Optics measurement algorithms and error analysis for the proton energy frontier

Optics measurement algorithms have been improved in preparation for the commissioning of the LHC at higher energy, i.e., with an increased damage potential. Due to machine protection considerations the higher energy sets tighter limits in the maximum excitation amplitude and the total beam charge, reducing the signal to noise ratio of optics measurements. Furthermore the precision in 2012 (4 TeV) was insufficient to understand beam size measurements and determine interaction point (IP) $\beta$-functions (${\beta }^{*}$). A new, more sophisticated algorithm has been developed which takes into account both the statistical and systematic errors involved in this measurement. This makes it possible to combine more beam position monitor measurements for deriving the optical parameters and demonstrates to significantly improve the accuracy and precision. Measurements from the 2012 run have been reanalyzed which, due to the improved algorithms, result in a significantly higher precision of the derived optical parameters and decreased the average error bars by a factor of three to four. This allowed the calculation of ${\beta }^{*}$ values and demonstrated to be fundamental in the understanding of emittance evolution during the energy ramp.

Figure 1

Illustration of the $\beta$-function measurement from phase. The phase advances ${\varphi }_{i,j}$ in between three positions $s_i$ are needed to derive the $\beta$-functions at those positions.

Figure 2

Expected error of a measured $\beta$-function at position $s_1$, depending on the phase advances to the other two BPMs. The six used phase advances (three BPM combinations each for horizontal and vertical plane) for a BPM position in IR4 from the neighboring BPM method are indicated by triangles. When an increased range of 7 BPM is used ($N$-BPM method), 15 different combinations of phase advances are possible per plane, including the ones that are indicated by triangles. Another six better suited combinations of phase advances from the range of 7-BPMs are indicated by circles.

Figure 3

In the arcs the phase advance between two consecutive BPMs is about $\pi /4$. If the blue BPM is probed, it is better to skip the grey BPM and use the two red BPMs. The resulting phase advances are approximately ${\varphi }_{1,2}=\pi /4$ and ${\varphi }_{1,3}=3\pi /4$, which is the optimum according to Eq. (3).

Figure 4

Relative uncertainty of the $\beta$-function derived in the error propagation compared to a fit of the variation of calculated $\beta$-functions.

Figure 5

Accuracy of the derived $\beta$-functions from simulations for different ranges of BPMs and different amount of BPM combinations. The oscillation amplitude was 1 mm in the arcs and a Gaussian noise of $200\mu \mathrm{m}$ was applied.

Figure 6

Precision of the derived $\beta$-functions from simulations for different ranges of BPMs and different amount of BPM combinations. The oscillation amplitude was 1 mm in the arcs and a Gaussian noise of $200\mu \mathrm{m}$ was applied.

Figure 7

Average error bar of the experimentally measured $\beta$-function for different optics configuration at 4 TeV, except for injection at 0.45 TeV. Error bars which are larger than 50% were disregarded in the calculation of the average.

Figure 8

Average error bar of experimentally measured $\beta$-functions separately for the two error contributions. Top: Error propagated from the uncertainty of the phase advance, cf. Eq. (7). Bottom: Systematic errors as described in Sec. 2b. For the neighboring BPM method instead the standard deviation of the three calculated $\beta$ from different BPM combinations is used. Beam energy was at 4 TeV, except for injection at 0.45 TeV.

Figure 9

Error from Eq. (19) of the phase advance in Segment-by-Segment. For IR1 starting at BPM10.L1.B2, the position of following BPMs is indicated.

Figure 10

$\beta$-beating for Beam 1 after local and global corrections at ${\beta }^{*}=0.6\mathrm{m}$.

Figure 11

$\beta$-beating for Beam 2 after local and global corrections at ${\beta }^{*}=0.6\mathrm{m}$.

Figure 12

Measured $\beta$-function at the wire scanners during the energy ramp. $\beta$ values from a k-modulation measurement at 0.45 TeV and 4 TeV are shown as a comparison. The dashed line connects the two points from the k-modulation measurement at injection and top energy.