1. Double RF system at IUCF Shaoheng Wang 06/15/04

2. Contents 1.Introduction of Double RF System 2.Phase modulation  Single cavity case  Double cavity case 3.Voltage modulation  Single cavity case  Double cavity case

3. Introduction: Double RF system : Harmonic number : RF peak voltages : RF phase of syn. particle : Orbit angle : Synchrotron tune at zero amplitude for primary cavity h2 V2 Primary cavity Secondary harmonic cavity h1 V1 Synchronous particle Other particles in the bunch Particle bunch

4. Introduction: Why Double RF system By reducing the voltage gradient at the bunch position, it will also increase the bunch length. Hence, lower the space charge effect. There is an increase in the spread of synchrotron frequencies within the bunch. This spread can help in damping coherent instabilities such as the longitudinal coupled bunch instabilities through an effect known as Landau damping which come from the non- linearity of the voltage along the bunch.

6. The voltage seen by the beam with a double RF system is The equation of synchrotron motion Make the integration  To maximize the bunch length, the first derivative of V should vanish at the center of the bunch.  To avoid having a second region of phase stability close by, the second derivative of V must also vanish. The peak value along a given trajectory Introduction: Working conditions

7. Computed distribution in synchrotron tune In the phase space, along the H contour, Period: Single RF System Zero Gradient Qs density Reduced voltage slope Shifting NonlinearitySpread Introduction: Synchrotron tune spread

8. Introduction: IUCF cooler ring Injection Cavity 1 Cavity 2

9. Introduction: Experiments at IUCF The bunched beam intensity was found to increase by about a factor of 4 in comparison with that achieved in operating only the primary rf cavity at same rf voltage.

10. Introduction: Equations of motion : Normalized momentum deviation : Synchrotron tune at zero amplitude for primary cavity Contribution from primary cavity Contribution from secondary cavity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

11. Introduction: Hamiltonian : Ratio of the amplitude of RF voltages : Ratio of harmonic numbers

12. Flattened potential Potential shape example r = 0 r = 0.5

13. Flattened potential Potential shape example At the equilibrium state, the particle distribution follows the shape of the potential

14. Synchrotron tune: Hamiltonian r > 0.5 r < 0.5 P P phi

15. Synchrotron tune: action-angle variables

16. Synchrotron tune: synchrotron tune

17. r > 0.5 r < 0.5 P P phi Synchrotron tune: graph Synchrotron tune variation w.r.t. phase amplitude for a double RF system.

18. Synchrotron tune compare with experiments J.Y. Liu et al, Phys. Rev. E 50, 3349 (1994) The synchrotron tune spread is maximized at r=0.5 for a given bunch area, which is provided to Landau damping. The effective tune spread is given by:

19. Synchrotron tune Compare with one RF cavity system Synchrotron tune measured as a function of phase amplitude at IUCF. M. Ellison et al, PRL 70, 591, 1993

20. Synchrotron tune: Synchrotron phase space measurement Synchrotron phase is measured with phase detectors. –By comparing the bunch arrival time with the RF cavity wave. Momentum deviation is measured according to the dispersion relation: FFT on to get synchrotron tune

21. Synchrotron tune: Phase shifter To have a certain phase amplitude, a phase shifter is used. 1.The RF signal is split into two channels, one of them is 90 degrees shifted. 2.Each of these two channels is multiplied with a signal proportional to sine or cosine of intended phase shift. 3.They are combined again.

22. Phase modulation Phase Modulation Signal With this phase modulation, the phase variation will be given by : Consider a sinusoidal RF phase modulation:

23. Single cavity: Equations of motion Hence, the equations of motion are given by : The corresponding Hamiltonian is : Perturbation potential

24. Single cavity expressed with action-angle variables can be transformed into Action-angle coordinates In this transformation, old coordinates are expressed as function of Further more, can be expanded in Fourier harmonics series of Perturbation potential can be expressed as: Note: since is an odd function of, only odd harmonics exist.

25. When is close to, stationary phase condition exists for a parametric resonance term. All non-resonance terms can be neglected. Single cavity: dipole mode When n=0 case, or dipole mode, is considered, the approximate effective Hamiltonian is: The effective Hamiltonian is dependent. We can go to resonance rotating frame to find the independent Hamiltonian.

26. Single cavity: resonance rotating frame In the phase space, the structure of resonant islands can be characterized by fixed points, which satisfy conditions: With the generating function: We can realize the transformation: And get the new Hamiltonian in resonance rotating frame:

27. Single cavity: Poincare surface of section : Outer SFP : Inner SFP : UFP : Seperatrix phase axis crossings

28. Single cavity: bifurcation When goes to from below, the fixed points move as arrows show When =, and coincide. This is the bifurcation point, beyond which, only exists

29. Single cavity: experiments

30. Double cavities: Equations of motion Hence, the equations of motion are given by : The corresponding Hamiltonian is : Perturbation potential

31. Double cavities: Perturbation analysis can expressed with action-angle variables of the unperturbed. can be expanded in Fourier series on,

32. Double cavities: Parametric resonances

33. Double cavities: numerical simulation (1) Simulations are based on the difference equations: with:

34. Double cavities: numerical simulation (2)

35. Double cavities: bifurcation

36. Double cavities: wave structure Figure (a) shows two beamlets obtained about 15 ms after the phase modulation was turned on, and Fig (b) shows the final beam profile captured after 25 ms, showing a wave structure. The beam profiles were extended from a half length of about 10 ns to 50 ns without beam loss.

37. Voltage modulation: Single cavity With the dot corresponds to the time derivative wrt θ. The equation of motion can be derived from the Hamiltonian: Unperturbed Hamiltonian: Perturbation: Action of the Unperturbed Hamiltonian: Synchrotron tune: Complete elliptic integral of the first kind

38. Single cavity: Action-angle variable Generating function: is zero except for n=even with RF voltage modulation contributes only even-order harmonics to the perturbation H 1

39. Single cavity: rotating frame Generating function: Including both ±n terms, the resonance Hamiltonian: The time-averaged Hamiltonian: When n=2: For simplicity, the tilde notation is dropped: Fixed points:

40. Single cavity: Fixed point

41. Single cavity: experiments The bunch was kicked longitudinally, all particles then were captured and dampened into one attractor, see fig. At the same time, rf voltage modulation was applied. A total of 16000 points at intervals of 50 revolutions, i.e. 800000 orbital revolutions, was recorded. Poincare surfaces in resonance processing frame, see fig, the particle damping paths and the island structure were clearly observed

42. Single cavity: experiments

43. Single cavity: beam profile The profile of the beam in a single pass.

44. Double cavity: Hamiltonian G n is not zero only for even harmonics. Time dependent part

45. Double cavity: Fixed point

46. Double cavity: numerical simulation

47. Double cavity: Bifurcation point Bifurcation point

48. Conclusion The benefits of double RF system –Longer bunch, less space charge effect –Landau damping from synchrotron tune spread Resonance structure when system is under phase and voltage modulation

49. reference A. Hoffman, SY Lee, M. Ellison JY Liu, D. Li, H. Huang …