1. FEL Simulations: Undulator Modeling Sven Reiche Start-end Workshop DESY-Zeuthen 08/20/03

2. FEL Simulations Start-end simulation aims to predict the FEL performance as realistic as possible: Modeling the electron beam, Modeling the undulator. VUV/X-Ray undulator with large array of undulator poles (>1000), grouped in modules and combined/interleaved with strong focusing elements (quadrupoles).

3. Resolving the Undulator Period Non-period averaged codes can use any arbitrary field profile but are bound to a sub- period length integration step size Integration step size of period averaged codes can scale with gain length, thus, independent of undulator length Example: Integration steps for LCLS 100 x 300010 x 20 Non-averagedAveraged PeriodsGain lengthsResolve harmonicsResolve gain

4. Resolving the Undulator Lattice Integration step sizes are limited due to: Undulator drift sections, Field Errors, Quadrupole length, Phase shifter… Integration step size of period-averaged codes scales rather with undulator period than with gain length.

5. Speeding-up the Calculation… Increasing length of quadrupoles, while reducing the field strength (thin lens approximation). Insert virtual phase shifter to control phase slippage in drift section. Replace effect of field errors by a single net kick per integration step.

6. Undulator Elements Essential Undulator field + taper, Quadrupoles + sextupole focusing of undulator field, Drifts with phase shifter. Optional (Exotic) Undulator endfield, Solenoid field, Higher multipoles of undulator field (e.g. for apple-type undulators), Coupling of motion in x and y.

7. Example: LCLS Undulator Superperiod of 6 quadrupoles & undulator modules Permanent magnet quadrupoles (107 T/m) FDFDFDF Module 3.42 m Long Gap 42.1 cm Short Gap 18.7 cm Quadrupole 5 cm Super Period 22.11 m

8. Effects of Field Errors Introduces steering in beam orbit. Reduced overlap between beam and radiation field Phase shake KiKi K i+1 pp Centroid motion zz

9. Correlated Errors Undulator assemble depends on measuring, sorting and arranging all permanent magnets with the goal of Minimizing 1st and 2nd field integral Minimizing phase shake Residual variation in the field strength of undulator field is highly correlated

10. Correlated vs Uncorrelated RMS variation not a good parameter to describe FEL performance

11. Quadrupole Misplacement Interface with BBA-Procedures. Similar effect as field errors. 2D codes overestimate effect.

12. Undulator Misplacement Possibility for undulator tapering, but Stronger alignment tolerances, Larger K-spread over beam size.

13. Undulator Misplacement (cont’) Impractical for LCLS as a possibility for a controlled field taper. LCLS alignment tolerance: 100  m tolerance K-spread

14. Undulator Taper Constant energy loss due to spontaneous radiation, compensated by taper. Enhancing FEL efficiency after saturation.

15. Wakefields Contributions by Resistive wall, Aperture changes (geometric wakes), Surface Roughness, 99% from resistive wall wakefields

16. Wakefields (cont’) FEL degradation by 30-50 % Weak dependence on chamber radius due to large transient at bunch head. Gap due to wakefields

17. Conclusion Non-averaged codes too CPU-intensive for full time-dependent simulations. Transverse and longitudinal motion decoupled in period averaged codes -> different solvers. All essential features of undulators covered by FEL codes. Standard support for arbitrary undulator lattices. 2D codes limited precision for certain effects such as transverse centroid motion.