Genesis Tolerance Simulations 27 January 2005 Sven Reiche

1. Genesis Tolerance Simulations27 January 2005Sven Reiche • Impact on FEL Performance • Error Classification • Error Sources • Tolerance Budget • Comments & Conclusion

2. FEL Performance - Overlap • Any ‘dilution’ of the radiation field reduces the amplification: • Missing overlap due to distortion in electron orbit. Requires about one gain length to build up field intensity from zero. • Diffraction negligible for LCLS-like parameters • Stringent requirement for straightness of orbit to less than the beam size.

3. FEL Performance - Overlap II • If trajectory is stronger than the critical angle then the build-up is significantly inhibited. Reduced Bunching (Tanaka et al, NIM A 528 (2004) 172) Red Shift + Separation Steering Separation Bunching

4. FEL - Phase Synchronization • For best amplification, electrons have to stay in phase with radiation field. • Splitting up the phase error (’=0’-’) Field detuning Betatron oscillation

5. FEL - Phase Synchronization • Constant term in ’ is compensated by automatic tuning of SASE process ’  Linear Trend = Frequency Shift z z Phase Shake  z

6. Example - Betatron Oscillation • No strong variation in ponderomotive phase. • No modulation with betatron phase Action variable = 1 rad/ Resulting phase shake

7. Error Classification • Field Detuning • Field error of each individual pole • Effective undulator parameter seen by electron beam (module alignment) • Orbit Distortion • Quadrupole and module alignment • Mismatch and initial offsets • Quadrupole and undulator field variation • Phase matching between modules

8. A Note on Orbit Distortion • Causes missing overlap and phase shake, but the effects are strongly correlated. • Orbit distortion tends to be rather like an excited betatron oscillation than a random trajectory. Variation in rms orbit distortion and phase shake can be large for a given rms error but different samples. Example: pole field errors (TTF Undulator) RMS errors in alignment and field strength is not necessarily a good parameter to describe FEL performance.

9. Tolerance Simulation • Time consuming, because a frequency range has to be scanned for a possible shift in resonance condition. • SASE simulations of 5 micron subsection of uniform bunch profile with design parameters • Saturation length and power difficult to be determined for cases with large errors • Average power at 90 m and 130 m instead.

10. Beam Mismatch and Alignment

11. Beam Offsets Rms values 29.8 microns @ 130 m 35.5 microns @ 90 m 130 m Tolerance (rms value/3) 90 m 9.9 microns @ 130 m 11.8 microns @ 90 m This corresponds to an action variable of about 1.10-7

12. Beam Mismatch Rms values (Gauss For x->log(x)) 0.356 @ 130 m 0.263 @ 90 m 130 m Tolerance (rms value/3) 0.119 @ 130 m 90 m 0.088 @ 90 m This corresponds to a variation of about 10% There is a slight asymmetry due to incomplete oscillation in the beam Envelope and the non-optimized beta-function.

13. Undulator Field Errors

14. Module Detuning Rms values 0.035% @ 130 m 130 m Advantageous phase shift in post saturation regime 0.035% @ 90 m Tolerance (rms value/3) 0.012% @ 130 m 0.012% @ 90 m 90 m This corresponds to a phase slippage of about 0.15 rad per module Using this phase slippage as the tolerance than the tolerance for the drift spaces can be estimated with 6 mm.

15. Module Offsets Rms values (x/y) 1.12/0.27mm @ 130 m x 0.78/0.27mm @ 90 m 130 m y Tolerance (rms value/3) 0.37/0.09mm @ 130 m x 90 m 0.26/0.09mm @ 90 m y This corresponds to a variation of about 10% There is a slight asymmetry due to incomplete oscillation in the beam Envelope and the non-optimized beta-function.

16. Pole Field Errors (to be done) • Variation in the individual pole field strength causes a transverse steering effect. • Two types of errors: • Correlated: the resulting orbit distortion is minimized • Uncorrelated: residual field error after chimming x’=(-1)j (Kj+1-Kj)/

17. Pole Field Errors (Old Results) • Correlated Errors • Perfect correlated errors not a problem • Residual uncorrelated defines performance Daw = 1% rms Dx<100 nm rms (correlated errors) 1% 0.1% 0.01%

18. Quadrupole Errors

19. Quadrupole Field Variation Rms values 8.76% @ 130 m 130 m 8.67% @ 90 m Tolerance (rms value/3) 90 m 2.92% @ 130 m 2.89% @ 90 m Quadrupole field errors acts like beta-mismatch, but with a spread in the resulting mismatches at the same rms variation.

20. Quadrupole Offsets 130 m 90 m Tolerance (rms value/3) Rms values (beam offset) The tolerance for the quadrupole is about a factor 3 larger 4.67 m @ 130 m 1.56 m @ 130 m 4.13 m @ 90 m 1.38 m @ 90 m

21. Comments on BBA • The beam orbit tolerances in BBA are more relaxed because the orbit tends to be flatter than with random quadrupole offsets. The case is closer too a systematic beam offset. BBA - Trajectory Random Offset - Trajectory Constant frequency freq1 freq2

22. Tolerance Budget

23. To Do • Pole Field Errors • Module spacing (drift length) • BBA results • Because other errors exits orbit distortion (e.g. Module offsets, Field Errors) they are included in the BBA algorithm to straighten the orbit. • Using this methods to combine errors.

24. Conclusion • Errors, causing orbit distortion, affect the FEL performance in the same way. The resulting tolerance on the orbit for truly random errors are 1.5 microns rms beam orbit. However these errors have often large local amplitude, which can be detected by the BPM and straighten by the BBA procedure. These resulting ‘correlated’ errors allows for a larger tolerance in the orbit.