In the main linac of the Compact Linear Collider (CLIC), longitudinal and transverse dynamic imperfections, such as RF phase jitter, variation of bunch length and movements of elements, can result in significant luminosity loss. The responses of local trajectory feedbacks to these imperfections are studied in this paper

Leros, Nicolas

Schulte, Daniel

Leros, N

Schulte, D

CERN Document Server

Feedback response in the CLIC main linac to transverse and longitudinal dynamic imperfections

  EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH  CERN - PS DIVISION       CERN/PS 2002-042 (AE) CLIC Note 524      FEEDBACK RESPONSE IN THE CLIC MAIN LINAC TO TRANSVERSE AND LONGITUDINAL DYNAMIC IMPERFECTIONS   N. Leros, D. Schulte            Abstract  In the main linac of the Compact Linear Collider (CLIC), longitudinal and transverse dynamic imperfections, such as RF phase jitter, variation of bunch length and movements of elements, can result in significant luminosity loss.  The responses of local trajectory feedbacks to these imperfections are studied in this paper.                 EPAC2002 Conference, 3-7 June, 2002, Paris, France  Geneva, Switzerland 3 July 2002   Feedback Response in the CLIC Main Linac to Transverse and LongitudinalDynamic ImperfectionsN. Leros, D. Schulte, CERN, Geneva, SwitzerlandAbstractIn the main linac of the Compact Linear Collider (CLIC),longitudinal and transverse dynamic imperfections, such asRF phase jitter, variation of bunch length and movementsof elements, can result in significant luminosity loss. Theresponses of local trajectory feedbacks to these imperfec-tions are studied in this paper.1 INTRODUCTIONIn the main linac of CLIC, trajectory feedbacks will beused to reduce the emittance growth and luminosity lossdue to dynamic effects. In practice, several local trajectoryfeedbacks along the linac will re-steer the beam back tothe reference trajectory. The purpose of this paper is firstto study the feedback responses and the emittance growthin the frequency domain due to transverse excitations fordifferent feedback configurations. Next, results concerningthe spectral emittance growth due to quadrupole vibrationsare given. Other dynamic imperfections will also be in-vestigated, such as RF phase and bunch length variationswhich may affect the feedback performances and changethe feedback spectral response.The results concerning these studies have been obtainedby using PLACET [1]. The simulations have been per-formed with a single bunch beam containing 4× 109 parti-cles, assuming a repetition frequency of 100 Hz. The bunchhas no energy dispersion and its length is 35 µm. The verti-cal emittance is y = 5 nm at the entrance of the linac. Onlyresults concerning the vertical emittance are presented.2 FEEDBACK PARAMETERSIn our simulations, a feedback is able to measure thebeam offset and to calculate the correction to apply in orderlocally to re-steer the beam back to the reference beamline.The feedbacks consist of 2 dipole correctors, placed in twoconsecutive vertically focusing quadrupoles, and 3 beamposition monitors (BPMs) with a resolution of 100 nm. Thefirst BPM is in front of the second feedback quadrupole andthe 2 other BPMs in front of the next two vertically focus-ing quadrupoles. The BPMs define the beam angle andposition offsets. The dipoles correct only a fraction g (thegain) of the calculated correction.We will consider 2 kinds of feedbacks. First, to keepthe emittance growth small, a chain of feedbacks along thelinac will operate at a gain g1. Secondly, the feedback atthe IP is considered through a simple model. For that, afeedback which is independent of the previous chain steersthe beam back to the reference beamline. It operates witha higher gain g2 and only 2 BPMs are used to monitor thebeam angle and position.In the linac, the feedbacks of the chain can be indepen-dent, which means they do not exchange information. Inanother feedback configuration, N feedbacks can also bedependent. In this case, they constitute a bin of length N ,where all the feedbacks of the bin communicate. A chain ofk independent feedback bins, each made of N feedbacks,can also be implemented along the linac.The feedback device configuration for the present sim-ulation is based on 40 feedbacks in the linac and 1 feed-back at the IP. In principle, the number of feedbacksand the gain optimisation settings depend on the noisespectrum. For example, it has been shown that for aground motion described by the ATL model [2] withA=0.5×10−6 (µm)2/(sm), 40 independent feedbacks in thelinac minimise the emittance growth for a gain g1 of 0.02.For the present studies, adopting g1=0.01 and g2=0.3 wasfound to be a reasonable choice.3 EMITTANCE AND FEEDBACKRESPONSE CALCULATIONDue to the dynamic imperfections, the beam is disturbedin the linac. When feedbacks operate, the beam is cor-rected. We consider disturbed and corrected beams com-posed of L slices at the end of the linac. A beam pulse ismade of Nb particles and the slice l has Nl particles. Eachbeam pulse p occurs at some time t. We define for eachslice l the position yl(t) and the angle y′l(t), which arefunctions of the time. The fraction of the beam particlesinside the slice is wl=Nl/Nb. The corresponding Fouriertransforms of yl(t) and y′l(t) are given by F [yl](f) andF [y′l ](f). For F [yl](f), we define the real part Fr[yl](f)and the imaginary part Fi[yl](f), with similar definitionsfor F [y′l ](f).The emittance at the pulse p, i.e. at some time t, is com-puted by :(t) ∝√√√√ L∑jwj (yj(t))2L∑kwk(y′k(t))2−(L∑lwlyl(t)y′l (t))2For a single mode with frequency f , the time depen-dent evolution of the emittance is (t) ∝ sin2(2pift). Inorder to average the emittance over 1 total oscillation foreach considered mode of our spectrum, the emittance in thefrequency domain F [](f) is computed from the Fouriertransform function as follows :Fr,i[](f) ∝ {L∑jwj (Fr,i[yj ](f))2L∑kwk(Fr,i[y′k](f))2−(L∑lwlFr,i[yl](f)Fr,i[y′l ](f))2}12F [](f) =12(Fr[](f) + Fi[](f))The power spectral density (PSD) of the emittance is de-fined as :P [](f) =√Fr[](f)2 + Fi[](f)2We will consider the spectral emittance F [](f) for thecorrected beam and the response function of the feedbacksP [](f)cor./P [](f)dist., obtained from the corrected anddisturbed beams.4 FEEDBACK CONFIGURATIONIn this section, we compare the feedback response fordifferent feedback configurations. As explained above,forty-one feedbacks are used. The first 40 feedbacks areorganised into k groups, each of them containing N depen-dent feedbacks. The following cases are investigated :• 40 independent feedbacks (k=40 and N=1)• 10 groups (k=10), 4 dependent feedbacks in eachgroup (N=4).• 4 groups (k=4), 10 dependent feedbacks in each group(N=10).• 1 single group (k=1), 40 dependent feedbacks in onegroup (N=40).We simulate an incoming beam jitter of 1 µm at the en-trance of the linac. It is a single beam step disturbance. Thefeedbacks need some time to compensate for this instanta-neous perturbation.The response function of 40 independent feedbacks(Fig. 1) shows a huge amplification effect between 0.25 Hzto 1.4 Hz. The behaviour of the response function comesfrom the competition of the IP feedback with the linac feed-backs. The spectral emittance growth, as seen at the bottomof the same figure, is huge around 0.8 Hz within a band-width of about 1 Hz. We also see that the correction iseffective below 0.2 Hz.Going to a configuration with less independent feed-backs, we first see that the amplification becomes muchsmaller, as shown in Fig. 2 for 10 groups (k=10). Thedamping now occurs at lower frequency (≤ 0.1 Hz). Inthe case of k=4 (N=10), we see that there is no longer anamplification and the response function is damped below≈ 0.5 Hz. The damping becomes more and more effec-tive when the feedbacks are settled into more groups. For02468101214160 0.5 1 1.5 2 2.5 3P[ε](f) cor. /P[ε](f) dist. Frequency in Hzk=400204060801001201400 0.5 1 1.5 2 2.5 3F[ε](f) in [nm]Frequency in Hzk=40Figure 1: Feedback response and emittance growth dueto the 1µm incoming beam jitter for 40 independent feed-backs.00.20.40.60.811.21.40 0.5 1 1.5 2 2.5 3P[ε](f) cor. /P[ε](f) dist. Frequency in Hzk=1k=4k=10k=40Figure 2: Feedback response to the 1 µm incoming beamjitter for different feedback configurations.a single group configuration where all the feedbacks com-municate, the excitation due to the pulse offset is dampedbelow ≈ 1 Hz.The resulting emittance growth in the frequency domainis shown in Fig 3. When the amplification effect is avoided,the emittance growth is damped below≈ 1 Hz.00.511.522.533.50 0.5 1 1.5 2 2.5 3F[ε](f) in [nm]Frequency in Hzk=1k=4k=10 k=40Figure 3: Emittance growth in the frequency domain dueto the 1 µm incoming beam jitter for different feedbackconfigurations.5 EMITTANCE GROWTH DUE TOQUADRUPOLE JITTERSAll the quadrupoles of the linac are set in motion in thewhole frequency range (0-50Hz) with a very small ampli-tude of 1 nm. For this, 500 frequency modes are used. Theexcitation phases vary randomly from mode to mode andfrom quadrupole to quadrupole. In order to get some statis-tics, we consider five different machines. The investigatedconfiguration consists of 40 feedbacks in a single group(g1=0.01). For a given machine, we simulate 2500 pulses.Comparing the plots of Fig. 4, we see that the time evolu-tion of the emittance growth starts to be periodic after 180pulses. In order to avoid aliasing problems in the Fourieranalysis and to get the feedback reponses in a stable regime,we consider for each machine the pulses 1500 to 2500.0501001502002500 200 400 600 800 1000Pulse numberε y [nm]0501001502002501000 1200 1400 1600 1800 2000Pulse numberε y [nm]Figure 4: Emittance growth during 2000 pulses when thefeedbacks are switched on during quadrupole jitters. Singlegroup configuration consisting of 40 feedbacks.The response function for the 5 machines are very sim-ilar to the corresponding one in Fig. 2. The average spec-tral emittance F[] per frequency mode, for a amplitudequadrupole jitter of 1 nm, is shown in Fig. 5. Above 1 Hz,the expected emittance for a single mode is 0.12 nm, whichis an emittance growth of 2.4 %.00.020.040.060.080.10.120.140 1 2 3 4 5F[ε](f) in [nm]Frequency in HzFigure 5: The average spectral emittance (in nm) by fre-quency mode for a quadrupole jitter amplitude of 1 nm.Single group configuration for the 40 feedbacks.6 RF PHASE AND BEAM LENGTHVARIATIONWe have seen that using communicating feedbacks helpsto reduce the emittance growth due to transverse excitationsin the low frequency range. For this, we have to calcu-late the transport matrices between the feedbacks which arein the same group. Longitudinal dynamic effects changethe transport matrix between two feedbacks and then affectthe feedback correction. Here we evaluate the emittancegrowth induced by bunch length and RF phase variationfor the configuration of a single group defined previously.The linac is pre-aligned through the complete proceduredescribed in [3].The drive beam linac is composed of 22 decelerating sec-tions, which provide the power in the RF structures. TheRF phase can vary separately in each drive beam decel-erator from pulse to pulse. We consider that the RF phasefollows a normal distribution around its nominal value withσ=1◦. We simulate 500 pulses with and without RF phasevariation. To avoid the transient regime, the Fourier analy-sis is performed after the pulse 400. We compute the emit-tance growth of 100 pulses taken as a whole. The emittancegrowth due to the RF phase variation is 4.7 %.PLACET has been modified in order to simulate bunchlength variation from pulse to pulse. We assume that thebunch length follows a flat distribution from 30 to 40 µm.The analysis is done as above. In this case, the emittancegrowth is 1.3 %.7 CONCLUSIONIn this paper we have studied the feedback response todynamic transverse imperfections. Different feedback con-figurations have been studied, composed of 40 feedbacksalong the linac and one separate feedback at the IP. For aconfiguration with the 40 communicating feedbacks alongthe linac, the emittance growth is reduced for excitationfrequency modes below 1 Hz. Then, we have seen thatthe emittance per mode for 1 nm quadrupole jitter gives anemittance growth of 2.4 % above 1 Hz. Below this fre-quency, the studied feedback configuration damps the vi-bration effects. Finally, the effects on this feedback config-uration correction of RF phase and beam length variationhave been investigated. We have found that these longitu-dinal imperfections contribute by an emittance growth of4.7 % and 1.3 % respectively. These studies provide a ba-sis to further optimisations of the feedback configurationfor different noise spectra.8 REFERENCES[1] E.T. d’Amico, G. Guignard, N. Leros, D. Schulte, Simula-tion Package based on PLACET, CERN/PS 2001-028(AE)and PAC01.[2] N. Leros and D. Schulte, Dynamic Effects in the Main Linacof CLIC, CERN/PS 2001-029(AE) and PAC01.[3] N. Leros and D. Schulte, Static Beam-Based Alignment ofthe RF Structures in the CLIC Main Linac, This conference.

Feedback response in the CLIC main linac to transverse and longitudinal dynamic imperfections

Abstract

Similar works

Full text

Available Versions

CERN Document Server

CERN Document Server