SCU Measurements at LBNL

1. SCU Measurements at LBNL Diego Arbelaez (LBNL) Superconducting Undulator R&D Review Jan. 31, 2014

2. Introduction • Undulators must meet the trajectory and phase shake error requirements for the FEL • Magnetic field error sources • Random and systematic machining errors • Assembly errors • Accurate fabrication methods will be used in order to minimize the initial device errors • End and central tuning methods will be incorporated on the prototypes • Sufficiently accurate measurement and tuning methods must be available to meet the requirements for: • 1st and 2nd field integral • Phase and phase shake • Keff

3. Error Sources and Analysis

4. Error Analysis for Coil and Pole Tolerances Pole Coil h d w l Second Field Integral Error (Coil) Second Field Integral Error (Pole) δ = 0.94 T-mm2 100 μm errors I1 = 0.19 T-mm δ = 0.21 T-mm2 I1 = 0.047 T-mm 100 μm errors I1 • Coil error • Produces no net kick (displacement does not grow with distance) • Produces a phase error • Pole error • Produces a net kick (displacement grows with distance) * Tolerance = 50 T-mm2

5. Trajectory Error Scaling Pole Errors Coil Errors • Trajectory errors scale with the undulator length to the power of 3/2 • Determine the standard deviation in the trajectory error for a random ensemble of undulator feature errors • Pole errors • Characterized by a kick error (I1) • Total trajectory error is given by the sum of kick errors (Ki) with a drift length (x-xi) (i.e. ); scales with N3/2 • Coil errors • Characterized by a displacement error (I2) • Total trajectory error is a simple random walk of individual displacement errors (i.e. ); scales with N1/2

6. Scaling of Trajectory and Phase Errors for Untuned Devices • Random pole and coil errors with a given standard deviation are introduced using a Monte Carlo simulation for an undulator with length Lu = 3.3 m • Calculations performed for as-built undulator with no field tuning • RMS machining errors of < 2μm were measured in the ½-m long LBL prototype • Second field integral can be reduced to meet the requirements with end and central field correction mechanisms Second Integral Error Phase Shake Lu = 3.3 m LCLS-II requirement End and central field tuning methods will be used to reduce the second integral error quadratic increase with error size LCLS-II requirement linear increase with error size

7. Simulated Trajectory with Field Correction • Random errors generated using CMM-measured distribution of machining errors • Corrector locations and excitation (same for all locations) of correctors is applied • On average 11 correctors are needed to reduce the first and second integral errors to negligible levels over 3.3 m • The trajectory requirement is met for the entire range of operation with the only adjustment being the amplitude of the corrector current (same through all correctors) Before correction After correction 11 correctors Lu = 3.3 m

8. Undulator Measurements at LBNL

9. Field Measurement Technology Approaches • Hall Probe (ANL) • Local field measurement • Need to know the location of the hall probe to high accuracy • Stretched wire or coil scan (ANL) • Obtain net first and second field integrals • Only length integrated information • Pulsed wire (LBNL) • Measure first and second field integrals • Measurements give integral values as a function of position along the length of the undulator

10. Pulsed Wire Method Description Traveling wave y x z I Observation point (z = 0) Bx(z) Tensioned wire between two points Part of the wire is in an external magnetic field A current pulse is applied to the wire The wire is subjected to the Lorentz force A traveling wave moves along the wire The displacement at a given point is measured The displacement of the wire as a function of time is related to the spatial dependence of the magnetic field

11. Analytical Solution (Dispersion Free) ρ: wire mass per unit length T: wire tension : wire position at z = 0 as a function of time c: wave speed General solution: Special cases: DC current: ct I1 ; δt 0: z Solution for the wire motion at a given location as a function of time A square current pulse with pulse width δt is assumed

12. Dispersion Euler-Bernoulli Beam General Solution Dispersive wave motion: Undispersive wave motion: • The flexural rigidity of the wire leads to dispersive behavior • Thin wires with lower flexural rigidity are less susceptible to dispersion • Dispersive behavior can be predicted using Euler Bernoulli theory for bending of thin rods

13. Experimental Validation Wire Positioning stages Wire position sensors (referenced to undulator fiducials) Wire motion detectors Echo-7 Undulator

14. Wave Speed Measurement Wire motion from magnet at two locations Wave Speed Fit to analytical expression Wave speed obtained by placing the motion sensor in two different locations and measuring the phase difference as a function of frequency in the two signal

15. ECHO-7 First and Second Integral Measurement Before Dispersion Correction After Dispersion Correction First Integral Second Integral

16. ECHO-7 Phase Error Wire damping introduces error in the field integral measurement which must be compensated in the calculation of phase errors Phase error calculation with upstream and downstream detectors Comparison of the calculated phase errors for Hall Probe and PW measurements

17. SCU Test System In-vacuum Pulsed Wire System Test Cryostat • Cryogen-free cryostat (two cryo-coolers) • Pulsed wire attachment at each end of the cryostat • In-vacuum pulsed wire measurement • Decreased air damping overcome with passive damping at the ends and pulse cancelling with reverse current

18. Measurement Plan Pulsed wire will be used as the main method during the R&D and commissioning phase for the field correction mechanism at LBNL The pulsed wire method will be incorporated and used as one of the measurement methods in the ANL measurement system Absolute Keff measurements will be performed using the ANL hall probe system