Phase advance constraint in $K$ modulation for precise $\beta$-function determination

A precise determination of the $\beta$ function at different locations of an accelerator is essential to allow accurate optics correction and to ensure high machine performance. The $\beta$ function can only be measured directly at locations where beam position monitors are installed. For other locations, we rely on a $K$-modulation technique. However, this technique presents some limitations resulting in imprecise $\beta$ values when the phase advance between the modulated quadrupole and the observation point is separated by 90°. To mitigate these limitations, we have introduced the same phase advance as an additional constraint in the $K$-modulation algorithm. In this paper, the improvement of the $\beta$ measurement uncertainty is quantified for different optics configurations for both, the LHC and the proton synchrotron (PS) booster. The new algorithm is used to reanalyze measured data during van der Meer scans for luminosity calibration providing significantly more accurate results than obtained previously. Moreover, in the PS booster, this improvement has also reduced the uncertainty of the $\beta$ function at different locations of the machine.

Figure 1

Illustration of an interaction region and optics function [3].

Figure 2

Error amplification when given by Eq. (6) when ${\beta }^{*}\approx L^{*}+L_Q/2$. For the LHC, $L^{*}=23\mathrm{m}$ and $L_Q/2=2.1\mathrm{m}$.

Figure 3

Difference between the two solutions given by Eq. (8) as a function of ${\beta }^{*}$.

Figure 4

Relative error in ${\beta }^{*}$ extracted from simulations as a function of the selected optics and a tune uncertainty $\delta Q=5\times {10}^{-5}$. We can see that for ${\beta }^{*}\sim L^{*}+L_Q/2$, the relative error in ${\beta }^{*}$ increases significantly.

Figure 5

Contribution of different errors to the total error in the ${\beta }^{*}$ determination from $K$ modulation for B1 vertical plane in IP1 for different optics configurations.

Figure 6

Histograms for ${\beta }^{*}$ beating and ${\varphi }_{\mathrm{IR}}$ extracted from the model after introducing magnetic errors in the quadrupoles for IP1 before $K$ modulation is applied.

Figure 7

Example of how the optimization algorithm can converge to the wrong solution (top) and how the introduction of the phase advance constraint [$\mathrm{\Omega }=0.1$ (middle) and $\mathrm{\Omega }=0.2$ (bottom)] helps in finding the real minimum for the case of vdM optics configuration.

Figure 8

Histogram of the reconstructed vertical ${\beta }^{*}$ (top) and vertical waist (bottom) $w$ using $K$ modulation in IP1 for different values of the weight $\mathrm{\Omega }$.

Figure 9

Fraction of the reconstructed ${\beta }^{*}$ values using $K$ modulation that converges to the wrong solution (top) and statistical uncertainty of the measurements (bottom).

Figure 10

Scans over $w$ and ${\beta }_w$ and the figure of merit value obtained for different phase advance weights: $\mathrm{\Omega }=0$ (top), $\mathrm{\Omega }=0.2$ (middle), and $\mathrm{\Omega }=0.5$ (bottom) for LHC injection optics.

Figure 11

Example of the optimization algorithm for low-${\beta }^{*}$ for $\mathrm{\Omega }=0$ (top) and $\mathrm{\Omega }=0.2$ (bottom). Regardless of the value of the weight, $\mathrm{\Omega }$ will always converge the right solution.

Figure 12

mad-x layout of the PS booster lattice in the vicinity of wirescanner (top) together with simulated phase advance (center) and $\beta$ functions (bottom). The wirescanner location is indicated by a red line.

Figure 13

Analytical uncertainty on extrapolated $\beta$, as determined by Eq. (6), from the PS booster QFO in the wirescanner insertions.

Figure 14

Histograms of the percentage error on the ${\beta }_y$ at the location of the PS booster wirescanner, as determined in simulation via extrapolation from the nearest quadrupoles. Results are shown for 1000 test cases of the PS booster model encompassing the operational range of working points. Results are shown for the standard K-mod extrapolation method via minimization of Eq. (5) (orange) and for the K-mod extrapolation including the phase constraint, Eq. (10), with $\mathrm{\Omega }=0.3$ (black).

Figure 15

Histograms of the percentage error on the ${\beta }_y$ at the location of the PS booster wirescanner, as determined in simulation via extrapolation from the nearest quadrupoles, including constraint on the phase advance of the closest BPMs. Results are shown for 1000 test cases of the PS booster model encompassing the operational range of working points and including a range of realistic measurement errors (light blue). A histogram of the applied test-case error on the average $\beta$ at the modulated quadrupoles is shown in dark blue. The value given is the mean of the error introduced to the left and right quadrupoles.