Optimization of Compact X-ray Free-electron Lasers

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

2. Free-Electron Lasers

3. Electron Beam Parameters (Ideal Case) 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):

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

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

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

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

8. Additional Effects Spreading Convection Diffraction Beam size variation Phase spread by electrons Phase spread by field TEM01 Growth rate FEL Theory only treats diffraction effects, but spreading is dominating effect in short wavelength FELs Rigid Emittance constraints is relaxed when realistic electron motion is included FODO Lattice • Growth of transverse coherence = spreading the phase information transversely.

9. Definition of Coherence (Goodman) Mutual Coherence Function Normalized Coherence Factor E0(r1,t) E0(r2,t-dt) E0(r2,t) • 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 (E0(r1) ≠E0(r2)). • The quasi-monochromatic approximation (E0(t)=E0(t+dt)) is not necessarily fulfilled. • Single spike pulse is not a stationary process and would require ensemble average.

10. Example: Coherence for SwissFEL en = 0.86 mm mrad Spot Size: 0.0023 mm2 Coherence Area: 0.015 mm2 Intensity en = 0.43 mm mrad Spot Size: 0.0018 mm2 Coherence Area: 0.021 mm2 Mutual Coherence Intensity Mutual Coherence z=0.69 z=0.86

11. Towards Compact FELs Approach A Approach B • 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. Field Emitter Source Plasma Injector 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.

12. Case Study:Laser Wiggler

13. From XFEL to Laser Wiggler Resonance Condition FEL Parameter lu: 5 cm  500 nm g: 15 GeV  20 MeV r: 5.10-4  5.10-5 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.

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.

15. Coherence Properties Pulse profile at saturation Less than 10% coherence 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.

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

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

18. Basic Strategy 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.) 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.

19. Electron Beam Parameters Most critical parameter is the energy spread, which requires a FEL parameter on the same order or larger. C. Rechatin et al, Phys. Rev. Lett., 102, 164801 (2009) Ambitious beam parameters, though the lower end has been successful demonstrated at LOA.

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

21. A More Reasonable Case Larger spread and emittance, less current Results for l = 1 nm

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 I1/3 dependence of the FEL parameter. Helps for very large energy spreads:

23. Summary Plasma Beam Laser Wiggler 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.