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

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CD3A review

LCLS-II Undulator TolerancesHeinz-Dieter NuhnLCLS-II Undulator Physics ManagerMay 8, 2013LCLS-II Physics Meeting, May 08, 2013Summary of project scopekey milestonesfunding profilecommissioning by LCLS operationspath to CD-2priority of longer tunnelurgency of getting CF bidsmention FAC

1OutlineSlide 2Tolerance Budget MethodExperimental Verification of LCLS-I SensitivitiesAnalytical Sensitivity Estimates for LCLS-IITolerance Budget ExampleSummary

LCLS-II Physics Meeting, May 08, 2013

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

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

LCLS-II Physics Meeting, May 08, 2013Tolerance BudgetCombination 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

tolerancessensitivities

Slide 4

LCLS-II Physics Meeting, May 08, 2013Individual 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 averageLoop over several random seeds and obtain mean and rms values of the x-ray intensity readings for the distribution for this error amplitudeLoop over allPlot 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 5LCLS-II Physics Meeting, May 08, 20135Segment x Position Sensitivity Measurement

Generate random misalignment with flat distribution of width => rms distribution

meanSensitivity:Slide 6rmsLCLS-II Physics Meeting, May 08, 2013Slide 7LCLS Error: Horizontal Module Offset

Horizontal Model Offset (Gauss Fit)LocationFit rmsUnit090 m0782m 130 m1121mAverage0952mSimulation 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 Simulations 90 m130 mLCLS-II Physics Meeting, May 08, 20137DK/K Sensitivity Measurement

Consistent with Dx sensitivity (sx=0.77 mm), because with dK/dx ~ 27.510-4/mm and K~3.5 one gets

sDK/K = sx (1/K) dK/dx ~ 610-4=r

Slide 8Sensitivity:

LCLS-II Physics Meeting, May 08, 2013Slide 9

LCLS Error: Module DetuningModule Detuning (Gauss Fit)LocationFit rmsUnit090 m0.042%130 m0.060%Average0.051%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 Simulations 90 m130 mExpected: 0.040 for en=1.2 m & Ipk = 3400 ALCLS-II Physics Meeting, May 08, 20139Quad Strength Sensitivity Measurement

Slide 10Sensitivity:LCLS-II Physics Meeting, May 08, 2013Slide 11LCLS Error: Quad Field Variation

Quad Field Variation (Gauss Fit)LocationFit rmsUnit090 m8.7% 130 m8.8%Average8.7%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. S. Reiche Simulations 90 m130 mLCLS-II Physics Meeting, May 08, 201311Horiz. Quad Position Sensitivity Measurement

Slide 12Expected: 8.0 m for en=0.45 m & Ipk = 3000 ASensitivity:LCLS-II Physics Meeting, May 08, 2013Slide 13LCLS Error : Transverse Quad Offset ErrorTransverse Quad Offset Error (Gauss Fit)LocationFit rmsUnit090 m4.1m 130 m4.7mAverage4.4mSimulation 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.

S. Reiche Simulations 90 m130 mHorz. Quad Offset: 4.4 m = 6.2 m

Expected: 6.9 m for en=1.2 m & Ipk = 3400 ALCLS-II Physics Meeting, May 08, 201313Sensitivity 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 m. This is consistent with a value of about 7 m for a random misalignment of all quadrupoles.Range too small for a good Gaussian fit.Offset parameter is too large.Slide 14LCLS-II Physics Meeting, May 08, 2013Analytical Approach*Slide 15For 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, 2013Undulator Parameter Sensitivity CalculationExample: Launch AngleSlide 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

LCLS-II Physics Meeting, May 08, 2013For 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 ErrorSlide 17In 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

LCLS-II Physics Meeting, May 08, 2013Undulator Parameter Sensitivity CalculationExample: Horz. Quadrupole MisalignmentSlide 18A 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

LCLS-II Physics Meeting, May 08, 2013Undulator Parameter Sensitivity CalculationExample: Vertical MisalignmentSlide 19The 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, 2013Model Detuning Sub-Budget

Parameter piTypical Valuerms dev. dpiNoteKMMF3.50.00030.015 % uniformaK-0.0019 C-10.0001 C-1Thermal CoefficientDT0 C0.32 C0.56 C uniform without compensationbK0.0023 mm-10.00004 mm-1Canting CoefficientDx1.5 mm0.05 mmHorizontal Positioning

Slide 20Some 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 Physics Meeting, May 08, 2013LCLS-II HXR Undulator Line Tolerance BudgetSlide 21nError Sourcerms valuesbudget calculationsUnitsCorrriValueTolUnits(DP/P)i1- Launch Angle x0,y01.88rad0.710.3600.480.48rad93.7%2- (DK/K)rms0.000601.000.4430.000260.0002690.6%3- Segment misalignment in x 17527998m21.000.145254048504m99.0%4- Segment misalignment in y 30915.8m21.000.262810090m96.6%5- Jaw Pitch [rad]201.7rad1.000.0992020rad99.5%6- Quad Position Stability x,y 4.77m0.710.0740.250.25m99.7%7- Quad Positioning Error x,y 4.77m0.710.2971.001.00m95.7%8- Break Length Error16.8mm1.000.0591.001.00mm99.8%9- Strongback deflection 79.0m1.000.13911.011.0m99.0%10- Phase Shake Error16.6degXray1.000.1813.03.0degXray98.4%11- Phase Shifter Error45.4degXray1.000.0663.03.0degXray99.8%12- Cell Phase Error45.4degXray1.000.0663.03.0degXray99.8%TotalDP/P:74.7%Total Loss1-DP/P:25.3%sensitivities

LCLS-II Physics Meeting, May 08, 2013LCLS-II SXR Undulator Line Tolerance BudgetSlide 22nError Sourcerms valuesbudget calculationsUnitsCorrriValueTolUnits(DP/P)i1- Launch Angles x0,y04.5rad0.710.3111.001.00rad95.3%2- (DK/K)rms0.001311.000.3450.000450.0004594.2%3- Segment misalignment in x 1932472m21.000.118228168478m99.3%4- Segment misalignment in y 264225m21.000.15140000200m98.9%5- Jaw Pitch [rad]85.4rad1.000.2932525rad95.8%6- Quad Position Stability x,y 11.88m0.710.2382.002.00m97.2%7- Quad Positioning Error x,y 11.88m0.710.1191.001.00m99.3%8- Break Length Error90.4mm1.000.0444.04.0mm99.9%9- Strongback deflection 310.0m1.000.14244.044.0m99.0%10- Phase Shake Error16.6degXray1.000.3015.05.0degXray95.6%11- Phase Shifter Error47.0degXray1.000.1708.08.0degXray98.6%12- Cell Phase Error47.0degXray1.000.1708.08.0degXray98.6%TotalDP/P:74.8%Total Loss1-DP/P:25.2%sensitivities

LCLS-II Physics Meeting, May 08, 2013SummarySlide 23A 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, 2013End of PresentationLCLS-II Physics Meeting, May 08, 2013

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