Further Studies on longitudinal dynamics in the CR

1. Further Studies on longitudinal dynamics in the CR T. Agoh*, D. Alesini, C. Biscari, R. Corsini**, A. Ghigo, LNF, INFN, Frascati *KEK, Japan ** CERN 11th CTF3 Collaboration Meeting – 16-17 January 2007

2. Contents • CSR in Combiner Ring (Tom Agoh, KEK) • Transverse effects of rf deflectors (David Alesini) • Possible Measurements (Roberto Corsini, Andrea Ghigo)

3. Previous studies on CSR in CR • Roberto (1999) analytical and simulations for first CR design • Mikhail analytical approach (2001) with present dipoles and chambers To be on the safe path Bunch length of the order of 2 mm to limit energy spread below 1% R56 tuning chicane

4. T. Agoh at Frascati for SUPERB studies – Nov 06 has dedicated some time to simulate the CR dynamics and validate the previous analytical approach results • CSR calculation using paraxial approximation • Parabolic equation • Comparison with analytic solutions • CSR effect in the CTF3 combiner ring • Wakefield, Impedance, Loss factor • Energy spread growth due to CSR

5. Time domain : (5) Inverse Fourier transform Back to the time domain Numerical method * • Begin with Maxwell equations in vacuum (E, B) (2) Fourier transform EM field w.r.t z Frequency domain : (3) Approximate these equations in Paraxial approximation (chamber cross section much smaller than radius of curvature) (4) Solve them by finite difference Beam pipe = boundary condition *T.Agoh, K.Yokoya, PRST-AB, 7, 054403 (2004)

6. The assumption is the condition so that the radiation field can be a paraxial ray. Role of beam pipe The light, emitted from a bunch, cannot deviate from the s-axis due to the reflection on the pipe wall. The radiation always propagates near around the axis. Equation to describe CSR: Equation of Evolution (parabolic equation)

7. Algorithm Solve equation of evolution with boundary condition Discretize the equation by central difference: Solve initial condition at the entrance of bending magnet (radius=∞) Proceed field evolution step by step along s-axis

8. Examples to which this approach cannot be applied • Free space or very large vacuum chamber • EM field is no longer a paraxial ray. • Chamber structure so that backward waves are produced • Bellows, Cavity  Chamber wall must be smooth. • Ultra-short bunch, or fine structure in the bunch • Fine mesh is required to resolve the field. (expensive) • The shortest bunch length I computed is 10 microns in 6cm pipe. • Bunch profile with sharp edge,e.g. rectangular, triangular, etc •  Bunch profile must be smooth.

9. Combiner ring parameters • Vacuum chamber (aluminium) height = 36mmwidth = 90mm Rectangular cross section in CSR calculation • Requirement sd < 1% after 4.5 turns

10. Longitudinal wakefield for single turn Solid lines = simulation (Agoh) Dotted lines = analytic solution (parallel plates model) sz=2mm sz=3mm Es ~ ±0.03 MeV/m DE ~ ±0.2 MeV/turn For 4.5 turns sz=2mm DE ~ ±0.9MeV (DE/E ~ ±0.5%) sz=1mm A.Ghigo, M.Zobov CTFF3-004 (2001)

11. CSR in transient states Analytic solution (steady state) black -> red -> magenta -> green -> blue-> cyan Lbend = 0.56m Simulation for Lbend = 1.5m Simulation (sz=2mm) DIFFERENT COLOURS CORRESPONDING TO DIFFERENT POSITIONS ALONG THE BENDING Magnet entrance Exit Static solution of parallel plates model works for CTF3 combiner ring because of the small bending radius, the wide chamber width.

12. Loss factor CSR+RW CSR RF deflector

13. sz=3mm sd0=5×10-3 Ne=2.33nC Longitudinal phase space distribution (z,d) Initial state 0.5 turn 1.5 turns 2.5 turns 3.5 turns 4.5 turns

14. sz=2mmsd0=5×10-3 Ne=2.33nC Longitudinal phase space distribution (z,d) Initial state 0.5 turn 1.5 turns 2.5 turns 3.5 turns 4.5 turns

15. Change of energy due to CSR Energy spread vs. turn Center of energy vs. turn sz=1mm DE/E = 1% sz=1.5mm sz=2mm sz=3mm (Initial energy spread sd0 = 5×10-3) Uo = 40 eV @150MeV (in 4.5 turns DE/E = 0.001 ‰ )

16. Longitudinal distribution at 4.5 turns sz=2.5mm sz=1mm sz=2mm sz=1.5mm Energy distribution Bunch length — 1mm — 1.5mm— 2mm— 3mm

17. Energy spread vs. Bunch length sz=1.2mm sd=9.4×10-3 DE/E = 1% 4.5 turns If the bunch length is shorter than 1.2mm, the energy spread exceeds 1% at 4.5 turns.

18. Summary (Agoh) • Parallel plates model works for the CTF3 combiner ring, • because of small bending radius, wide chamber width. • A preliminary study by A.Ghigo and M.Zobov (CTFF3-004) goes on well. • A short bunch is affected by CSR in the combiner ring. • If the bunch length is 1.2mm, the energy spread is 0.94% (< 1%) after 4.5 turns, however, the energy distribution is significantly deformed. • Considering a bunch compression after the combiner ring, minimum acceptable length may be around 2mm.

19. All previous simulations have been done with uncorrelated longitudinal distribution. Tom is now simulating the effects with different longitudinal phase space correlations obtained by tuning the chicane R56 Work in progress

20. Missing: start to end simulations for overall beam dynamics behaviour • Longitudinal plane : linac acceleration, csr, wake fields, RF deflector effects, non-linear isochronicity, CSR • Transverse plane: transverse non linearities, betatron beating from mismatch, chromaticity, high order terms, …

21. 10 bunches microstructure at CR output From linac 20 cm – 667 psec Passage through Delay Loop # turns 5 4 3 2 15 4 3 2 1 2 cm – 67 psec

22. 2003 From Linac exit To Decelerating section input 1 6 2 7 3 8 4 9 10 5 L H V L H V

23. Variation of transverse and longitudinal emittances along the bunch train (single particle effects)

24. t Vz PRELIMINARY ANALYSIS OF THE RFD INDUCED ENERGY SPREAD IN THE CR Vx • The Panofsky-Wenzel theorem relates the RFD transverse deflecting voltage and the longitudinal electric field gradient; • The transverse deflecting voltage and the longitudinal one are 90 deg out-of-phase t The stored bunches that passes out-of-phase with respect to the deflecting field “see” longitudinal electric field off axis that induces an energy spread. The deflected bunch does not “see”, to the first order, any longitudinal electric field. FIRST ORDER ESTIMATION x Induced energy spread Average transverse beam dimension at the deflector CR single passage in one RFD • Other effects to be evaluated (tracking): • multi-passage case; • RFD beam loading effects; D. Alesini

25. x’ 1st RFD t x Vz 2nd RFD • Multipassage: 180° betatron phase between RFDs: the effect is compensated at first order in each passage X

26. Is it possible to measure these effects, discriminate among them, and consequently optimize the drive beam structure? Tools • Runs with different currents (single particle against collective effects) • Tuning R56 of chicane and/or TL1, keeping the changes transparent (need of optimum optical model for every element) • Tuning of bunch logitudinal correlation with Linac • Measure of bunch length and energy spread in as many possible points along the whole system Do we have enough resolution in the diagnostics?

27. Beam diagnostic WCM = Wall Current Monitor BPM = Beam Position M. ( 40 mm) BPI= BPM from INFN (90 x 40 mm) BPR= BPM(for bunch length behavior) MTV = Ensemble camera & mirrors (for synchrotron light or Transition radiation) PHM = Phase Monitor (Frequency meas.) NCR = Nearly Confocal Resonator TL1 CR WCM BPR. MTV NCR BPR PHM

28. SOME EXAMPLE Measure of beam size with synchrotron radiation monitors in two dipoles Isochronous configuration MTV 0796 MTV 0751

29. Measure of energy loss by max # of turns in the cr without extraction Switch off the rf deflectors (time needed of the order of one turn) and let the beam spiralize inside # turns before hitting the vacuum chamber (if the bunch length were kept constant but… ) 5 % including betatron dimensions

30. Sext on: residual non isochronicity Sext off : 2° order terms : Non-isochronicity 10 turns -0.5 mm 20 turns -1.1 mm After few tens of turns at high current all bunches will have similar energy spread – will behave in a similar way. (6D simulations with CSR included must still be done) 50 turns - 3 mm

31. Measure of energy loss with BPMs: dE/E = 10-3 (sz = 1mm, 1 turn) Dmax = 0.75 m => Dx = 0.75 mm from turn to turn Measure difference in orbit between BPMs with dispersion and those with zero dispersion for the different parameter sets (low charge, high charge, short bunch, long bunch, etc)

32. Conclusions • Beam dynamics calculations in progress (Cern, KEK, Frascati, …) • Main aim: find the best configuration for power production • Opportunity to use the 2007 commissioning as a test bench for emittance preservation, csr studies, .. applicable to other accelerators