Abstract
SuperKEKB is an asymmetric energy electron positrion collider currently under commissioning in Japan. It aims to achieve a record high luminosity of
8
×
10
35
cm
−
2
s
−
1
, for which accurate control of
β
∗
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
β
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.
8
×
10
35
cm
−
2
s
−
1
, for which accurate control of
β
∗
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
β
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.