1. William Lou Supervisor: Prof. Georg Hoffstaetter
BBU simulation
William Lou
Supervisor: Prof. Georg Hoffstaetter

2. Outline BBU overview BBU theory Bmad simulation 2) CBETA BBU results
What is CBETA
HOM assignment
statistics (1-pass and 4-pass)
Aim for higher
Results with additional phase-advances
Results with x-y coupling

3. Beam breakup Instability (BBU)
Higher Order Modes (HOM) in the cavities give undesired kick.
Off-orbit bunch returns to the cavities and excite more HOMs…
BBU limits the maximum achievable current in an ERL threshold current
decreases with # recirculation turns.

4. BBU theory: the mathematics
Elementary case: 1-dipole-HOM + 1 recirculation pass
Wake function [N/C^2]
Transverse
offset
Current
Transverse momentum gained from HOM
Wake function
(Cavity HOM property)
An HOM is characterized by
( , , )

5. HOM voltage T12 of the transfer matrix is a lattice property
Transverse offset
after recirculation
Measured
current
Bunch time
spacing
Current as a train of (dirac) beam bunches

6. After some math, we obtain the dispersion relation between and
To solve this difference equation,
we write HOM voltage (retaining
all possible frequencies ) as:
After some math, we obtain the dispersion relation between and
At a given , multiple (complex) can satisfy the relation.
At a stable , all must have negative imaginary part. At the threshold current, one crosses the real axis!!

7. Find the critical which gives the maximum real
1-pass 1-HOM thin-lens cavity
(1) General analytic formula
Find the critical which gives the maximum real
The corresponding is

8. BBU theory

9. BBU theory

10. BMAD software Developed by Cornell LEPP
Open source, free, compiled in C and Fortran
Multi-pass lattice design, lattice optimization, multi-particle tracking algorithms, wakefields, Taylor maps, real-time control “knobs”…

11. Theory v.s simulation

12. Theory v.s simulation
DR scan for 1 dipole-HOM
2-pass (Np = 4) lattice

13. Find the critical which gives the maximum real eigenvalue of M
BBU theory
General formula (Np-pass, N-cav) for
Find the critical which gives the maximum real eigenvalue of M
The corresponding is

14. BBU simulation on Bmad
Given a complete lattice with multi-pass cavities and HOMs assigned…
Starts with a test current…
tracks off-orbit bunches through lattice
computes bunch-HOM momentum exchanges
determines stability of all HOM voltages
Attempts different test currents to pin down

15. CBETA

16. Simulated HOM data for one CBETA cavity
Cavity construction error: ± 125 μm, ± 250 μm…
400 unique cavities provided per error case.
The “10 worst dipole HOMs” (large figure of merit) provided per cavity.

17. CBETA 1-pass Cavity shape error: 125 μm
499 out of results above 40mA

18. CBETA 4-pass Cavity shape error: 125 μm
Before: 70 out of 300
494 out of results below 40mA

19. CBETA 4-pass With various cavity shape errors
125 μm
250 μm
Before: 33 out of 300
500 μm
1000 μm

20. For the current CBETA design, the
Summary on Statistics
For the current CBETA design, the
can usually reach the high goal of 40 mA.
Cavity shape matters!

21. Aim for better
Potential ways to improve :
1) Change bunch frequency
2) Introduce additional phase advance
3) Introduce x-y coupling

22. Does vary with bunch frequency?

23. Two cases: Vary the optics in Bmad Introduce either T matrix
at the end of LINAC 1st pass

24. CBETA 1-pass v.s additional phase advances (decoupled optics)
Min = 140 mA
Max = 611 mA
nominal = 342 mA
The valley location depends on which cavity has the most dominant HOM
results can improve significantly

25. with different HOM assignments…
1-pass decoupled
with different HOM assignments…
Valley location
Min = 175 mA
Max = 640 mA
Nominal = 218 mA
Min = 165 mA
Max = 595 mA
Nominal = 248 mA

26. CBETA 1-pass with x-y coupling
Min = 140 mA
Max = 520 mA
nominal = 342 mA
Nominal ( no coupling ) = 342mA
(0,0) at 277mA
results can improve significantly

27. with different HOM assignments…
1-pass coupled
with different HOM assignments…
Min = 136 mA
Max = 436 mA
Nominal = 218 mA
Min = 166 mA
Max = 678 mA
Nominal = 248 mA

28. CBETA 4-pass v.s additional phase advances (decoupled optics)
Min = 61 mA
Max = 193 mA
Nominal = 69 mA
Location of the valley depends on which cavity has the most dominant HOM.
results can improve

29. with different HOM assignments…
4-pass decoupled
with different HOM assignments…
Min = 37 mA
Max = 176 mA
Nominal = 38 mA
Min = 39 mA
Max = 205 mA
Nominal = 49 mA

30. CBETA 4-pass with x-y coupling
Min = 89 mA
Max = 131 mA
Nominal = 69 mA
Nominal = 69mA
(0,0) = 99mA
Location of
results can improve

31. with different HOM assignments…
4-pass coupled
with different HOM assignments…
Min = 47 mA
Max = 100 mA
Nominal = 38 mA
Min = 57 mA
Max = 107 mA
Nominal = 49 mA

32. For both CBETA 1-pass and 4-pass, introducing additional phase advances is more promising than x-y coupling
In reality the phase variations are restricted by other optical constraints

33. Summary Bmad simulation agrees well with BBU theory
For both CBETA 1-pass and 4-pass , 98% simulated are above the high goal (40mA)
Introducing additional phase advances gives promise to improve the for CBETA

34. Acknowledgment References
Special thanks to:
Prof. Georg Hoffstaetter
Christopher Mayes
Nick Valles
David Sagan
References
BBU papers by Georg and Ivan
CBETA review meeting documents
Bmad manual
Nick Valles’s dissertation

35. THE END

37. RFC cavity HOMs from Nick Valles
Helper Slide #2
RFC cavity HOMs from Nick Valles