Measuring β* in SuperKEKB with K modulation

SuperKEKB is an asymmetric energy electron positrion collider currently under commissioning in Japan. It aims to achieve a record high luminosity of $8\ifmmode\times\else\texttimes\fi{}{10}^{35}\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}2}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$, for which accurate control of ${\ensuremath{\beta}}_{y}^{*}$ is needed. The advanced final focus system is also relevant for studies related to future linear colliders, which similarly need to achieve very small beam sizes. To work for SuperKEKB, the $K$ modulation technique is generalized to allow known quadrupole fields between the modulated magnets and the interaction point. Initial measurements taken in HER agree with simulations, and show that $K$ modulation is suitable for measuring the minimum of the $\ensuremath{\beta}$ function at the interaction point within 1%, however it is too uncertain to be used for measuring the displacement of the beam waist away from the interaction point. In addition, tune shift equations for a modulated quadrupole are derived without assuming a thin lens perturbation, giving a simple method to calculate the tune shift to second order in the quadrupole strength modulation.