1. Optimization of Compact X-ray Free-electron Lasers Sven Reiche May 27 th 2011

2. Free-Electron Lasers

3. Electron Beam Parameters Almost everything scales with the FEL parameter (1D Model): The characteristic length is the power gain length: Increasing the current and/or reducing the emittances increase the performance and reduce the overall required length of the FEL. Impact of energy spread and emittance needs to be small (Both spreads out any bunching and thus act against FEL process): (Ideal Case)

4. Saturation Power and Brilliance Saturation power: For a given wavelength it favors a higher beam energy and thus a longer undulator period. Peak Brilliance: At saturation: SwissFEL (transverse coherence)

5. Optimizing the Focusing Decreasing the  -function (increase focusing), increases the FEL parameter . Too strong focusing enhances the emittance effect and increasing the FEL gain length. From 1D Theory: 3D Optimization for SwissFEL

6. Transverse Coherence (2D FEL Theory) Diffraction Parameter: Assuming electron size as radiation source size: Rayleigh Length Char. Scaling of FEL (=2k u  in 1 D model) x pxpx Photon Emittance Diffraction Limited Photon Emittance Constraint for emittance to be smaller than photon emittance for all electrons to contribute on the emission process Electron

7. Transverse Coherence (Saldin et al) Growth rates for FEL eigenmodes (r,  -decomposition): Increased gain length due to strong diffraction Mode competition and reduced coherence Optimum growth rate of 1D model Optimum: Note: fundmental FEL Eigenmode has intrinsic wavefront curvarture and thus correspond to a larger photon emittance

8. Growth of transverse coherence = spreading the phase information transversely. Additional Effects 8 Diffraction Convection Spreading Phase spread by field Phase spread by electrons Beam size variation Emittance constraints is relaxed when realistic electron motion is included FEL Theory only treats diffraction effects, but spreading is dominating effect in short wavelength FELs TEM 01 Growth rate FODO Lattice Rigid

9. Definition of Coherence (Goodman) Coherence is a stochastic quantity and correlates the phase relation between two transverse location: In relation to diffraction experiments: A pulse can be fully coherent without 100% contrast in interference pattern (E 0 (r 1 ) ≠E 0 (r 2 )). The quasi-monochromatic approximation (E 0 (t)=E 0 (t+dt)) is not necessarily fulfilled. Single spike pulse is not a stationary process and would require ensemble average. Mutual Coherence Function Normalized Coherence Factor E 0 (r 2,t-dt) E 0 (r 2,t) E 0 (r 1,t)

10. Example: Coherence for SwissFEL 10  n = 0.43 mm mrad Spot Size: 0.0018 mm 2 Coherence Area: 0.021 mm 2  n = 0.86 mm mrad Spot Size: 0.0023 mm 2 Coherence Area: 0.015 mm 2  =0.86  =0.69 Intensity Mutual Coherence Intensity Mutual Coherence

11. Towards Compact FELs Maximize the FEL parameter by maximizing the peak current. Large FEL parameters allows for relaxed values for energy spread and emittance. “Conventional” undulator design though with a short undulator period. Reduce undulator period by using a laser wiggler as radiation device. Provide a low emittance, low energy spread beam. Moderate peak current which are consistent with the low emittance and energy spread values. Approach AApproach B Plasma InjectorField Emitter Source

12. Case Study: Laser Wiggler

13. From XFEL to Laser Wiggler Due to the scaling, the FEL parameter is always smaller for laser wiggler than for conventional SASE FELs. In addition, the extreme constraints on the beam quality limits the current to a much smaller value than for SASE FELs. u : 5 cm  500 nm  : 15 GeV  20 MeV Resonance ConditionFEL Parameter  : 5. 10 -4  5. 10 -5

14. Case Study for Laser Wiggler Emittance and spread has been artificially chosen to allow the FEL to laser with the given peak current. Energy spread is way beyond state-of-the-art electron beam sources. Electron emittance much larger than photon emittance. Energy6.5 MeV Current500 A Energy Spread60 eV (!!!) Emittance0.1 mm mrad (!) Laser Period500 nm Laser Field Strength0.5 FEL Wavelength2 nm

15. Coherence Properties An estimate of the coherence can be obtained by the fluctuation in the radiation power. As expected from the high beam emittance value, only a poor degree of coherence is achieved. Less than 10% coherence Pulse profile at saturation

16. Field stability requirement for all electrons along the interaction length: Most stringent requirement for transverse dimension over at least 2  of the bunch, assuming a fundamental Gauss mode (alternative is transverse mode stacking): Required radiation power: Comments on the Laser Field Interaction Length Waist size Bunch For the example given the required power is 0.5 PW with a pulse duration of 15 ps !!!! Incident Laser

17. Case Study: High Current Electron Beam and Short Period Undulator

18. Basic Strategy Very high peak currents yield a very large FEL parameter, which allows relaxed constraints for the electron beam quality (energy spread, emittance). The beam energy is still reasonable large (>100 MeV) to allow for short wavelength while fulfilling the coherence condition. Possible sources are plasma injector with an ultra short drive laser to drive a non-linear trapping mechanism of electrons. Currently an active field of research (LOA, LMU, LBNL etc.)

19. Electron Beam Parameters Ambitious beam parameters, though the lower end has been successful demonstrated at LOA. Energy0.1 - 2 GeV Current10 - 150 kA (!) Emittance1 mm mrad Energy spread<1% Charge10 – 1000 pC C. Rechatin et al, Phys. Rev. Lett., 102, 164801 (2009) Most critical parameter is the energy spread, which requires a FEL parameter on the same order or larger.

20. Case Study Overly optimistic case (LMU Munich, courtesy of A. Maier/A. Meseck). E = 2 GeV  E /E = 0.04 % (!!!)  n = 0.5 mm mrad (!) I = 100 kA (!) Undulator: u = 5 mm, K = 0.65 = 2.5 Å FEL parameter is ~10 -3, thus requiring good beam parameters to make the FEL work. Also coherence is reduced.

21. A More Reasonable Case Energy1 GeV Current10 kA Emittance1 mm mrad Energy spread5 MeV Effective FEL Parameter0.1% Gain Length0.23 m Saturation Length4.8 m Saturation Power9 GW Bandwidth0.23 % FWHM Results for = 1 nm Larger spread and emittance, less current

22. Optimization for Very High Energy Spreads Use dispersion to stretch the bunch. Reduces bunch current and energy spread Energy spread conditions improves due to the I 1/3 dependence of the FEL parameter. Helps for very large energy spreads: Current10 kA1 kA Energy Spread20 MeV2 MeV Sat. Power1 GW0.4 GW Sat. Length19 m14 m

23. Summary Coherence and peak brightness favors GeVs electron beams with high peak currents. Plasma injectors are promising candidates for compact x-ray FELs, though the performance seems to be feasible only down to 1 nm Laser wigglers yield only poor performance (poor coherence) even if the unrealistic beam and laser parameters could be met. Laser Wiggler Plasma Beam