LCLS-II Undulator Tolerances

1. LCLS-II Undulator Tolerances Heinz-Dieter Nuhn LCLS-II Undulator Physics Manager May 8, 2013

2. Outline • Tolerance Budget Method • Experimental Verification of LCLS-I Sensitivities • Analytical Sensitivity Estimates for LCLS-II • Tolerance Budget Example • Summary

3. Undulator Tolerances affect FEL Performance FEL power dependence exhibits half-bell-curve-like functionality that can be modeled by a Gaussian in most cases. Functions have been originally determined with GENESIS simulations through a method developed with Sven Reiche. Several have been verified later with the LCLS-I beam: Effect of undulator segment strength error randomly distributed over all segments. Goal: Determine the rms of each performance reduction (Parameter Sensitivity si) Slide 3

4. Tolerance Budget Combination of individual performance contribution in a budget. tolerances sensitivities Calculate sensitivities Set target value for Select tolerances , calculate resulting , compare with target. Iterate: Adjust , such that agrees with target. Target used in budget analysis Slide 4

5. Individual Studies (Example: Segment Position x) • Start with a well aligned undulator line with each segment at position • Choose a set of values (error amplitudes) to be tested, for instance { 0.0 mm, 0.2 mm, …, 1.8 mm, 2.0 mm} • For each choose 32 random values, , from a flat-top distribution within the range of ± • Move each undulator segment to its corresponding error value, • Determine the x-ray intensity from one of {YAGXRAY, ELOSS, GDET} as multi-shot average • Loop over several random seeds and obtain mean and rms values of the x-ray intensity readings for the distribution for this error amplitude • Loop over all • Plot the mean and average values vs. , i.e. vs. { 0.000 mm, 0.115 mm, …, 1.039 mm, 1.155 mm} • Apply Gaussian fit, , to obtain rms-dependence (sensitivity) for this ith error parameter Slide 5

6. Segment x Position Sensitivity Measurement Sensitivity: mean rms Generate random misalignment with flat distribution of width ± => rmsdistribution Slide 6

7. LCLS Error: Horizontal Module Offset Simulation and fit results of Horizontal Module Offset analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point. 130 m 90 m S. Reiche Simulations Slide 7

8. DK/K Sensitivity Measurement Sensitivity: • Consistent with Dx sensitivity (sx=0.77 mm), because with dK/dx ~ 27.5×10-4/mm and K~3.5 one gets • sDK/K = sx (1/K) dK/dx ~ 6×10-4=r Slide 8

9. LCLS Error: Module Detuning Simulation and fit results of Module Detuning analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point. 130 m 90 m • Expected: 0.040 for en=1.2 µm & Ipk = 3400 A Z. Huang Simulations Slide 9

10. Quad Strength Sensitivity Measurement Sensitivity: Slide 10

11. LCLS Error: Quad Field Variation Simulation and fit results of Quad Field Variation analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point. 130 m 90 m S. Reiche Simulations Slide 11

12. Horiz. Quad Position Sensitivity Measurement Sensitivity: Slide 12 • Expected: 8.0 µm for en=0.45 µm & Ipk = 3000 A

13. LCLS Error : Transverse Quad Offset Error Simulation and fit results of Transverse Quad Offset Error analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point. 130 m 90 m • Horz. Quad Offset: 4.4 µm = 6.2 µm • Expected: 6.9 µm for en=1.2 µm & Ipk = 3400 A S. Reiche Simulations Slide 13

14. Sensitivity to Individual Quad Motion Range too small for a good Gaussian fit.Offset parameter is too large. Correlation plot for different horizontal and vertical positions of QU12. The sensitivity of FEL intensity to a single quadrupole misalignment comes out to about 34 µm. This is consistent with a value of about 7 µm for a random misalignment of all quadrupoles. Slide 14

15. Analytical Approach* • For LCLS-I, the parameter sensitivities have been obtained through FEL simulations for 8 parameters at the high-energy end of the operational range were the tolerances are tightest. • LCLS-II has a 2 dimensional parameter space (photon energy vs. electron energy) and two independent undulator systems. • Finding the conditions where the tolerance requirements are the tightest requires many more simulation runs. • To avoid this complication, an analytical approach for determining the parameter sensitivities as functions of electron beam and FEL parameters has been attempted. • *H.-D. Nuhn et al., “LCLS-II UNDULATOR TOLERANCE ANALYSIS”, SLAC-PUB-15062

16. Undulator Parameter Sensitivity Calculation Example: Launch Angle • As seen in eloss scans, the dependence of FEL performance on the launch angle can be described by a Gaussian with rmssQ. • Comparing eloss scans at different energies reveals the energy scaling. • This scaling relation agrees to what was theoretically predicted for the critical angle in an FEL: • * • When calculating B using the measured scaling, we get the relation • *T. Tanaka, H. Kitamura, and T. Shintake, Nucl. Instr. Methods Phys. Res., Sect. A 528, 172 (2004). Slide 16

17. Undulator Parameter Sensitivity Calculation Example: Phase Error • In order to arrive at an estimate for the sensitivity to phase errors, we note that the launch error tolerance, discussed in the previous slide, corresponds to a fixed phase delay per power gain length • Path length increase due to sloped path. • Now, we make the assumption that the sensitivity to phase errors over a power gain length is constant, as well. • For LCLS-I we obtained a phase error sensitivity of to phase errors in each break between undulator segments based on GENESIS 1.3 FEL simulations. • In these simulations, the section length corresponded roughly to one power gain length. Therefore we can write the sensitivity as • The same sensitivity should exist to all sources of phase errors. Slide 17

18. Undulator Parameter Sensitivity Calculation Example: Horz. Quadrupole Misalignment • A horizontal misalignment of a quadrupole with focal length by will cause a the beam to be kicked by • The sensitivity to quadrupole displacement can therefore be related to the sensitivity to kick angles as derived above • The square root takes care of the averaging effect of many bipolar random quadrupole kicks (one per section). Slide 18

19. Undulator Parameter Sensitivity Calculation Example: Vertical Misalignment • The undulator K parameter is increased when the electrons travel above or below the mid-plane: • This causes a relative error in the K parameter of • In this case, it is not the parameter itself that causes a Gaussian degradation but a function of that parameter, in this case, the square function. Using the fact that the relative error in the K parameter causes a Gaussian performance degradation we can write • The sensitivity that goes into the tolerance budget analysis is • resulting in a tolerance for the square of the desired value, which can then easily be converted Slide 19

20. Model Detuning Sub-Budget • Some parameters can be introduced in the form of a sub-budget approach as first suggested by J. Welch for the different contributions to undulator parameter, K. The actual K value of a perfectly aligned undulator deviates from its tuned value due to temperature and horizontal slide position errors: • The total error in K can be calculated through error propagation • The combined error is the sensitivity factor used in the main tolerance analysis Slide 20

21. LCLS-II HXR Undulator Line Tolerance Budget sensitivities Slide 21

22. LCLS-II SXR Undulator Line Tolerance Budget sensitivities Slide 22

23. Summary • *H.-D. Nuhn, “LCLS-II Undulator Tolerance Budget”, LCLS-TN-13-5 • A tolerance budget method was developed for LCLS-I undulator parameters using FEL simulations for calculating the sensitivities of FEL performance to these parameters. • Those sensitivities have since been verified with beam based measurements. • For LCLS-II, the method has been extended to using analytical formulas to estimate the sensitivities. LCLS-I measurements have been used to derive or verify these formulas.* • The method, extended by sub-budget calculations is being used in spreadsheet form for LCLS-II undulator error tolerance budget management.

24. End of Presentation