Transverse Dynamics – Optics Correction, Optimization, & Betatron Coupling

1. Transverse Dynamics – Optics Correction, Optimization, & Betatron Coupling [MCCPB, Chapter 2] 0. addenda: ICA & Fourier theorem 1. gradient error detection and cures 2. numerical optimization using evolutionary & sociological algorithms 3. betatron coupling & correction

2. 1. gradient error detection & cures difference from model might be b beat due to gradient error Dk at s0 (A) 1st turntrajectories excite steering correctors, or injection kick,… fit difference orbit (or difference trajectory) to model should be betatron oscillation! e.g., fit Dx to the model using signal from a few BPMs then propagate along the beam line; point of deviation from model is suspect! confirm location by fitting to region downstream and propagate backwards

3. illustration: x1 orbit 1 x2 orbit 2 Dx=x2-x1 difference s

4. discrepancy starts here Dx=x2-x1 fit region propagate according to model fit region propagate backwards Dx=x2-x1 confirm location

5. practical problems: • beam loss • BPM spray • kicker noise • BPM timing & readout error • BPM electronic noise • change in beam parameters, e.g., current • …

6. fixed reversed BPMs Measured difference orbit in RHIC (circles) together with a fit to the online model (dashed line) [Courtesy S. Peggs, 2000].

7. variant of this method used in LHC tests to find polarity errors “difference of difference orbits” DoubleDifferenceOrbit = “BaseCorrQuad”-”BaseCorr” -(“BaseQuad”-”Baseline”) removes effect of initial orbit offset at quadrupole

8. single matching quadrupole

9. skew quadrupole circuit

10. another variant of this method: (B) closed orbit distortion evolves like betatron oscillation except for location of the kick; apply same fitting and propagation technique as above change strength of suspect quadrupole and repeat the measurement gradient errors (~0.1%) in this region propagate fit to the right finding quadrupole gradient errors by fitting betatron oscillations to closed-orbit distortions: example from PEP-II HER commissioning using the codes LEGO and RESOLVE [Courtesy Y. Cai]

11. one can considerably extend this simple scheme e.g., measure response of all BPMs to all steering correctors corrector excitations BPM readings big matrix statistical fitting program LOCO (J. Safranek) varying all quadrupole gradients to find best fit

12. (C) phase advance determine b from phase advance (harmonic analysis of multi-turn BPM reading); then change either actual magnets or magnet strength in model so as to improve agreement and identify source of discrepancy earlier example from PEP-II (M. Donald): BQF5=98.1027 kG BQF5=98.1518 kG BQF5=98.2498 kG

13. phase beating induced by a single quadrupole QE604 in the SPS; red: measurement, blue: model prediction (J. Klem, 2000) from phase of harmonic analysis

14. beta beating induced by a single quadrupole QE604 in the SPS; red: measurement, blue: model prediction (J.Klem, 2000) from amplitude of harmonic analysis

15. interlude: Singular Value Decomposition (SVD) M,V: mxn matrices V,W: nxn matrices only invert di’s for di above some cutoff; set other diagonal elements to zero in the SVD “inversion” with with with SVD “pseudo-inversion” finds ‘optimum’ solution without exciting correctors strongly; eliminates ‘corrector fighting’ [see example in book on pages 88-89]

16. more systematic approach to gradient errors phase beating introduced by gradient error at location s0 DK for all quadrupoles and phase changes Df at all BPMs vectors can be solved by standard minimization procedure (Micado, SVD,…) in case of SVD need to introduce additional set of constraints: restrains the magnitude of the DK values optimize weight factor l and SVD cut-offs to get the correct solution [gradient error also causes b beating: - used later]

17. SPS test with quadrupole QE603, DQ~0.05, varying weight l for DK in SVD solution weight low – many quadrupoles excited to get perfect fit reasonable weight correct quadrupole is identified F.Z. et al., EPAC04 weight high – all quadrupole changes a small

18. corresponding fit results versus measured phase shift

19. (D) beta functions at all BPMs beta beat inferred (with errors) from multi-turn BPM readings “gradient” errors to be determined again solved with standard minimization procedures (Mikado, SVD,…)

20. 1st LHC beta beating correction 29 November 2009 – beam 1 R. Tomas et al

21. iterative matching of quadr. gradient errors to reproduce measured data M. Aiba, R. Tomas et al 2008 LHC data (red) compared with design model (blue)

22. iteration #1 M. Aiba, R. Tomas et al

23. iteration #2 M. Aiba, R. Tomas et al

24. iteration #3 M. Aiba, R. Tomas et al

25. iteration #4 M. Aiba, R. Tomas et al

26. iteration #5 M. Aiba, R. Tomas et al

27. forward and back propagation of measured local b functions compared with (full) measurement allows localization of quadrupole errors (simulation for LHC) location of quadrupole error between BPMs #8 and 9 forward propagation back propagation Glenn Vanbavinckhove; Rogelio Tomas

28. (E) “jumps” in oscillation invariants J. Cardona

29. simulation for RHIC J. Cardona

30. (F) p bump has been used at TRISTAN, ATF, … nominally closed bump leakage due to corrector imbalance ~dq1 cos f q1 q2 K leakage due to quadrupole gradient error dK ~dK q1 sin f different phase allows the distinction of gradient & corrector errors

31. principle of p bump method applied at TRISTAN MR for measuring quadrupole errors and sextupole misalignments (S. Kamada, 1994)

32. optics tuning multiknobs combination of quadrupoles and/or skew quadrupoles and/or sextupoles that are excited (with constant ratio) so that only one tuning parameter is varied (bx,by,ax,ay,coupling,Dx,Dx’Dy,Dy’,xx,xy,…) • knobs should be orthogonal to each other for fast convergence of tuning • knobs should be reversible (SLC’s SVD knobs were not) • knobs should not cause additional nonlinear aberrations that blow up the beam size • knobs only work over a limited (linear) range • at the SLC there was an attempt to make nonlinear ‘Irwin’ knobs (built from many infinitesimal linear steps)

33. how to construct a multiknob (1) matching functions of optics program like MAD or • singular value decomposition of, e.g., quadrupole response matrix B unless there is some symmetry that can be exploited, we need can solve above equation for vectors (DK1,DK2,…,DKN) which change only 1 of the variables on the left, leaving all others constant

34. aberration scans at the SLC interaction point; shown is the square of the measured horizontal or vertical convoluted spot size of the colliding electron and positron beams in mm2 as function of multiknob setting; 10 aberrations were scanned and optimized using orthogonal multiknobs; each aberration scan was fitted to a parabola and the corresponding multiknob set to the minimum spot size; in last years replaced by dither feedback (up-down changes) using the same multiknobs

35. Optics tuning at KEKB • Waist of by at IP. RF phase. • Vertical dispersion at IP • X-y coupling R1-R4 at IP. • These are tuned everyday. Courtesy K. Ohmi, 2004

36. Courtesy K. Ohmi, 2004 • Optics parameters are controlled by local bumps

37. Courtesy K. Ohmi, 2004 • x-y coupling LER R4

38. Wednesday Homework #2 Waist Shift For optimum luminosity at a circular collider, like RHIC or the LHC, the beams should collide a the center of the particle physics experiment at a location with minimum beta function (“waist”). The beam size measured 20 m upstream and 20 m downstream of the collision point is su=601 mm and sd=664 mm, respectively. A) How much is the waist shifted from the ideal position, for which the upstream beam size should equal the downstream beam size? Hint: In the drift space around the collision point b(s)=(bmin+s2/bmin)~s2/bmin where s denotes the longitudinal distance from the actual waist location and bmin the beta function at this actual waist. B) Assume that the beta function at the waist bmin equals the nominal beta function at the collision point b*, how large a change in a* (i.e. a at the collision point) would be needed to shift the waist to the desired position and to achieve b*=bmin (i.e. to make the waist coincide with the collision point)? Hint: remember that <xx’>=-ae. C) Consider a multiknob based on a linear relation between optical parameters at the collision point and the strength of two quadrupoles; the coefficients b, c, e, f are computed by an optics program Db*=b Dk1+c Dk2 ; Da*=e Dk1+f Dk2 Which changes (Dk1,Dk2) ae required to accomplish the required waist shift? D) Assume an emittance e=0.5 nm. What is the beam size 20 m upstream and downstream from the IP after successful correction? What is b*?

39. optics monitoring “non-invasive” continuous or regular check; may use tuning-knobs scans examples: • SLC IP tuning knob scans at regular intervals • KEKB IP tuning knobs (coupling, waist, dispersion) scans essential for reaching & maintaining high luminosity • SLC linac diagnostics pulse (phase advance along the linac) • linac rf phase dithering • LEP K modulation

40. 10 pulses with 2 different initial phases were sent down the linac every 15 minutes to monitor optics stability a large day-night variation was observed, that could subsequently be corrected

41. “flight simulator” test beds for new control and tuning schemes (ex. HPSim) 22 quads, 2 buncher phases A. Scheinker, et al. PRSTAB, 16, 102803, 2013. A. Scheinker, et al. NIM-A, 756, 30, 2014 Xiaoying Pang, IPAC’15, Advanced in Proton Linac Online Modeling • Testing on real machine can be risky and costly • HPSim  virtual experimental environment • resembles the real accelerator • safe, inexpensive, efficient and productive • Model Independent Accelerator Tuning first tested with HPSim before carrying out real experiments

42. 2. numerical optimization & modelling • multivariate multi-objective optimization using a genetic algorithm • swarm optimization

43. numerical accelerator optimization • Traditional gradient-based methods: • may get stuck in a local minimum/maximum • (and never come out) • require local gradients • work if initial guess is already close to the optimum • Parameter scans: • Only applicable for 1D and 2D parameter spaces • Accelerator problems: multidimensional, nonlinear, multi-objective, • several “optimum” solutions (requiring “choice” of the accelerator designer/operator) • Genetic algorithms: search for solutions using techniques inspired by natural evolution, such as inheritance, mutation, selection, and cross over. • Can be combined with fast converging gradient-based methods (e.g. for refinement) and slowly converging random search methods. O. Boine-Frankenheim, oPAC School 2014

44. genetic algorithms (GA) R. Hajima et al., NIMA318 (1992) 822 S. Ramberger, S. Russenschuck, EPAC’98 (1998) I. Bazarov and C. Sinclair, PRST-AB 8, 034292 (2005) A. Hofler et al. PRST-AB 16, 010101 (2013) Correspondence between evolution and multidimensional optimization The 2006 NASA ST5 spacecraft antenna. This complicated shape was found by an evolutionary computer design program to create the best radiation pattern (from Wikipedia) Fitness Value of objective function(s) A. Hofler et al. PRST-AB 16, 010101

45. a practical example: travel cost optimization O. Boine-Frankenheim, oPAC School 2014

46. typical structure of GA optimization O. Boine-Frankenheim, oPAC School 2014

47. multi-objective genetic algorithms (MOGA) example two objectives: Search space (red and green) and Pareto-optimal front (green) for the minimization of the system. Pareto-optimal front estimates found after 1, 10, and 20 generations for 16 individuals are marked. The overlap between the search space and the rectangles with individuals A, B, and C at the top right vertices contain the points, if any, in the search space that dominate the respective individuals. A. Hofler et al. PRST-AB 16, 010101

48. Pareto efficiency, or Pareto optimality, is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off. The term is named after Vilfredo Pareto (1848–1923), an Italian economist.

49. genetic algorithms at work maximizing luminosity A. Hofler et al. PRST-AB 16, 010101 maximizing dynamic aperture

50. recent GA accelerator papers • Simultaneous optimization of beam emittance and dynamic aperture for electron storage ring using genetic algorithm, Weiwei Gao, Lin Wang, and Weimin Li, Phys. Rev. ST Accel. Beams 14, 094001 (2011) • Multiobjective genetic algorithm optimization of the beam dynamics in linac drivers for free electron lasers, R. Bartolini, M. Apollonio, and I. P. S. Martin, Phys. Rev. ST Accel. Beams 15, 030701 (2012) • Magnet sorting in a storage ring. Chen, J., Wang, L., Li, W.-M., & Gao, W.-W., Optimization of magnet sorting in a storage ring using genetic algorithms, Chinese Physics C (2013) • Linac settings for high intensity Pang, X., & Rybarcyk, L. J., Multi-objective particle swarm and genetic algorithm for the optimization of the LANSCE linac operation. NIMA 741 (2013) • Minimization of the energy consumption of an accelerator facility, Ripp, C., Boine-Frankenheim, O., Hanson, J., Stadlmann, J., Spiller, P., Lindenberg, J., Zimmer, H. Electric energy consumption of an accelerator facility. IYCE, IEEE 2013 • Real machine based optimization in a storage ring, Tian, K., Safranek, J., & Yan, Machine based optimization using genetic algorithms in a storage ring, Phys. Rev. ST Accel. Beams 17, 020703 (2014) • Simultaneous optimization of the cavity heat load and trip rates in linacs using a genetic algorithm, Balša Terzić, et al., Phys. Rev. ST Accel. Beams 17, 101003 (2014)