Geometrical interpretation of nonlinearities from a cylindrical pick-up

In many accelerators, cylindrical pick-ups are used to measure transverse beam positions. Theoretically, signals from these pick-ups are related to infinite power series of the beam position but, in practice, only finite number of terms are considered and the position measurements degrade when a beam is far from the center of a pick-up. This paper shows there is actually a simple geometrical relation between a beam position and induced signals. With help of the geometrical relation, the beam position can be written in a compact function of signals. The paper is concluded with numerical simulations and a test to show this geometry based expression can calculate a beam position better than the conventional methods.