLIC bunch by two charges .2 mm apart the minimmn2 V 2V- z'““” g [1-cos(Aq0)- 7[9] G. Guignard, G. Parisi private communication. OCR Output[8] H. Henke, CLIC note 40.Energy Accelerators, 1983.[7] V.Balakin, A Novokhalsky, V. Smirnov, Proc. 12th Int. Conf on High1985.[6] K. Bane, IEEE Transactions on Nuclear Science, Vol NS-32, No. 5, October[5] H. Henke, W. Schnell, CLIC note 22.[4] G. Parisi, Minutes of the CLIC meeting held on 3 March 1994.[3] B. Montague, Minutes of the CLIC meeting held on 16 October 1992.[2] ]. P. H. Sladen, I Wilson, W. Wuensch, CLIC note 189.[1] G. Guignard, CLIC note 215 and references contained within.ReferencesI. Wilson for their help.The author wishes to thank G. Guignard, G. Parisi,]. Sladen, W. Schnell andAcknowledgementsfinite length to measure a bunch of finite length.emittance approximating measurement which uses a detection element ofcritical issue for the sensitivity. This issue is general to any emittance orParallelism of the tilt monitor to beam trajectory seems to be the mostenough.Beam simulations must confirm that the tilt monitor sensitivity is goodso some trajectory corrections could be made in parallel.so could be installed as often as needed. The measurement is not destructivetilt monitor. Such a tilt monitor would be relatively simple and inexpensiveCLIC BPMs mounted on PZTs may be sensitive enough to fulfil the role of adamping in the CLIC main linac.A beam tilt measurement is appropriate for adjusting the level of BNSConclusions- 8Figure 2: Corresponding emmitances OCR Output0 500 1000 1500 2000cv 0 . 0;,"'· M. iw05LOciao Q c<=.42.5me5.1)quadrupole.of oz. The data is given for the end of the 4th sector of the linac in a focusingFigure 1: Beam displacement as a function of slice number for various values40 5010 302010..+——# r '?·<=04*.510Figure 3: Beam tilt monitor response as a function of on.alpha0.1 0.2 0.3 0.4 0.50.050.10.150.20.250.30.35Tilt vs. alpha

Measurements for adjusting BNS damping in CLIC

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