LCLS-II Undulator Tolerances
Heinz-Dieter Nuhn
LCLS-II Undulator Physics Manager
May 8, 2013
LCLS-II Physics Meeting, May 08, 2013
Outline
•
•
•
•
•
Tolerance Budget Method
Experimental Verification of LCLS-I Sensitivities
Analytical Sensitivity Estimates for LCLS-II
Tolerance Budget Example
Summary
LCLS-II Physics Meeting, May 08, 2013
Slide 2
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:
Pi  P0e

ti2
Effect of undulator segment strength error K K
randomly distributed over all segments.
2s i2
Goal: Determine the rms of each performance reduction (Parameter Sensitivity si)
LCLS-II Physics Meeting, May 08, 2013
Slide 3
Tolerance Budget
Combination of individual performance contribution in a budget.
tolerances
sensitivities
P
 e
P0
i

ti2
2s i2
 e
1
 ri2
2
i
Calculate sensitivities s i
Set target value for P P0
Select tolerances ti , calculate resulting P P0 , compare with target.
Iterate: Adjust ti , such that P P0 agrees with target.
Target used in budget analysis P P0  0.75
LCLS-II Physics Meeting, May 08, 2013
Slide 4
Individual Studies (Example: Segment Position x)
• Start with a well aligned undulator line with each segment at position x j
• Choose a set of xm values (error amplitudes) to be tested, for instance
•
•
•
•
•
•
{ 0.0 mm, 0.2 mm, …, 1.8 mm, 2.0 mm}
For each xm choose 32 random values, xs , j , from a flat-top distribution
within the range of ± xm
Move each undulator segment to its corresponding error value, x j  xs , j
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 xm
Loop over all xm
1
xm , i.e. vs.
Plot the mean and average values vs.
3
{ 0.000 mm, 0.115 mm, …, 1.039 mm, 1.155 mm}
• Apply Gaussian fit, Pi  x   P0 e
x2

2s i2
, to obtain rms-dependence
LCLS-II
Physics Meeting,
Maythis
08, 2013
(sensitivity)
for
ith error parameter
Slide 5
Segment x Position Sensitivity Measurement
Pi  x   P0 e
x2

2s i2
Sensitivity:
mean
rms
Generate random misalignment with flat distribution of width ±xm => rms distribution
LCLS-II Physics Meeting, May 08, 2013
1
xm
3
Slide 6
LCLS Error: Horizontal Module Offset
Simulation and fit results of Horizontal
Module Offset analysis. The larger
amplitude data occur at the 130-mpoint, the smaller amplitude data at the
90-m-point.
130 m
Horizontal Model Offset (Gauss Fit)
90 m
LCLS-II Physics Meeting, May 08, 2013
Location
Fit rms
Unit
090 m
0782
µm
130 m
1121
µm
Average
0952
µm
S. Reiche Simulations Slide 7
K/K Sensitivity Measurement
Sensitivity:
Consistent with x sensitivity (sx=0.77 mm),
because with dK/dx ~ 27.5×10-4/mm and K~3.5
one gets
sK/K = sx (1/K) dK/dx ~ 6×10-4=r
LCLS-II Physics Meeting, May 08, 2013
Slide 8
LCLS Error: Module Detuning
Simulation and fit results of Module
Detuning analysis. The larger
amplitude data occur at the 130-mpoint, the smaller amplitude data at the
90-m-point.
130 m
Module Detuning (Gauss Fit)
90 m
Location
Fit rms
Unit
090 m
0.042
%
130 m
0.060
%
Average
0.051
%
Expected: 0.040 for en=1.2 µm & Ipk = 3400 A
LCLS-II Physics Meeting, May 08, 2013
Z. Huang Simulations Slide 9
Quad Strength Sensitivity Measurement
Sensitivity:
LCLS-II Physics Meeting, May 08, 2013
Slide 10
LCLS Error: Quad Field Variation
Simulation and fit results of Quad
Field Variation analysis. The larger
amplitude data occur at the 130-mpoint, the smaller amplitude data at the
90-m-point.
130 m
Quad Field Variation (Gauss Fit)
90 m
LCLS-II Physics Meeting, May 08, 2013
Location
Fit rms
Unit
090 m
8.7
%
130 m
8.8
%
Average
8.7
%
S. Reiche SimulationsSlide 11
Horiz. Quad Position Sensitivity Measurement
Sensitivity:
LCLS-II Physics Meeting, May 08, 2013
Expected: 8.0 µm for en=0.45 µm & Ipk = 3000 A
Slide 12
LCLS Error : 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.
130 m
Transverse Quad Offset Error (Gauss Fit)
90 m
Location
Fit rms
Unit
090 m
4.1
µm
130 m
4.7
µm
Average
4.4
µm
Horz. Quad Offset: 4.4 µm 2 = 6.2 µm
Expected: 6.9 µm for en=1.2 µm & Ipk = 3400 A
LCLS-II Physics Meeting, May 08, 2013
S. Reiche SimulationsSlide 13
Sensitivity to Individual Quad Motion
Range too small for a good Gaussian fit.
Offset parameter is too large.
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.
LCLS-II Physics Meeting, May 08, 2013
Slide 14
Analytical Approach*
• 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
LCLS-II Physics Meeting, May 08, 2013
Slide 15
Undulator Parameter Sensitivity Calculation
Example: Launch Angle
As seen in eloss scans, the dependence of
FEL performance on the launch angle can
be described by a Gaussian with rms sQ.
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:
*
When calculating 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).
LCLS-II Physics Meeting, May 08, 2013
Slide 16
Undulator Parameter Sensitivity Calculation
Example: Phase Error
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.
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.
In these simulations, the section length corresponded roughly to one power gain length.
Therefore we can write the sensitivity as
The same sensitivity should exist to all sources of phase errors.
LCLS-II Physics Meeting, May 08, 2013
Slide 17
Undulator Parameter Sensitivity Calculation
Example: Horz. Quadrupole Misalignment
A horizontal misalignment of a quadrupole with focal length
beam to be kicked by
by
will cause a the
The sensitivity to quadrupole displacement can therefore be related to the sensitivity to
kick angles as derived above
The square root takes care of the averaging effect of many bipolar random quadrupole
kicks (one per section).
LCLS-II Physics Meeting, May 08, 2013
Slide 18
Undulator Parameter Sensitivity Calculation
Example: Vertical Misalignment
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
LCLS-II Physics Meeting, May 08, 2013
Slide 19
Model Detuning Sub-Budget
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:
K  K MMF   K T   K x
The total error in K can be calculated through error propagation
 K 
K 
2
 K

 
 pi 
i  pi

 KMMF    T K    KT    x K     Kx 
2
2
Typical Value
rms dev. pi
3.5
0.0003
K
-0.0019 °C-1
0.0001 °C-1
T
0 °C
0.32 °C
K
0.0023 mm-1
0.00004 mm-1
x
1.5 mm
0.05 mm
Parameter pi
KMMF
2
2
2
2
Note
±0.015 % uniform
Thermal Coefficient
±0.56 °C uniform without compensation
Canting Coefficient
Horizontal Positioning
The combined error is the sensitivity factor used in the main tolerance analysis
LCLS-II Physics Meeting, May 08, 2013
 K / K  0.020%
Slide 20
LCLS-II HXR Undulator Line Tolerance Budget
n
Error Source
sensitivities
rms values
Units
1
2
3
4
5
6
7
8
9
10
11
12
- Launch Angle x’0,y’0
- (K/K)rms
- Segment misalignment in x
- Segment misalignment in y
- Jaw Pitch [µrad]
- Quad Position Stability x,y
- Quad Positioning Error x,y
- Break Length Error
- Strongback deflection
- Phase Shake Error
- Phase Shifter Error
- Cell Phase Error
1.88
0.00060
17527998
30915.8
201.7
4.77
4.77
16.8
79.0
16.6
45.4
45.4
µrad
µm2
µm2
µrad
µm
µm
mm
µm
degXray
degXray
degXray
budget calculations
Corr
ri
Value
Tol
Units
0.71
1.00
1.00
1.00
1.00
0.71
0.71
1.00
1.00
1.00
1.00
1.00
0.360
0.443
0.145
0.262
0.099
0.074
0.297
0.059
0.139
0.181
0.066
0.066
0.48
0.00026
254048
8100
20
0.25
1.00
1.00
11.0
3.0
3.0
3.0
0.48
0.00026
504
90
20
0.25
1.00
1.00
11.0
3.0
3.0
3.0
Total
Total
Loss
P
 e
P
i
LCLS-II Physics Meeting, May 08, 2013

ti2
2s i2
µrad
P/P)i
µm
µm
µrad
µm
µm
mm
µm
degXray
degXray
degXray
93.7%
90.6%
99.0%
96.6%
99.5%
99.7%
95.7%
99.8%
99.0%
98.4%
99.8%
99.8%
P/P:
74.7%
1-P/P:
25.3%
 e
1
 ri2
2
i
Slide 21
LCLS-II SXR Undulator Line Tolerance Budget
n
1
2
3
4
5
6
7
8
9
10
11
12
Error Source
- Launch Angles x’0,y’0
- (K/K)rms
- Segment misalignment in x
- Segment misalignment in y
- Jaw Pitch [µrad]
- Quad Position Stability x,y
- Quad Positioning Error x,y
- Break Length Error
- Strongback deflection
- Phase Shake Error
- Phase Shifter Error
- Cell Phase Error
sensitivities
rms values
Units
4.5
0.00131
1932472
264225
85.4
11.88
11.88
90.4
310.0
16.6
47.0
47.0
µrad
µm2
µm2
µrad
µm
µm
mm
µm
degXray
degXray
degXray
Corr
ri
0.71
1.00
1.00
1.00
1.00
0.71
0.71
1.00
1.00
1.00
1.00
1.00
0.311
0.345
0.118
0.151
0.293
0.238
0.119
0.044
0.142
0.301
0.170
0.170
budget calculations
Value
Tol
Units
1.00
0.00045
228168
40000
25
2.00
1.00
4.0
44.0
5.0
8.0
8.0
P
 e
P
i
LCLS-II Physics Meeting, May 08, 2013
1.00
0.00045
478
200
25
2.00
1.00
4.0
44.0
5.0
8.0
8.0

µrad
µm
µm
µrad
µm
µm
mm
µm
degXray
degXray
degXray
P/P)i
95.3%
94.2%
99.3%
98.9%
95.8%
97.2%
99.3%
99.9%
99.0%
95.6%
98.6%
98.6%
Total P/P:
74.8%
Total
Loss 1-P/P:
25.2%
ti2
2s i2
 e
1
 ri2
2
i
Slide 22
Summary
• 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
LCLS-II Physics Meeting, May 08, 2013
Slide 23
End of Presentation
LCLS-II Physics Meeting, May 08, 2013