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

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

3. 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)

4. 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

5. 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).

6. 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.

7. 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

8. 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
0.560 ±
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.

9. Tolerances Effects are Energy Dependent
Horizontal Launch Angle
±0.249
µrad
Vertical Launch Angle
(DK/K)rms ±
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)

10. 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

11. 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.

12. Main Undulator Tolerance Summary
(DK/K)eff (K Reproducibility) ±
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

13. 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.

14. End of Presentation