1. Undulator Physics, Tolerances and Simulations Heinz-Dieter Nuhn, SLAC / LCLS February 7, 2006
Introduction [Documents and Parameters]
Components Motion Monitoring and Control
Error Tolerance Budget
Wakefield Budget 1

2. Physics Requirement Documents
Doc Type
Number
Title
GRD
Global Requirement Document
PRD
LCLS Start-Up Test Plan
Conventional Alignment System Requirements
General Undulator System Requirements
Magnetic Measurement Facility Requirements
Undulator Beam Based Alignment System Requirements
Undulator Beam Finder Wire 2

3. Engineering Specification Documents
Doc Type
Number
Title
ESD
Quadrupole Magnet Specifications
Diagnostics System Specifications
Wire Position Monitor System Specifications
Hydrostatic Leveling System Specifications
Vacuum System Specifications
Controls Specifications
Undulator Protection System
Technical Specifications for the Undulator OTR Beam Profiler (Long Break)
Wire Scanner Specifications for the Undulator System
Undulator Engineering Requirements
LCLS Undulator Support/Mover System Engineering Specifications
Undulator Tunnel Survey Monument Positions 3

4. Summary of Nominal Undulator Parameters
Undulator Type planar hybrid
Magnet Material NdFeB
Wiggle Plane horizontal
Gap 6.8 mm
Period Length 30.0±0.05 mm
Effective On-Axis Field T
Range of Effective Undulator Parameter K (0.6 %) [ 0]
K Tuning Tolerance ± 0.015%
Accumulated Segment Phase Error Tol. ± 10 degrees (at 1.5 Angstroms)
Segment Length 3.40 m
Number of Segments 33
Undulator Magnetic Length m
Standard Break Lengths cm
Nominal Total Device Length m
Quadrupole Magnet Technology EMQ
Nominal Quadrupole Magnet Length 7.4 cm
Integrated Quadrupole Gradient 3.0 (4.0 max) T 4

5. Strategies for Controlling Component Motion
Girder motion will be caused by
Ground Motion
Temperature Changes
CAM Rotation
Girder motion will be monitored in 2 ways.
Directly, through the Component Monitoring Systems
Indirectly, through impact on electron beam trajectory (as detected by BPMs)
Girder Positions will be frequently corrected using the CAM movers.
Both monitoring venues complement each other but each will be sufficient for maintaining trajectory straightness. 5

6. Component Position Monitoring Systems (Alignment Diagnostics System – ADS)
Wire Position Monitor system (WPM)
Resolution < 100 nm in X & Y direction
Instrument Drift < 100 nm per day
Moving Range ±1.5 mm in X & Y direction
Accuracy 0.1 % of full Scale
Availability Permanent, no interrupts
X and Y, can be measured Roll, Jaw & Pitch can be calculated.
See presentation by Franz Peters
Hydrostatic Leveling System (HLS)
Capacitive Sensor
Precision < 1 mm
Instrument Drift ~1-2 mm / month
Accuracy < 0.1 % of full Scale
Roll Y
Ultrasound Sensor
Precision < 0.1 mm
Instrument Drift potentially no drift
Accuracy < 0.1 % of full Scale
Pitch
See presentation by Franz Peters 6

7. *Worst Case - no reliance on position monitoring system
Correction Zones*
Zone 1 (non-invasive correction)
120-Hz traj-feedback (LTU BPM’s)
0.1-Hz traj-feedback (und. BPM’s)
Zone 2 (Dt > 1 hr, P/P0 > 90%, non-invasive)
MICADO (best 5) steering within undulator
Zone 3 (Dt > 24 hr, P/P0 > 75%, invasive)
Weighted steering of undulator traj. (1 min.)
... or quadrupole gradient scans - fast BBA (10 min.)
Possible x-ray pointing (few min.)
Zone 4 (Dt > 1 wk, P/P0 > 50%, machine time)
One iteration of BBA (<1 hr)
Zone 5 (Dt > 6 mo, shut-down)
Reset movers set to zero and manual realignment (1 wk)
Full 3 iterations of BBA (~3 hrs)
*Worst Case - no reliance on
position monitoring system 7

8. Correction Zones * (Illustration)
BBA- Light
*Worst Case - no reliance on
position monitoring system 8

9. LCLS Undulator Tolerance Budget Analysis
Analysis based on time dependent SASE simulations with Genesis 1.3
Eight individual error sources considered:
Beta-Function Mismatch,
Launch Position Error,
Module Detuning,
Module Offset in x,
Module Offset in y,
Quadrupole Gradient Error,
Transverse Quadrupole Offset,
Break Length Error.
The ‘observed’ parameter is the average of the FEL power at 90 m (around saturation) and 130 m (undulator exit)
The Results are combined into the Error Budget 9

10. Example – Error 3: 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
Module Detuning (Gauss Fit)
Location
Fit rms
Unit
090 m
0.042 %
130 m
0.060
Average
0.051
90 m
Budget Tolerance
Z. Huang Simulations 10

11. Example – Error 5: Vertical Module Offset
Simulation and fit results of Vertical Module Offset analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point.
130 m
Vertical Model Offset (Gauss Fit)
Location
Fit rms
Unit
090 m
268 µm
130 m
Average
90 m
Budget Tolerance
S. Reiche Simulations 11

12. Tolerance Budget
Gaussian fit yields functional dependence of power reduction on error amplitude:
Assuming that each error is independent on the others other, i.e. each error source causes a given fraction power reduction independent of the presence of the other sources:
tolerance
fitted rms
fi=xi/si 12

13. Tolerance Budget (cont’)
Error Source si fi
< si> fi
Units
@ 130 m
(24.2% red.)
Hor/Ver Optics Mismatch (z-1)0.5
0.71
0.452
0.32
Hor/Ver Transverse Beam Offset 30
0.176
3.7 µm
Module Detuning DK/K
0.060
0.400
0.024 %
Module Offset in x
1121
0.125
140
Module Offset in y
268
0.298 80
Quadrupole Gradient Error
8.8
0.029
0.25
Transverse Quadrupole Offset
4.7
0.214
1.0
Break Length Error
20.3
0.049 mm
z < <b/b0<1.56 13

14. Model Detuning Sub-Budget
Parameter pi
Typical Value
rms dev. dpi
Note
KMMF
3.5
0.0003
±0.015 % uniform aK
°C-1
°C-1
Thermal Coefficient DT
0 °C
0.32 °C
±0.56 °C uniform without compensation bK
mm-1
mm-1
Canting Coefficient Dx
1.5 mm
0.05 mm
Horizontal Positioning
< %
=> Sub-Budget is well within allowance of overall budget 14

15. Simulation of Temperature Error Effect On Gain
Z (meter)
Power (W)
± 0.5 ºC (K/K = ±5×10-4) at undulator-end causes only 5-6% power drop if we have the taper set optimally initially.
Power degradation for a linear taper error is about the same as for a random modular-K error of the same magnitude
Courtesy of Zhirong Huang 15

16. Wakefield Budget Undulator Wakefield Sources:
Resistive Wall Wakefields (ac conductivity) => Main Contributor
Mitigation: Aluminum Surface, Rectangular Cross Section
Surface Roughness Wakefields
Mitigation: Limit roughness aspect ration to larger than 300.
Total contribution small compared to resistive wall wakefields
Geometric Wakefields
Sources:
Rectangular to Round Transition
BFW Replacement Chamber Mis-Alignment
RF Cavity BPMs
Bellows Shielding Slots
Flanges
Pump Slots 16

17. Short Break Section Chamber Profile
Bellows Shielding Slots 35 mm 10 mm dia 20% azimuth
BFW Replacement Chamber
Flange (5) Gaps .5 mm
RF Cavity Length 10 mm
Pump Slot
Chamber Diameter 8 mm
Chamber Diameter 10 mm
Undulator Chamber 5x10 mm
Undulator Chamber 5x10 mm 17

18. Long Break Section Chamber Profile
BFW Replacement Chamber
Flange (7) Gaps 2 mm
Bellows (2) Shielding Slots 35 mm 10 mm dia 20% azimuth
RF Cavity Length 10 mm
Chamber Diameter 8 mm
Chamber Diameter 10 mm
Undulator Chamber 5x10 mm
Undulator Chamber 5x10 mm
Pump Slot 18

19. Beam Finder Wire (BFW)
Wire Replacement
Wire
BFW
A misaligned undulator will not steer the beam. It will just radiate at the wrong wavelength. The BFW allows the misalignment to be detected. (allows beam size measurements)
Beam Direction 19

20. Undulator Wakefield Parameters
Transition Model Parameters
Segment to Break Chamber
5 mm x 10 mm  8 mm dia; 153 V/pC
BFW Replacement Chamber
0.5 mm displacement; 10 mm dia; 69.8 V/pC
Diffraction Model Parameters
RF Cavity BPMs
35 mm 10 mm dia., 20 % azim.
Flanges
2 mm 10 mm dia.
Pump Slots
10 mm 10 mm dia., 68 % azim.
Surface Roughness Parameters
Segment Chambers
5 mm dia.; Aspect Ratio 300;
Short Break Chambers
10 mm dia.; Aspect Ratio 100;
Long Break Chambers
Resistive Wall Parameters
par plates 5 mm sep, Al AC;
Shielded Bellows
10 mm dia. cir.; Cu AC; 20

21. Total Longitudinal Wake Field Summary
Beam Energy = GeV
Undulator Length = 132 m
Bunch Core (0.45 nC):
<Wc> = keV/m ( %) Wc,rms = keV/m ( 0.12 %)
Total Bunch (1 nC):
<Wt> = keV/m (-0.27 %) Wt,rms = keV/m ( 0.22 %)
r  52.0 keV/m (0.50 %) 21

22. Total Longitudinal Wake Budget Summary
core
Wakefield Component
<d> sd
[%]
Transition Model
-0.062
0.039
-0.033
0.003
Diffraction Model
-0.032
0.032
-0.028
Surface Roughness
-0.029
0.029
0.008
0.012
Resistive Wall
-0.150
0.150
-0.023
0.120
Total
-0.274
0.222
-0.075
0.116
Beam Energy = GeV
Undulator Length = 132 m
Total Charge = 1 nC
Core Charge = 0.45 nC
Transverse Wakefields Negligible ! De/e < 0.1 % for 100 µm beam offset 22

23. GENESIS Simulation of Wake Contributions
Average Power vs z
Start-to-End Simulation
All Wakefields Included
Total Charge: 1 nC
Beam Energy: GeV
Wavelength: 1.5 Å
head
tail
Power Profile at 100 m 23

24. Summary
Two complementary ways of mitigating the effect of component motion have been identified incorporating a Correction Zone approach.
Undulator tolerance parameters have been combined in a tolerance budget. The analysis is based on GENESIS simulations. Tolerances were balanced to allow relaxation of critical tolerances:
Temperature Stability is now 0.5oC (was 0.2oC)
Vertical Segment Alignment is now 80 µm (was 70 µm) rms
Short Term (1hr ) Quadrupole/BPM Stability 2 µm (was 1 µm in 10 hrs)
Long Term (24hrs ) Quadrupole/BPM Stability 5 µm
A comprehensive undulator wakefield budget keeps track of the various wakefield sources during the component design phase. GENESIS simulations confirm that the expected wakefield amplitudes are consistent with FEL performance expectations. 24

25. End of Presentation 25