Wakefields & Impedances

1. 1 1 Wakefields & Impedances Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU

2. 2 2 Panofsky Wenzel Theorem 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

3. Wakes from Ultrarelativistic Beams 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

4. Maxwell’s Equations Maxwell’s Equation with a steadily Moving charge as the only source: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

5. Approximations Rigid Beam / Impulse Approximation: The beam does not change trajectory as a result of the beamline element under consideration. The EM fields themselves are not important, what matters is the total impulse (change in momentum) delivered as a result. Ultra-relativistic Approximation: Take the limit that β -> 1 and γ -> ∞. The E-field and B-field are fully perpendicular to the beam orbit. 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

6. Calculating the PW Impulse Relations (Curl) We take the curl of the impulse. Maxwell’s equations: 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

7. Calculating the PW Impulse Relations (Div) We take the divergence of the impulse. Maxwell’s equations: Ultrarelativistic: 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

8. Calculating the PW Impulse Relations We have calculated: There is a trick- we can simplify expressions by observing: Then: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

9. Panofksy Wenzel Impulse Relations Panofsky Wenzel Relations: PW Ia: Transverse Rotational PW Ib: Longitudinal Rotational PW II: Divergence 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

10. Example: Sextupole Multipole How can we apply this to find the sextupole kick? from PW II: from PW Ia: 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

11. Example: Off-Axis Cavity Forces Acceleration from TM010: Panofsky Wenzel Ib: Which is exactly the right answer! Yet it will also violate Panofsky Wenzel II: 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

12. 12 12 Cylindrical Wakes 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

13. The Wake Potential We can solve a boundary value problem for W W(x,y,D) is the wake potential Wm(D) is the wake function 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

14. The Wake Potential From the wakes, we can calculate the impulses: 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

15. Shape of Wake functions 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

16. Shape of Wake functions Decaying cosine-like Decaying sine-like

17. Wake Potential & Beam Moments There will be many distributions that share the same moment: 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

18. Wake Potential & Beam Moments The modes generally act on the beam in the same manner in which they are excited. Here the azimuthal m is the same we had on Monday, 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

19. 19 19 Impedances 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

20. Wakes in Frequency-Domain If we know the wake functions, and the moments of the beam, we can calculate the impulses on a trailing bunch (or trailing part of the bunch) as a function of its distance D from the wake source. But what we really want to know is often frequency-domain information – what bunch length and bunch spacing will excite wakes. Therefore the Fourier transforms of the wakes are often more useful. 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

21. Wakes in Frequency-Domain Inverse Fourier Transform: Fourier Transform: There is a clear circuit analogy here: Voltage = Current x Impedance 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

22. Wakes in Frequency-Domain Imaginary impedance, such as space-charge, will induce a frequency shift in the betatron or synchrotron motion. Real impedance, such as a “resistive wall” or cavity modes, will introduce an energy loss or a deflecting force. Voltage = Current x Impedance 22 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

23. Calculated Impedances Many common sources of wakefield impedances – space-charge, resistive wall, RF cavities, BPMs, beampipe transitions, etc. – have been already been calculated. Alex Chao’s Handbook of Accelerator Physics & Engineering MAFIA, CST are able to perform simulations for more complicated scenarios. 23 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

24. Broad vs. Narrow Impedances 24 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

25. Impedance Compensation Impedance from one source can be compensated by impedances from another source. Some impedances are capacitive and others are inductive and consequently the imaginary part of the impedance can be canceled in the complex plane. Example: IMPEDANCE CALCULATION FOR FERRITE INSERTS Ferrite inserts to compensate space-charge driven impedances without introducing undesired new impedances. 25 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

26. 26 26 Two-Particle Model of Instability 26 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

27. Two-Particle Model Consider two particles, one leading the other. (Actually, two bunch-slices with centroid motion.) The first particle is oscillates normally under external fields: The second particle also oscillates, but sees the wake from the first: The second particle is being driven exactly at resonance! 27 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

28. Balakin, Novokhatsky, and Smirnov (BNS) Damping Now consider the effect on the trailing particle is detuned from the leading particle is detuned by Δk: The particle motion is no longer unbounded: In addition, if , then the trailing particle oscillation will be exactly in phase with the leading particle oscillation! 28 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

29. 29 29 Robinson Instability 29 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

30. Robinson Instability For standard synchrotron motion, the equations of motion can be written: Now including the beam-loading / wakefield effect: 30 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

31. Robinson Instability Linearize the W’ dependence on z to calculate the stability: And write z as a complex oscillation: Now we have: 31 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

32. Robinson Instability Use the Poisson Sum identity to express in terms of impedance The expression for z, , means the imaginary part of Ω leads to the instability: 32 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

33. Robinson Instability 33 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

34. 34 34 Landau Damping 34 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

35. Vlasov Equation See Lecture 11. 35 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

36. Ensemble of Oscillators Consider a single driven harmonic oscillator: For initial conditions 0,0 we have: Now consider a distribution of such oscillators, each with its own oscillation frequency ω. Clearly there will be some oscillator with Ω = ω, which will be lost. But what will happen to the ensemble average ? Remember, the collective effects are driven by beam moments. 36 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

37. Ensemble Average The ensemble average will be given by: Which can be shown to be: Where PV is the principal value. 37 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

38. Example: Lorentz Spectrum For example the Lorentz Spectrum The principal value of the integral is calculated: Where the PV is given by: 38 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

39. Example: Lorentz Spectrum The principal value of the Lorentz spectrum integral: And we find the ensemble average: The ensemble average is damped by frequency spread! 39 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU