LCLS-II Physics Meeting, May 08, 2013 LCLS-II Undulator Tolerances Heinz-Dieter Nuhn LCLS-II Undulator Physics Manager May 8, 2013

LCLS-II Physics Meeting, May 08, 2013

LCLS-II Undulator Tolerances

Heinz-Dieter Nuhn

LCLS-II Undulator Physics ManagerMay 8, 2013


Outline

Slide 2

• Tolerance Budget Method• Experimental Verification of LCLS-I Sensitivities• Analytical Sensitivity Estimates for LCLS-II• Tolerance Budget Example• Summary


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:

Goal: Determine the rms of each performance reduction (Parameter Sensitivity si) Slide 3

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Effect of undulator segment strength error randomly distributed over all segments.

KK


Tolerance Budget

Combination of individual performance contribution in a budget.

Calculate sensitivitiesSet target value for Select tolerances , calculate resulting , compare with target.Iterate: Adjust , such that agrees with target.Target used in budget analysis

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tolerancestolerances

sensitivitiessensitivities

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Individual Studies (Example: Segment Position x)

• Start with a well aligned undulator line with each segment at position • Choose a set of values (error amplitudes) to be tested, for instance

{ 0.0 mm, 0.2 mm, …, 1.8 mm, 2.0 mm}• For each choose 32 random values, , from a flat-top distribution

within the range of ± • Move each undulator segment to its corresponding error value, • 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• Loop over all• Plot the mean and average values vs. , i.e. vs.

{ 0.000 mm, 0.115 mm, …, 1.039 mm, 1.155 mm}

• Apply Gaussian fit, , to obtain rms-dependence (sensitivity) for this ith error parameter

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Slide 5


Segment x Position Sensitivity Measurement

Generate random misalignment with flat distribution of width ± => rms distribution mx mx3

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Sensitivity:

Slide 6

rms

LCLS-II Physics Meeting, May 08, 2013 Slide 7

LCLS Error: Horizontal Module Offset

Horizontal Model Offset (Gauss Fit)

Location Fit rms Unit

090 m 0782 µm

130 m 1121 µm

Average 0952 µm

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.

S. Reiche SimulationsS. Reiche Simulations

90 m

130 m


DK/K Sensitivity Measurement

Consistent with Dx sensitivity (sx=0.77 mm), because with dK/dx ~ 27.5×10-4/mm and K~3.5 one gets

sDK/K = sx (1/K) dK/dx ~ 6×10-4=r

Slide 8

Sensitivity:


LCLS Error: Module Detuning

Module Detuning (Gauss Fit)


090 m 0.042 %

130 m 0.060 %

Average 0.051 %

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.

Z. Huang SimulationsZ. Huang Simulations

90 m

130 m

Expected: 0.040 for en=1.2 µm & Ipk = 3400 A


Quad Strength Sensitivity Measurement

Slide 10

Sensitivity:


LCLS Error: Quad Field Variation

Quad Field Variation (Gauss Fit)


090 m 8.7 %

130 m 8.8 %

Average 8.7 %

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.


90 m

130 m


Horiz. Quad Position Sensitivity Measurement

Slide 12 Expected: 8.0 µm for en=0.45 µm & Ipk = 3000 A

Sensitivity:


LCLS Error : Transverse Quad Offset Error

Transverse Quad Offset Error (Gauss Fit)


090 m 4.1 µm

130 m 4.7 µm

Average 4.4 µm

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.


90 m

130 m

Horz. Quad Offset: 4.4 µm = 6.2 µm

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Expected: 6.9 µm for en=1.2 µm & Ipk = 3400 A


Sensitivity to Individual Quad Motion

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.

Range too small for a good Gaussian fit.Offset parameter is too large.

Slide 14


Analytical Approach*

Slide 15

• 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


Undulator Parameter Sensitivity CalculationExample: Launch Angle

Slide 16

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:

*T. Tanaka, H. Kitamura, and T. Shintake, Nucl. Instr. Methods Phys. Res., Sect. A 528, 172 (2004).

*

When calculating B using the measured scaling, we get the relation


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.

Undulator Parameter Sensitivity CalculationExample: Phase Error

Slide 17

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.

The same sensitivity should exist to all sources of phase errors.

In these simulations, the section length corresponded roughly to one power gain length. Therefore we can write the sensitivity as


Undulator Parameter Sensitivity CalculationExample: Horz. Quadrupole Misalignment

Slide 18

A horizontal misalignment of a quadrupole with focal length by will cause a the beam to be kicked by

The square root takes care of the averaging effect of many bipolar random quadrupole kicks (one per section).

The sensitivity to quadrupole displacement can therefore be related to the sensitivity to kick angles as derived above


Undulator Parameter Sensitivity CalculationExample: Vertical Misalignment

Slide 19

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


Model Detuning Sub-Budget

MMF K KK K T x

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Parameter pi Typical Value rms dev. dpi Note

KMMF 3.5 0.0003 ±0.015 % uniform

aK -0.0019 °C-1 0.0001 °C-1 Thermal Coefficient

DT 0 °C 0.32 °C ±0.56 °C uniform without compensation

bK 0.0023 mm-1 0.00004 mm-1 Canting Coefficient

Dx 1.5 mm 0.05 mm Horizontal Positioning

2 2 2 2 2

MMF K K K KK K T T x x

/ 0.020%K K Slide 20

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:

The combined error is the sensitivity factor used in the main tolerance analysis

The total error in K can be calculated through error propagation


LCLS-II HXR Undulator Line Tolerance Budget

Slide 21

n Error Source rms values budget calculations

Units Corr ri Value Tol Units(DP/P)i

1 - Launch Angle x’0,y’0 1.88 µrad 0.71 0.360 0.48 0.48 µrad 93.7%

2 - (DK/K)rms 0.00060 1.00 0.443 0.00026 0.00026 90.6%

3 - Segment misalignment in x 17527998 µm2 1.00 0.145 254048 504 µm 99.0%

4 - Segment misalignment in y 30915.8 µm2 1.00 0.262 8100 90 µm 96.6%

5 - Jaw Pitch [µrad] 201.7 µrad 1.00 0.099 20 20 µrad 99.5%

6 - Quad Position Stability x,y 4.77 µm 0.71 0.074 0.25 0.25 µm 99.7%

7 - Quad Positioning Error x,y 4.77 µm 0.71 0.297 1.00 1.00 µm 95.7%

8 - Break Length Error 16.8 mm 1.00 0.059 1.00 1.00 mm 99.8%

9 - Strongback deflection 79.0 µm 1.00 0.139 11.0 11.0 µm 99.0%

10 - Phase Shake Error 16.6 degXray 1.00 0.181 3.0 3.0 degXray 98.4%

11 - Phase Shifter Error 45.4 degXray 1.00 0.066 3.0 3.0 degXray 99.8%

12 - Cell Phase Error 45.4 degXray 1.00 0.066 3.0 3.0 degXray 99.8%

Total DP/P: 74.7%

Total Loss 1-DP/P: 25.3%

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LCLS-II SXR Undulator Line Tolerance Budget

Slide 22

n Error Source rms values budget calculations

Units Corr ri Value Tol Units (DP/P)i

1 - Launch Angles x’0,y’0 4.5 µrad 0.71 0.311 1.00 1.00 µrad 95.3%

2 - (DK/K)rms 0.00131 1.00 0.345 0.00045 0.00045 94.2%

3 - Segment misalignment in x 1932472 µm2 1.00 0.118 228168 478 µm 99.3%

4 - Segment misalignment in y 264225 µm2 1.00 0.151 40000 200 µm 98.9%

5 - Jaw Pitch [µrad] 85.4 µrad 1.00 0.293 25 25 µrad 95.8%

6 - Quad Position Stability x,y 11.88 µm 0.71 0.238 2.00 2.00 µm 97.2%

7 - Quad Positioning Error x,y 11.88 µm 0.71 0.119 1.00 1.00 µm 99.3%

8 - Break Length Error 90.4 mm 1.00 0.044 4.0 4.0 mm 99.9%

9 - Strongback deflection 310.0 µm 1.00 0.142 44.0 44.0 µm 99.0%

10 - Phase Shake Error 16.6 degXray 1.00 0.301 5.0 5.0 degXray 95.6%

11 - Phase Shifter Error 47.0 degXray 1.00 0.170 8.0 8.0 degXray 98.6%

12 - Cell Phase Error 47.0 degXray 1.00 0.170 8.0 8.0 degXray 98.6%

Total DP/P: 74.8%

Total Loss 1-DP/P: 25.2%

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Summary

Slide 23

• 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


End of Presentation

Documents

LCLS-II Physics Meeting, May 08, 2013 LCLS-II Undulator Tolerances Heinz-Dieter Nuhn LCLS-II Undulator Physics Manager May 8, 2013