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

2. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 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. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 3 Low-gain FEL Estimated spectrum of spontaneous undulator radiation and FEL radiation in LCLS (Stanford) FEL radiation (H.-D. Nuhn, SLAC)

4. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 4 Low-gain FEL  undulator radiation  low gain FEL  high gain FEL electrons Interaction EM field Model FEL radiation electrons

5. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 5 Low-gain FEL Upon each passage the light intensity grows only by small gain factor 

6. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 6 Energy exchange

7. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 7 Energy exchange Field on the axis Undulator parameter Electron motion trajectory

8. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 8 Energy exchange in frame moving with averaged velocity trajectory Electron motion

9. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 9 Energy exchange - the electron is slower than the light A steady energy transfer? Laser field is approximated by a plane EM wave

10. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 10 Energy exchange - the electron is slower than the light A steady energy transfer?

11. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 11 Energy exchange - the electron is slower than the light A steady energy transfer?

12. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 12 Energy exchange - the electron is slower than the light A steady energy transfer?

13. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 13 Energy exchange the electron should be slower by one wavelength - the electron is slower than the light A steady energy transfer?

14. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 14 Energy exchange the electron should be slower by one wavelength A steady energy transfer?  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. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 15 Energy exchange A steady energy transfer? The electron energy changes as described by the equation

16. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 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. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 17 Energy exchange Energy transfer equation - makes 2 oscilations per undulator period and cancels out

18. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 18 Energy exchange Refinement (see, for example, P. Schmüser et al)

19. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 19 FEL pendulum equations Ponderomotive phase where we have used the coherence condition

20. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 20 FEL pendulum equations Phase equation

21. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 21 FEL pendulum equations Hamiltonian

22. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 22 FEL pendulum equations Separatrix

23. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 23 FEL pendulum equations Change of the independent variable

24. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 24 FEL pendulum equations Asymptotic expansion No impact of the EM field on the particles in the lowest order

25. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 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. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 26 FEL pendulum equations Second order There is a mean energy exchange between the particles and the EM field in this order

27. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 27 FEL gain and Madey theorem The change in the electron energy density reads 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 energy gain is

28. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 28 FEL gain and Madey theorem The energy deviation (of electron) is equivalent to the wavelength deviation (of EM wave)

29. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 29 FEL gain and Madey theorem J.M.J. Madey, Nuovo Cimento, B50, 64 (1979) spontaneous radiation spectrumenergy gain energy deviation ηfrequency deviation Δω

30. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 30 FEL gain and Madey theorem Low-gain FEL  no energy gain at the resonance energy  the electron energy has to be higher

31. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 31 Microbunching FLASH in low-gain model FLASH (Hamburg)

32. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 32 Microbunching FLASH in low-gain model (Exercise 5)

33. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 33 Microbunching Low-gain model

34. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 34 Microbunching Low-gain model

35. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 35 Microbunching Low-gain model

36. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 36 Microbunching experimental evidence of microbunching in Stanford K.N. Ricci ant T.I Smith, PR- STAB 3, 032801 (2000)

37. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 37 High-gain FEL equations in 1D data from FLASH W. Ackermann et al, Nature Photonics 1, 336 (2007) The amplification is very high

38. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 38 High-gain FEL equations in 1D The laser field can not be considered as constant 1D model

39. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 39 High-gain FEL equations in 1D Representation with slowly changing amplitude

40. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 40 High-gain FEL equations in 1D

41. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 41 High-gain FEL equations in 1D  numerical methods are required  very high number of electrons  calculations with the help of macroparticles. High-gain FEL model

42. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 42 High-gain FEL equations in 1D

43. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 43 High-gain FEL equations in 1D saturation: beam fully modulated

44. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 44 High-gain FEL equations in 1D

45. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 45 High-gain FEL equations in 1D

46. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 46 High-gain FEL equations in 1D

47. PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 47 Outlook linear saturation