CERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH STUDY OF PRE-ALIGNMENT TOLERANCES IN THE RTML

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1 CERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CLIC Note 943 STUDY OF PRE-ALIGNMENT TOLERANCES IN THE RTML Thibaut Lienart, CERN, Geneva, Switzerland Abstract CERN-OPEN /5/212 In this document a study of the impact of jitter and static misalignment on the elements of the Ring To Main Linac transport (RTML) on the emittance growth through Monte Carlo simulations using the tracking code PLACET is presented. Tolerances are proposed for the dynamic alignment requirements of the RTML in order to meet the budget emittance growth. A study of the impact of static misalignment and correction thereof with basic Beam-Based Alignment techniques is also presented and resulting tolerances are proposed for the prealignment phase of the machine. Geneva, Switzerland May 212

2 CERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CLIC Note XXXXXX STUDY OF PRE-ALIGMENT TOLERANCES IN THE RTML Thibaut Lienart CERN Abstract In this document a study of the impact of jitter and static misalignment on the elements of the Ring To Main Linac transport (RTML) on the emittance growth through Monte Carlo simulations using the tracking code PLACET is presented. Tolerances are proposed for the dynamic alignment requirements of the RTML in order to meet the budget emittance growth. A study of the impact of static misalignment and correction thereof with basic Beam-Based Alignment techniques is also presented and resulting tolerances are proposed for the pre-alignment phase of the machine. Geneva, Switzerland April 2, 212

3 1 Contents 1 Introduction The RTML Method I Dynamic effects 5 1 Quadrupoles Transverse Jitter Global Jitter Individual Jitter Power Ripples Injection 9 3 Cavities (Coherent) Gradient jitter Global jitter Individual Jitter (Coherent) Phase jitter Global jitter Individual Jitter II Static misalignment 14 1 One-to-one correction General approach Tolerances obtained Dispersion-free steering General approach Tolerances obtained Additional tests Conclusion 22 Acknowledgments 22 REFERENCES 22

4 2 APPENDIX 23 A Correction algorithms 23 A.1 One-to-One correction (1:1) A.2 Dispersion Free Steering B Complementary plots 28 B.1 Dynamic B.2 Static

5 3 1 Introduction 1.1 The RTML As stated in [Stu1], the ring to main linac transport (RTML) connects the damping rings and the main linacs. Its aims are multiple: 1. spin rotation of the electrons (Spin Rotator), 2. bunch compression (Bunch Compressor 1 and 2), 3. acceleration (Booster), 4. transport. In summary, it has to match beam properties (... ) from the values delivered by the damping rings to the values required by the main linacs. A sketch of the RTML is shown in figure 1 below. 1 Bunch Compressor 2 Main Linac Main Linac Bunch Compressor 2 e- Central Arc & e- Long Transfer Line Vertical Transfer Long Transfer Line e+ Turn Around Loop Spin Rotator Booster Bunch Compressor 1 e+ Central Arc & Vertical Transfer Turn Around Loop Figure 1: Sketch of the RTML. In blue the electron line, in red the positron line. Cavities are illustrated in green. In the sequel, we will use acronyms to indicate the different main parts of the RTML as follows: SR for Spin Rotator, BC1 and BC2 for the two bunch compressors, BOO for the booster, CA for the Central Arc, VT for the Vertical Transfer, LTL for the Long Transfer Line and TAL for the Turnaround Loop. As can be seen by looking at the figure 1, the e+ line is simpler than the e line which has a spin rotator and a central arc. This is the reason why we will mostly focus on thee line assuming that if the specifications are met with this line, they should also be met by the other line. The specifications [Sch1],[Stu1], are presented in table 1 and table 2 below. In this report we will mostly focus on the tolerances needed to meet the budget emittance-growth in the presence of jitter and static misalignment. 1 This sketch was adapted from [Stu1]. It is not to scale.

6 4 Property Symbol Unit start end Particle energy E GeV Bunch charge q nc.65 >.6 RMS bunch length σ s µm RMS energy spread σ E /E %.13 < 1.7 Normalized emittance ǫ x nmrad 5 < 6 ǫ y nmrad 5 < 1 Table 1: Nominal beam properties at both ends of the RTML [Stu1]. Property Symbol Unit Design Static Dynamic Horizontal Emittance Growth ǫ x nm Vertical Emittance Growth ǫ y nm Table 2: Emittance budgets for the RTML [Sch1]. As can be seen in the table above, the budget for the emittance growth is rather tight, especially in the vertical plane, and are split in three parts. The design part designates the natural emittance growth of the machine when it is perfectly aligned (due to non-linearities and stochastic effects), the dynamic part corresponds to the jitter and the static part to the static misalignment. 1.2 Method This document is split in two main parts: the first part studies the effect of jitter or dynamic perturbation on the emittance growth in the RTML, the second part studies the application of correction algorithms to counter emittance growth in the presence of static misalignment and the tolerances obtained after application of these algorithms. The budget for each of these two parts is computed by adding the budget for the design emittance growth and the respective budget for either the emittance growth tolerated for jitter or for static misalignment. Finally, the simulations were run with the tracking code PLACET [Sch8], based on previous work by F. Stulle and on the latest lattice design available. 2 2 This design is namedbaseline_3tev_211-6_v1 and can be found athttp://clicr.web.cern.ch/ CLICr/MainBeam/RTML/Baseline_3TeV_211-6_v1/.

7 5 Part I Dynamic effects In this first part, the aim is to present the impact of dynamic jitter on the emittance growth in the RTML. The dynamic jitter is the noise that cannot be corrected from one train to another as its frequency is higher than the repetition rate (of5hz). We assume that the machine remains static within one train as the bunches within a train are only.5ns apart so we study the effect from train to train. Typically, between 1 and 5 particles are used for the simulations with between 1 and3 seeds for each perturbationσ considered. As the emittance of the beam in the horizontal and vertical plane is linked to the determinant of the covariance matrix of thexx andyy ellipse, a quadratic evolution of the average emittance growth with the perturbationσ is expected. The budget emittance growth in this part is of 3nm (1nm design, and 2nm dynamic) in the vertical plane and8nm (6nm design, and2nm dynamic). 1 Quadrupoles 1.1 Transverse Jitter Global Jitter In this case the jitter is applied on all elements of the RTML and we consider the global emittance growth at the end. The results are illustrated at the figure 2 below. 16 e- line e+ line Budget e- line e+ line Budget 1 x [nm] 12 y [nm] Perturbation [nm] Perturbation [nm] Figure 2: Average horizontal and vertical emittance growth ( ǫ x,y ) in thee /e+ lines of the RTML when the quadrupole jitter RMS goes fromto1 nm.

8 6 In the previous figure, it can be observed that the evolution of the emittance growth goes quadratically with the perturbation as was expected. One can also observe that the impact of transverse jitter is much more harmful in the vertical plane than in the horizontal plane. This is due to the fact that the input beam emittance in the vertical plane is extremely small and hence very sensitive to perturbations. The figure 3 below is a magnified version of the previous plot in order to better show the crossing of the budget line in the vertical plane. e- line e+ line Budget e- line e+ line Budget 8 5 x [nm] 7 6 y [nm] Perturbation [nm] Perturbation [nm] Figure 3: Zoom on the average horizontal and vertical emittance growth in thee /e+ lines of the RTML. The global average tolerance is thus of12nm (taking the tightest value, i.e., the one corresponding to the vertical plane). We will now investigate which subsystem produces most emittance growth. A similar study for the multipoles can be done but it seems that the impact of transverse jitter of multipoles on the emittance growth is negligible as can be seen on figure 17 in Appendix B Individual Jitter In this case, each subsystem is considered separately assuming a perfect machine except in the subsystem. We focus on the emittance growth in the vertical plane of the electron line as it seemed to be the tightest case. The figure 4 below illustrates the results obtained.

9 7 y [nm] 12 8 y [nm] 6 3 SR BC1 BOO CA VT LTL TAL BC2 Budget Perturbation [nm] Perturbation [nm] 12 6 y [nm] 8 y [nm] Perturbation [nm] Perturbation [nm] Figure 4: Average horizontal (left) and vertical (right) emittance growth of the e (top) and e+ lines in each subsystem of the RTML when the individual jitter RMS of each subsystem goes fromto1 nm (values well above twice the budget are not displayed). On the previous plots, one can easily observe that the arcs are the most susceptible to generate emittance growth under transverse jitter. This is due to the fact that their lattice is complex and contains many non-linear elements (multipoles). Also, the curve corresponding to the central arc of thee+ line is obviously very flat as it is not a full arc but merely a match from the Booster to the Vertical Transfer. The tolerances obtained for the tightest systems are illustrated in the table 3 below.

10 8 Subsystem Upper bound forσ [nm] TAL 2 CA 25 VT 65 BC2 9 Table 3: Average tolerances for transverse jitter in critical subsystems (based on ǫ y ine line). The subsystems that that are not included in the table above have an emittance growth below the budget with a perturbation RMS below1nm. 1.2 Power Ripples In this case, each magnet is powered assuming independent power ripples. 3 Typical values for the power ripples are of the other of It can be verified at the figure 5 below that the budget is met for power ripples of that order. 16 e- line e+ line Budget 4 e- line e+ line Budget x [nm] 12 y [nm] Perturbation [%].25.5 Perturbation [%] Figure 5: Average horizontal and vertical emittance growth in the e /e+ lines of the RTML when the power ripples RMS goes from.1% to.5% (values well above twice the budget are not displayed on the left graph). 3 In practice it is expected that quadrupoles be powered in sectors and hence not independently which could account for additional emittance growth. As the sectoring was not known during the study, it was not considered. 4 This information was kindly provided by A. Dubrovskiy.

11 9 2 Injection In this case we consider that the beam enters the RTML with an offset linked to second moments (denoted σx in and σy in ) of the distribution of particles at the injection. The figure 6 illustrates the emittance growth with the offset fraction. 2 e- line e+ line Budget e- line e+ line Budget x [nm] y [nm] Perturbation [offset fraction] Perturbation [offset fraction] Figure 6: Average horizontal and vertical emittance growth in the e+/e lines of the RTML when the offset at injection goes from.1 σx in RMS to1 σx in RMS (resp. from.1 σy in RMS to1 σy in RMS). As can be observed in the figure above, a fraction above.3 σ in x in the horizontal plane seems to be the limit that can be accepted before reaching the budget emittance-growth in the horizontal plane. Typical injection errors are of the order of 1% of the transverse beam size.

12 1 3 Cavities Cavities are present in the three following subsystems: the Bunch Compressors (BC1 and BC2) and the Booster. In the first two, the particles are not supposed to be accelerated and the bunch is supposed to pass at the zero crossing of the cavities. In the last, the particles are accelerated from 2.86 GeV to9gev by passing on crest. Having a bunch crossing the cavities with a phase offset or a perturbed gradient will result in an acceleration or deceleration of the particles resulting in a bunch shaped differently than the nominal bunch at the end of the RTML. One thing to note is that there is a coupling between the two effects. A fluctuation in gradient in the BC1/BOO will imply an energy offset in the BC2 which will have a rather big impact in the offset at the output of the RTML. In this section, we look at two properties: the arrival time or offset of arrival with respect to the nominal bunch and the bunch length. The first property is not too difficult to correct but not the second which is important for the luminosity and should not wander too far from the nominal 44µm. The jitter is assumed to be coherent i.e., the same offset is applied on all the cavities which corresponds to the fact that, in principle, the cavities are powered by the same source. 3.1 (Coherent) Gradient jitter Global jitter In this case, we consider that each of the three subsystems undergoes a coherent gradient jitter (the jitter is assumed independent between subsystems). The range of relative gradient jitter RMS in percent studied goes from1to1 percent. The results obtained are illustrated at the figure 7 below. 46 e- line e+ line 6 e- line e+ line s [ m] s [ m] Perturbation [%] 5 1 Perturbation [%] Figure 7: Average offset and bunch length at the end of the RTML when the gradient in the cavities fluctuates around the nominal value with an RMS going from1to1%.

13 Individual Jitter In this case, we consider that each of the three subsystem individually undergoes a coherent gradient jitter with, as before, the range of gradient jitter RMS from 1 to 1 percent. The results obtained are illustrated at the figure 8 below. BC1 Booster BC2 BC1 Booster BC s [ m] 45 s [ m] Perturbation [%] 5 1 Perturbation [%] BC1 Booster BC2 BC1 Booster BC s [ m] 41 4 s [ m] Perturbation [%] 5 1 Perturbation [%] Figure 8: Average offset and bunch length at the end of the RTML in thee (top) ande+ (bottom) lines when the gradient in the cavities of each of the three subsystems fluctuates around the nominal value with an RMS going from1to1%. One can see on the figure above that the BC1 seems to have the biggest impact, this is due to the fact that the effect of jitter in the BC1 results in an induced effect in both the Booster and the BC2 resulting in a increased impact at the end of the RTML. It is not so much the case for the Booster which, individually, has a smaller impact (because the jitter is around the peak value) and hence the cumulated impact in the BC2 is not too big.

14 (Coherent) Phase jitter Global jitter In this case, we consider that each of the three subsystems undergoes a coherent phase jitter (the jitter is assumed independent between subsystems). The range of phase jitter RMS in degrees goes from.5 to5degrees. The results obtained are illustrated at the figure 9 below. 2 e- line e+ line 52 e- line e+ line s [ m] s [ m] Perturbation [deg ] Perturbation [deg ] Figure 9: Average offset and bunch length at the end of the RTML when the phase in the cavities fluctuates around the nominal value with an RMS going from.5 to5deg. The fluctuations in arrival time due to the phase jitter are very irregular as one can see and should probably be investigated further. However, as mentioned before, the arrival time of the bunch can be corrected easily and hence this property is not crucial.

15 Individual Jitter In this case, we consider that each of the three subsystems individually undergoes a coherent phase jitter with, as before, the range of phase jitter RMS from.5 to 5 degrees. The results obtained are illustrated at the figure 1 below. 15 BC1 Booster BC2 BC1 Booster BC s [ m] 5 s [ m] Perturbation [deg ] Perturbation [deg ] BC1 Booster BC2 BC1 Booster BC s [ m] s [ m] Perturbation [deg ] Perturbation [deg ] Figure 1: Average offset and bunch length at the end of the RTML in the e (top) and e+ (bottom) lines when the phase in the cavities of each of the three subsystems fluctuates around the nominal value with an RMS going from.5 to5deg. For the bunch length, the conclusion is the same as for the individual gradient jitter: there is a cumulated effect for the BC1 resulting in a bigger impact.

16 14 Part II Static misalignment In this part, we consider the effect of the static misalignment of the quadrupoles on the emittance growth and the correction of that effect using two methods of Beam-based Alignment: one-to-one correction (1 : 1) and Dispersion Free Steering (DFS). In the sequel, we will denote by σ pos the RMS of the misalignment of both the quadrupoles and the BPM. For each subsystem of the RTML a range of σ pos was considered looking for the point where the budget emittance growth was reached. This information can then be used in a pre-alignment phase of the construction of the RTML. The RMS of the resolution of the BPM will be denoted byσ res and, by default, is set toσ res = 1µm. The VT is assumed to have similar or slacker tolerances than the CA and was hence not considered. All other subsystems however have been tested with one or both correction algorithms to find the tolerances required to meet the budget emittance growth of8nm (6nm design,2nm static) in the horizontal plane and3nm (1nm design,2nm static) in the vertical plane. Finally, in this part, we focus on the emittance growth in the vertical plane of the electron line which is clearly what constrains most the tolerances on the pre-alignment of each subsystem. 1 One-to-one correction 1.1 General approach For each subsystem, the response matrix of the subsystem was computed with 1 particles or more assuming a perfect machine. The response matrix obtained should thus be close to the theoretical one which could be obtained directly from the lattice. In an imperfect machine, a bunch is generated and tracked in the subsystem. Corrections based on the Beam Positioning Monitors (BPM) readings and using the response matrix are then applied (cf. Appendix A.1 for a mathematical description of the algorithm). A second bunch is then tracked in the subsystem in order to obtain the emittance growth at the end of the subsystem in the corrected machine. This is done for a hundred seeds for each static misalignment RMS tested such as to obtain an average emittance growth after correction. The tolerance is then interpolated from the results by observing where the graph of the emittance growth against static misalignment RMS crosses the budget value. The effect of the correction on the orbit is illustrated in the case of the LTL in the figure 11 below.

17 15 O rbit ex cursion [mm] 2-2 N o correction 1:1 correction BPM positions BPM ind ex Figure 11: Orbit in the vertical plane before and after one-to-one correction in the LTL (σ pos = 2µm). For the LTL, the system is perfectly linear and the response matrix obtained is well-conditioned which accounts for the near-perfect matching of the corrected orbit to the BPM. The plot of the emittance growth in the vertical plane with respect to the misalignment RMS of the quadrupoles and BPM in the LTL is represented at the figure 12 below. 15 1:1 (Averag e) 1:1 (+9%) y-- [nm] Perturbation -- [ m] Figure 12: Average and 9-th percentile of the vertical emittance growth in the LTL after 1 : 1 correction when σ pos goes from5µm to3µm.

18 16 In the arcs, this method is not sufficient to even have the tracked beam passing through the machine let alone to have a low emittance growth. It is thus necessary to use a binning technique. In that case, the arcs are separated in bins of a given number of arc cells which are overlapping. The tracking and the correction can then be done in each bin one after the other in order to have shorter parts where the non-linearity of the machine (due to the presence of multipoles) is not overwhelming. This binning cannot be made arbitrarily small as it should allow for a sufficient sampling to have a good correction in both planes. Typically a minimum of 4 full betatronic oscillations are required. There is a tradeoff between proper sampling and avoiding non-linearity. In [Sch1], it is specified that the phase advance in one arc cell is of 2.8π (resp..8π) in the horizontal (resp. vertical) plane. Here we have used and compared the three following splitting scheme (4 : 2) i.e. 4 cells/bin with 2 cells/overlap. The phase advance of a bin is(11.2π,3.2π), (6 : 3) i.e. 6 cells/bin with 3 cells/overlap. The phase advance of a bin is(16.8π,4.8π), (8 : 4) i.e. 8 cells/bin with 4 cells/overlap. The phase advance of a bin is(22.4π,6.4π). A slight improvement in the tolerances can be observed when using the dispersion free-steering method and increasing the number of cells per bin but here, for tolerances obtained for the 1 : 1 corrections the results presented assume the first splitting scheme. 1.2 Tolerances obtained The table 4 below summarizes the average tolerances needed at the pre-alignment step in order to respect the budget emittance growth of 8nm in the horizontal plane and 3nm in the vertical plane. Subsystem Horizontal Vertical SR > 25 (> 25) 1 (75) BC1 8 (55) 17 (11) BOO > 1 (9) 31 (21) CA (VT) 26 (16) 1 (5) LTL > 3 (> 3) 127 (82) TAL > 2 (17) 6 (4) BC2 > 2 (> 2) 1 (< 1) Table 4: Average tolerance inµm obtained after applying1 : 1 correction. Between brackets is the value corresponding to the 9-th percentile curve. In the above table, some of the values are marked with a > or < sign indicating that the corresponding curve did not meet the budget line in the range ofσ pos considered. A value can be

19 17 extrapolated assuming a quadratic evolution but it is not necessary as the vertical value is the one which effectively matters and for which a value was obtained in the range studied. As can be observed, both the Spin Rotator and the Long Transfer Line seem to accept rather slack tolerances in the alignment of their quadrupoles and BPM. For the other subsystems however the1 : 1 correction alone does not seem to provide a sufficient correction and unreasonably tight tolerances are required to meet the budget emittance growth. In the next point, we consider the additional application of the Dispersion-Free Steering method hoping it will further slacken the tolerances. Remark: the TAL is twice as long as the CA and, assuming a quadratic dependence of the emittance growth withσ pos, a factor 2 should, in principle, link the tolerance of the CA and the TAL. This can indeed be roughly verified in the simulations when comparing tolerances. 2 Dispersion-free steering 2.1 General approach For each susbsysten, the response matrix of the subsystem for a test beam at altered energye (1+δ) was computed with1 particles or more assuming a perfect machine (cf. Appendix A.2 for a mathematical description of the algorithm). In the case of the arcs, several sets of response matrices were computed corresponding to test beams at different energies and to different splitting schemes. In an imperfect machine, a bunch is generated and tracked in the subsystem. Corrections based on the BPM readings and on the dispersion measured are applied using the different response matrices. A second bunch is then tracked in the subsystem in order to obtain the emittance growth at the end of the subsystem in the corrected machine. This is done for a hundred seeds for each static misalignment RMS tested such as to obtain an average emittance growth after correction similarly to what was presented in the1 : 1 case. The figure 13 below illustrates the orbit after 1 : 1 and DFS correction in the first bin of the Central Arc.

20 18 O rbit ex cursion [mm] 4-4 N o correction 1:1 correction D FS correction BPM positions BPM ind ex Figure 13: Orbit in the vertical plane before and after corrections in the first bin of the CA (σ pos = 2µm, ω = 1). On the above picture, one can easily observe that the orbit after applying DFS is much closer to the ideal orbit than the orbit after having just applied the1 : 1 correction. The energy difference is assumed to be generated by the Booster (this can be done by tracking a beam with the cavities of the Booster working slightly off-crest). However, as the arcs correspond to fairly complex lattice, a maximum relative energy difference δ can be used without the test-beam touching the aperture. Talks with F. Stulle suggested a maximum δ of half a percent. Additional quick tests showed that this value was a bit pessimistic and could be increased three folds to1.5%. Both cases were tested. In the Bunch Compressors and in the Booster, one can track a test-beam with the cavities having a slight phase advance (around ±5deg, here 5deg were used 5 ) such as to pass slightly off the zero-crossing. An energy increase or decrease (depending on the sign of the phase perturbation) is then propagated in the subsystem. Finally, for eachσ pos considered, a range of weightω were tested and the best value retained. A typical plot of the improvement over 1 : 1 with respect to the weight parameter is presented in the Appendix A.2, figure 16. The figure 14 below illustrates the case of the Turn Around Loop comparing the emittance growth after only1 : 1 and after DFS considering the bestω for each point. 5 One could maybe consider larger offsets resulting in a larger energy difference and hence in a better resolution on the measured dispersion.

21 19 y-- [nm] :1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) Perturbation -- [ m] Figure 14: Average and 9-th percentile of the vertical emittance growth in the TAL after DFS correction and comparison with 1 : 1 when σ pos goes from5µm to 5µm. Symmetric test-beams at δ = ±1.5% are considered with the 8 : 4 splitting scheme. The optimal weight factor for the different σ pos considered is aroundω = 1. On the above figure, one can infer for example the average tolerance based on DFS at around 15µm. All the plots corresponding to the various cases with the best settings tested are presented in the Appendix B.2. The tolerances obtained off these various plots are summarized in the next point.

22 2 2.2 Tolerances obtained The table 5 below summarizes the average tolerances needed at the pre-alignment step in order to respect the budget emittance growth of 8nm in the horizontal plane and 3nm in the vertical plane. The Spin Rotator is not present as a test cannot be generated at its entrance. In this table, the default4 : 2 splitting scheme is assumed for the arcs and an energy difference ofδ =.5% is used except in the Bunch Compressors and the Booster where a phase advance of5deg is used. Subsystem Horizontal Vertical Improvement [%] BC1 > 1 (> 1) 6 (35) 25 BOO > 1 (> 1) 5 (3) 61 CA (VT) > 5 (> 5) 14 (7) 4 LTL > 3 (> 3) 28 (15) 12 TAL > 2 (> 2) 9 (6) 33 BC2 > 2 (> 2) 4 (2) 3 Table 5: Average tolerance in µm obtained after applying DFS correction. Between brackets is the value corresponding to the 9-th percentile curve. The last column corresponds to the relative improvement of the average tolerance of the vertical plane with respect to1 : 1 correction. As can be observed in the above table, even though a big improvement relatively to the tolerances obtained with the 1 : 1 correction, the tolerances remain very tight especially for the arcs and the BC2. One can also observe that the arcs having a similar design, the gain over 1 : 1 is similar. The improvement is very big for both bunch compressors which can be explained by the greater energy offset achieved through tracking off the zero crossing. In the Booster, this effect is less present as we run slightly off crest and the energy difference is smaller. 2.3 Additional tests A couple of parameters can in principle be modified to improve further the performances of DFS. The resolution of the BPM can be improved, two test-beams can be used, different splitting schemes can be tested and a higher energy differences can be applied. Here we present briefly the improvement observed in two cases. Improvement of the resolution If the resolution of the BPM improves by a factor 1 from 1µm to.1µm, the dispersion is more accurately measured and hence, in principle, can be more accurately corrected. The table 6 below

23 21 summarizes the results obtained with better BPM for both arcs and the BC2. Subsystem Vertical Improvement [%] CA 19 (1) 35 TAL 12 (7) 33 BC2 4.5 (2.5) 12 Table 6: Average tolerance inµm obtained after applying DFS correction with BPM with a ten times better resolution. The last column corresponds to the relative improvement of the average tolerance with respect to the previous tolerances obtained with less precise BPM. Two test-beams, bigger bins and higher energy offset If we use two test-beams, we gain a factor 2 in resolution over the measured dispersion. 6 If we useδ = 1.5%, we gain an additional factor3and bigger bins can, in principle, improve the sampling (here the8 : 4 splitting scheme is used). The table 7 below summarizes the results obtained for both arcs. Subsystem Vertical Improvement [%] CA 24 (12) 7 TAL 15 (9) 66 Table 7: Average tolerance in µm obtained after applying DFS correction with two test-beams, the 8 : 4 splitting scheme and δ = 1.5%. The last column corresponds to the relative improvement of the average tolerance with respect to the previous tolerances obtained with one test-beam, using 4 : 2 splitting scheme andδ =.5%. The effect of having a different splitting scheme is empirically negligible. Hence, using two test-beams with a larger energy difference must have very big impact. Indeed, using BPM with a 1 time better resolution has an impact 1/(3 2) 2.3 times bigger on the improvement of the resolution than the current case but the tolerance improvement is only half as good. 6 cf. Appendix A.2.

24 22 3 Conclusion In this part, the results corresponding to the application of two class of correction algorithms for the misalignment of the quadrupoles and BPM in the subsystems were presented. The subsystems can be split in the three following groups: 1. SR and LTL: the tolerances are moderate (of the order of 1µm or more after applying correction algorithms) 2. BC1 and BOO: the tolerances are tight (of the oder of5µm or more after applying correction algorithms) 3. Arcs and BC2: the tolerances are too tight (below25µm after applying correction algorithms) For the second group, one can reasonably think that using two test-beams and a bigger phase offset (especially in the Booster) with a fine tuning of the weight parameter could bring the tolerances to around1µm. For the third group however, other techniques and tests should be considered as fine tuning of the current parameters is unlikely to bring a factor 5 improvement in tolerances. For this group, looking for points in the lattice of each subsystems which could generate a bump in the emittance growth would also be interesting. References Acknowledgments, After six months spent at CERN thanks to the CERN Technical Student program, I would like to particularly thank my supervisor, Andrea Latina, for his friendly help, his patience and his availability. I would also like to sincerely thank Daniel Schulte for his precise and useful comments and for making me feel welcome in the CLIC Beam Physics team. Finally I would like to thank Cedric Hernalsteens and Nicolo Biancacci for their kind support and useful discussions. [Sch8] [Sch1] [Stu1] Daniel SCHULTE, The Tracking Code PLACET. Available online athttp://isscvs.cern.ch/ cgi-bin/viewcvs-all.cgi/placet-documentation/placet.pdf Daniel SCHULTE, Highest Luminosity: Preservation of ultra-low emittances, CLIC Conceptual Design Report draft. Available online athttp://project-clic-cdr.web.cern.ch/ project-clic-cdr/drafts/highlumi.pdf. Frank STULLE, Ring To Main Linac transport (RTML), CLIC Conceptual Design Report draft. Available online at Drafts/rtml.pdf.

25 23 A Correction algorithms We consider a part of an accelerator structure (e.g. a transfer line) with a set of n Beam-Position- Monitors (BPM) and of m dipole correctors. We consider the following linear model relating the variation of the BPM readings u R n with a variation of the dipole kick applied at the correctors θ R m through the application of a response matrixr R n m : 7 u = R θ. (A.1) Assuming this (forward) model, the response matrixrcan be measured or, in our case, be computed in an system in a given configuration. In the sequel we will neglect the error on this response matrix but, in practice, there is a possibly non negligible error due chiefly to BPM imperfections. In the sequel, we will denote byr 1 the response matrix obtained for a beam at design energy E and byr 2 the response matrix obtained for a beam at energye (1+δ) for a givenδ.the reason for considering a beam at a slightly different energy will be made clear in the point about the Dispersion-Free-Steering (DFS) method. A.1 One-to-One correction (1:1) We consider a misaligned transfer line where both quadrupoles and BPMs are (supposedly independently) misaligned in the xy plane with a misalignment drawn from a Gaussian distribution centered at zero and of standard deviation σ pos. Each BPM also has a certain resolution with standard deviationσ res (we don t consider the scale error of the BPM here). Figure 15: Illustration of a reading (ũ) of a misaligned BPM (static misalignmentm) with a resolution error (e). Let us denote byu the BPM readings observed after tracking a beam at nominal energy in a perfect machine (i.e., the ideal or target orbit 8 ). Let us denote by ũ 1 the BPM readings observed 7 This model assumes a perfectly linear subsystem. In a subsystem were non-linearity appears (due to multipoles), one can consider smaller subdivisions where it can then be assumed that the departure from the model is not too critical. 8 Usually, u =, corresponding to a beam centered with respect to the machine elements.

26 24 after tracking a beam at nominal energy in the misaligned machine. Let m R n denote the misalignment vector of the BPM and e i R n the random vector corresponding to the error due to their resolution for thei th measurement, we have: ũ 1 = u 1 m e 1, (A.2) which is illustrated at figure 15. The problem is then to find the best 9 kick variation θ 1 to correct the difference u 1. = (u1 u ). Using the forward model, this amounts to solving the following problem: 1 θ 1 = arg min θ R m u 1 R 1 θ 2 2, (A.3) where the minus sign indicates that this kick corrects the effect. The solution of this optimization problem is simply given by θ 1 = R 1 u 1 (A.4) where R 1 denotes the pseudo-inverse of R As we obviously don t know the misalignment vectormnor the resolution errore 1, one can only approach this ideal correction by applying θ 1 = R 1 ũ 1, (A.5) with then θ 1 θ 1 = R 1 (m+e 1). (A.6) This difference between the estimated correction and the ideal one can be big if some components of m are big which can be harmful for the correction (note that usually σ pos σ res and hence m 2 e 2 ). This will bring us further on to consider an additional correction which is not sensitive to this misalignment vector. Before that, we define another vectorũ 1 corresponding to the BPM readings after tracking a beam in the corrected machine with then: ũ 1 = u 1 m e 1, (A.7) where the true orbit after correction is given by u 1 = u 1 +R 1 θ 1 (A.8) = u 1 +R 1 θ 1 +R 1 R 1 (m+e 1) (A.9) = Au +Bu 1 +A(m+e 1 ), (A.1) 9 Here, best is to be understood in the l 2 -sense. 1 If the response matrix used is very ill-conditioned, indicating possible redundancy in the effect of some dipoles or in the readings of some BPMs, one might want to add a regularization term penalizing candidates with large kicks or using a SVD decomposition of the matrix in order to drop some of the BPMs. 11 R 1 = ( R t 1R 1 ) 1 R t 1. In Octave or Matlab one should use the solving operator: θ 1 = R 1\ u 1.

27 25 where, at the last line, we have defined the two matricesa,b R n n as follows: A. = R 1 R 1 I n (A.11) B. = I n A n. (A.12) The two matrices are respectively equal to the identity matrix and the null matrix if R 1 is of full rank. This is the case for the LTL and as the system is perfectly linear, the corrected orbit is nearly perfectly matched to the position of the BPM as was illustrated at figure 11 above. A.2 Dispersion Free Steering Once the machine has been corrected with the one-to-one algorithm, 12 we can track a beam at a different energye (1+δ) (test-beam) and define the vectorũ 2 corresponding to the BPM readings of this new beam. As before, the corresponding true orbit is given by ũ 2 = u 2 m e 2. (A.13) We thus have the two orbitsu 1 andu 2 in the one-to-one corrected machine. If we denote byu the BPM readings corresponding to the tracking of a beam at energye (1+δ) in the perfect machine, the difference (u u ) is proportional to the dispersion d taken at the BPM: 13 In a similar fashion, for the misaligned machine, we can define η. = u u = δd. (A.14) η. = u 2 u 1 = δd (A.15) where d is then the dispersion of the misaligned machine at the BPM. We can then apply a kick variation at the correctors in order to correct the difference between the dispersion in the misaligned machine and the target dispersiond. Using a similar development as in the 1:1 correction case, we obtain the ideal correction: θ 2 = D η (A.16) withd. = (R 2 R 1 ) and η. = (η η ). As the BPM are not perfect, we actually have: η = η η (A.17) = ũ 2 ũ 1 η (A.18) = η e 3 (A.19) 12 Or even without this preliminary step. u 1 is then simply the true orbit when tracking a beam at nominal energy in the misaligned and uncorrected machine. 13 It corresponds to the so-called R 16 term taken at the BPM. See [Stu1] for an illustration of this term in an arc cell.

28 26 where e 3. = e2 e 1. Hence the error term is normally distributed around zero with variance2σ2 res and does not depend on the misalignment of the BPM. We can actually estimate the accuracy of the measured difference of dispersion by noting that ) ( η η δ N ( ) d d, 2σ2 res δ 2 I n, (A.2) and hence that the resolution on the measurement of the difference of dispersion is proportional to 2σ res / δ. This indicates that the relative energy difference δ of the test-beam should not be too small and the resolution of the BPM not too bad in order to hope for a positive effect of the correction. Coming back to the description of DFS, if we add an additional correction kick θ 2 with θ DFS 2 = D η (A.21) to the machine in order to match the dispersion in the corrected machine with the target dispersion, there will also be an impact on the orbit measured. Hence, the optimal correction should be such that the orbit measured stays close to the target orbit (1-to-1 correction) and such that the dispersion matches the target dispersion (DFS) with a balance factorω R +. The formulation of this optimization problem is then θ 2 = arg min θ R m ũ 1 R 1 θ 2 2 +ω2 η D θ 2 2, (A.22) with possibly the addition of a regularization factor β 2 θ tot 2 2 which constrains the total kicks applied at the correctors to not be too large. To obtain an estimation of the regularization factor, we note that the measured difference ũ 1 has a variance(σpos 2 +σres) 2 around the true difference whilst the measured difference θ 2 has a variance 2σres 2 around the true difference. To balance the two,ω should then be close to ω 2 th. = σ2 pos +σres 2, (A.23) 2σ 2 res which can be interpreted as the relative confidence we have in the respective mismatch terms measured. For example, in the case where the BPM have an extremely good resolution, a very big weight should be attached at accurately matching the dispersion. The figure 16 below illustrates the relative improvement in emittance growth over only using 1 : 1 correction with various weights ω in the case of the BC2 with a phase offset in the cavity of 5deg.

29 27 Relativ e improv ement [%] x(av g.) x(+9%) y(av g.) y(+9%) W eig ht parameter Figure 16: Average and 9th percentile relative improvement in % of the emittance growth obtained using DFS compared to the one obtained using only 1 : 1 correction. The theoretical weight parameterω th is represented as a vertical line (here ω th 14.2). On the figure above, one can see that the theoreticalω is, in this case (and it is true in the other cases as well), a good guess or starting point. Also one can see that the relative impact of DFS in the vertical plane is much more pronounced in the vertical plane which is obviously linked to the fact that the emittance constrain in the vertical plane is extremely tight. Finally, one should note that, even though we observe a huge relative improvement over1 : 1, the method still does not seem sufficient to meet the budget emittance growth in the vertical plane as was stated earlier on in the report. Coming back to the general description of the method, one can seek to further improve results using a second test-beam tracked in the machine with the symmetric energy difference δ. The principle stays the same but it permits to avoid possible asymmetries and to have a better resolution on the dispersion difference. We define η +. = u + 2 u 1 = δd (A.24) η. = u 2 u 1 = δd, (A.25) with then ( η + η ) δ = (ũ+ 2 ũ 2 δ ) N ( ) d d, σ2 res δ 2 I n. (A.26) We thus gain a factor 2 on the resolution on the measurement of the difference of dispersion. The response matrix to use is obviouslyd 2 = (R + 2 R 2 ). Remark: another advantage of using two test-beams is that, analogously to the centered difference numerical derivation scheme, one gain an order of precision on the dispersion with respect to the use of only one test beam. In practice, using two test-beams improves rather dramatically the orbit correction as was previously shown for the arcs.

30 28 B Complementary plots B.1 Dynamic Global transverse jitter of multipoles e- line e+ line e- line e+ line 8 4 x [nm] 6 y [nm] Perturbation [nm] 5 1 Perturbation [nm] Figure 17: Average horizontal and vertical emittance growth in the e /e+ lines of the RTML when the jitter RMS of the multipoles goes fromto1 nm. B.2 Static Misalignment in the SR 1:1 (Averag e) 1:1 (+9%) 15 1:1 (Averag e) 1:1 (+9%) x-- [nm] 12 8 y-- [nm] Perturbation -- [ m] Perturbation -- [ m] Figure 18: Average and 9-th percentile of the horizontal and vertical emittance growth in the SR after 1 : 1 correction when σ pos goes from1µm to25µm.

31 29 Misalignment in the BC1 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) x-- [nm] 12 y-- [nm] Perturbation -- [ m] Perturbation -- [ m] Figure 19: Average and 9-th percentile of the horizontal and vertical emittance growth in the BC1 after DFS correction and comparison with 1 : 1 when σ pos goes from 1µm to 1µm. One test beam with a phase offset of 5deg is considered. The optimal weight factor used for the different σ pos considered is aroundω = 1. Misalignment in the Booster 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) x-- [nm] 12 y-- [nm] Perturbation -- [ m] Perturbation -- [ m] Figure 2: Average and 9-th percentile of the horizontal and vertical emittance growth in the Booster after DFS correction and comparison with 1 : 1 when σ pos goes from 1µm to 1µm. One test beam with a phase offset of 5deg is considered. The optimal weight factor used for the different σ pos considered goes fromω = 1 toω = 5.

32 3 Misalignment in the CA 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) x-- [nm] 12 y-- [nm] Perturbation -- [ m] Perturbation -- [ m] Figure 21: Average and 9-th percentile of the horizontal and vertical emittance growth in the CA after DFS correction and comparison with 1 : 1 when σ pos goes from 5µm to 1µm. Two test beams atδ = ±1.5% are considered and the8 : 4 splitting scheme is used. The optimal weight factor used for the differentσ pos considered is aroundω = 1. Misalignment in the LTL 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) 1:1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) x-- [nm] 12 y-- [nm] Perturbation -- [ m] Perturbation -- [ m] Figure 22: Average and 9-th percentile of the horizontal and vertical emittance growth in the LTL after DFS correction and comparison with 1 : 1 when σ pos goes from 5µm to 3µm. One test beam at δ =.5% is considered. The optimal weight factor used for the differentσ pos considered is aroundω = 5.

33 31 Misalignment in the TAL x-- [nm] :1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) y-- [nm] :1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) Perturbation -- [ m] Perturbation -- [ m] Figure 23: Average and 9-th percentile of the horizontal and vertical emittance growth in the TAL after DFS correction and comparison with1 : 1 whenσ pos goes from5µm to5µm. Two test beams atδ = ±1.5% are considered and the8 : 4 splitting scheme is used. The optimal weight factor used for the differentσ pos considered is aroundω = 1. Misalignment in the BC2 x-- [nm] :1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) y-- [nm] :1 (Averag e) 1:1 (+9%) D FS (Averag e) D FS (+9%) Perturbation -- [ m] Perturbation -- [ m] Figure 24: Average and 9-th percentile of the horizontal and vertical emittance growth in the BC2 after DFS correction and comparison with 1 : 1 when σ pos goes from 1µm to 2µm. One test beam with a phase offset of 5deg is considered. The optimal weight factor used for the different σ pos considered is aroundω = 5.