Vertical emittance reduction and preservation in electron storage rings via resonance driving terms correction

In this paper the influence of betatron coupling on the transverse beam emittances is described using the resonance driving terms formalism. Betatron coupling and vertical dispersion generated by magnetic and installation errors are major sources of vertical emittance. A new scheme for minimizing the latter is presented here, together with results from measurements carried out in 2010 at the ESRF electron storage ring, which provided vertical emittance of about 4.4 pm, a record low for this machine. Two schemes for the automatic compensation of coupling introduced by insertion devices are also presented with results from the first implementation tests. This paper is also an attempt to clarify the various definitions and meanings of vertical emittance in the presence of coupling.

Figure 1

Example of RMS apparent and projected emittances plotted along the ESRF storage ring. Betatron coupling is driven by three skew quadrupoles whose location is indicated by the vertical dashed lines. In the top and center plots, the horizontal and vertical emittances are evaluated via Eqs. (4, 5, 13, 14). The bottom plot shows the results for the vertical plane as computed by AT. The corresponding equilibrium emittances are ${\mathcal{E}}_u=4.0\mathrm{nm}$ and ${\mathcal{E}}_v=18.4\mathrm{pm}$.

Figure 2

${\mathbb{E}}_y$ (red) from Eq. (5) and ${ϵ}_y$ (green) from Eq. (14) plotted along the ESRF storage ring, for two different amounts of coupling inferred from ORM measurements. The black triangles indicate the positions where ${\mathbb{E}}_y$ would overestimate the average projected emittance ${\overline{ϵ}}_y$, that on the other hand would be instead underestimated by measurements at the blue circles.

Figure 3

Example of comparison between the apparent emittances ${\mathbb{E}}_y$ (before coupling correction) measured at ten available in-air x-ray detectors (blue) and the predictions of AT (red) after creating the lattice error model from the ORM measurement of January 16, 2010.

Figure 4

Left column: Vertical apparent emittances ${\mathbb{E}}_y$ measured along the ESRF storage ring with all correctors off (upper plot), after first correction with minimization of coupling RDTs and vertical dispersion (center plot), and after a second measurement and correction (bottom plot). The names in the abscissa refer to the used monitors: ten (out of 11) in-air x-ray monitors (from C05 to C31) and two pinhole cameras (D09 and D25). Right column: Amplitude of the corresponding coupling RDTs.

Figure 5

Record low vertical apparent emittances ${\mathbb{E}}_y$ measured along the ESRF storage ring on July 22, 2010 at ID gaps open. The horizontal dashed line denotes the mean values between the ten (out of 11) available x-ray monitors. The D09 and D25 were taken out of the average, see main text.

Figure 6

Example of vertical apparent emittance abrupt jumps (top) following two ID gap movements (bottom). In the upper graph the emittance is measured at 12 monitors (red lines) and the corresponding mean value ${\overline{ϵ}}_y$ is depicted by the blue line. The corresponding apertures of ID6 and ID13 are reported in the lower plot.

Figure 7

Drawings of an ESRF straight section housing two undulators. Corrector steerers (blue ellipse) are placed at both ends.

Figure 8

Mean vertical emittance ${\overline{ϵ}}_y$ measured against the vertical aperture of the ID6 in-vacuum undulator without any correction (red circles) and with automatic coupling compensation (blue diamonds). Error bars corresponds to the spread $\delta {ϵ}_y$ of Table .

Figure 9

Snapshot of the software application driving the trim values for the 32 skew quadrupole correctors. Amplitude $A$ (already normalized in terms of magnet current) and phase $\Theta$ for both resonances may be varied either by hand or by a coupling feedback software loop.

Figure 10

Comparison between the mean vertical apparent emittance ${\overline{ϵ}}_y$ measured during beam delivery without (top) and with (bottom) the coupling feedback. Data acquired during injections are not displayed.

Figure 11

Comparison between the coupled Twiss parameters as computed by MADX-PTC (Twiss module) and the ones from Eqs. (a27, a28, a29) (zoom of the entire ring length of 844 m). Parameters refer to the ESRF storage ring lattice with the same coupling of Fig. 1. The uncoupled Twiss parameters, $\beta$, $\alpha$, and $\gamma$, are obtained from the MADX by using the ideal uncoupled lattice.

Figure 12

Comparison between the vertical eigenemittance ${\mathcal{E}}_v$ computed by MADX (blue diamonds) and via Eqs. (26, 27) (red plus) of Table .

Figure 13

Comparison between the vertical eigenemittance ${\mathcal{E}}_v$ computed by MADX (blue diamonds) and via Eqs. (26, 27) (red plus) of Table .