1. Lecture 4. High-gain FELs X-Ray Free Electron Lasers Igor Zagorodnov Deutsches Elektronen Synchrotron TU Darmstadt, Fachbereich 18 19. May 2014

2. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 2 Contents  Review  Exponential growth and saturation  Bunching factor and space charge field  Linearized model  FEL parameters  FEL bandwidth  3D effects  Outlook

3. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 3 Review  undulator radiation  low gain FEL  high gain FEL electrons Interaction EM field Model FEL radiation electrons

4. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 4 Review What is shown? (Exercise 7)

5. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 5 Review Electron motion trajectory

6. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 6 Review Exercise 7Exercise 4

7. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 7 Review the electron should be slower by one wavelength - the electron is slower than the light Resonance condition

8. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 8 Review Low gain FEL model

9. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 9 Review data from FLASH W. Ackermann et al, Nature Photonics 1, 336 (2007) The amplification is very high Microbunching

10. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 10 Review High-gain FEL model

11. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 11 Exponential growth and saturation Exercise 6

12. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 12 Exponential growth and saturation saturation: beam fully modulated Exercise 6

13. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 13 Exponential growth and saturation Exercise 6

14. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 14 Exponential growth and saturation Exercise 6

15. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 15 Exponential growth and saturation Exercise 6

16. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 16 Exponential growth and saturation linear saturation Exercise 6

17. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 17 Bunching factor and space charge field Fourier series of real periodic function Dirac delta-function

18. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 18 Bunching factor and space charge field

19. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 19 Bunching factor and space charge field

20. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 20 Bunching factor and space charge field

21. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 21 Bunching factor and space charge field Revision of energy equation

22. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 22 Bunching factor and space charge field High-gain FEL model with space-charge

23. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 23 Linearized model

24. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 24 Linearized model Vlasov-Maxwell equations - monoenergetic beam According to the Liouville Theorem the phase space volume occupied by an ensemble of particles is conserved along the particle trajectory.

25. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 25 FEL parameters Gain parameter Detuning Space charge parameter Pierce parameter

26. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 26 FEL parameters Normalized high-gain FEL model with space-charge

27. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 27 FEL parameters Power gain length

28. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 28 FEL parameters Power gain length Field gain length

29. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 29 FEL parameters Power amplification

30. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 30 FEL parameters “lethargy regime”

31. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 31 FEL parameters Saturation

32. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 32 Exponential growth and saturation Exercise 6

33. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 33 FEL parameters Saturation power

34. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 34 FEL parameters Saturation power

35. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 35 FEL parameters Saturation

36. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 36 FEL Bandwidth Detuning for resonance frequency Detuning for resonance energy

37. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 37 FEL Bandwidth

38. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 38 FEL Bandwidth

39. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 39 FEL Bandwidth

40. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 40 FEL Bandwidth

41. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 41 FEL Bandwidth  amplification at resonance energy  small positive detuning increases amplification

42. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 42 3D Effects Diffraction Diffraction reduces the field amplitude on the axis

43. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 43 Outlook Electrons produce spontaneous undulators radiation How to obtain a useful external field? SASE Self-Amplified Spontaneous Emission (SASE) A. Kondratenko, E. Saldin, Part. Accelerators 10, 207 (1980) R.Bonifacio et al, Opt. Comm.50, 373 (1984)