Low-emittance tuning for circular colliders

Transcription

1 Low-emittance tuning for circular colliders Tessa Charles With thanks to: S. Aumon, B. Härer, B. Holzer, K. Oide, T. Tydecks, F. Zimmerman eefact2018, Hong Kong

2 Challenges & constraints for FCC-ee (and CEPC) emittance tuning IP y =1.6 mm x,max = m y,max = m Large emittance ratio, y / x =0.201% Tessa Charles, eefact2018, Hong Kong 2

3 Vertical dispersion & betatron coupling dominate εy growth Horizontal emittance: x = C g J x 2 3 F F FODO = 1 2sinψ 5+ 3cosψ L 1 cosψ l B L: cell length l B : dipole length : phase advance/cell Vertical emittance: 2 dp y = ( Dy 2 +2 D y Dy 0 + Dy 02 ) p Sources of vertical emittance growth: vertical dispersion D y betatron coupling opening angle ~ 1/ (here negligible) Tessa Charles, eefact2018, Hong Kong 3

4 Correction methods used: Orbit correction: MICADO & SVD from MADX Hor. corrector at each QF, Vert. corrector at each QD 1598 vertical correctors / 1590 horizontal correctors BPM at each quadrupole 1598 BPMs vertical / 1590 BPMs horizontal Vertical dispersion and orbit: Orbit Dispersion Free Steering (DFS) Linear coupling: Coupling resonant driving terms (RDT) 1 skew at each sextupole + skews correctors at the IP Beta beating correction & Horizontal dispersion via Response Matrix: Rematching of the phase advance at the BPMs 1 trim quadrupole at each sextupole µ 1 2a u ad 1 u µ 1 2a A u 0, ab ~f 1001 ~f 1010! meas ¼ M ~J c ; ð Þ ð Þ ð Þ Þ ( xy, D x, y) =R k 1 Tessa Charles, eefact2018, Hong Kong 4

5 Initial assessment Sextupoles: y = 10 µm y, rms (mm) D y,rms (mm) Quadrupoles: y =2µm Tessa Charles, eefact2018, Hong Kong 5

6 Correction methods applied to Vertical Quadrupole Misalignments (σ y =100 μm) Sextupoles turned off: Dispersion correction during the coupling correction Switch on sextupoles Initial D y After orbit correction - factor 2e4 improvement DFS - factor 50 improvement Factor ~10 Initial D y After orbit correction - factor 2e4 improvement DFS - factor 50 improvement Tessa Charles, eefact2018, Hong Kong 6

7 Coupling matrix elements Tessa Charles, eefact2018, Hong Kong 7

8 Beta-beating Correction Weighted SVD Beta-beat introduced through: Arc quads: Δx = 100 μm, Δy = 100 μm, Δθ = 100 μm Sextupoles: Δx = 100 μm, Δy = 100 μm, Δθ = 100 μm Tessa Charles, eefact2018, Hong Kong 8

9 Correction Strategy 7-8h up to one day of Sextupoles strengths set to 0 simulation/seed x-y orbits correction Coupling correction Tune matching Beat-beat correction Loop 20 times 1 step Dispersion Free Steering wo sextupole (Dy correction) + 1 step coupling correction (kicker strength change the coupling configuration) Save x,x,y,y at the beginning of the machine Set sextupoles strength to 10% of their design current orbit corrections coupling correction, tune matching beta beat correction coupling + Dy correction increase by 10% the sextupole strength This avoid the tunes run of to resonance and maximize the number of seeds Final correction Coupling corrections Weighted beat-beat correction Tessa Charles, eefact2018, Hong Kong 9

10 Corrected Lattice - Misaligned arc quads & sextupoles x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles IP quads perfectly aligned (for now) 436 out of 500 seeds converged ε y = pm +/ ε x = nm +/ ε y /ε x = 0.006% (limit 0.1%) Tessa Charles, eefact2018, Hong Kong 10

11 - Misaligned arc and IP quads & sextupoles x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles Corrected Lattice 700 out of 1000 seeds converged ε y = pm +/ ε x = 1.52 nm +/ ε y /ε x = % (limit 0.1%) Tessa Charles, eefact2018, Hong Kong 11

12 Vertical dispersion highly susceptible misalignments Dispersion introduced through: quadrupoles: Δx = 100 μm, Δy = 100 μm, Δθ = 100 μm Sextupoles: Δx = 100 μm, Δy = 100 μm, Δθ = 100 μm D y,rms = m D y,rms =1.02 mm Tessa Charles, eefact2018, Hong Kong 12

13 Beta beat after correction horizontal Δβ/β (%) vertical Δβ/β (%) Tessa Charles, eefact2018, Hong Kong 13

14 Corrected Lattice - Misaligned arc and IP quads & sextupoles x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles out of 1000 seeds converged ε y = 0.11 pm +/ ε x = 1.52 nm +/ ε y /ε x = % (limit 0.1%) Tessa Charles, eefact2018, Hong Kong 14

15 To increase the number of successful seeds: Add more tune matching to strategy. Start with relaxed optics, before reducing β*. Place limit on maximum trim and skew quad strength that can be applied any given iteration step.

16 Tracking done by T. Tydecks Dynamic / Momentum aperture with radiation damping only y0 / µm x0 / mm x 0 / mm Apertures sufficient for beam storage and injection. x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles

17 Tracking done by T. Tydecks Dynamic / Momentum aperture with radiation damping and quantum excitation y0 / µm 3 2 x0 / mm x 0 / mm Apertures sufficient for beam storage and injection. x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles

18 Conclusions FCC-ee poses a unique challenge for emittance tuning With 100 μm, 100 μrad misalignments in arc quads & sextupoles and 50μm and 50 μrad misalignments in IP quads, the mean vertical emittance achieved after correction schemes applied is ε y = 0.11 pm rad ε y = 0.11 pm +/ ε x = 1.52 nm +/ ε y /ε x = % Tessa Charles, eefact2018, Hong Kong 18

19 Back up slides top v213 lattice x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles dipoles x (µm) y (µm) (µrad) arc quadrupoles IP quadrupoles sextupoles dipoles

20 Back up slides - DFS Build numerically a matrix for vertical orbit (u) & dispersion (D u ) response under a corrector kick (al) µ 1 2a u Orbit response Dispersion response a D u A i,j = p 1 i j µ 1 2a A ab 2 sin( Q y ) cos( µ i µ j Q y ) u 0, quad K l L l β l B ij = { 4sin(πQ) cos( µ 2 i µ l πq)cos( µ l µ j πq) l sext SVD analysis to solve the system and find a solution m K 2,m D x,m L m β m 4sin(πQ) 2 cos( µ i µ j πq) } β l β j sin(πq) cos( µ i µ m πq)cos( µ m µ j πq) Emittance optimization with dispersion free steering at LEP R. Assmann et al. Phys. Rev. ST Accel. Beams 3,

21 arcs [1]. where ΔQ phases q ) oscillate withstrength. the betatron pha normalised quadrupole The cons each k the where x is the normalized horizontal position and min is the closest the tunes can approach with The following definitions (all s dependent) apply: When a askew quadrupole kick ismomentum met, bot other, N is the number of BPMs and Δ is the fractionalterm tune k yhorizontal transverse momentum. The provides vertical dipole component and there split. A more precise relation was published in [23] but and phase execute abrupt jumps. This be directly measurable quantity with a BPM but nee C ¼ cosh ð2p Þ; (6) dispersion. vertical Sextupole offsets produce coupling 13/10/16 never applied to data. The nomenclature used in this article reconstructed using two BPMs. The momentum a y tapered option - or sectorwise version- the machine mitted to since, the RMS emittances: As shown vertical dipole kick is different and we therefore derive the relation in the BPM can be written as [22] vides a tapering to the dipoles only, leaving therefore a apparent emittances oscillate around the r sinh ð2p Þ Appendix for clarity. The relation is described as Coupling RDT f -f are related to the coupling parameter via: jf S ¼ j; (7) 1001 = k x ys+$ksxþ cosðq yx þx ) cos $as 1010 q $Δϕ Þ. This os thebxterms aining horizontal orbit shownpin Fig [5]. iþ1 i x!! I p ¼ ; References: xi 2!! 4Δ course more important for the vertical appa 2 2 herefore, with targeted emittances of the order of nm sin Δϕ iðϕ ϕ ÞþisΔ=R!! B = k (x y ) 2k y ( y ) x y x y ΔQmin ¼ jc j ¼! ; ð2þ dsf 1001 e -Vertical emittance reduction and preservation in as it is proportional to the larger horizontal 2πR sinh ð2p Þ performances of! pm, FCC-ee is a collider with foreseen electron storage rings via resonance driving terms jf1010 j; Sþ ¼ (8) where Δϕ is the horizontal phase advance betwee xplots The refer toathe ideal ESRF stora E u.roll Quadrupole angles produce skew strength, genera t sources (ESRF, SLS). P correction, A.BPM Franchiunder et al, PRSTAB 14, that th where R is the radius of the machine, ϕx is the horizontal and ði þ 1Þth the assumption errors but localized skew emittanc quadrup betatron coupling andthree transfering horizontal between the two BPMs is free of coupling sou phase, ϕy be is the phase, and s is the longitudinal ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qvertical f1001-fdistance. can computed via analytical formulas, projectedthe emittance constant in the vertical emittance. resultingstays vertical dispersion cha 1010 nonlinearities contributing to the main and the The¼integral extends over the in j2 þ jf1010 j2 ;entire ring but(9) P $jf1001 AMPLIFICATION FACTOR BY ERROR the Equation two coupling sources, the appo or viapractice a matrix formalism with theatcoupling matrix: due strength componant is awhile line. (4) indicates that phase advance it will only be evaluated the locations of theto a skew TYPE keeps oscillating. When a skew quadrupo pffiffiffiffiffiffiffiffiffiffiffiffiffi PW p w!w eið!"w;x (!"w;y Þ J! w;1 x y y y0 the projected emittance jumps, while the ; (10) ¼ w f D y = ( Jw )D x cos( Q y0 n this section, amplification factors2#iðq on uthe orbit and/or y ) ( Q Þ 1010 v 4ð1 $ e Þ tance changes in oscillation amplitude 2sin( Q) vertical dispersion are computed by errors type. For (because both S $ and S þ change abruptly ttance tuning purposes, any source of vertical dispersion where J is the skew strength, D x is the horizontal RVATION..q. $ Phys. Rev. ST Accel. Beams 14, (2011) w ¼ arg ff1001 g; q þ ¼ arg ff1010 g: (11) coupling has to be identified and should be corrected as persion, y and y0 are respectively vertical beta func JwA, found w ¼ 1; 2; 3..., W are the skew quadrupole integrated response matrix can be written to measure the of the RDTs to a to minimize as uniformly as possible both couplingthe response4.003 strengths present the ring and originated bysvd quadrupole RDTs along thein ring. The invert via readsbe inverted via skew quadrupole field, Jsystem system, which can SVD: c. The to tilts, sextupole misalignments, insertion devices, and corapparent emittan! ~ f projected emitta rector skew quadrupoles Qu;v are 1001 already powered. ~ 4 (30)the meas ¼ "MJ c ; ~ eigentunes, which are equal to the measured tunes up to f w the first order inð1þstrengths, Qu;v ¼ Qx;y þ OðJw;1 Þ.!r where J~c ¼ ðj1 ;... ; J1ðcÞ ;... ; J1ð32Þ Þ are the integrated 45 denotes the Twiss parameter corresponding to the location strengths of the 32 corrector skew quadrupoles to be 40 of the skew quadrupole kick, whereas!" is its phase [pm] horizontal plane [nm] Back up slides - coupling

22 Back up slides beta-beating For n trim quadrupoles which can exercise a small field strength k 1,the weighted SVD can be applied through adding weighting factors f to each measurement of the beta-beat. 0 f 1 f 2 f m 1 y0 y0 2 y0... y0 m y0 y0 1 C A meas = 0 f 1 (R 11,R 12,R 13,...,R 1n ) f 2 (R 21,R 22,R 23,...,R 1n )... f m (R m1,r m2,r m3,...,r mn ) 1 C A 0 k 1 k 2... k n 1 C A where y0 is the ideal beta function at the given BPM, R i,j are elements in the response matrix.