Longitudinal Dynamics - Longitudinal Phase Space Manipulation -

# Longitudinal Dynamics - Longitudinal Phase Space Manipulation -

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## Longitudinal Dynamics - Longitudinal Phase Space Manipulation -

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1. Longitudinal Dynamics - Longitudinal Phase Space Manipulation - Introduction Bunch Length Manipulation Energy Spread Manipulation Transverse and Longitudinal Emittance Exchange and Control Summary

2. Introduction e+/-: =momentum compaction factor p: -1/γt2 2πfrf single-particle equation of motion energy deviation E=beam energy T=revolution period phase deviation derivatives wrt time, t  harmonic oscillator equations (=s,=0) =t ellipses in phase space (for linear rf; i.e. small amplitude particle motion) synchrotron frequency synchrotron tune

3. Analogy between transverse and longitudinal motion transverse longitudinal kx .. px + kpx = 0 equations of motion k  (/c)^2 x’  (xco+x=0, x’co+x’=0) (=s,=0) x =t phase space second moments beam matrix (taking <x>=0, <x’>=0) (taking <>=0, <>=0) emittance

4. Homework Calculate the Fourier spectrum for the transverse dipole moment D()=x()i() for a single bunch executing synchrotron oscillations in a storage ring assuming that the particle distribution of the bunch may be represented as a d-function; e.g. The current distribution in the time domain is given by where T is the revolution period, ta is the synchrotron oscillation amplitude, ws is the angular synchrotron frequency and the sum is over turns with index n. (The results can be compared with the computer lab assignment from last week). Useful relations:

5. Longitudinal phase space – bunch manipulation topics parameters of most interest (first) applications in manipulation method bunch splitting bunch coalescing N circular accelerators I linear accelerators and linac FELs z , I bunch length compression colliders z , I bunch “rebucketing” z, σ bunch length pre-compression and/or energy acceptance matching transfer between circular accelerators synchrotron light sources z, I bunch lengthening using harmonic cavities linear accelerators and transfer lines energy compression (single and multibunch)  linacs energy spread minimization using harmonic cavities 

6. Bunch Splitting

7. Bunch splitting (for the LHC) motivation: using existing accelerators, produce multiple high-current bunches produce ~40 bunch trains of 72 bunches with 1011 protons and 25 ns bunch spacing (LHC) History (< about year 2000): debunching of 6-7 high intensity bunches in the CERN PS + capture in higher-f rf system (microwave instability observed in the process leading to non-uniform beam distributions) concept: application of higher-harmonic rf cavities layout of the LHC including the preinjectors one bunch from the PS booster gets split into twelve bunches in the CERN PS

8. Split factor 1 3 time example: simulation of bunch triple-splitting in the CERN PS(courtesy R. Garoby, 1999) one of 6 bunches from the booster in the CERN PS time example: measurement of bunch triple-splitting in the CERN PS (courtesy R. Garoby, 2001) Issues: preservation of longitudinal beam emittance stability of initial conditions complicated (then) by B-field drift requires careful synchronization control of longitudinal coupled-bunch instabilities bunch intensity fluctuations stability of initial conditions

9. Split factor 1 4 time (cumulative split factor: 112) example: simulation of bunch quadrupole-splitting in the CERN PS (courtesy R. Garoby, 1999) time example: measurement of bunch triple-splitting in the CERN PS (courtesy R. Garoby, 2001)

10. Recent results with rf gymnastics for the LHC Triple splitting following single-batch injection and controlled longitudinal blow-up. “Single-Batch filling of the CERN PS for LHC-Type Beams”, C. Carli et al, IPAC10 Comparison of bunch splitting in the case of double-batch injection (top, which is followed by a longitudinal blow-up step) and single-batch injection (bottom, which has blow-up before splitting).

11. BCMS (bunch compression, merge and splitting) scheme in the CERN PS “A new higher brightness option”, S. Cettour et al, Chamonix Wksp (2012). BCMS results in much smaller transverse emittance! (due to less initial charge in the booster needed per final bunch, thanks to the merger). Transverse emittance results in higher luminosity, like 1/emit or even 1/emit~2 . However the low emittance beam is more dangerous (injection limits on the maximum number of trains per injection, machine protection,…) Longitudinal emittance is not critical (anyway increased in the SPS and LHC to provide Landau damping.

12. Bunch splitting arising from modulation of the beam near the synchrotron frequency (from a study of the effects of ground motion for the SSC) Example (IUCF): bunch deformations resulting from modulation of a trans- verse dipole (at nonzero dispersion) near the synchrotron frequency

13. Bunch Coalescing

14. Bunch coalescing motivation: combine many bunches into 1 bunch for high peak intensity (and luminosity)  concept: 1) initial condition with multiple bunches in different high frequency rf buckets   2) lower (vector sum) of cavity voltages  bunches “shear” due to longitudinal mismatch   3) turn on a subharmonic rf system bunches rotate with new synchrotron frequency  4) restore initial rf (with appro- priate phase), turn off the lower frequency rf system 

15. example: bunch coalescing in the Fermilab Main Ring (courtesy P. Martin, 1999) initial condition: 11 bunches captured in 53 MHz rf buckets “paraphasing” – adiabatic reduc- tion of the vector sum rf voltage by shift of the relative phases between rf cavities application of higher voltage 2.5 MHz rf system (in practice, a 5 MHz rf system was used to help linearize the rotation) capture of bunches in a single 53 MHz rf bucket time peak intensity monitor with successive traces spaced by 6.8 ms intervals “snap coalescing” – fast change in voltage amplitude applied (instead of adiabatic voltage reduction) observed advantage: avoidance of high-current beam instabilities during paraphasing observed disadvantage: reduced capture efficiency (~10%)

16. Bunch Length Compression

17. Bunch length compression and “rebucketing” motivation: produce short bunches (at expense of increased energy spread) to allow subsequent acceleration whereby bunch experiences only linear part of accelerating voltage (compression) to minimize luminosity reduction in a collider due to the hour-glass effect (rebucketing) to consequently allow for smallest possible β* (β-function at the interaction point) so maximizing luminosity (rebucketing) maximize single-bunch peak intensity (for lasing or for high luminosity) Concept, bunch compression: introduce an E-z correlation across the bunch combined with an energy-dependent path length Dz = hd = R56d + T566d2 + U5666d3 +…

18. Types of Bunch Compressors from “Bunch Compressors”, ILC Accelerator School, Eun-San Kim (2006) which includes an excellent overview of associated dynamics including incoherent and coherent synchrotron radiation

19. example: SLC bunch compressor main linac 3 2 R56≠0 E=eV damping ring H compressor cavity, V s 1 T upstream of compressor downstream of cavity at injection into linac 2 3 1    H H z z z T T =-z/c I(z) I(z) z z

20. Again, combining the equations: in the limit of a linear rf; i.e. bunch length short compared to rf wavelength, so sin ~ =-z1/c final bunch length: ( assuming < z1 > = 0 ) if the compressor voltage is adjusted so that then the final bunch length is independent of the initial bunch length: z,f = |R56|,0

21. Phase errors in bunch compressors at the SLC the tolerance on the phase of the beam injected into the linac was <0.1 with frf=2856 MHz (barely measurable!) here we consider the sources of phase error for the single-stage compressor just described, letting =-z/c compressor phase error (deviation of beam from the zero crossing of the rf) assume: the errors in the injected beam phase i and the compressor phase: c are independent and that the initial momentum deviations are independent of these phases; i.e. d1/d1=d1/dc=0 then with combining the two contributions in quadrature: in particular, for =-1 (full compression), the final phase error is independent of the initial phase error

22. Bunch compression in the SLC From “Bunch Compression at the Stanford Linear Collider”, R. Holtzapple et al, SLAC-PUB-7014 (1995)

23. example: two-stage bunch compression scheme for the NLC (compressor cavity) E-z correlation (R56≠0) using wiggler 2 rf “sections” 180° arc magnet chicane for E-Z correlation πn (=360˚, NLC) to minimize sensitivity to phase errors BC1 (L-band rf + wiggler) : σz = 5mm500 μm BC2 (arc, S-band rf, chicane): (100-150) μm (16 ps -> 1.6 ps -> 330-500 fs)

24. Bunch compression at FLASH (DESY) σzfinal = 300 μm (achieved – L. Froehlich, PhD thesis (2005) (maybe shorter later?) Bunch compression at SACLA (Riken) σzfinal = 15 μm (achieved – K. Togawa et al, FEL2011) 9 μm (achieved – H. Maesaka et al, IBIC2012)

25. Bunch compression the European X-FEL (DESY) σzfinal = 20 μm Bunch compression for LCLS-II (SLAC) σzfinal = 9 μm

26. Homework Design an energy compressor consisting of an energy-dependent path length (R56≠0) and a compression cavity. a) Assuming no correlation of the incoming beam (<1z1>=0), what condition must be satisfied in order that the resulting bunch energy spread is independent of the initial energy spread? b) Suppose <1z1>≠0. Is there a solution which would lead to an even smaller energy spread? How might <1z1>≠0 be realized in practice?

27. Rebucketing

28. Rebucketing concept: (1) stretch the bunch along the phase (time) axis by lowering the cavity voltage, or shifting the phase of the rf placing the beam on an unstable fixed point on the separatrix so that the beam shears longitudinally (2) wait an appropriate time (~ ¼ synchrotron oscillation) to allow the bunch to rotate in phase space (3) capture beam in higher harmonic rf system (and either turning off lower harmonic rf system or at a minimum reverting the phase shift if that option used to induce shearing Plots from “Rebucketing after tran- sition in RHIC”, D.-P. Deng (PAC95) RHIC (100 GeV p+p, 04/26/15)

29. Bunch Length Pre-Compression

30. Bunch length pre-compression motivation: produce short bunches (at “expense” of increased energy spread) for longitudinal matching from one accelerator to a downstream accelerator for reducing beam loss in downstream transfer line with high dispersion for longitudinal phase space matching (energy spread to bucket in receiving accelerator) concept: induce longitudinal quadrupole-mode oscillation by variation of rf voltage amplitude  initial state: bunches matched in longitudinal phase space  matched phase space ellipse in receiving accel- erator   longitudinal phase space after 1/4 synchrotron oscillation period mismatch in longitudinal phase space after raising cavity voltage  

31. Equation of motion for the bunch length (pp. 178-179) equations of motion: notation: a few equalities: for example: seek then omit derivative, p. 178, before “using (7.1)” first result [A] omit minus signs (2 occasions) in Eq. 8.12

32. express <2> in terms of  and the longitudinal emittance  here next result [B] combine [A] and [B] using gives the result [C] equation of motion for bunch length noting that combining (iv) and [C] while noting the cancellation of the terms equation of motion for energy spread omit minus sign in Eq. 8.15

33. example: bunch pre-compression at the SLC (variant when higher V not an option) measured cavity voltage Vc 50 kV (800 kV max), 10 μs per division Vc t measured peak current I~z-1 10%, 5μs per division measured beam centroid energy with =50μm, 0.77%, 2.3 μs per division (not shown: cancellation of dipole-mode oscillation)

34. effect in downstream compressor section: phase space for case of long bunches wrt compressor cavity wavelength: energy aperture given by trans- verse aperture (=x/) avoided       I(z) I(z) also, eliminate “tails” by short-ening bunch shorten the bunch at extraction, energy apertures are avoided less particle loss in downstream transfer line (~25% more Ie-Ie+ at SLC IP) side-benefit: reduced energy “tails” in the final focus (chromatic aberrations…)

35. Bunch Lengthening Using Harmonic Cavities

36. Bunch lengthening using harmonic cavities (in a storage ring) cross section for scattering beyond the energy accep- tance (given by rf or by physical apertures, which- ever is smaller) motivation: increase beam lifetime (reduce the loss rate) by reducing the probability for Touschek scattering (large angle intrabeam scattering) concept: increase the bunch length (i.e. reduce the volume density) by adding a higher harmonic rf system so that the vector sum of the voltages seen by the beam is constant no. particles per bunch particle bunch density phase of primary rf (wrt zero crossing) n=ratio of rf frequencies=h/c relative phase of the two rf systems k=amplitude ratio=|Vh|/|Vc| example: harmonic cavity design for the ALS (courtesy J. Byrd, 1999) nominal rf rf of the third har- nomic rf system vector sum of the two rf systems

37. Again, “Boundary conditions”: i.e. energy loss per turn (due to radiation) is compensated voltage profile across bunch is flat without curvature optimum amplitude optimum relative phase optimum phasing of the primary rf example: harmonic cavity design for the ALS (courtesy J. Byrd, 1999) potential with primary rf bunch profile with primary rf potential with both rf systems bunch profile with both rf systems 1 rad ~ 9.5 mm @ 500 MHz

38. Experience with the ALS harmonic (5 single-cell, passively driven) cavities: factor of 2 increase in beam lifetime using uniform current distribution (with minimum ~2% gap for clearing ions and for allowing for dump kicker rise and fall times) 50% increase in beam lifetime during normal operation with 20% gap Example: streak camera images from the ALS (courtesy J. Byrd, 2000) 17% gap 15 mm rapid variations in the beam current produce transient loading in the h.c. variable harmonic voltage across the bunch train and variation in synchrotron phase (increased Landau damping) 2.4% gap =(2πf), t=18º (t=100 ps) @500 MHz; ~54º at 1.5 GHz

39. Longitudinal Phase Space Energy Spread Manipulation

40. Energy spread circular accelerators e+/-  naturally damped to limit of quantum fluctuations p, pbar  given by accelerating rf or controlled using electron and stochastic cooling transport lines  can be modified using an energy compressor (“backwards” bunch length compressor) linear accelerators given by accelerating rf and controlled using compressors and bunch shaping Example: energy spread in the SLC damping ring vs beam current wire scanner data made in the downstream transfer line after adjustment of the optics to make high dispersion at the wire evidence of current-dependent energy spread increase attributed to a microwave instability compared to (x , x’ , y , y’ , ),  is perhaps the most difficult to measure and control

41. particle energy in a linear accelerator energy gain energy loss injected beam energy longitudinal density distribution energy gain from each klystron longitudinal wake function [eV/Cm] phase of beam wrt rf crest spacing between successive klystrons phase of particle wrt i Energy spread E is given by averaging over the particle distribution after subtracting out the mean energy <E>. Normalized to this mean energy of the bunch mean energy of the bunch distribution:

42. Again, In pictures: effective energy gain energy spread (obtained by projecting onto the energy axis) s minimum energy spread (and maximum energy) obtained by placing beam at crest low current limit: beam does not take away energy a low current, mis- phased beam has higher energy spread accelerating rf voltage accelerating rf voltage a high current beam placed on crest has higher energy spread high current limit: beam takes away energy and the two terms in [ ] above should be balanced minimum energy spread obtained by placing beam off-crest beam-induced voltage

43. Bunch shaping using bunch compressors (upstream of linac)  We have seen that the energy spread in a linac is given not by the incoming beam energy spread but rather by the incoming beam bunch length Recall the resulting beam distribution following the bunch compressor for the case of long bunches  Minimizing the bunch length therefore results (with appropriate linac phasing) in the smallest energy spread Example: bunch “over-compression” (courtesy F.-J. Decker, 1999) normal compression over- compression measured beam profile at end of linac with under-compression and over-compression (z,f=R56,0) long. phase space:  s bunch length I y x= s

44. Bunch shaping in the linac proper (G. Loew and J.W. Wang) 0phase of the head of the bunch wrt rf crest same equation as before expressed here in terms of the phase of the beam wrt crest k energy gain of particle within a bunch integration over all preceeding bunches Minimum energy spread when V(k) is independent of k: Taking the derivative, it has been shown that an optimal bunch charge distribution () exists which can be found by numerically solving () please replace x with  in Eq. (8.27), p. 194 general trend: the higher the bunch charge, the more forward-peaked the charge distri- bution for minimum energy spread

45. Example: conceptual illustration for optimizing the relative phase of the beam for the case of very long (or high current) bunches -- trade-off between beam energy and energy spread in the SLC linac (courtesy J. Seeman, 1999) accelerating rf voltage longitudinal wakefield loading from front of bunch compensates the curvature of the rf perfectly energy spread of core small but large energy “tails” are present energy spread of core slightly larger with fewer particles at extreme high or low beam energy undercompen- sation of the rf overcompen- sation of the longitudinal wake energy spread of core large, many particles at extreme high or low beam energy, mean energy also lower (worst case) remark: visualizing such projections is very helpful in interpreting data  next slide

46. Example: bunch energy spread measurements at the linac-based SASE FEL (data courtesy F. Stulle, 2003) y 

47. Bunch shaping using harmonic cavities (in a linear accelerator) motivation: minimize bunch energy spread by removing the effect of the rf curvature on beam energy distribution concept: by adding a 3rd harmonic cavity the net accelerating voltage is more uniform across the bunch length first results  z References “3.9 GHz SRF Cavity Module Update”, H. Edwards (2010) http://www.fnal.gov/directorate/program_planning/all_experimenters_meetings/special_reports/Helen_Flash-Desy_12_06_10.pdf “Test and Commissioning of the Third Harmonic RF System for FLASH”, E. Vogel et al IPAC10

48. Multi-bunch energy compensation 1. t-method (for the NLC) Vk Vk+Vb Vb concept: change (advance) the relative time of arrival of the bunch train with respect to the accelerating voltage from the pulsed power source so that the vector sum Vk+Vb is constant note: if the compensation is not perfect, while the projected energy spread is minimized, each bunch could have a different energy spread

49. 2. f-method (for the JLC and ATF) concept: detune some structures positively, some later structures negatively so that the position-energy correlations introduced by the slope of the rf cancels V0 = cost V1 = sin(t+t)  sin(t)cos(t) + tcos(t) V2 = sin(t-t)  -sin(t)cos(t) + tcos(t) compensating voltages add single-bunch energy spread cancelled bunch energy spread and projected energy spread of train both minimized

50. Summary The (linearized) equations of motion in longitudinal phase space (LPS) were reviewed Small-amplitude motion in LPS obeys that of a simple harmonic oscillator Many techniques using rf systems to manipulate the LPS were presented: bunch length compression for shortening the bunch length in linear accelerators (to subsequently minimize the energy spread and hence the projected transverse beam emittances) for producing maximum peak intensities (e.g. as required for SASE-FELs) bunch length precompression for better matching in LPS (to avoid longitudinal emittance dilution, HERA) for minimizing beam loss in downstream transfer lines (SLC) bunch lengthening using harmonic cavities in synchrotron light sources for minimizing the particle density (to improve the beam lifetime) bunch coalescing (FNAL), re-bucketing (BNL) and bunch splitting (LHC) to change the current distributions (i.e. number of bunches) energy compression single-bunch for avoiding “longitudinal apertures” multi-bunch as a method for compensating beam loading in linacs e+/- e+/-,p e+/- p e+/-