X-Ray Free Electron Lasers

1. X-Ray Free Electron Lasers Lecture 3. Low-gain and high-gain FELs Igor Zagorodnov DeutschesElektronen Synchrotron TU Darmstadt, Fachbereich 18 12. May 2014

2. Contents • Low-gain FEL • Energy exchange between electrons and EM wave • FEL pendulum equations • FEL gain and Madey theorem • Microbunching • High-gain FEL equations in 1D

3. Low-gain FEL FEL radiation (H.-D. Nuhn, SLAC) Estimated spectrum of spontaneous undulator radiation and FEL radiation in LCLS (Stanford)

4. Low-gain FEL FEL radiation Model Interaction • undulator radiation EM field electrons electrons • low gain FEL EM field EM field • high gain FEL electrons

5. Low-gain FEL Upon each passage the light intensity grows only by small gain factor 

6. Energy exchange

7. Energy exchange Electron motion Field on the axis Undulator parameter trajectory

8. Energy exchange Electron motion in frame moving with averaged velocity trajectory

9. Energy exchange A steady energy transfer? - the electron is slower than the light Laser field is approximated by a plane EM wave

10. Energy exchange A steady energy transfer? - the electron is slower than the light

11. Energy exchange A steady energy transfer? - the electron is slower than the light

12. Energy exchange A steady energy transfer? - the electron is slower than the light

13. Energy exchange A steady energy transfer? - the electron is slower than the light the electron should be slower by one wavelength

14. Energy exchange A steady energy transfer? the electron should be slower by one wavelength • Slippages by odd number of wavelength is also possible (higher harmonics) • Spontaneous radiation has the same wavelength and can serve as a seed radiation

15. Energy exchange A steady energy transfer? The electron energy changes as described by the equation

16. Energy exchange A steady energy transfer? The first term in the first equation provides a continuous energy transfer from the electron to the light wave, if the coherence condition is hold.

17. Energy exchange Energy transfer equation - makes 2 oscilations per undulator period and cancels out

18. Energy exchange Refinement (see, for example, P. Schmüser et al)

19. FEL pendulum equations Ponderomotive phase where we have used the coherence condition

20. FEL pendulum equations Phase equation

21. FEL pendulum equations Hamiltonian

22. FEL pendulum equations Separatrix

23. FEL pendulum equations Change of the independent variable

24. FEL pendulum equations Asymptotic expansion No impact of the EM field on the particles in the lowest order

25. FEL pendulum equations First order No mean energy exchange between the particles and the EM field, but there are energy modulation in the particle beam

26. FEL pendulum equations Second order There is a mean energy exchange between the particles and the EM field in this order

27. FEL gain and Madey theorem The net decrease in particle energy results in an increase in the EM energy The energy density of the seed EM field (plane wave) reads The change in the electron energy density reads The energy gain is

28. FEL gain and Madey theorem The energy deviation (of electron) is equivalent to the wavelength deviation (of EM wave)

29. FEL gain and Madey theorem J.M.J. Madey, NuovoCimento, B50, 64 (1979) spontaneous radiation spectrum energy gain frequency deviation Δω energy deviation η

30. FEL gain and Madey theorem Low-gain FEL • no energy gain at the resonance energy • the electron energy has to be higher

31. Microbunching FLASH in low-gainmodel FLASH (Hamburg)

32. Microbunching FLASH in low-gainmodel (Exercise 5)

33. Microbunching Low-gainmodel

34. Microbunching Low-gainmodel

35. Microbunching Low-gainmodel

36. Microbunching K.N. Ricci ant T.I Smith, PR-STAB 3, 032801 (2000) experimental evidence of microbunching in Stanford

37. High-gain FEL equations in 1D datafrom FLASH The amplification is very high W. Ackermann et al, Nature Photonics1, 336 (2007)

38. High-gain FEL equations in 1D The laser field can not be considered as constant 1D model

39. High-gain FEL equations in 1D Representation with slowly changing amplitude

40. High-gain FEL equations in 1D

41. High-gain FEL equations in 1D High-gain FEL model • numerical methods are required • very high number of electrons • calculations with the help of macroparticles.

42. High-gain FEL equations in 1D

43. High-gain FEL equations in 1D saturation: beam fully modulated

44. High-gain FEL equations in 1D

45. High-gain FEL equations in 1D

46. High-gain FEL equations in 1D

47. Outlook saturation linear