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Beam dynamics for ultra low emittance rings

Beam dynamics for ultra low emittance rings. R. Bartolini Diamond Light Source and John Adams Institute, University of Oxford. TW DULER Diamond, 19 April 2018. Outline. Ultra low emittance rings Single particle dynamics Collective beam dynamcis. TW DULER Diamond, 19 April 2018.

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Beam dynamics for ultra low emittance rings

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  1. Beam dynamics for ultra low emittance rings R. Bartolini Diamond Light Source and John Adams Institute, University of Oxford TW DULER Diamond, 19 April 2018

  2. Outline • Ultra low emittance rings • Single particle dynamics • Collective beam dynamcis TW DULER Diamond, 19 April 2018

  3. Ultra low emittance rings TW DULER Diamond, 19 April 2018

  4. Survey of low emittance lattices for light sources Old DBA/TBA concept MBA On axis injection Antibend … Full symmetry TW DULER Diamond, 19 April 2018

  5. Low emittance lattices: Multi-Bend Achromats 7BA DBA Simplified explanation – Emittance is driven by randomness of photon emission in presence of dispersive (energy-dependent) orbits – electron recoils randomly – Breaking up dipoles and putting focusing (quadrupoles) between the parts allows reducing the amplitude of dispersive orbits – smaller electron recoils TW DULER Diamond, 19 April 2018

  6. From Theoretical Minimum Emittance (TME) to MBA cells Low emittance lattices are built using the concept of Multibend Achromats (MBA) Minimise the contribution of the dipoles to the emittance These conditions provide the theoretical minima for the emittance generated in dipoles (for constant B) TW DULER Diamond, 19 April 2018

  7. Low emittance lattices: from TME to MBA cells TME DBA TBA ~TME QBA ~TME ~TME S.Y. Lee, PRE 54, 1940, (1996) S.Y. Lee, Accelerator Physics, World Scientific A. Streun, CAS, (2003)

  8. DIFfraction Limited light source (DIFL) ~20 years from the first proposal • D. Einfeld et al. NIMA 1993 PAC1995 to the first beam • M. Eriksson et al. IPAC 2016 Presently 2 new machines under constructions SIRIUS, ESRF-EBS about 15 new / upgrade R&D projects DIFL PAC95 MAX IV TW DULER Diamond, 19 April 2018

  9. e.g. MAX IV-like TME unit cell TME cells generate strong focussing and large chromaticity The natural chromaticity increses sharply for low emittance (chromaticity wall) Courtesy D. Einfeld

  10. Nonlinear dynamics optimisation strong sextupoles are required to correct the chromaticity large natural x,yfrom arcs and SS linear optics Low x Low x lattice strong chromatic sextupoles, low Dx correction/mitigation strategies large aberrations small DA MA Injection efficiency, beam lifetime (Touschek), control of beam losses Off axis injection ~ 10 mm Dynamic Aperture few per cent Momentum Aperture On axis (swap-out) injection allows more aggressive desing – (like Top Up) reduced DA requirements but still needs a good MA

  11. Single particle dynamics TW DULER Diamond, 19 April 2018

  12. Linear optics Not too difficult to match the linear optics of an ultra low emittnace Original DBA 7BA Emittance 45 pm on 561 m Large natural chromaticity (–349; – 119) Small dispersion make chromaticity correction very hard (strong sextupoles) DA < 1 mm Must control design of linear optics to ease the nonlinear optics optimisation TW DULER Diamond, 19 April 2018

  13. Non-linear beam dynamics • detuning with amplitude • trajectory distortion • resonances (fixed points and islands) • regular stable trajectories (quasi periodic decompositions) • chaotic trajectories (generally unstable) • regular unstable trajectories • limited stable phase space area available to the beam – DA and MA

  14. Design and optimisation tools Based on perturbative theory of betatron motion • resonance driving terms (and detuning terms) Based on numerical tracking • detuning with amplitude, momentum, driving terms, FMA, ... • direct evaluation of DA, MA • direct evaluation of injection efficiency • effect of errors (magnets, misalignment, ...) and IDs Tracking tools: MADX-PTC, elegant, AT, Tracy-II(-III), OPA, ... Iteration with the linear optics design • control natural chromaticity • exploit cancellation rules – symmetries • while achieveing small emittance with compact designs involving GD, LGB, RB, ... TW DULER Diamond, 19 April 2018

  15. Hamiltonian of betatron motion j+k+l+m+p = N A. Schoch, CERN 57-23 (1958); R. Ruth, SLAC-pub 4103, (1986); J. Bengtsson, CERN 88-04 (1988); Resonant driving terms complex coefficients s-dependent each multipole contributes to definite resonant driving terms e.g. sextupoles contribute to 2 chromaticities, 3 chromatic and 5 geometric driving terms (unavoidable 16 additional terms after correcting the chromaticity) phase relations may enhance or cancel their contribution all  b3 …then higher order multipoles  b4;  b5; … …then higher perturbative orders  b3b3;  b3b4; …

  16. Deterministic algorithms Based on perturbative theory of Hamiltonian of betatron motion • sextupole resonance driving terms • first order cancellations in a cell over N cells via symmetry • sensivity matrices driving terms  sextupoles (and SVD) • extension to higher order magnetic multipoles • extension to higher order perturative terms TW DULER Diamond, 19 April 2018

  17. Compensation rules in first order Simple rules can be establish to compensate the contribution of two sextupoles to a given resonant driving term, e.g. for (3,0) S1 S2 S2 3x S1 We can set the phase advance to target specific resonances or, even better, to cancel the contribution to all normal sextupoles geometric resonances terms by x = (2n+1) y= n Normal and skew sextupoles resonances are cancelled by x = (2n+1) y= (2n+1) 3x=  S1 S2 TW DULER Diamond, 19 April 2018

  18. Compensation rules in N cells If the contribution of two sextupoles cannot be compensated within a cell some compensation can be obtained after N cells, e.g. for (3,0) e.g. for N = 5 we can choose Qx = 2/5 and Qy = 1/10 e.g. for N = 7 we can choose Qx = 3/7 and Qy = 1/7 and so on… will cancel all third order resonances Qx = x/2 Qy = y/2 Role of symmetry: Cancellation of imaginary part of driving terms occur if the cell is symmetric S1 x –x x –x S2 Re-adapetd from A. Streun

  19. sensitivity matrices (and SVD inversion) Linear relations between resonance driving terms and multipole gradients using Msext sextupoles for N driving terms J. Bengtsson, SLS 97-9, (1997) Linear relations can be extended to octupoles, to compensate resonances or amplitude detuning terms, e.g. using Noct octupoles for 5 detuning terms S. Leemann et al., PRSTAB 14, (2011) TW DULER Diamond, 19 April 2018

  20. Numerical algorithms Based on numerical search of parameter space • GLASS (GLobal Search of All Stable Solutions) • gradient search, symplex, least square, ... • genetic algorithms, MOGA, particle swarm, (or just random search ) Accurate numerical calculation of the objectives (tracking-based) • dynamic aperture, momentum aperture (s-dependent) • FMA, detuning terms, diffusion rates, RDT, lifetime, injection eff. • verified experimentally: Diamond, SOLEIL, ESRF, SPEARIII, NSLS-II, ... Realistic models can be used directly in the optimisation stage • engineering apertures, IDs, full 6D motion with RF, radiation damping • errors in magnets: fringe fields, systematic and random multipoles • misalignments: girders, individual magnets, BPMs Parallelised on clusters (large throughput) TW DULER Diamond, 19 April 2018

  21. Objectives: simultaneous optimisation of multiple objectives, particularly apt to cases where objectives are conflicting Parameters: suitable for large dimension parameter spaces ensuring the search of global minimum Can track directly DA, MA, injection efficiency, touschek lifetime, even with errors Multi objective optimisation: MOGA injection tracking in phase space and captured particles vs turns correlation DA  RDT full simulation of the injection process (6D) takes into account DA and MA for injection Not perfect but large DA always corresponds to small RDT Y. Li et al, PRSTAB, 14, 045001,(2011)

  22. MAX IV is the first low emittance light source based on 7BA lattice MBA: MAX IV 528 m, 330 pm bare lattice (20 cells) 5 detuned TME cells and two matching cells distributed sextupoles in the arc for chromatic correction Nonlinear dynamics optimisation based on minimisation of detuning with amplitude terms  weak octupoles minimisation of resonance driving terms via SVD of sensitivity matrix numerical evaluation of lattices based on FMA and direct tracking TW DULER Diamond, 19 April 2018

  23. Driving terms minimisation with 5 sextupoles families distributed sextupoles 3 octupoles families to control amplitude dependent tuneshift MAX IV – detuning and diffusion maps 5 sextupoles only with octupoles FMA used to study tune footprint, FM folding, resonance crossing and diffusion rates. Direct tracking with errors for robustness check

  24. Nat. emittance 29 pm-rad (4.5 GeV) 5TME unit cells Cell phase advances: mx=(2+1/8) x 3600, my=(1+1/8) x 3600 Compensation of driving terms achieved after 8 cells 10 mm DA in high beta section Pep-X: 7BA achromat All Geometrical 3rd and 4th Resonances compensated except 2nx-2nytargeted explicitly using additional sextupoles(Y. Cai et al. PRSTAB 15, (2012))

  25. Hybrid 7BA cell pioneered at ESRF-EBS (130pm) and adapted to APS U (67 pm) HEPS (60 pm) • Dispersion bump for chromatic sextupoles; • 3 / phase advance for cancellation of sextupole driving terms; • Longitudinal gradient bend for emittance minimisation • high beta section for injection at ESRF-EBS – swap out inj. at APS-U, HEPS Hybrid MBA ESRF-EBS • longitudinal gradient dipoles: • to reduce the emittance via bending where Hinv is small • increase the dispersion in • bump APS-U

  26. 12 cells 9BA lattice (100 pm) has quad components in all bendings and 2 families of chromatic + 2 families of harmonic sextupoles MBA: ALS-U y • Lattice design work still in progress • Options to use antibends are considered (20% reduction in x) • Small DA requires on-axis injection and small-emittance injected beam with full coupling: x Courtesy C. Steier, C. Sun, M. Venturini

  27. Implementation low emittance: linear optics ORM based methods: LOCO (Linear optics from closed orbit) measure the orbit response matrix compare with the model adjust quadrupoles, skew quads, BPM gains, corrector gain gigantic SVD to provide the machine model ...and correct it based on orbit measurements (FA or SA data) TBT based methods: Linear optics from excited betatron oscillations measure the TBT oscillations at all BPMS compare with the model adjust quadrupoles, skew quads, BPM gains, fit or deterministic corrections based on TBT data of excited oscillations (TBT data) Beside BBA TW DULER Diamond, 19 April 2018

  28. Many examples Hor.  - beating Ver.  - beating Quadrupole gradient variation LOCO-type of analysis successfully implemented in almost all light sources Remarkable agreement machine / model on linear optics SPEAR III, ALS, SOLEIL, ASP, ALBA, SSRF, … e.g. Diamond Modified version of LOCO with constraints on gradient variations (see ICFA Newsl, Dec’07)  - beating reduced to 0.4% rms Quadrupole variation reduced to 2% Results compatible with mag. meas. and calibrations

  29. Skew quadrupoles can be simultaneously zero the off diagonal blocks of the measured response matrix and the vertical disperison Linear coupling correction with LOCO (II)

  30. BPMs coupling and V dispersion correction Correction of V dispersion After LOCO correction r.m.s. Dy = 700 μm (2.2 mm if BPM coupling is not corrected) LOCO fits also the BPM gain and coupling BPM coupling includes mechanical rotation and electronics cross talk These data are well reproducible over months TW DULER Diamond, 19 April 2018

  31. Measured emittances Coupling without skew quadrupoles off K = 0.9% (at the pinhole location; numerical simulation gave an average emittance coupling 1.5% ± 1.0 %) Emittance [2.78 - 2.74] (2.75) nm Energy spread [1.1e-3 - 1.0-e3] (1.0e-3) After coupling correction with LOCO (2*3 iterations) 1st correction K = 0.15% 2nd correction K = 0.08% V beam size at source point 6 μm Emittance coupling 0.08% → V emittance 2.2 pm Variation of less than 20% over different measurements

  32. LOCO based on harmonic excitation Diamond tests (see e.g. BIW10, IPAC 14): low frequency excitation on correctors (tens of Hz) rather than step functions (FA orbit data taken – 10 KHz sampling rate) Multifrequency exctiation and detection (up to 23 frequencies tried at NSLSII – other labs as well ALBA, ESRF) Corrector (t) = u0 * sin(t) BPM x(t) = x0 * sin(t + ) ORM from u0, x0 and 

  33. Linear optics from TBT data Excite betatron oscillations via pingers or AC excitation Take FFT of BPM single at all BPMs Tried successfully in hadron machine (see review by R. Tomas et al PRAB18) Also for linear coupling some attempts at ESRF (see Franchi’s talk) Comparison of TBT vs ORM methods, similar results except for ultra low coupling where S/N is very small

  34. Implementation low emittance: nonlinear optics TBT based methods: non linear optics from excited betatorn oscillations measure the TBT oscillations at all BPMS compare with the model adjust sextupoles, octupoles, ... fit or deterministic corrections based on TBT data of excited oscillations (TBT data) ORM based methods: few tests, not fully developed yet!? fast ORM intial test at ALBA and NSLS II chromatic terms at ESRF (see Franchi) TW DULER Diamond, 19 April 2018

  35. Example: SpectralLinesfortracking data for the Diamond lattice Spectral Lines detected with SUSSIX (NAFF algorithm) Frequency Analysis of betatron motion e.g. in the horizontal plane: • (1, 0) 1.10 10–3horizontal tune • (0, 2) 1.04 10–6 Qx + 2 Qz • (–3, 0) 2.21 10–7 4 Qx • (–1, 2) 1.31 10–7 2 Qx + 2 Qz • (–2, 0) 9.90 10–8 3 Qx • (–1, 4) 2.08 10–8 2 Qx + 4 Qz Each resonance driving term can be associated to a spectral line

  36. Nonlinear dynamics from betatron oscillations All BPMs have turn-by-turn capabilities • excite the beam diagonally • measure tbt data at all BPMs • colour plots of the FFT H BPM number QX = 0.22 H tune in H Qy = 0.36 V tune in V V All the other important lines are linear combination of the tunes Qx and Qy BPM number m Qx + n Qy frequency / revolution frequency A. Ando Part. Acc (1984), J. Bengtsson (1988): CERN 88–04, (1988). R. Bartolini, F. Schmidt (1998), Part. Acc., 59, 93, (1998). R. Tomas, PhD Thesis (2003)

  37. Generalisation of LOCO Frequency Maps and amplitudes and phases of the spectral line of the betatron motion can be used to compare and correct the real accelerator with the model LOCO Closed Orbit Response Matrix from model fitting quadrupoles, etc Linear lattice correction/calibration Closed Orbit Response Matrix measured Nonlinear calibration and correction Spectral lines + FMA + … from model fitting sextupoles and higher order multipoles Nonlinear lattice correction/calibration Spectral Lines + FMA + … measured Combining the complementary information from FM and spectral line should allow the calibration of the nonlinear model and a full control of the nonlinear resonances

  38. Spectral lines (-1,1) in V measured vs modela first fit of the sextupole gradient Blue model; red measured A first attempt to fit the spectral line (-1,1), determined by the resonance (-1,2), improved the agreement of the spectral line with the model However the lifetime was worse by 15% The fit produced non realistic large deviation in the sextupoles (>10%); The other spectral lines were spoiled BPM number TW DULER Diamond, 19 April 2018

  39. Simultaneous fit of (-2,0) in H and (1,-1) in V (-1,1) (-2,0) sextupoles start iteration 1 iteration 2 Both resonance driving terms are decreasing

  40. Studies at ESRF TW DULER Diamond, 19 April 2018

  41. Frequency map and detuning with momentum comparison machine vs model (I) detuning with momentum model and measured FM measured FM model Sextupole strengths variation less than 3% The most complete description of the nonlinear model is mandatory ! Measured multipolar errors to dipoles, quadrupoles and sextupoles (up to b10/a9) Correct magnetic lengths of magnetic elements Fringe fields to dipoles and quadrupoles Substantial progress after correcting the frequency response of the Libera BPMs

  42. Frequency map and detuning with momentum comparison machine vs model (II) Synchrotron tune vs RF frequency DA measured DA model The fit procedure based on the reconstruction of the measured FM and detunng with momentum describes well the dynamic aperture, the resonances excited and the dependence of the synchrotron tune vs RF frequency R. Bartolini et al. Phys. Rev. ST Accel. Beams 14, 054003 TW DULER Diamond, 19 April 2018

  43. Limits with TBT data • Resolution of position measurment (single bunch or short trains) better instrumentation N BPMs method • Leaking of BPM turn position readings to neighbouring turns understanding of BPM filtering in TBT mode • Nonlinearites of BPM geometry linearisation is required • Decoherence of excited signal (especially at large chromaticity) force to AC excitation or wobbling orbits • Intrinsic limit in perturbative theory Nevertheless : many brilliant results at CERN, ESRF, ALBA, Diamond, ... TW DULER Diamond, 19 April 2018

  44. Non-linearities of BPM readings The relation between BPMs reading and beam position is linear only in a reduced region around the BPM block centre PBPM Nonlinearites in the BPM response can generate spurious spectral lines, which behave like the ordinary nonlinear terms (sextupole-like , octupole like, …) TW DULER Diamond, 19 April 2018

  45. Implementation of low emittance optics Implementation of the optics is not a show stopper Many machine already operate at 8 pm emittance or less! Challanges can be met with state-of-the-art alignment and beam based techniques BBA on quad centred type of routine extended to sextupole centre BBA based on LOCO to fit feed-down components from sextupoles (e.g. APS) analysis based on ORM and TBT TW DULER Diamond, 19 April 2018

  46. NSLSII- tolerances (not dissimilar to Diamond tolerance) Optics and linear coupling numerical studies: sensitivity to machine errors Magnet based in situ alignment (stretched wire) can achieve better than 30 um at least in straight part of the MBA cell Ultra low emittance requires similar constraints on the alignment tolerances: but use first turn trajectory simulations (see APS-U)

  47. Tolerances on alignment and field quality Over-specifying tolerances can be very expensive There was a general consensus that the tolerances assigned to magnet alignment in the field quality are computed with a pessimistic approach. The possibility of beam based correction should be considered in specifying the tolerances. This is true for alignment errors both for orbit and for optics corrections. Beam based correction tools like BBA, LOCO, coupling free steering can significantly relax the tolerances. It as suggested to set tolrances based on first-turn just to store the beam. Swap out injection allows operating with small dynamic aperture ad might help likewise to relax also the tolerances on harmonics J. Safranek, R. Bartolini DLSR 2013 TW DULER Diamond, 19 April 2018

  48. Collective dynamics TW DULER Diamond, 19 April 2018

  49. Ultra low emittance rings require operation with much smaller apertures: Experience gained with the DDBA upgrade at diamond Problems with ultra low emittance rings plus specific issues related to the small emittance: sensitivity to IBS, and intrinsically small momentum compaction factor (and short bunch length)

  50. Each bunch oscillates with the betatron frequency x,y In a multi-bunch train without wakefields all bunches are independent The motion can be expanded in Fourier modes, (M bunches = M modes, and in full fill, M = harmonic number) However, without wakefields there is no reason to prefer this set to any other “basis” (i.e. modes are degenerate) Transverse multibunch oscillations - no wakefields TW DULER Diamond, 19 April 2018

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