Undulator Tolerances for LCLS-II using SCUs Heinz-Dieter Nuhn (SLAC) Superconducting Undulator R&D Review Jan. 31, 2014

Outline Tolerance Budget Method Tolerance Budget Energy Dependence of Performance Predictions Beam Heating Estimates Summary

Undulator Errors Affect FEL Performance FEL power dependence modeled by Gaussian. Sensitivities originally determined with GENESIS simulations developed with Sven Reiche. Several sensitivities have been verified experimentally with LCLS-I beam. Goal: Determine rms of each performance reduction (Parameter Sensitivity si) Effect of undulator segment strength error randomly distributed over all segments. FEL Power (Pi)

Analytical Approach* For LCLS-I, parameter sensitivities were obtained by FEL simulations at max. energy, where tolerances are tightest. LCLS-II has a 2-dimensional parameter space (photon energy vs. electron energy). Finding the conditions where tolerance requirements are tightest requires many simulation runs. To avoid this, an analytical approach to determine sensitivities, as functions of e-beam and FEL parameters, has been developed. *H.-D. Nuhn et al., “LCLS-II UNDULATOR TOLERANCE ANALYSIS”, SLAC-PUB-15062

Undulator Parameter Sensitivity Calculation Example: Launch Angle As seen in E-loss scan, dependence of FEL performance on launch angle can be described as Gaussian with rms sQ. Comparing E-loss 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 coefficient 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).

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

Undulator Parameter Sensitivity Calculation Example: Undulator Vertical Misalignment The undulator K parameter is increased when electrons travel above or below mid-plane: Note the dependence on the inverse square of the undulator period. This causes a relative K error of Here, it is not the parameter itself that will be modeled by a Gaussian, but a function of that parameter. Using the fact that the relative K error causes a Gaussian performance degradation we 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

LCLS-II HXR Tolerance Budget (SCU/Cu Linac) Ee = 15 GeV Ep = 25 keV lu = 2.0 cm, gmag = 7.5 mm, for Nb3Sn DK/K rms tolerance n error source sensitivities budget calculations values units ri ti rms Tol range Units (P/P0)i 1 Horizontal Launch Angle 1.24 µrad 0.116 0.144 ±0.249 99.3% 2 Vertical Launch Angle 3 (DK/K)rms 0.00031   0.560 0.00017 ±0.00030 85.5% 4 Segment misalignment in x 154772 µm2 0.070 10800 104 ±180 µm 99.8% 5 Segment misalignment in y 6273 0.191 1200 35 ±60 98.2% 6 Horz. Quad Position Stability 4.57 0.126 0.577 ±1.0 99.2% 7 Vert. Quad Position Stability 8 Horz. Quad Positioning Error 0.379 1.73 ±3.0 93.1% 9 Vert. Quad Positioning Error 9.46 10 - Break Length Error mm 0.061 11 - Phase Shake Error 16.6 degXray 0.174 2.89 ±5.0 98.5% 12 - Cell Phase Error 0.145 5.77 ±10.0 99.0% Total P/P0: 67.8% Total Loss 1-DP/P0: 32.2% These tolerances are challenging, but quite similar to the successful LCLS-I tolerances.

Tolerances Effects are Energy Dependent Horizontal Launch Angle ±0.249 µrad Vertical Launch Angle (DK/K)rms ±0.00030   Segment misalignment in x ±180 µm Segment misalignment in y ±60 Horz. Quad Position Stability ±1.0 Vert. Quad Position Stability Horz. Quad Positioning Error ±3.0 Vert. Quad Positioning Error Break Length Error mm Phase Shake Error ±5.0 degXray Cell Phase Error ±10.0 Ee = 15 GeV Ep = 25 keV P/P0=67%  FEL Power Reduction lu = 2.0 cm gmag = 7.5 mm Nb3Sn Proposed Operational Range has Excellent Performance Photon Energy (keV) Electron Energy (GeV)

Performance Sensitivity to Main Tolerances Same as on previous slide: lu = 2.00 cm Dy= ±60 mm Dfrms= ±5 deg DK/K= ±3.0×10-4 Significant violation of tolerances does not cause catastrophic failure. lu = 2.00 cm Dy= ±120 mm lu = 2.00 cm Dfrms= ±23 deg lu = 2.00 cm DK/K= ±6.5×10-4

Chamber Heating There are two main beam related sources that can heat the LCLS-II vacuum chamber: (1) Resistive Wall Wakefields, (2) Spontaneous Radiation. Beam Parameters: Electron Energy: 4 GeV Bunch Charge: 300 pC Bunch Repetition Rate: 100 KHz => Average Electron Beam Power: 120 kW (1) Total Spontaneous Radiation Produced (ignoring microbunching) SC-HXU Undulator gap: 7.5 mm SC-HXU Undulator Period: 1.85 cm SC-HXU K: 3.31 <dP/dz> = 1.1 W/m. (2) Resistive Wall Wakefields Beam Pipe Radius: 2.5 mm Beam Pipe Profile: parallel plates Ipk = 1000 A Chamber Material: Al Conductivity: 37.7×106W-1m-1 <dP/dz> = 0.26 W/m Only a fraction of this power will contribute to vacuum chamber heating.

Main Undulator Tolerance Summary (DK/K)eff (K Reproducibility) ±0.00030   Segment misalignment in x ±180 µm Segment misalignment in y ±60 Phase Shake Error ±5.0 degXray Cell Phase Error ±10.0 Horizontal and Vertical First Field Integral ±40.0 µTm Horizontal and Vertical Second Field Integral ±50.0 µTm2

Summary A tolerance budget method was developed for the LCLS-I undulator (PMU) Those sensitivities have since been verified with beam based measurements The method is being used for LCLS-II SCU undulator error tolerance budget The SCU tolerances are challenging, but similar to LCLS-I Radiation based vacuum chamber heating appears modest.

End of Presentation