Measuring $\beta^*$ 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 $8 \times 10^{35} \ \mathrm{cm}^{-2} \mathrm{s}^{-1}$, for which accurate measurements of $\beta_y^*$ are needed. The advanced final focus system is also relevant for studies related to future linear colliders, which similarly need to achieve very high luminosities. Such studies also require accurate measurements of $\beta_y^*$. Using both simulations and experimental measurements from SuperKEKB, K modulation is found to be suitable for measuring the $\beta$ 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. To work for SuperKEKB, the K modulation technique is generalized to allow known quadrupole fields between the modulated magnets and the interaction point. In addition, the tune shift from a modulated quadrupole is found without assuming a thin lens perturbation, giving a simple method to calculate the tune shift to second order in the quadrupole strength modulation.