1. Temporal Characteristics of CSR Emitted in Storage Rings – Observations and a Simple Theoretical Model P. Kuske, BESSY „Topics in Coherent Synchrotron Radiation (CSR) Workshop : Consequences of Radiation Impedance“ Nov. 1st and 2nd 2010, Canadian Light Source

2. Outline of the Talk I.Motivation II.Observations – at BESSY II III.Theoretical Model – μ-wave Instability „numerical solution of the VFP-equation with BBR-wake“ III. 1BBR-Impedance III. 2Vlasov-Fokker-Planck-Equation – „wave function“ approach III. 3Numerical Solution of this VFP-equation IV.Results IV. 1Comparison to other Solutions IV. 2New Features and Predictions IV. 3Comparison of Experimental and Theoretical Results V.Conclusion and Outlook 2

3. I.Motivation 3 open questions and issues related to CSR directly: spectral characteristic? steady-state radiation vs. emission in bursts – temporal characteristics? how can we improve performance? more generally: some kind of impedance seems to play a role potential instability mechanism very often we operate with single bunch charges beyond the stability limit How does instability relate to CSR?

4. II.Observations at BESSY II 4 time dependent CSR-bursts observed in frequency domain:  0 =14 ps, nom. optics, with 7T-WLS Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005, Frascati Spectrum of the CSR-signal: CSR-bursting threshold Stable, time independent CSR 

5. II.Observations at BESSY II 5 time dependent CSR-bursts observed in frequency domain:  0 =14 ps, nom. optics, with 7T-WLS Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005, Frascati Spectrum of the CSR-signal: CSR-bursting threshold Stable, time independent CSR 

6. II.Observations at BESSY II 6 time dependent CSR-bursts observed in frequency domain:  0 =14 ps, nom. optics, with 7T-WLS Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005, Frascati Spectrum of the CSR-signal: CSR-bursting threshold Stable, time independent CSR 

7. II.Observations at BESSY II 7 time dependent CSR-bursts observed in frequency domain:  0 =14 ps, nom. optics, with 7T-WLS Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005, Frascati Spectrum of the CSR-signal: CSR-bursting threshold Stable, time independent CSR 

8. 8 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

9. 9 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

10. 10 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

11. 11 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

12. 12 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

13. 13 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

14. 14 II.Observations at BESSY II with 4 sc IDs in operation Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

15. 15 II.Observations at BESSY II without sc IDs Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

16. 16 II.Observations at BESSY II Investigation of the temporal structure of CSR-Bursts at BESSY II, Peter Kuske, PAC’09, Vancouver

17. 17 III. 1Theoretical Model – BBR-Impedance

18. 18 III. 1Theoretical Model – CSR-Impedance ‚featureless curve with local maximum and cutoff at long wavelenght‘

19. 19 III. 1Theoretical Model – CSR-Impedance

20. 20 III. 2Vlasov-Fokker-Planck-Equation (VFP) Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005, Frascati numerical solution based on:R.L. Warnock, J.A. Ellison, SLAC-PUB-8404, March 2000 M. Venturini, et al., Phys. Rev. ST-AB 8, 014202 (2005) S. Novokhatski, EPAC 2000 and SLAC-PUB-11251, May 2005 RF focusingCollective ForceDampingQuantum Excitation solution for f(q, p,  ) can become < 0 requires larger grids and smaller time steps (M. Venturini)

21. 21 III. 2Vlasov-Fokker-Planck-Equation (VFP) „wave function“ approach Ansatz – “wave function” approach: Distribution function, f(q, p,  ), expressed as product of amplitude function, g(q, p,  ) : original VFP-equation: f (q, p,  ) >= 0 and solutions more stable

22. 22 III. 3Numerical Solution of the VFP-Equation and BESSY II Storage Ring Parameters F syn = 7.7 kHz ~60 T syn per damping time Energy  1.7 GeV Natural energy spread   /  7  10 -4 Longitudinal damping time  lon 8.0 ms Momentum compaction factor  7.3 10 -4 Bunch length  o 10.53 ps Accellerating voltage V rf 1.4 MV RF-frequency  rf 500  2  MHz Gradient of RF-Voltage  V rf /  t 4.63 kV/ps Circumference C 240 m Revolution time T o 800 ns Number of electrons 5  10 6 per µA solved as outlined by Venturini (2005): function, g(q, p,  ), is represented locally as a cubic polynomial and a time step requires 4 new calculations over the grid distribution followed over 200 T syn and during the last 160 T syn the projected distribution  (q) is stored for later analysis: determination of the moments and FFT for the emission spectrum grid size 128x128 and up to 8 times larger, time steps adjusted and as large as possible M. Venturini, et al., Phys. Rev. ST-AB 8, 014202 (2005)

23. 23 Theoretical Results VI. 1Comparison with other Theories and Simulations K. Oide, K. Yokoya, „Longitudinal Single-Bunch Instability in Electron Storage Rings“, KEK Preprint 90-10, April 1990 K.L.F. Bane, et al., „Comparison of Simulation Codes for Microwave Instability in Bunched Beams“, IPAC’10, Kyoto, Japan and references there in

24. 24 Theoretical Results VI. 1Comparison with other Theories and Simulations

25. 25 Theoretical Results VI. 2New Features broad band resonator with R s =10 kΩ

26. 26 Theoretical Results VI. 2New Features broad band resonator with R s =10 kΩ

27. 27 Theoretical Results VI. 2New Features broad band resonator with R s =10 kΩ

28. 28 Theoretical Results VI. 2New Features – Islands of Stability

29. 29 Theoretical Results VI. 2New Features – Islands of Stability

30. 30 Theoretical Results VI. 2New Features – Hysteresis Effects

31. 31 Theoretical Results - burst VI. 3 BBR: R s =10kΩ, F res =200GHz, Q=4 and I sb =7.9mA

32. 32 Theoretical Results – lowest unstable mode VI. 3BBR: R s =10kΩ, F res =100GHz, Q=1 and I sb =2.275mA

33. 33 Theoretical Results VI. 3BBR + R + L in Comparison with Observations BBR: R s =10 kΩ, F res =40 GHz, Q=1; R= 850 Ω and L=0.2 Ω

34. 34 Theoretical Results VI. 3BBR + R + L in Comparison with Observations

35. 35 V.Conclusion and Outlook a simple BBR-model for the impedance combined with the low noise numerical solutions of the VFP-equation can explain many of the observed time dependent features of the emitted CSR: typically a single (azimuthal) mode is unstable first frequency increases stepwise with the bunch length at higher intensity sawtooth-type instability (quite regular, mode mixing) at even higher intensitiy the instability becomes turbulent, chaotic, with many modes involved Oide‘s and Yokoya‘s results are only in fair agreement with these calculations do not show such a rich variety of behavior – why? are better approaches available? simulation can serve as a benchmark for other theoretical solutions need further studies with more complex assumption on the vacuum chamber impedance the simple model does not explain the amount of CSR emitted at higher photon energy – much higher resonator frequencies required is this produced by a two step process, where a short wavelength perturbation is first created by the traditional wakefield interaction and in a second step another wake with even higher frequency comes into play? (like bursting triggered by slicing)

36. 36 Acknowledgement K. Holldack – detector and THz-beamline J. Kuszynski, D. Engel – LabView and data aquisition U. Schade and G. Wüstefeld – for discussions