Transverse dynamics: Orbit Corrections

1. Transverse dynamics: Orbit Corrections Introduction overview of methods at some (large) accelerators (SLC, HERA, RHIC, LHC) Effects of magnet alignment errors Lattice Diagnostics and R-matrix Reconstruction Online Orbit Fitting as a Diagnostic for Errors Beam-based Alignment of Localized Errors quadrupoles sextupoles accelerating structures Beam Steering Algorithms one-to-one steering global beam-based steering singular valued decomposition “wakefield bumps” dispersion-free steering Summary

2. Stanford Linear Collider (SLC) 50 GeV single linac, double-arc electron-positron collider, L=3 km global BBA 1:1, SVD, DFS steering 1:1, SVD steering BBA in IP 1:1, SVD steering + s,e,c, & m – orbit feedback Acronyms: BBA = beam-based alignment SVD = singular valued decomposition DFS = dispersion free steering (IP = interaction point) s = orbit/steering feedback e = energy feedback c = collision feedback

3. Hadron Elektron Ring Anlage (HERA) 2-ring, 920 GeV proton on 27.5 GeV lepton collider, L=6.3 km 1:1 steering, HERA-E/P global BBA in arcs, HERA-E (for polarization) BBA in IP DFS, HERA-E orbit feedback, HERA-E/P BBA in IP

4. Relativistic Heavy Ion Collider, RHIC 2-ring, 250 * 250 GeV proton-proton collider; 100 GeV/u (Au) ion-ion collider, L=6.8 km 1:1 and SVD steering orbit feedback tune/coupling feedback mean radius / energy feedback (chromaticity feedback) BBA in IP BBA in IP

5. Large Hadron Collider (LHC) 2-ring, 7 * 7 TeV proton-proton collider 2.8 *2.8 TeV/u (Pb) ion-ion collider, L=27 km 1:1 and SVD steering orbit feedback (tune/coupling feedback) mean radius feedback (energy feedback)

6. Effects of magnet alignment errors

7. Effects of magnet alignment errors types of relative alignment errors and their effect on the beam: z multipole field expansion (Ex. 2.3) x bn  0, an= 0: bn=0, an  0: “skew” (e.g. rotated by 45° component) “normal” (e.g. up- right) component p. 65 ,below Eq. 2.106 Lorentz force example: (normal) quadrupole n=1: Bz+iBx=(b1+ia1)(x+iz) Bz=b1x Bx=b1z Fx = -qvBz = -qvb1x Fz = qvBx = qvb1z B F for x0, the field experienced is that of a dipole the off-axis beam sees the next-lower order field (“feeddown effect”)

8. Bz/b0 normal dipole Bz=b0 Re(x+iz)0=b0 Bx=b0 Im(x+iz)0=0 x Bz=a0 Im(x+iz)0=0 Bx=a0 Re(x+iz)0=a0 “skew” dipole Bz/b1 Bz=b1 Re(x+iz)=b1x Bx=b1 Im(x+iz)=b1z normal quad x Bz=a1 Im(x+iz)=a1z Bx=a1 Re(x+iz)=a1x “skew” quad

9. Bz/b2 Bz=b2 (x2-z2) Bx=2b2xz normal sextupole (z=0) x skew sextupole Bz=2a2xz Bx=a2(x2-z2) x = xco + xβ + x dispersive orbit orbit defined by magnetic focussing closed orbit off-axis beam sees sextupole, quadrupole, and dipole fields example: z=0, x=0: Bz/b2=(xc02+2xxco+x2xco0)

10. In addition to perturbing the design optics, off-axis in quadrupole  dipole field: generates dispersion D~1/ρ leads to dispersive emittance growth when  ≠ 0 since x =η and x ~ <x2>1/2 (here x=transverse position offset) causes sampling of nonlinear fields, contributes to emittance growth leads to beam depolarization when magnetic moment of particle μ experiences a transverse deflecting field B off-axis in sextupole  quadrupole and dipole fields: introduces additionally β-beats (gradient errors), vertical dispersion, betatron coupling, … off-axis in accelerating structures  wakefield kicks: dilution of projected beam emittance since different particles within the bunch follow different trajectories solutions: alignment (by surveying) steering beam-based alignment feedback (not) covered focus of this lecture covered later

11. famous question: “where is zero?” beam centroid position with respect to the reference axis xco = xd + xb + xm quadrupole misalignment measured beam position BPM electronic and/or misalignment offset

12. the “answer” depends on application chosen: orbit correction method consequence(s) BBA of single elements and global BBA one-to-one steering SVD steering dispersion-free steering absolute position of beam wrt magnet center - best solution (but invasive, time consuming, and for global BBA, complicated) BPM values zeroed, corrector values (and ) nonzero - beam may be still off-axis BPM readings and corrector values minimized using empirical weighting of the unknown offsets BPM reading and corrector values minimized with additional constraints given by dispersive orbits

13. Homework #1. The multipole field expansion for a magnet of arbitrary order may be calculated using a useful trick (Laplace’s tree) for avoiding lots of algebra. Consider the example sketched below: order 0 1 1 1 1 2 1 2 1 (x+y)3=x3+3xy2+3x2y+y3 3 1 3 3 1 4 1 4 6 4 1 (etcetera) Make a table showing the position dependence (on x and z) of the vertical (Bz) and horizontal (Bx) magnetic fields for upright magnets up to order 4 (octupole magnet). multipole field expansion: #2 Considering a nondispersive orbit (x=xc0+x, y=yco+y), show that a beam off-axis in x in a normal sextupole experiences a normal quadrupole field while if off-axis in y, a skew quadrupole field.

14. Lattice Diagnostics and R-matrix Reconstruction

15. Lattice diagnostics and R-matrix reconstruction simple FODO lattice quadrupole with BPM corrector dipole point-to-point transfer matrix: measure R12=dx/dx’ plot x=R11x+R12x’max method I “jitter data” plus careful 2-dimensional error propagation, parasitic method II 2-point fit x(x’=0), x(x’≠0) or, for better measurement (smaller error), x vs x’

16. example: linac-to-ring transfer line (SLC) angular deflection (“kick”) due to dipole: since then Bρ=magnetic rigidity =E(GeV)/0.3 [T-m] =p (MeV)/300 [T-m] slope: measurement at a single location as the strength of an upstream corrector is varied to probe all elements in the beamline, need a second corrector dipole separated in betatron phase by π/2

17. example: linac-to-ring transfer line (SLC), continued: “fitted” R12 concept: acquire data using all available BPMs while one upstream corrector is varied then minimize calc fitting for the best incoming orbit given by x1, x1’ data (solid line): measured trajectory fitted (dashed line): x1 + R1jxj’ (using R1j measurements and x1 fitted) y(s) x(s) 60 m, 20 BPMs 60 m, 20 BPMs Comparison allows for localization of optics errors and BPM errors (these BPMs may then be excluded in subsequent steering procedures)

18. Suppose there is still unsatisfactory agreement between expectation and measurement. In the absence of hardware errors, this may result, for example, from: systematic measurement errors (e.g. in corrector and/or BPM gains) an incomplete model example: taking into account additional error sources / lattice diagnostics in the main SLC linac (courtesy T. Himel and K. Thompson, 1999) expanded R-matrix: energy dependence of the point-to-point transfer matrix obtained using a Taylor series expansion to first order in the energy with Nk(=240)=total number of klystrons, Ek=energy gain per klystron, or with Nk(=30)=total number of sectors, Ek=energy gain per sector data (solid line): measured trajectory fitted (dashed line) as in previous example and taking into account an assumed linear scaling of beam energy 180° cumulative phase shift in center of linac? ~ 1 GeV ~ 50 GeV 1500 m, 300 BPMs

19. Possible causes for discrepancy chromatic phase advance in a FODO cell: with (90° FODO cell) if not taken into account, expect a 90° cumulative error with =2.5%, =30sectors●2π/sector unknown energy errors Procedure used for analysis: 1. measure xm(s) versus the applied deflection 2. select a set of Ek’s 3. compute computed measured 4. iterate for minimum 2

20. aside: the method of least squares (aka “least-squares fitting”) From Wikipedia: The method of least squares is a standard approach in regression analysis to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation. The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. example: data (N blue dots): (x,y) y “least squares fit”: y = ax + b, where a (slope) and b (offset) are found by minimizing N S [yi2-(axi+b)2] x i = 1 (with N > number of unknowns) reminder: if errors in x not negligible, then “eigenfit” is better (ref. FZ’s lectures 6/16/15)

21. Back to lattice diagnostics in the SLC linac: equations to be solved: number of unknowns: 30 if Ek = energy gain in each of the 30 sectors 240 if Ek = energy gain per klystron (8 klystrons per sector) number of BPMs (measurements): ~ 300  number of measurements is greater than number of unknowns  LS-fit approach ok (note: can limit data set to a few sectors provided number of BPMs still > number of unknowns) solution: 1. measure xm(s) versus the applied deflection 2. select a set of Ek’s 3. compute figure of merit this is the fit in the least squares sense computed measured 4. iterate for minimum figure of merit

22. result before the energy fit data (solid line) fitted (dashed line) model still lacks all physics input (i.e. energy spread variation along the linac not taken into account) the estimated energy error ~30%, greatly exceeding the estimated uncertainty (bad BPMs) after the energy fit data (solid line) fitted (dashed line) After localization of BPM and klystron phase errors, the fitted energy errors were incorporated into the accelerator model leading to improved “predictive power”; i.e. ability to predict beam dynamics (in this case beam trajectories) and build better bumps

23. Online Orbit Fitting as a Diagnostic for Errors

24. Online Orbit Fitting In essence: - the online model is used to fit the measured beam trajectory - in a given plane (cross-plane coupling is neglected) a set of initial coordinates (x, x’) which results in best agreement between the orbit calculated with these initial conditions and the measured orbit is determined. Types of errors that can be found and localized: - component malfunction - deficiencies in the model (including magnet “end-effects” and/or feeddown effects) - anomalies of unknown origin (e.g. stray fields, sensitivities to localized thermal variations) Method Beam position (z=transverse displacement, x or y – here, uncoupled transport) at location s given upstream measurements (z1, z1’) (optical functions from model) Construct a difference orbit (as resulting e.g. from some closed orbit distortion) and express as with

25. Again Select a fit range from the mth to the nth BPM, then For m.ne.n, this gives an overconstrained set of equations with two unknowns, Dz1 and Dz1’ Define a merit function - measured difference in position at BPM k calculated difference in position at BPM k Minimize merit function, e.g. using least-squares algorithm, to determine Dz1 and Dz1’ Then the position difference (and angle difference) can be computed at any location along the accelerator Next: examples Remark: application is exact in linear transport lines, but works surprisingly well in circular accelerators (as examples will show)

26. Application: Example: unexpected orbit oscillations problem: fluctuations in orbit rms (rms = rms value of the absolute Measurement from all BPMs) expanded view and selection of two times for further analysis (maximum orbit deviations)

27. 2 orbits (as measured) difference orbit zoom fit range difference orbit (blue) and fitted orbit (red) un-zoom difference: fit less fitted (difference) orbit zoom, forward-propagated solution (fit range: 800-1100 m) zoom, backward-propagated solution (fit range: 1500-1800 m) Error localized to between 1250-1270 m and found to be due to oscillating large dipole at ~ 1255 m.

28. Example: model imperfections

29. Beam-based Alignment

30. Beam-based alignment of single elements: quadrupoles typical residual alignment errors after surveying: ~ (100-200) μm typical quadrupole alignment tolerances in storage rings (i.e. synchrotron light sources): ~ 10 μm circular collider IPs (i.e. B-factories, HERA, LHC) ~ 5 μm linear collider IPs (i.e. NLC, TESLA) < 1 μm Beam-based alignment determines the relative offset between magnet centers and nearby BPMs. If the offsets are sufficiently stable, then orbit correction (steering) can be used to maintain a well-centered orbit.

31. Beam-based alignment of quadrupoles using quadrupole excitation (xb and xq are unknown) dipolekick xb≠0 xb=0 xb=0 s s s xq≠0 xq=0 xq=0 ideal case (with misalignments) ideal case (no misalignments) result of “steering “flat” (zeroing BPM readings) Note: (a) and (b) assume electrical offsets in BPMs are zero, case (c) does not.

32. single-pass measurement (transport line or linac): dipole kick (for horizontally focussing quadrupole) Θ=K·xq (in 1/m) change in integrated field  B(I), where B(I) depends on magnet properties quadrupole misalignment ρ(eVs/m) m Vs/m3 example from the TESLA test facility (courtesy P. Castro, 2000) total quad current variation of ~ 5% triplet quad center corrector s x (mm) slope ~ R12 since x~ (Icor) BPM (set field to zero) independent of absolute BPM position (and electronic offsets) K minimum displacement Icor (A)

33. multiple-pass measurement (circular accelerator): s Θ≈K·xq- K x + O (K· x) xq BPM #1 BPM #2 (as before) change in the closed orbit offset at the quadrupole solve iteratively using expression for the c.o.d. at the location of the kick (Eq. 2.34): solving for x: and inserting back into first eq. above:

34. example from SPEAR (courtesy J. Corbett, 1998) quadrupole shunt circuitry (allowing variation of single quad given multiple magnets per power supply) BPM #1 quad center x=difference orbit measured at BPM BPM #2 xq=orbit offset at quadrupole (varied using local bump)

35. example from RHIC (J. Beebe-Wang, 2013) (Suspect systematic errors associated with incoming angle errors)

36. quadrupole gradient modulation (LEP) modulation of single quadrupoles with (non-resonant) discrete frequencies example from LEP(courtesy I. Reichel, 1998) response of beam to modulation of four different quadrupoles natural orbit drift and correction at a single quad (time scale, 6 hrs) orbit correction ~ 1 μm FS orbit variation measured at BPM close to modula- ted quad using position jitter (time scale short) relative offset FFT of turn-by-turn BPM measurements minimum gives relative BPM offset (~200 μm)

37. Beam-based alignment of single elements: sextupoles recall (normal sextupole): Θx~Bz=b2 (x2-z2) Θz~Bx=2b2xz Possible approaches (using sampled quadrupole field): 1. use additional quadrupole trim windings, mounted on the sextupoles, quadrupole-based BBA (KEK) – (assumes coinciding magnetic centers of trim windings and sextupole) 2. measure shift in betatron tune vs sextupole strength (PEP-II, HERA,…): θx~ ( xβ+ xco )2 - ( zβ+ zco )2 ~ … + xβxco + … zβzco + … need for each sextu- pole an indi- vidual power supply off-axis in x (xco≠0)  θx~xβ (quadrupole field); similarly in z 3. measure tune separation near the difference resonance vs sextupole strength (NLC): θz ~ … + xβzco + … off-axis in z (zco≠0)  θz~xβ (skew quadrupole field)

38. Possible approaches (using sampled dipole field): 1. using local orbit bumps (KEK,DESY): horizontal sextupole alignment: θx = -0.5 Ks (xbump-xs)2 θz = -0.5 Ks (xbump-xs)zs vertical sextupole alignment: θz = Ks (x0-xs)(zbump-zs) xs and zs are the sextupole offsets change strength of all sextupoles by Ks measure induced orbit change repeat for different bump amplitudes independent of the number of sextupoles driven by a single power supply x0 is a preferably large horizontal orbit offset (for enhanced sensitivity)

39. 2. using induced orbit kick vs sextupole position (KEK B-factory, SLAC FFTB) example from the FFTB (courtesy P. Tenenbaum, 1998) sextupoles mounted on precision movers, measure quadratic change in closed orbit (transfer line or linac) center at dxb/dxm=0

40. Beam-based alignment of accelerating structures s x couple out power at natural resonant frequencies of the structure (higher-order modes) if x,y≠0 using output coupler example from the ASSET experiment at SLAC (courtesy M. Seidel and C. Adolphsen, 1998) signal amplitude at fc = 16 GHz ~ 7.5 MHz (after mixing) T (structure central axis to beam position infered from BPMs) 90 relative position offsett phase (deg) please fix scale on bottom plot of Fig. 3.10, p. 79 -90

41. Homework #3. Design a 2-corrector “-bump” (phase advance between correctors is ). a) determine the ratio of the applied kick angles 1/2 in terms of the beta functions at the correctors 1and 2, b) supposing 1=2, what is the position offset in the middle of the bump (take =1/2) assuming a 1 mrad applied deflection angle? # 4. Using 4 correctors, determine the coefficients of so-called symmetric (x≠0 and x’=0) and asymmetric bumps (x=0 and x’≠0) at the middle of the bump assuming a symmetric optic about this point and that =0 there. Hint: evaluate the required kick angles using two correctors and apply symmetry arguments to determine the kick angles at the other correctors. With =1/2=2/2, what are the deflections angles required to produce x=1mm and x’=1 mrad at the symmetry point? symmetric bump: asymmetric bump: s s Hint for both exercises: apply linear superposition

42. Global beam-based steering Take into account now that the electrical center of the BPMs ≠ center of quadrupole xm xbpm xq beam position wrt quad center sum over all upstream quadrupoles beam position wrt reference axis beam position wrt reference axis transported between quad (j) and quad (j+1) beam position wrt quad center transported between quads (j) and (j+1) = beam position at quad k

43. again rearrange terms sum over upstream quadrupoles function to be minimized: measurements fitting function with unknowns xq, xbpm, and (x0,x0’) Since the number of measurements is about ½ the number of unknowns, an independent set of measurements is required. One possibility is to scale the lattice (all quadrupole and corrector strengths) and repeat the measurement.

44. Beam Steering Algorithms

45. Beam Steering Algorithms review: so far we have discussed beam-based alignment of single elements which, while accurate, is time consuming and so used most often during early stages of accelerator commissioning and in interaction regions, where single-element alignment tolerances are most critical lattice diagnostics which is often needed prior to being able to implement automated steering algorithms, which may be model-dependent and are significantly less prone to errors if “bad BPMs” are already excluded from the data samples until now (in the discussion of BBA and lattice diagnostics) only the coherent motion of the bunch centroid (center of charge) has been considered Beam steering is applied over long regions of the accelerator and is significantly more time-efficient than BBA of single elements. Note that while the center of charge is measured by the BPMs, the steering algorithms generally do minimize the projected beam emittance. With x = horizontal or vertical, x = xβ + x dispersive orbit orbit defined by magnetic focussing closed orbit The emittance εx ~ σx2/β, where σx = <x2>1/2 = <(xβ + x)2>1/2 which contains for example a dispersive term σ=<x2>1/2=2<2>1/2

46. “One-to-One” steering Beam is steered to zero the transverse displacements measured by the BPMs. The BPMs are typically mounted inside the quadrupoles (where the β-functions are largest). closed bump that would minimize the BPM readings (but would also generate dispersion) indices i denote each corrector kick Beam position measured at a downstream BPM (denoted by subscript, j): m is the total number of BPM measurements In matrix form with n is the total number correctors Solving the matrix equation: x is the vector containing the BPM measurements or Θ is the vector containing the unknown kick angles

47. Again , the solution is given by Here, θ is an (n1) matrix [n=number of correctors] M is an (nm) matrix x is an (m1) matrix [m=number of BPM readings (per plane of interest)] If m>n, the matrix M is overdetermined m=n, the matrix M is square and the solution is simply θ=M-1x m<n, the matrix M is underdetermined and more measurements are needed (to ensure that the number of unknowns is at least equal to the number of measurements). To constrain the solution in this case, some parameter (e.g. the beam energy) needs to be varied and the measurement repeated (however, the measurement fails to be noninvasive)

48. minimization procedure: (using linear least-squares algorithm) The solution to the matrix equation can equivalently be found by minimizing fitting function with unknowns Θi (i = number of unknown kicks) measurements (j = number of BPM measurements) i.e. vary the unknowns θi given the BPM measurements xj and the assumed transfer elements Mji so that the relation holds (as closely as possible) i.e. (algebra) (reorganizing terms) This equation, expressed in matrix form, is equivalent to the result from before:

49. Homework # 5. One-to-One steering a) Consider a transport line with a single BPM. Given a measured transverse displacement of x1, give an expression for the required kick angle of a single upstream corrector to zero the displacement. The answer may be expressed in terms of the beta functions at the corrector βc and at the BPM βb and of the relative phase advance b-c. b) Consider a transport line with the geometry, lattice functions, and symbol definitions shown below: x c1 = π/2 correctors c2 = π s 1 = 3π/2 BPMs 2 = 2π • Plot the required kick angles θ1 and θ2 as a function of the measured displacements x1 and x2 assuming that the beta functions at the correctors and the BPMs are (for purposes of this illustration) all equal and equal to 5 m. Express the slopes in both graphs numerically. Remark: this example, while easy to solve numerically, is well adapted for extension to more BPMs and correctors if solved using MATLAB. As a reminder, if then (ii) Suppose x1=x2=10 mm. Assuming a beam energy of 1 GeV and corrector dipole lengths of 6 cm, what are the magnetic fields required to provide the corrections? Reminder: where Bρ =magnetic rigidity = p/c = E (GeV)/0.3 [T-m]

50. Singular valued decomposition (SVD) Global orbit correction using the SVD algorithm offers the advantage of reducing not only the rms of the BPM measurements, but also the rms of the applied corrector strengths. Correctors compensating (or “fighting”) one another generate unwanted dispersion. This powerful algorithm may also be used in application to orbit feedback, dispersion-free steering, and in the computation of multiknobs matrix of corrector kick angles, (n1) matrix Matrix equation to be solved: measured orbit displacements, (linear) orbit response matrix, (m1) matrix (mn) matrix m=number of BPM readings n=number of correctors If m>n, the matrix A may be decomposed (easily using matlab, for example) U is an (mn) matrix V is an (nn) matrix The column vectors of U and V are orthonormal satisfying (In is the (nn) identity matrix)