1. Analysis of Multipole and Position Tolerances for the ATF2 Final Focus Line James Jones ASTeC, Daresbury Laboratory

2. Field Tolerances Use as.mad lattice (original NLC-like solution from M. Pivi) Track 1000 particle beam from beginning of extraction line to IP Calculate beam size and beam position offset at IP in both planes Calculate tolerances for the following cases: Individual Multipole for each magnet separately Individual Multipoles for all magnets together with the same amplitude Individual Multipoles for all magnets with amplitude relative to maximum strength of design field Results given in terms of K-values: Factorials accounted for!

3. Field Tolerances – Individual Quads Multipole Errors Skew Normal Tolerance for 10% beam growth due to the multipole field in an individual magnet Multipoles from: Order 10 (20 pole) : Red.. Order 5 (10 pole): Light Green.. Order 2 (Quad): Orange Absolute values of Multipole strength Small asymmetry between +ve and –ve of ~±20%

4. Tightest Tolerances are: Order Normal (Quad) Skew (all on QF1) 20 pole  4.78 10 18 (QD10) 1.70 10 18 m -10 18 pole  5.35 10 15 (QD0) 1.60 10 15 m -9 16 pole  6.33 10 12 (QF1) 1.76 10 12 m -8 14 pole  9.76 10 9 (QF1) 2.07 10 9 m -7 12 pole  1.39 10 7 (QF1) 2.98 10 6 m -6 10 pole  33826 (QF1) 4467 m -5 8 pole  45.54 (QF1) 8.06 m -4 6 pole  0.096 (QF1) 0.0161 m -3 4 pole  4.5 10 -5 (QF1) 2.64 10 -5 m -2

5. Field Tolerances – All Quads Multipole Errors Skew Normal Tolerance for 10% beam growth due to the multipole field in all Quads together Multipole order from Quadrupole(2) up to 20pole (10) Multipole strength is the same in each of the quadrupoles

6. Field Tolerances – All Quads Multipole Errors Relative Skew Normal Tolerance for 10% beam growth due to each multipole component in all Quads together, with the strength relative to the design quadrupole strength Maximum K2 is -3.5 m -2

7. Field Tolerances – Sextupole Multipole Errors All Normal Normal Same Data for the Sextupoles! Indiviual Sextupoles All Sextupoles at same multipole strength All Sextupoles with each multipole component relative to the design sextupole field All Normal Relative

8. Field Tolerances - Summary Have tolerances for all multipole components up to 20pole Can be used to understand the requirements from the magnet designs Data is available over a wide range of values so it is very simple to analyse the beam size increase from each multipole component separately Analysis of the effects of combined multipole errors is more ambiguous Requires dedicated tracking studies Already set-up for the original Hitachi Type 5 quadrupoles (cf Cherrill Spencer!) All data available as excel S/Sheets, data files ….

9. Position Tolerances Calculate increase in beam size and change in spot position in both planes for the following cases: Individual Errors on each magnet All magnets with the same static error All magnets with random errors, averaged over 10 seeds The effects of the correction system were also included: No correction at all Correction using 3 correctors in the FF-line with a BPM at every quad – fast correction Correction using linear tuning knobs (  waists, horiz. and vert. dispersion) – static correction

10. Position Tolerances – Correction System Features 3 correctors: Modelled as 20cm long – no physical reality to this! All 3 are dual plane correctors Positioned where there was space BPMs assumed at every quadrupole Did not include tolerances on BPM accuracy etc. Correction ratio ~10:1 Not optimised very well… Heavily over-constrained No weighting for the IP position No angle correction Horizontal Vertical Red: Uncorrected Blue: Corrected

11. Position Tolerances – No Correction  Jitter There is a difference between tolerance for change in beam size and change in (either beam size, or position as a function of beam size): Factor of 10 3 difference! Of course, since this is jitter, need both position and beam size… 2% Increase in beam size OR 2% change in position[beamsize]2% Increase in beam size ONLY Tolerance [  m -1 ]

12. Position Tolerances – No Correction  Jitter Analyse the beam line with random errors truncated @ 3  Average change in beam size or position over 10 random seeds Limited by time… Estimates the random jitter levels required in a timescale less than the correction system can operate Quadrupoles only (2% increase) X-plane: 11nm Y-plane: 0.46nm Roll Angle: 1.9  rad Vertical BS Horizontal BS Increasing Horizontal Error

13. Position Tolerances – No Correction  Jitter Look at the results without the final doublet as these have the tightest tolerances More likely to be specially mounted and aligned Quadrupoles only (2% increase) X-plane: 14.5nm Y-plane: 0.87nm Roll Angle: 6.9  rad Improves the tolerances from ~2/3 in x plane to a factor of 3 in roll angle…

14. Position Tolerances – 3 iteration Correction  Jitter Same basic analysis as with no correction (but with FD) Run the SVD algorithm 3 times per random seed No attempt to correct the dispersion Quadrupoles only (2% increase) X-plane: 11nm Y-plane: 0.51nm Roll Angle: 1.5  rad Doesn’t significantly improve matters...

15. Position Tolerances – 3 iteration Correction  Jitter If we assume that the correction system will maintain the beam at the correct position (which in this case it doesn’t), and assume problem is only due to increase in beam size: Compare with and without correction: Obvious that dispersion correction is very important! Vertical dispersion reaches the mm level Quadrupoles only (No Correction) X-plane: 585nm Y-plane: 197nm Roll Angle: 1.48  rad Quadrupoles only (3 x Correction) X-plane: 589nm Y-plane: 143nm Roll Angle: 1.5  rad

16. Position Tolerances – Tuning Knobs Tuning knobs created for:  x waist,  y waist,  x and  y All created using 3 sextupole magnets and horizontal or vertical displacements Optimised the linearity and the ratio of primary to secondary terms of the tuning knobs using a Simplex based optimisation routine Beta waist shifts calculated by varying the length of the final drift until the beta function is at a minimum Used Brent’s method in code to find the minimum Done for both planes separately Drift length returned to normal afterwards!

17. Position Tolerances – Tuning Knobs Performed 3 iterations of tuning knobs along with 4 iterations of the correction algorithm Order: Orbit,  x,  y,  x waist,  y waist + 1 extra orbit iter. Results do not include beam position as these are effectively static errors Quadrupoles only (2% increase) X-plane: 16  m Y-plane: 141nm Roll Angle: 3.5  rad No coupling correction yet – hence tight tolerance on Roll angle

18. Position Tolerances – Tuning Knobs Tolerance on the vertical position is no better @ 141nm Data shows that it is not  y as this maintained to ~10 -8 m (maybe) need to include second order tuning knobs: Already created, just need to work out which ones are the most important! To approximate reality, also need way of measuring these values, or A generalised optimiser that can operate on the change in beam size These tolerances are also the tolerances for after beam- based alignment Do not necessarily give the physical alignment tolerance

19. Conclusions Finally have a generalised method of analysing the tolerances on the ATF magnets in terms of field or position errors Can be used with real errors to analyse effects on beam very simply Have produced a set of specifications for the multipole components and for the position tolerances for all of the final focus line quadrupoles and sextupoles Data is not specific to a given tolerance specification (i.e. 2% or 10% beamsize increase) Analysis using tuning knobs is ongoing, and linear correction works well Next step to include 2 nd -order and generalised optimiser Analysis of other tuning scenarios, as well as model of BBA may also be useful

20. Questions / Further Work Would like to include: New ATF2 lattice Have started to convert to TRACY, but the lattice is very complicated! Upstream extraction line in analysis – Already in code, just needs some tweaking Dipole tolerances A more optimised correction system Dispersion and angle correction Better choice of BPMs Some relationship with reality, in terms of location etc

21. Questions / Further Work What tolerance levels should we design to? Is a 10% beam size increase too big? What about 2% It doesn’t take many independent errors for the beam to blow up in either case… How do the extraction and final focus lines interact? Can we use the extraction line correctors for the FF? What are the error sources in the extraction line? This could have major implications on the analysis in the FF line… Up to now, assumed ideal beam at entrance of FF line What can we actually measure? Tuning relies on observables, but what can we really expect to see out of the Extraction and FF sections?