Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings

1. ICFA Workshop on High Brightness Beams Sardinia, July 1-5, 2002 Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings M. Venturini and R. Warnock Stanford Linear Accelerator Center M. Venturini

2. Outline • Review of recent observations of CSR in electron storage rings. Radiation bursts. • Two case studies: • Compact e-ring for a X-rays Compton Source. • Brookhaven NSLS VUV Storage Ring. • Model of CSR impedance. • Modelling of beam dynamics with CSRin terms of 1D Vlasov and Vlasov-Fokker-Planck equation. • Linear theory CSR-driven instability. • Numerical solutions of VFP equation. Effect of nonlinearities. • Model reproduces main features of observed CSR. M. Venturini

3. Observations of CSR - NSLS VUV RingCarr et al. NIM-A 463 (2001) p. 387 Spectrum of CSR Signal (wavelength ~ 7 mm) Current Threshold for Detection of Coherent Signal M. Venturini

4. Observations of CSR - NSLS VUV RingCarr et al. NIM-A 463 (2001) p. 387 Detector Signal vs. Time • CSR is emitted in bursts. • Duration of bursts is • Separation of bursts is of the order • of few ms but varies with current. M. Venturini

5. NSLS VUV RingParameters Energy 737 MeV Average machine radius 8.1 m Local radius of curvature 1.9 m Vacuum chamber aperture 4.2 cm Nominal bunch length (rms) 5 cm Nominal energy spread (rms) Synchrotron tune Longitudinal damping time 10 ms M. Venturini

6. X-Ring Parameters (R. Ruth et al.) Can a CSR-driven instability limit performance? Energy 25 MeV Circumference 6.3 m Local radius of curv. R=25 cm Pipe aperture h~ 1 cm Bunch length (rms) cm Energy spread Synchrotron tune Long. damping time ~ 1 sec Filling rate 100 Hz # of particles/bunch RADIATION DAMPING UNIMPORTANT! M. Venturini

7. When Can CSR Be Observed ? • CSR emissions require overlap between (single particle) radiation spectrum and charge density spectrum: • What causes the required modulation on top of the bunch charge density? Radiated Power: incoherent coherent

8. Dynamical Effects of CSR Presence of modulation (microbunches) in bunch density CSR may become significant Collective forces associated with CSR induce instability Instability feeds back, enhances microbunching

9. Content of Dynamical Model • CSR emission is sustained by a CSR driven instability [first suggested by Heifets and Stupakov] • Self-consistent treatment of CSR and effects of CSR fields on beam distribution. • No additional machine impedance. • Radiation dampingand excitations. M. Venturini

10. Model of CSR Impedance • Instability driven by CSR is similar to ordinary microwave instability. Use familiar formalism, impedance, etc. • Closed analytical expressions for CSR impedance in the presence of shielding exist only for simplified geometries (parallel plates, rectangular toroidal chamber, etc.) • Choose model of parallel conducting plates. • Assume e-bunch follows circular trajectory. • Relevant expressions are already available in the literature[Schwinger (1946), Nodvick & Saxon (1954), Warnock & Morton (1990)]. M. Venturini

11. Parallel Plate Model for CSR: Geometry Outline M. Venturini

12. AnalyticExpression for CSR Impedance(Parallel Plates) By definition: Impedance With , Argument of Bessel functions , beam height M. Venturini

13. Collective Force due CSR FT of (normalized) charge density of bunch. Assume charge distribution doesn’t change much over one turn (rigid bunch approx). M. Venturini

14. Parallel Plate Model : Two Examples X-Ring NSLS VUV Ring M. Venturini

15. Properties of CSR Impedance • Shielding cut off • Peak value • Low frequency limit of impedance energy-dependent term curvature term M. Venturini

16. Longitudinal Dynamics • Zero transverse emittance but finite y-size. • Assume circular orbit (radius of curvature R). • External RF focusing + collective force due to CSR. • Equations of motion : RF focusing collective force is distance from synchronous particle. is relative momentum (or energy) deviation. M. Venturini

17. VlasovEquation • Scaled variables • Scale time 1 sync. Period. M. Venturini

18. Equilibrium Distribution in the Presence of CSR Impedance Only (Low Energy) • Haissinski equilibriai.e. • Only low-frequency part of impedance affects equilibrium distribution. • For small n impedance is purely capacitive • If energy is not too high imaginary part of Z may be significant (space-charge term ). M. Venturini

19. Haissinski Equilibrium for X-Ring • If potential-well distortion is small, Haissinski can be approximated • as Gauss with modified rms-length: Haissinski Equilibrium (close to Gaussian with rms length ) =>Bunch Shortening. I=0.844 pC/V corresponding to 2 cm M. Venturini

20. Linearized Vlasov Equation • Set and linearize about equilibrium: • Equilibrium distribution: • Equilibrium distribution for • equivalent coasting beam • (Boussard criterion): M. Venturini

21. (Linear) Stability Analysis • Ansatz • Dispersion relation with , and • Look for for instability. Error function of complex arg M. Venturini

22. Keil-Schnell Stability Diagram for X-Ring (stability boundary) Most unstable harmonic: Threshold (linear theory): Keil-Schnell criterion: M. Venturini

23. Numerical Solution of Vlasov Equation coasting beam –linear regime Amplitude of perturbation vs time (different currents) Initial wave-like perturbation grows exponentially. Wavelength of perturbation:

24. Validation of Code Against Linear Theory (coasting beam) Growth rate vs. current for 3 different mesh sizes Theory

25. Coasting Beam: Nonlinear Regime(I is 25% > threshold). Density Contours in Phase space 2 mm Energy Spread Distribution M. Venturini

26. Coasting Beam: Asymptotic Solution Density Contours in Phase Space Energy Spread vs. Time Energy Spread Distribution Large scale structures have disappeared. Distribution approaches some kind of steady state.

27. Bunched BeamNumerical Solutions of Vlasov Eq. - Linear Regime. Amplitude of perturbation vs. time Wavelength of initial perturbation: RF focusing spoils exponential growth. Current threshold (5% larger than predicted by Boussard). M. Venturini

28. Bunched Beam: Nonlinear Regime(I is 25% > threshold). Density Plots in Phase space 2 cm Charge Distribution M. Venturini

29. Bunched Beam: Asymptotic Solutions Bunch Length and Energy Spread vs. Time Quadrupole-like mode oscillations continue indefinitely. Microbunching disappears within 1-2 synchr. oscillations M. Venturini

30. X-Ring: Evolution of Charge Density and Bunch Length Charge Density Bunch Length (rms) (25% above instability threshold) M. Venturini

31. Inclusion of Radiation Damping and Quantum Excitations • Add Fokker-Planck term to Vlasov equation damping quantum excit. Case study NSLS VUV Ring Synch. Oscill. frequency Longitudinal damping time M. Venturini

32. Keil-Schnell Diagram for NSLS VUV Ring Current theshold: Most unstable harmonic: G. Carr et al., NIM-A 463 (2001) p. 387

33. NSLS VUV Model Bunch Length vs. Time 10 ms Incoherent SR Power: CSR Power: vs. Time current =338 mA, (I=12.5 pC/V)

34. Bunch Length vs. Time vs. Time 1.5 ms M. Venturini

35. Snapshots of Charge Density and CSR Power Spectrum 5 cm current =311 mA, (I=11.5 pC/V)

36. NSLS VUV Storage Ring Charge Density Bunch Length (rms) Radiation Power Radiation Spectrum M. Venturini

37. Conclusions • Numerical model gives results consistent with linear theory, when this applies. • CSR instability saturates quickly • Saturation removes microbunching, enlarges bunch distribution in phase space. • Relaxation due to radiation damping gradually restores conditions for CSR instability. • In combination with CSR instability, radiation damping gives rise to a sawtooth-like behavior and a CSR burstingpattern that seems consistent with observations. M. Venturini