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

A precise determination of the β function at different locations of an accelerator is essential to allow accurate optics correction and to ensure high machine performance. The β 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 β 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 β 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 β function at different locations of the machine.