1. Undulator Physics Update Heinz-Dieter Nuhn, SLAC / LCLS October 27, 2005
Response to Recommendations
Tolerance Budget based on Genesis Simulations
Electron Beam Parameter ‘Tolerances’
Wakefield Budget 1

2. Response to FAC Recommendations
FAC April 2005 Recommendation:
The radiation produced by scattering from OTR foils in the undulator is a concern. The Committee recommends that a plan be developed to minimize risk of damage to undulators from OTR screen use.
Response:
With regards to the undulator, Radiation Physics simulations have shown that OTR foils are not likely to cause a problem if designed and used properly. A foil of 10 microns thickness or less used for a few shots at a time will not cause a problem. The use of the foil will be interlocked to the MPS system. Also, bunches will not be allowed to enter the undulator area while the OTR foil is performing an insert or remove motion (indeterminate position).
Presently, the plan for the undulator OTR foils is being reduced down to an R&D project. We are removing the funds for actually building and installing OTR foils in the undulator area from the base line. We will still have the ability to measure the x and y beam sizes at every undulator break by using the secondary function of the Beam Finder Wire (BFW). 2

3. Response to FAC Recommendations
FAC April 2005 Recommendation:
The procedure to align the undulator appears to be feasible and offers additional redundancy; however, the justification for an upstream beam monitor was not made clear.
Response:
The need for the upstream beam monitor, i.e. the Beam Finder Wire (BFW), comes from the tight tolerances for positioning the electron beam on the undulator axis as defined during the tuning procedure. While this alignment can be achieved using a portable wire position monitor system, using such a system requires extended tunnel access during the commissioning process after a straight electron beam trajectory has been established with the beam-based alignment procedure. The BFW will provide a beam-based measurement, and allow this alignment task to be accomplished from the control room without the need for tunnel access. The portable wire position monitor system will serve as a backup. 3

4. Response to FAC Recommendations
FAC April 2005 Recommendation:
Concern remains about the ground settlement and stability of the undulator hall floor. The Committee recommends that LCLS project physicists quantify the allowable ground motion given the range of instrumentation available, and provide specifications on ground motion based on realistic day-to-day alignment and periodic beam-based alignment. The physics analysis should include study of the extent to which the systems can accommodate movements beyond the survey tolerances.
Response:
We have studied more carefully the tolerances for alignment variations over both short and long term time-scales, and have devised an escalating series of beam-based correction levels, each with an associated time-scale and tolerable FEL power loss, as was suggested by the FAC in April The ‘bulls-eye’ diagram proposed by the FAC has been tagged “Kem’s Zones” and has been described in some detail in Paul Emma’s presentation. Briefly, the correction levels extend from shot-to-shot trajectory feedback systems, to hourly ‘micado’ steering algorithms, to daily weighted steering or ‘BBA-light’, to weekly BBA, and finally to semi-annual conventional alignment. The outcome of these studies has also served to define the tolerable trajectory drift errors over short term (BBA execution duration: 1 hr) and longer term (diurnal variations: 1 day). These tolerances are incorporated into the undulator Physics Requirements Document (PRD) and serve as a guideline for the design of supports, temperature regulation, and BPM systems. 4

5. Response to FAC Recommendations
FAC April 2005 Recommendation:
The very tight temperature tolerances in the undulator tunnel (+/- 0.2 C) have severe implications on controls. There are plans to put electronics in the ceiling air return duct where it will be difficult to maintain and concerns that the stepping motors will give off more heat than allowed. The air conditioning system necessary to maintain that temperature stability is also very expensive. The accelerator physicists should have a hard look to see if there is a way to increase this tolerance.
Response:
The temperature stability tolerances for the undulator tunnel have been re-examined both with respect to their influences on the undulator magnetic field as well as to the positional stability of the quadrupoles and BPMs. GENESIS simulations of the effects of errors of the average K values for each undulator segment, both random and systematic, show that temperature errors from a uniform distribution with a width of ±1 degree F (±0.56 degrees C) are consistent with a total overall error budget for a 25% reduction in FEL power (but not taking credit for simple undulator x-position adjustments to compensate temperature variations). In parallel, a thermal expansion study was carried out at the APS with the result that for temperature changes of ±0.5 degree C the critical components will stay with in the position tolerances (±5 microns over 24 hours). Based on these analyses, which will be presented during the next FAC meeting, the temperature tolerances for the undulator tunnel have been relaxed. The requirement specification says now: “The absolute temperature along the Undulator will stay within a range of 20±0.6 °C at all times.” 5

6. LCLS Undulator Tolerance Budget Analysis
Based On Time Dependent SASE Simulations in 2 Phases
Simulation Code: Genesis 1.3
Simulate Individual Error Sources
Combine Results into Error Budget 6

7. Parameters for Tolerance Study
The following 8 errors are 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) 7

8. Step I - Individual Study
Time-dependent runs with increasing error source (uniform distribution) and different error seeds. Gauss fit to obtain rms-dependence.
Detailed Analysis Description 8

9. Step I – Error 1b: Optics Mismatch
Simulation and fit results of Optics Mismatch analysis. The larger amplitude data occur at the 114-m-point, the smaller amplitude data at the 80-m-point.
Transformation from negative exponential to Gaussian:
Optics Mismatch (Gauss Fit)
Location
Fit rms
Unit
080 m
0.58
114 m
0.71
Average
0.64
z < 1.41
Y. Ding Simulations 9

10. Comparison of z vs. b/b0 Simplifies at waist location: + -
or, resolved for b
1-s value 10

11. Step I – Error 2: Transverse Beam Offset
Simulation and fit results of Transverse Beam Offset (Launch Error) analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point.
Transverse Beam Offset (Gauss Fit) /
Location
Fit rms
Unit
090 m
25.1 µm
130 m
21.1
Average
23.1
S. Reiche Simulations 11

12. Step I – 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.
Module Detuning (Gauss Fit)
Location
Fit rms
Unit
090 m
0.042 %
130 m
0.060
Average
0.051
Z. Huang Simulations 12

13. Step I – Error 4: 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.
Horizontal Model Offset (Gauss Fit)
Location
Fit rms
Unit
090 m
0782 µm
130 m
1121
Average
0952
S. Reiche Simulations 13

14. Step I – 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.
Vertical Model Offset (Gauss Fit)
Location
Fit rms
Unit
090 m
268 µm
130 m
Average
S. Reiche Simulations 14

15. Step I – Error 6: 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.
Quad Field Variation (Gauss Fit)
Location
Fit rms
Unit
090 m
8.7 %
130 m
8.8
Average
S. Reiche Simulations 15

16. Step I – Error 7: 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.
Transverse Quad Offset Error (Gauss Fit)
Location
Fit rms
Unit
090 m
4.1 µm
130 m
4.7
Average
4.4
S. Reiche Simulations 16

17. Step I – Error 8: Break Length Error
Simulation and fit results of Break Length Error analysis. The larger amplitude data occur at the 130-m-point, the smaller amplitude data at the 90-m-point.
Break Length Error (Gauss Fit)
Location
Fit rms
Unit
090 m
13.9 mm
130 m
20.3
Average
17.1
S. Reiche Simulations 17

18. Step II - Tolerance Budget
Assuming that each error is independent on each other (validity of this assumption is limited)
Each should yield the same degradation
Tolerance is defined for a given power degradation
tolerance
fitted rms
fi=xi/si
unit weights
n = 8
1 - P/P0 f
20 %
0.236
25 %
0.268 18

19. Step III - Correlated Error Sources
For the simplest approach, the tolerance budget assumes uncorrelated errors of 8 different sources.
Some tolerances (e.g. the break length error) are very relaxed and can be reduced to relax other tolerances, i.e. use individual tolerances.
Next step is to combine all error sources in the simulation.
Include BBA and other correction scheme in the runs 19

20. Step II - Tolerance Budget (cont’)
Error Source
< si>
< si> f fi
< si> fi
Units
f=0.268 (25% red.)
(24.2% red.)
Hor/Ver Optics Mismatch (z-1)0.5
0.64
0.19
0.453
0.32
Hor/Ver Transverse Beam Offset 23
5.7
0.177
3.7 µm
Module Detuning DK/K
0.051
0.016
0.402
0.024 %
Module Offset in x
952
301
0.125
140
Module Offset in y
268 72
0.298 80
Quadrupole Gradient Error
8.7
2.3
0.028
0.25
Transverse Quadrupole Offset
4.4
1.3
0.215
1.0
Break Length Error
17.1
5.4
0.048 mm
z < 1.1
Can be mitigated through steering. 20

21. 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 21

22. e- beam Tolerances Parameter Fits Parameter s Param (rms ) Unit en
1.05 µm
Ipk
1.66 kA
Dp/p
0.025 % 22

23. e--Beam Quality ‘Tolerance Budget’
Beam Parameter
< si> fi
< si> fi
Units
(18.8% red.) en
1.03
0.385
0.4 µm
Ipk 23
0.329
0.5 kA
dpp
0.051
0.400
0.01 % 23

24. 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 24

25. Short Break Section Chamber Profile
BFW Replacement Chamber
Flange Gaps .5 mm
RF Cavity Length 10 mm
Bellows Shielding Slots Gaps 20 mm / 10%
Pump Slot
Chamber Diameter 8 mm
Chamber Diameter 10 mm
Undulator Chamber 5x10 mm
Undulator Chamber 5x10 mm 25

26. Long Break Section Chamber Profile
BFW Replacement Chamber
Flange Gaps .5 mm
Bellows Shielding Slots Gaps 20 mm / 10%
RF Cavity Length 10 mm
Chamber Diameter 8 mm
Chamber Diameter 10 mm
Undulator Chamber 5x10 mm
Undulator Chamber 5x10 mm
Pump Slot 26
Courtesy of Dean Walters

27. Geometric Wakefield Budget Summary
Beam Energy = GeV
Undulator Length = 132 m
Charge = 1 nC
total
core
Component
Characterization
Count
<d> sd
[%]
Transitions
5mm x 10mm <=> 8 mm dia 33
-0.043
0.027
-0.022
0.002
BFW Replacement
0.5 8 mm dia
-0.036
0.022
-0.018
Total Transition
-0.080
0.049
-0.041
0.004
Shielded Bellows
20 mm 10 mm dia 48
-0.004
RF Cavity BPM
10 mm 8 mm dia.
-0.009
0.003
0.000
Flanges
0.5 mm 8 mm dia
148
-0.007
0.001
Pump Slots
10 mm dia
-0.003
Total Diffraction
-0.026
0.010 27

28. Transition Model Wake Field Summary
Total Bunch:
<Wt> = keV/m Wt,rms = keV/m
Bunch Core:
<Wc> = keV/m Wc,rms = keV/m 28

29. Diffraction Model Wake Field Summary
Total Bunch:
<Wt> = keV/m Wt,rms = keV/m
Bunch Core:
<Wc> = keV/m Wc,rms = keV/m 29

30. Surface Roughness Wake Field Summary
Aspect Ratio 300
Total Bunch:
<Wt> = keV/m Wt,rms = keV/m
Bunch Core:
<Wc> = keV/m Wc,rms = keV/m 30

31. Resistive Wall Wake Field Summary
AC Conductivity
Al, parallel plates
Total Bunch:
<Wt> = keV/m Wt,rms = keV/m
Bunch Core:
<Wc> = keV/m Wc,rms = keV/m 31

32. Total Wake Field Summary
Total Bunch:
<Wt> = keV/m Wt,rms = keV/m
Bunch Core:
<Wc> = keV/m Wc,rms = keV/m 32

33. Total Wake Budget Summary
Beam Energy = GeV
Undulator Length = 132 m
Charge = 1 nC
total
core
Wakefield Component
Parameters
<d> sd
[%]
Transition Model
-0.080
0.049
-0.041
0.004
Diffraction Model
-0.026
0.010
-0.022
0.002
Surface Roughness
-0.013
0.026
Resistive Wall
0.085
-0.018
0.060
Total
-0.198
0.123
-0.077
0.057 33

34. Summary
An undulator tolerance budget analysis based on GENESIS simulations was presented.
Several critical tolerances have been relaxed:
Temperature Stability is now 0.56oC (was 0.1oC)
Vertical Segment Alignment is now 80 µm (was 70 µm) rms
Short Term (1hr ) Quadrupole Stability 2 µm (was 1 µm in 10 hrs)
Long Term (24hrs ) Quadrupole Stability 5 µm
An undulator wakefield budget analysis is used to keep track of the various wakefield sources during the component design phase. 34

35. End of Presentation 35