1. LCLS-II Physics Meeting, May 08, 2013 LCLS-II Undulator Tolerances Heinz-Dieter Nuhn LCLS-II Undulator Physics Manager May 8, 2013

2. LCLS-II Physics Meeting, May 08, 2013 Outline Slide 2 Tolerance Budget Method Experimental Verification of LCLS-I Sensitivities Analytical Sensitivity Estimates for LCLS-II Tolerance Budget Example Summary

3. LCLS-II Physics Meeting, May 08, 2013 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: Goal: Determine the rms of each performance reduction (Parameter Sensitivity  i ) Slide 3 Effect of undulator segment strength error randomly distributed over all segments.

4. LCLS-II Physics Meeting, May 08, 2013 Tolerance Budget Combination of individual performance contribution in a budget. 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 tolerances sensitivities Slide 4

5. LCLS-II Physics Meeting, May 08, 2013 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 i th error parameter Slide 5

6. LCLS-II Physics Meeting, May 08, 2013 Segment x Position Sensitivity Measurement Generate random misalignment with flat distribution of width ± => rms distribution mean Sensitivity: Slide 6 rms

7. LCLS-II Physics Meeting, May 08, 2013Slide 7 LCLS Error: Horizontal Module Offset Horizontal Model Offset (Gauss Fit) LocationFit rmsUnit 090 m0782µm 130 m1121µm Average0952µm 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. S. Reiche Simulations 90 m 130 m

8. LCLS-II Physics Meeting, May 08, 2013  K/K Sensitivity Measurement Consistent with  x sensitivity (  x =0.77 mm), because with dK/dx ~ 27.5×10 -4 /mm and K~3.5 one gets   K/K =  x (1/K) dK/dx ~ 6×10 -4 =  Slide 8 Sensitivity:

9. LCLS-II Physics Meeting, May 08, 2013Slide 9 LCLS Error: Module Detuning Module Detuning (Gauss Fit) LocationFit rmsUnit 090 m0.042% 130 m0.060% Average0.051% 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. Z. Huang Simulations 90 m 130 m  Expected: 0.040 for  n =1.2 µm & I pk = 3400 A

10. LCLS-II Physics Meeting, May 08, 2013 Quad Strength Sensitivity Measurement Slide 10 Sensitivity:

11. LCLS-II Physics Meeting, May 08, 2013Slide 11 LCLS Error: Quad Field Variation Quad Field Variation (Gauss Fit) LocationFit rmsUnit 090 m8.7% 130 m8.8% Average8.7% 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. S. Reiche Simulations 90 m 130 m

12. LCLS-II Physics Meeting, May 08, 2013 Horiz. Quad Position Sensitivity Measurement Slide 12  Expected: 8.0 µm for  n =0.45 µm & I pk = 3000 A Sensitivity:

13. LCLS-II Physics Meeting, May 08, 2013Slide 13 LCLS Error : Transverse Quad Offset Error Transverse Quad Offset Error (Gauss Fit) LocationFit rmsUnit 090 m4.1µm 130 m4.7µm Average4.4µm 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. S. Reiche Simulations 90 m 130 m  Horz. Quad Offset: 4.4 µm = 6.2 µm  Expected: 6.9 µm for  n =1.2 µm & I pk = 3400 A

14. LCLS-II Physics Meeting, May 08, 2013 Sensitivity to Individual Quad Motion 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. Range too small for a good Gaussian fit. Offset parameter is too large. Slide 14

15. LCLS-II Physics Meeting, May 08, 2013 Analytical Approach* Slide 15 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. LCLS-II Physics Meeting, May 08, 2013 Undulator Parameter Sensitivity Calculation Example: Launch Angle Slide 16 As seen in eloss scans, the dependence of FEL performance on the launch angle can be described by a Gaussian with rms  . 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: *T. Tanaka, H. Kitamura, and T. Shintake, Nucl. Instr. Methods Phys. Res., Sect. A 528, 172 (2004). * When calculating B using the measured scaling, we get the relation

17. LCLS-II Physics Meeting, May 08, 2013 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. Undulator Parameter Sensitivity Calculation Example: Phase Error Slide 17 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. The same sensitivity should exist to all sources of phase errors. In these simulations, the section length corresponded roughly to one power gain length. Therefore we can write the sensitivity as

18. LCLS-II Physics Meeting, May 08, 2013 Undulator Parameter Sensitivity Calculation Example: Horz. Quadrupole Misalignment Slide 18 A horizontal misalignment of a quadrupole with focal length by will cause a the beam to be kicked by The square root takes care of the averaging effect of many bipolar random quadrupole kicks (one per section). The sensitivity to quadrupole displacement can therefore be related to the sensitivity to kick angles as derived above

19. LCLS-II Physics Meeting, May 08, 2013 Undulator Parameter Sensitivity Calculation Example: Vertical Misalignment Slide 19 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

20. LCLS-II Physics Meeting, May 08, 2013 Model Detuning Sub-Budget Parameter p i Typical Value rms dev.  p i Note K MMF 3.50.0003±0.015 % uniform KK -0.0019 °C -1 0.0001 °C -1 Thermal Coefficient TT 0 °C0.32 °C±0.56 °C uniform without compensation KK 0.0023 mm -1 0.00004 mm -1 Canting Coefficient xx 1.5 mm0.05 mmHorizontal Positioning Slide 20 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 combined error is the sensitivity factor used in the main tolerance analysis The total error in K can be calculated through error propagation

21. LCLS-II Physics Meeting, May 08, 2013 LCLS-II HXR Undulator Line Tolerance Budget Slide 21 nError Sourcerms valuesbudget calculations UnitsCorrriri ValueTolUnits  P/P) i 1- Launch Angle x’ 0,y’ 0 1.88µrad0.710.3600.48 µrad93.7% 2 - (  K/K) rms 0.00060 1.000.4430.00026 90.6% 3- Segment misalignment in x17527998µm 2 1.000.145254048504µm99.0% 4- Segment misalignment in y30915.8µm 2 1.000.262810090µm96.6% 5- Jaw Pitch [µrad]201.7µrad1.000.09920 µrad99.5% 6- Quad Position Stability x,y4.77µm0.710.0740.25 µm99.7% 7- Quad Positioning Error x,y4.77µm0.710.2971.00 µm95.7% 8- Break Length Error16.8mm1.000.0591.00 mm99.8% 9- Strongback deflection79.0µm1.000.13911.0 µm99.0% 10- Phase Shake Error16.6degXray1.000.1813.0 degXray98.4% 11- Phase Shifter Error45.4degXray1.000.0663.0 degXray99.8% 12- Cell Phase Error45.4degXray1.000.0663.0 degXray99.8% Total P/P:P/P: 74.7% Total Loss1-  P/P:25.3% sensitivities

22. LCLS-II Physics Meeting, May 08, 2013 LCLS-II SXR Undulator Line Tolerance Budget Slide 22 nError Sourcerms valuesbudget calculations UnitsCorrriri ValueTolUnits  P/P) i 1- Launch Angles x’ 0,y’ 0 4.5µrad0.710.3111.00 µrad95.3% 2 - (  K/K) rms 0.00131 1.000.3450.00045 94.2% 3- Segment misalignment in x1932472µm 2 1.000.118228168478µm99.3% 4- Segment misalignment in y264225µm 2 1.000.15140000200µm98.9% 5- Jaw Pitch [µrad]85.4µrad1.000.29325 µrad95.8% 6- Quad Position Stability x,y11.88µm0.710.2382.00 µm97.2% 7- Quad Positioning Error x,y11.88µm0.710.1191.00 µm99.3% 8- Break Length Error90.4mm1.000.0444.0 mm99.9% 9- Strongback deflection310.0µm1.000.14244.0 µm99.0% 10- Phase Shake Error16.6degXray1.000.3015.0 degXray95.6% 11- Phase Shifter Error47.0degXray1.000.1708.0 degXray98.6% 12- Cell Phase Error47.0degXray1.000.1708.0 degXray98.6% Total P/P:P/P: 74.8% Total Loss1-  P/P:25.2% sensitivities

23. LCLS-II Physics Meeting, May 08, 2013 Summary Slide 23 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. *H.-D. Nuhn, “LCLS-II Undulator Tolerance Budget”, LCLS-TN-13-5

24. LCLS-II Physics Meeting, May 08, 2013 End of Presentation