Vertically polarizing undulator with dynamic compensation of magnetic forces

As part of the R&D program of the LCLS-II project, a novel 3.4-meter-long undulator prototype with horizontal magnetic field and dynamic force compensation has recently been developed at the Advanced Photon Source (APS). Previous steps in this development were the shorter 0.8-meter-long and 2.8-meter-long prototypes. Extensive mechanical and magnetic testing were carried out for each prototype, and each prototype was magnetically tuned using magnetic shims. The resulting performance of the 3.4-meter-long undulator prototype meets all requirements for the LCLS-II insertion device, including limits on the field integrals, phase errors, higher-order magnetic moments, and electron-beam trajectory for all operational gaps, as well as the reproducibility and accuracy of the gap settings.

Figure 1

LCLS-II horizontal gap vertically polarizing undulator (HGVPU).

Figure 2

Design rendering of the HGVPU gap drive mechanism. The linear slides are at the same positions along the undulator as the drive actuators. The blue and red coloring on the springs in the spring units indicates weaker and stronger, respectively. Combining their response curves and adjusting the engagement points using the spring calibration buttons (see Fig. 3) allows a better match to the gap dependence of the magnetic force.

Figure 3

Cross-sectional view of the undulator strongback, magnet/pole holder, and spring unit.

Figure 4

Sketch of a quarter period of the magnetic design; only half of the thicknesses of the magnet and pole are shown. Dimensions are mm. Z chamfer dimensions and the magnet recess are given in Table 2.

Figure 5

Analytical calculation of the strongback deformation for optimized actuator locations assuming ideal spring units. The vertical lines mark the actuator locations.

Figure 6

Simulation results of the strongback deformation, calculated using COMSOL and using actual forces from the spring units. The deflection shown in the upper panel corresponds to a gap of 7.2 mm. Unlike the analytical result of Fig. 5, which included only the stiffness of the strongback, this simulation includes stiffness from the strongback and the sections of magnet/pole holder. The deformation is therefore somewhat smaller at the same gap.

Figure 7

The strongback deformation (peak to peak) that results in an rms phase error matching the 4° requirement is shown as a function of gap for two different shapes of deformation. At the minimum gap of 7.2 mm, a deformation that exceeds 7 or $8\mu \mathrm{m}$ (depending on deformation shape) will result in excessive rms phase error. The deformation can be allowed to be larger at larger gaps. Note that this only includes mechanical shape. On a real undulator, magnetic tuning (see below) will further improve the rms phase error, but the mechanical shape must reproduce to this accuracy.

Figure 8

Schematic of systematic and random errors in spring unit forces (in kg). The magnetic force is uniform along the undulator except for the weaker ends (not shown but accounted for in modeling). The spring unit forces are applied at points F1–F18. The uniform spacing between F1–F4 and F15–F18 is slightly larger than the uniform spacing between F5–F14. “Random” shows the random errors between $\pm 5\mathrm{kg}$ that were added to the systematic 2 kg error. The bottom line shows the sorting of the units according to their random errors. The smallest error is in the undulator’s middle; the largest errors are close to the actuator since the actuator compensates for any overall force error to hold the desired gap. (The errors shown are symmetric left to right because the model assumed that symmetry.)

Figure 9

Deformation of the strongback (top) and the resulting phase errors (bottom) vs distance along the undulator, for the set of random and systematic errors in spring unit strength of Fig. 8, and for the same set of errors after sorting. The vertical lines mark the actuator locations.

Figure 10

Gap dependence of the magnetic force attracting the HGVPU strongbacks toward one another.

Figure 11

The two types of springs used in the spring units. The weaker type is on the left and the stronger on the right.

Figure 12

Spring unit calibration system. The side panel and the actuator motor have been removed in the photo on the left and are in place in the photo on the right.

Figure 13

An example of how the load characteristic of a spring unit can be tuned by adjusting the engagement point of the individual springs. A pair of springs—one strong, one weak—has its engagement points shifted one way in “Tuning #1” and the other way in “Tuning #2.” The resulting total force from the pair of springs in both cases is shown in the bottom panel, along with the total force from the untuned pair of springs.

Figure 14

The difference between the force of the worst of the spring cages and $1/18$ of the magnetic attractive force of the undulator jaws is shown as a function of gap. Some bumpiness arising from imperfectly uniform springs is visible, but the force of the tuned spring unit matches the desired force within $\pm 2\mathrm{kg}$ over the relevant gap range, as shown by the dashed lines.

Figure 15

The lower left panel shows one of the Capacitec sensors used to measure the gap profile. The Capacitec holder is shown in the top left panel. On the right, the sensors are shown installed and ready for use in the APS magnetic measurement facility.

Figure 16

Measured total gap in mm (left) and distance in mm between the gap center line and the pole tips on the left and right undulator jaws (right) vs z for several values of the gap. The horizontal dashed lines show that the mechanical tolerance of $\pm 10\mu \mathrm{m}$ straightness has been met.

Figure 17

Spring unit force at maximum compression (left), recorded over 30000 cycles. The force from the spring unit and the $\sigma$ deviation are shown at right vs gap.

Figure 18

Relative ${\mathrm{K}}_{\mathrm{eff}}$ change at a gap of 7.2 mm, measured after every 50 cycles of the gap during a total of more than 2000 cycles. The requirement of $\pm 0.023\%$ is indicated by the horizontal lines.

Figure 19

Photograph of the types of magnetic shims used in tuning the HGVPU. (a) and (b) are side shims, placed on the sides of the pole for strong trajectory adjustments. Magnetic forces hold them in place; the screws are used to adjust their distance from the side of the pole. (c) is a phase shim, placed to cover the entire gap side of the magnet (the tip of the magnet is recessed from the pole tips by 0.5 mm so there is space for a 0.1- or 0.2-mm-thick shim). The bent edges shape the shim to conform to the magnet chamfers. (g) is a flat shim, placed to span from the center in z of one magnet to the center in z of the next magnet, immediately adjacent to the outside edge of the pole so the minimum gap is unaffected. It is a weaker version of the side shim. (d), (e), (f), and (h) are trajectory shims of various lengths in y that cover only half of the z-distance of the magnet face, with the bend to conform to the magnet chamfer. They may or may not be centered on the magnet in y depending on the correction needed. Different lengths and placements allow for different trajectory and multipole corrections, as do asymmetric (only one side) placements of the side and flat shims.

Figure 20

Horizontal (top) and vertical (bottom) trajectories calculated from the magnetic field profiles at several gaps after magnetic tuning. Electron energy is taken to be 14 GeV. The limits on the graph are the $\pm 2\mu \mathrm{m}$ straightness requirement.