Quadrupolar Pick-ups to Measure Space Charge Tune Spreads of Bunched Beams

Size: px

Start display at page:

Download "Quadrupolar Pick-ups to Measure Space Charge Tune Spreads of Bunched Beams"

April Watts
6 years ago
Views:

1 Quadrupolar Pick-ups to Measure Space Charge Tune Spreads of Bunched Beams First MD Results from the PS Adrian Oeftiger, Malte Titze Acknowledgements: Simon Albright, Marcel Coly, Heiko Damerau, Marek Gasior, Tom Levens, Elias Métral, Guido Sterbini, Panagiotis Zisopoulos Space Charge 2017, GSI, Germany 4. October 2017

2 Goals of Study Motivation In the context of strong space charge regime with LHC Injectors Upgrade beam parameters: determine beam brightness (or incoherent space charge (SC) tune shift) directly via coherent quadrupolar modes Basic principle: coherent dipolar (centroid) oscillation: no influence from SC (Newton s third law, actio = reactio) coherent quadrupolar (envelope) oscillation: transverse SC reduces frequency = quantify SC from envelope mode shift in quadrupolar spectrum Content of this talk: peculiarities of CERN proton synchrotrons w.r.t. earlier experiences: 1 tunes close to coupling resonance, no far off coupling approximation 2 bunched beams, large incoherent tune spreads from strong SC first observations from PS set-up with a quadrupolar RF kicker 1 of 15 Adrian Oeftiger QPU Studies for SC Measurement

3 Schematic Quadrupolar Pick-up image taken from [1] In the PS, re-cabling the Long Pick-up 72 as (L + R) (T + B) results in the time signal S QPU (t) x 2 y 2 = σ 2 x (t) σ2 y (t) + x 2 (t) y 2 (t). (1) 2 of 15 Adrian Oeftiger QPU Studies for SC Measurement

4 Some Historical Perspective QPU in time domain for emittance measurements: 1983, R. H. Miller et al. at SLAC [2] 2002, A. Jansson at CERN in PS [3] 2007, C.Y. Tang at Fermilab [4] QPU in frequency domain for space charge measurements: 1996, M. Chanel at CERN in LEAR [5] 1999, T. Uesugi et al. at NIRS in HIMAC [6] 2000, R. Bär at GSI in SIS-18 [7] 2014, R. Sing et al. at GSI in SIS-18 [1] = all far off coupling and coasting beams CERN s proton synchrotrons: close to coupling = quadrupolar mode frequencies change bunched beam 3 of 15 Adrian Oeftiger QPU Studies for SC Measurement

5 GSI results at SIS-18 QPU measurements at GSI by R. Singh, M. Gasior et al. [1] particle type N 7+ E kin (MeV/u) I beam (ma) ɛ x,ɛ y (mm-mrad) 8, Q x0,q y0 4.21, 3.3 Q f x = Q x Q f y = Q y Q coh = Q ± far off coupling resonance coasting beam = sharp envelope peak 4 of 15 Adrian Oeftiger QPU Studies for SC Measurement

6 Far Away vs. On the Coupling Resonance 2 eigenmodes for coherent quadrupolar oscillation: far away from coupling full coupling (a) horizontal mode (b) vertical mode (a) breathing mode (b) antisym. mode Relation of mode frequencies to incoherent KV tune shift: Q ± = 2Q x,y ( Q KV x,y 3 σ ) x,y /2 σ x + σ y (2) Relation of mode frequencies to incoherent KV tune shift: Q + = 2Q 0 Q KV x,y (3a) Q = 2Q 0 3 Q KV x,y (3b) 2 (assuming round beams, Q x,y Q 0 ) 5 of 15 Adrian Oeftiger QPU Studies for SC Measurement

7 Peculiarity 1: Near Coupling Resonance At vanishing lattice coupling, keep constant incoherent SC tune shift and fixed Q x. Vary Q y for a coasting round beam: analytic expression simulation results envelope tunes Q x = 6. 22, N = Q Q + Q x, off Q y, off Qx = Qy Q y envelope tunes fixed Q x = 6. 22, N = Qx = Qy simulations Q (theory) Q + (theory) Q y (N refers to intensity of coasting beam within total bunch length B L = 180ns) 6 of 15 Adrian Oeftiger QPU Studies for SC Measurement

8 Peculiarity 2: Bunched Beam Envelope Signal Assumption: synchrotron motion much slower than betatron motion, Q s Q x0,y0 3D RMS envelope equation (Sacherer) decouples to 2D + 1D = for a given longitudinal bunch slice, the coherent transverse quadrupolar oscillation depends on local line charge density λ(z), longitudinal motion is quasi-stationary and independent (see e.g. [6]) = envelope tune spread with longitudinal bunch shape imprinted line charge density λ(z) = envelope spectral power 2Qx0 2Qy0 longitudinal position z envelope tune Q ± Figure: sketch of envelope detuning scaling with local line charge density Expectation: lower end of envelope tune spread strong SC at bunch centre = RMS-equivalent (maximal) KV tune shift Qx,y KV from envelope spread 7 of 15 Adrian Oeftiger QPU Studies for SC Measurement

9 Incoherent KV Tune Shift The uniform (Kapchinskij-Vladimirskij / KV) beam distribution has all particles at same incoherent space charge tune shift: Q KV x,y. K SC R 2 = 4σ x,y (σ x + σ y )Q x,y (4a). = 1 + σ x,y/σ y,x 2Q x,y Λ (4b) Connect Λ quantity to general envelope mode expressions in terms of observables: Λ = Q2 + +Q2 4(Q2 x +Q2 y ) 4 + 3(σ x /σ y + σ y /σ x ) (5) (Gaussian tune spread = 2x the RMS-equivalent KV tune shift!) space charge perveance K SC. qλ = 2πɛ 0 βγ 2 p 0 c 8 of 15 Adrian Oeftiger QPU Studies for SC Measurement

10 Experimental Set-up Ingredients: small time window of 15 ms with decoupled optics to single out space charge influence on envelope tune separation chirped excitation of beam via transverse feedback: external waveform generator appropriately connected to kicker plates 12 ms long frequency sweep with 1 ms return harmonic h = 5 with frequency range 2.19 MHz to 2.4 MHz single bunch in PS with a factor 5 smaller incoherent SC tune shift compared to currently operational LHC beams, off coupling intensity transverse emittance average betatron function average dispersion momentum deviation spread bunch length synchrotron tune KV space charge tune shift N ppb ɛ x,y 2.3mmmrad β x β y 16m D x 3m σ δ B L 180ns Q s = 1/600 = Q KV x,y of 15 Adrian Oeftiger QPU Studies for SC Measurement

11 Quadrupolar Excitation: Chirp beam response (via QPU) excitation signal spectrogram for quadrupolar excitation frequency quadrupolar spectrogram for N = fractional tune f/frev fractional tune f/frev 0.4 dipolar excitation due to feed-down from quadrupolar excitation, seen in beam response at lower frequencies f < 0.25 f rev 10 of 15 Adrian Oeftiger time from excitation start [ms] turn from QPU signal start 5000 time from excitation start [ms] 6000 turn from QPU signal start QPU Studies for SC Measurement 0 0.5

12 Extracting the Beam Response... (a) FFT across up-chirp time is not such a useful idea... FFT spectrum over chirp 1.6 1e fractional tune f/frev band filtered amplitude along chirp I beam response along chirp, amplitude 2 + Q 2 1e excitation harmonic f/frev (b)... instead project and band filter along local excitation frequency 11 of 15 Adrian Oeftiger QPU Studies for SC Measurement

13 Approach: In-phase and Quadrature Components Take a) QPU time signal S QPU (t) b) excitation signal S exc (t) (sine wave with increasing frequency) c) 90 deg shifted excitation signal C exc (t) = S exc (t) φ φ+90deg Assume immediate beam response to chirp: 1 correlation: find excitation start in S QPU (t) by correlation with S exc (t) 2 demodulation of measured QPU time signal into I (t) = S QPU (t) S exc (t) (in-phase component) and Q(t) = S QPU (t) C exc (t) (quadrature component) 3 band filter original S QPU (t) around time-varying excitation frequency by low pass filtering I (t) and Q(t) 4 amplitude of beam response along chirp amounts to I 2 (t) +Q 2 (t) 12 of 15 Adrian Oeftiger QPU Studies for SC Measurement

14 First Results: Parabolic Bunch of fractional tune f/frev fractional tune f/frev fractional tune f/frev 0.4 q beam response along chirp, amplitude I 2 + Q e time [ns] Adrian Oeftiger band filtered amplitude along chirp 0.1 bunch current [A] 0.0 Qxdip Qydip 2Qxdip 2Qydip turn from QPU signal start 5000 time from excitation start [ms] 6000 turn from QPU signal start turn from QPU signal start 7000 quadrupolar spectrogram for N = Qydip 1.5 Qxdip Qydip 2Qxdip 2Qydip Q spread Q+ spread time from excitation start [ms] vertical BBQ spectrogram for N = Qxdip time from excitation start [ms] horizontal BBQ spectrogram for N = excitation harmonic f/frev QPU Studies for SC Measurement

15 First Results: Flat Bunch fractional tune f/frev fractional tune f/frev 0.4 q beam response along chirp, amplitude I 2 + Q e time [ns] Adrian Oeftiger band filtered amplitude along chirp bunch current [A] fractional tune f/frev of Qxdip Qydip 2Qxdip 2Qydip turn from QPU signal start 5000 time from excitation start [ms] 6000 turn from QPU signal start turn from QPU signal start 7000 quadrupolar spectrogram for N = Qydip Qxdip Qydip 2Qxdip 2Qydip 2.0 Q spread spread 1.5 Q time from excitation start [ms] vertical BBQ spectrogram for N = Qxdip time from excitation start [ms] horizontal BBQ spectrogram for N = excitation harmonic f/frev QPU Studies for SC Measurement

16 Summary and Outlook In conclusion: transverse feedback (TFB) strong enough for quadrupolar excitation observed envelope tune spread in SC depressed bunched beam = Exciter + QPU = potentially very powerful diagnostic tool for SC Next steps: QPU hardware: planned to have all H,V,Q channels simultaneously after this coming technical stop (for PS as well as SPS!) TFB: generate synchronous and clean signals for quadrupolar excitation with internal waveform generator improve QPU set-up and spectral analysis (e.g. subtract dipolar contribution from quadrupolar signal 3 channels required!) dedicated space charge experiments (e.g. resonance studies) more realistic space charge simulations with bunched distributions 15 of 15 Adrian Oeftiger QPU Studies for SC Measurement

17 Thank you for your attention!

18 Appendix

19 References I [1] R Singh et al. Observations of the quadrupolar oscillations at GSI SIS-18. In: (2014). [2] R H Miller et al. Nonintercepting emittance monitor. Tech. rep. Stanford Linear Accelerator Center, [3] Andreas Jansson. Noninvasive single-bunch matching and emittance monitor. In: Physical Review Special Topics-Accelerators and Beams 5.7 (2002), p [4] Cheng-Yang Tan. Using the quadrupole moment frequency response of bunched beam to measure its transverse emittance. Tech. rep. Fermi National Accelerator Laboratory (FNAL), Batavia, IL, [5] Michel Chanel. Study of beam envelope oscillations by measuring the beam transfer function with quadrupolar pick-up and kicker. Tech. rep [6] T Uesugi et al. Observation Of Quadrupole Mode Frequency And Its Connection With Beam Loss. In: KEK (1999). url: 16 of 15 Adrian Oeftiger QPU Studies for SC Measurement

20 References II [7] R C Baer. Untersuchung der quadrupolaren BTF-Methode zur Diagnose intensiver Ionenstrahlen. Universitaet Frankfurt, Germany, of 15 Adrian Oeftiger QPU Studies for SC Measurement

BPM): By scanning the orbit location one can minimise the

21 TFB: Impact of Orbit Set up a local bump through the TFB and measure the induced beam signal on the plates (effectively a BPM): By scanning the orbit location one can minimise the difference signal: 18 of 15 Adrian Oeftiger QPU Studies for SC Measurement

22 TFB: Static Quadrupole on h = 1 TFB pulsing at f rev becomes a static quadrupole to the beam varying the phase of the pulsing RF quadrupole changes the tune impact Q x Q y Influence of TFB on/off (in anti-symmetric quadrupolar mode) on centroid tunes (measured via all 43 PS BPMs) TFB on TFB off # shot Q x Q y Tune Influence of TFB (in anti-symmetric quadrupolar mode) with respect to TFB phase TFB off TFB off TFB phase [deg] 19 of 15 Adrian Oeftiger QPU Studies for SC Measurement

23 Simulations for Tune Scan Simulations with KV beams for N = confirm theory: envelope tunes fixed Q x = 6. 22, N = Qx = Qy simulations Q (theory) Q + (theory) Q y coupling parameter D fixed Q x = 6. 22, N = Qx = Qy Q y r.m.s. equivalent Gaussian beams (with same σ x,y like KV beams) exhibit same quadrupolar tunes as KV Gaussian spectra broaden quickly 20 of 15 Adrian Oeftiger QPU Studies for SC Measurement

24 Intensity Scan envelope tunes With slightly split tunes, approach full coupling by increasing bunch intensity: Q x = 6. 22, Q y = approximations: Q ±, on Q x, off Q Q + eigenmodes rot. angle [deg] Q x = 6. 22, Q y = Q y, off N [10 12 ppb] N [10 12 ppb] = scan space charge tune shift Q KV x,y and verify theory coupling parameter D 21 of 15 Adrian Oeftiger QPU Studies for SC Measurement

25 Envelope Equations Envelope equations of motion (e.o.m.) r x + K x(s)r x ɛ2 x,geo r 3 x r y + K y(s)r y ɛ2 y,geo r 3 y K SC 2(r x + r y ) = 0, (6a) K SC 2(r x + r y ) = 0 (6b) for transverse r.m.s. beam widths r x,y = σ x,y have equilibrium Q 2 x R 2 r x,m ɛ2 x,geo r 3 x,m Q 2 y R 2 r y,m ɛ2 y,geo r 3 y,m K SC 2(r x,m + r y,m ) = 0, (7a) K SC 2(r x,m + r y,m ) = 0 (7b) 22 of 15 Adrian Oeftiger QPU Studies for SC Measurement

26 Linear Perturbation in Smooth Approximation Constant focusing channel K x,y = 1 β 2 x,y = Q2 x,y = const (8) R2 gives linearised e.o.m. for perturbation around equilibrium r = r m + δr with d 2 ( ) ( ) ( ) δrx κx κ ds 2 = SC δrx δr y κ SC κ y δr y }{{}. =(κ) κ x,y = 4 Q2 x,y κ SC. κ SC = R 2 2σ x,y +3σ y,x σ x,y (9) (10) K SC 2(σ x +σ y ) 2 23 of 15 Adrian Oeftiger QPU Studies for SC Measurement

27 Definitions D. = κ y κ x 2κ SC Coupling Parameter = 4 Q2 y Q2 x K SC R 2 (σ x + σ y ) ( σy σ ) x 2 σ x σ y (11) Rotation Into Decoupled Eigensystem tan(α) = 1 [ ] κ y κ x + 4κ 2 2κ SC + (κ y κ x ) 2 SC = D D 2 (12) 24 of 15 Adrian Oeftiger QPU Studies for SC Measurement

28 Incoherent Tune Shifts KV Space Charge Tune Shift Q KV x,y = K SC R 2 4σ x,y (σ x + σ y )Q x,y (13) with K SC. = qλ 2πɛ 0 βγ 2 p 0 c (14) R.m.s. Equivalent Gaussian Space Charge Tune Spread linearised Gaussian e-field = twice r.m.s. equivalent KV e-field { } = max Q spread x,y = 2 Q KV x,y (15) 25 of 15 Adrian Oeftiger QPU Studies for SC Measurement

29 Gaussian vs. R.m.s. Equivalent KV Ex [kv/m] Ex [kv/m] x [mm] y [mm] x [mm] y [mm] (a) Gaussian beam (b) r.m.s. equivalent KV beam Figure: Electric fields in r.m.s. equivalent distributions with same σ x,y 26 of 15 Adrian Oeftiger QPU Studies for SC Measurement

$C. min. Gaussian Q 6.24 6.23 6.22 6.21 6.20 6.18 6.20 6.22 6.24 Q x 4.0 2.0 0.0 fraction of particles [%] 27 of 15 Adrian Oeftiger QPU Studies for SC Measurement$

30 Incoherent Tunes and R.m.s. Equivalence Incoherent tune spread of a coasting, transversely Gaussian distribution: Qy bare tune Q r.m.s. equivalent Q KV S. C. min. Gaussian Q Q x fraction of particles [%] 27 of 15 Adrian Oeftiger QPU Studies for SC Measurement

31 Quadrupolar Mode Formulae Quadrupolar Mode Tunes (General Formula) Q 2 ± = R2 2 [ ] κ x + κ y ± 4κ 2 SC + (κ y κ x ) 2 = 2(Q 2 x +Q2 y ) K SC R 2 (σ x + σ y ) 2 [ ( σy + σ ) x σ x σ y ] 1 + D 2 2 (16) 28 of 15 Adrian Oeftiger QPU Studies for SC Measurement

32 Quadrupolar Mode Formulae Off-resonance D 1 With Round Beam for Q y > Q x otherwise exchange x y Q + = 2Q y 5 4 QKV y, (17a) Q = 2Q x 5 4 QKV x. (17b) 28 of 15 Adrian Oeftiger QPU Studies for SC Measurement

33 Quadrupolar Mode Formulae On-resonance D 0 With Round Beam. for Q 0 = Qx = Q y and Q KV. = Q KV x = Q KV y Q + = 2Q 0 Q KV, (18a) Q = 2Q QKV. (18b) 28 of 15 Adrian Oeftiger QPU Studies for SC Measurement

34 QPU Simulations in SPS simulation parameters: machine: SPS at injection γ = 27.7 ɛ x = ɛ y = 2.5mm mrad N b = turns macro-particles longitudinally matched Gaussian-type distribution betatron mismatch by 10% in both x, y = injection oscillations 29 of 15 Adrian Oeftiger QPU Studies for SC Measurement

35 QPU Spectrum: Only Betatron Mismatch need beam mismatched to both β x,β y to see clear peaks P [a.u.] Q x0 Q y0 2Q x0 2Q y f/f rev = 2Q x0, 2Q y0 from undepressed envelope oscillations = including synchrotron motion: same spectrum (no coupling!) 30 of 15 Adrian Oeftiger QPU Studies for SC Measurement

36 QPU Spectrum: Include Dispersion smooth approximation: constant D x = 2.96 around the ring P [a.u.] Q x0 Q y0 2Q x0 2Q y f/f rev = peak at Q x0 comes from dispersion 31 of 15 Adrian Oeftiger QPU Studies for SC Measurement

37 Reason for Dispersion Peak x 2 σ x (i turn ) = i beam 2 x i beam with x i (i turn ) = β x ɛ s.p. x,i cos(2πq x0 i turn + Ψ 0 ) + D x δ i 32 of 15 Adrian Oeftiger QPU Studies for SC Measurement

38 Reason for Dispersion Peak x 2 σ x (i turn ) = i beam 2 x i beam with x i (i turn ) = β x ɛ s.p. x,i cos(2πq x0 i turn + Ψ 0 ) + D x δ i = x 2 i =...cos2 (2πQ x0 i turn +...) +...D }{{} x δ i cos(2πq x0 i turn +...) cos(2π2q x0 i turn +...) due to: 2cos 2 (α)=cos(2α)+1 i.e. only for D x 0 = peak at Q x0 32 of 15 Adrian Oeftiger QPU Studies for SC Measurement

39 QPU Spectrum: Include Synchrotron Motion synchrotron motion couples to betatron motion through non-zero D x = 29.6m (smooth approximation!) P [a.u.] Q x0 Q y0 2Q x0 2Q y f/f rev = peak separation at Q x0 from synchrobetatron coupling Q s = at injection for V = 5.75MV 33 of 15 Adrian Oeftiger QPU Studies for SC Measurement

40 QPU Spectrum: Slower Synchrotron Motion synchrotron motion couples to betatron motion through non-zero D x = 29.6m (smooth approximation!) P [a.u.] Q x0 Q y0 2Q x0 2Q y f/f rev Q s = changing γ tr = (while γ = 27.7) = peak separation shrinks 34 of 15 Adrian Oeftiger QPU Studies for SC Measurement

41 Reason for Peak Separation with Q s x 2 i = D x δ i cos(2πq x0 i turn +...) +... with δ i (i turn ) = ˆδ i cos(2πq s i turn +...) 35 of 15 Adrian Oeftiger QPU Studies for SC Measurement

42 Reason for Peak Separation with Q s x 2 i = D x δ i cos(2πq x0 i turn +...) +... with δ i (i turn ) = ˆδ i cos(2πq s i turn +...) = x 2 i = cos(2πq x0i turn +...)cos(2πq s i turn +...) +... }{{} cos(2π(q x0 Q s )i turn +...)+cos(2π(q x0 +Q s )i turn +...) due to: 2cos(α)cos(β)=cos(α β)+cos(α+β) i.e. for D x 0 and Q s 0: one peak at Q x0 = two peaks located at Q x0 ±Q s 35 of 15 Adrian Oeftiger QPU Studies for SC Measurement

TUNE SPREAD STUDIES AT INJECTION ENERGIES FOR THE CERN PROTON SYNCHROTRON BOOSTER

TUNE SPREAD STUDIES AT INJECTION ENERGIES FOR THE CERN PROTON SYNCHROTRON BOOSTER B. Mikulec, A. Findlay, V. Raginel, G. Rumolo, G. Sterbini, CERN, Geneva, Switzerland Abstract In the near future, a new