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LCLS-II Undulator Tolerances

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LCLS-II Undulator Tolerances

Heinz-Dieter Nuhn

LCLS-II Undulator Physics Manager

May 8, 2013

- 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:

Effect of undulator segment strength error

randomly distributed over all segments.

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

Slide 3

Tolerance Budget

Combination of individual performance contribution in a budget.

tolerances

sensitivities

Calculate sensitivities

Set target value for

Select tolerances , calculate resulting , compare with target.

Iterate: Adjust , such that agrees with target.

Target used in budget analysis

Slide 4

- 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

Slide 5

Sensitivity:

mean

rms

Generate random misalignment with flat distribution of width ± => rmsdistribution

Slide 6

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.

130 m

90 m

S. Reiche Simulations

Slide 7

Sensitivity:

- 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

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

90 m

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

Z. Huang Simulations

Slide 9

Sensitivity:

Slide 10

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.

130 m

90 m

S. Reiche Simulations

Slide 11

Sensitivity:

Slide 12

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

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

90 m

- Horz. Quad Offset: 4.4 µm = 6.2 µm

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

S. Reiche Simulations

Slide 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.

Slide 14

- 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 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 rmssQ.
- 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).

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.

Slide 17

Undulator Parameter Sensitivity Calculation

Example: Horz. Quadrupole Misalignment

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

- 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).

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

Slide 19

- 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 total error in K can be calculated through error propagation

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

Slide 20

sensitivities

Slide 21

sensitivities

Slide 22

- *H.-D. Nuhn, “LCLS-II Undulator Tolerance Budget”, LCLS-TN-13-5

- 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.

End of Presentation