2. Modelling shot noise and coherent spontaneous emission in the FEL Brian McNeil*, Mike Poole & Gordon Robb* CCLRC Daresbury Laboratory, UK *Department of Physics, University of Strathclyde Glasgow, UK ICFA Future Light Sources Sub-Panel Mini Workshop on Start-to-End Simulations of X-RAY FELs August 18-22, 2003 at DESY-Zeuthen (Berlin, GERMANY)

3. Outline New model of electron shot-noise - derived from first principles Combined shot-noise/CSE numerical model – simulations New model for ultra high power/ultra short pulse radiation Simulation of such pulse propagation

4. New model of electron shot-noise

5. Previous Shot-Noise Model Previously with the averaged equations…  bMbM V(b M ) = V(b R ).... ….... …... …...... …... ….......... …........... …. bRbR -θj-θj

6. z Current   Spatial distribution Density variation over a radiation wavelength acts as a source. This is not modelled in the averaged equations. Coherent Spontaneous Emission - No SVEA approximation

7. Now with the un-averaged equations… There is now no reference to any averaging over radiation wavelength scale – previous method is not valid. Assume the arrival of electrons in a time interval is a Poisson process:.... ….... …... …...... …... ….......... …........... …. tt Some radiation period Mean rate of e - arrival Mean number of e - Poisson prob. of N n e - Mean arrival time of an e - Var. arrival time of an e -.... ….... …... …...... …... ….......... …........... ….

9. Equating real and macro variances: Macroparticle mean arrival time:.... ….... …... …...... …... ….......... …........... …. t (<< some radiation period tt.... ….... …... …...... …... ….......... …........... …. So, in the interval t :.... ….... …... …...... …... ….......... …........... ….

10. Results for cold mono-energetic electron distribution:

11. Schematic of model for higher dimensional phase space - e.g. including energy spread

12. Macroparticle charge weighting: Macroparticle position in phase space: - Mean # electrons in phase space cell - Charge weight assigned to jth macroparticle in cell is Poisson random variate Uniform random variates:

13. Results for electron distribution with top-hat energy spread after a drift of

14. The following equations were solved using Finite Element Method: - 1-D Pierce parameter l e - scaled electron pulse duration N – Expectation of total # electrons in pulse FEL Simulation * * *Random variables

15. z1z1 1 z1z1 element S 2 ( ) z1z1 S 1 ( ) z1z1 A(z, ) = a n (z) + a n+1 (z)S 1 ( ) z1z1 S 2 ( ) z1z1 z1z1 Finite elements: anan a n+1 a n-1 a n+2

16. Simulation Results In the following sequence of graphs: z is the scaled distance through the wiggler z 1 is the scaled position within the pulse in radiation periods:  = 0.1/4  |A| 2 is the scaled radiation power  j /  0 is the jth electron’s relativistic parameter scaled with respect to its initial value P is the scaled spectral power f is the frequency scaled with respect to the resonant fundamental A x,y are the scaled x-y components of the electric fields

17. Numerical solution including shot-noise and CSE: Top-hat electron pulse current Gaussian energy spread of FEL parameter Total charge Average macroparticle 0.1 0.008

20. Averaged quantities Bunching parameter |b| (x) and scaled intensity ( ) at a fixed position in the electron pulse Shot-noise SACSE

21. Numerical solution including shot-noise and CSE: Gaussian electron pulse current Gaussian energy spread of FEL parameter Total charge Average macroparticle 0.1 0.008

23. New model for ultra high power/ultra short pulse radiation propagation in the FEL

24. Theory predicts that very short intense pulses of radiation can be generated in an FEL when the electrons emit superradiantly. Electrons N N NS S S S S N N Wiggler magnet Radiation pulses We want to know : Is there a saturation mechanism in I pk ? – missing from current theory Is there a limit to the duration of the high power radiation pulses? - Current theory breaks down when the  ~ (frequency) -1 Why do we need another model for the FEL?

25. The Coupled 1-D Maxwell-Lorentz equations: These equations are rewritten in a scaled form with the minimum number of assumptions necessary to model: large energy exchanges between electrons and radiation - >>1 pulses of very short duration - The 1-D (plane wave) approximation is made and space-charge effects are neglected. - Wave equation - Lorentz equation - Current density

26. The fields in a helical wiggler FEL: ONLY assumption outwith 1D and neglect of space charge => Neglect backward wave - Wiggler magnetic - Radiation electric - Radiation magnetic

27. Scaled parameters With previous assumptions & fields we obtain: Working equations = 2ρp j

28. ‘Old’ model Averaged equations SVEA valid

29. Numerical solution including shot-noise and CSE: Top-hat electron pulse current Gaussian energy spread of FEL parameter Total charge Average macroparticle 0.1 0.008

31. New Model

34. Averaged modelNon-averaged model Scaled intensity |A| 2

35. For Planar, short, cold electron pulse: Gaussian electron pulse FEL parameter

37. Conclusions: New model of electron shot-noise derived from first principles Model used to simulate simultaneous FEL start-up from CSE and shot-noise Demonstrated model for simulating ultra high power pulse propagation in Free Electron Laser Sub-wavelength radiation pulses are seen to propagate with quasi-unipolar fields No saturation effects yet observed for this preliminary study Exciting prospects for future analytical and numerical work in generation of exotic FEL radiation spikes and post-saturation modelling

38. Things we would like to develop: Changes in statistical nature of radiation ? Search for saturation mechanisms in radiation spiking Introduce 2-D – diffraction, space charge Can this method be adapted to model CSR in bending magnets ?