Beam position monitor sensitivity for low-β beams

At low velocities, the EM field of a particle in a conducting beam tube is no longer a TEM wave, but has a finite longitudinal extent. The net effect of this is to reduce the coupling of the high-frequency Fourier components of the beam current to BPM (beam position monitor) electrodes, which modifies the BPM sensitivity to beam displacement. This effect is especially pronounced for high-frequency, large-aperture pickups used for low-β beams. Non-interceptive beam position monitors used in conjunction with high frequency RFQ (radio-frequency-quadrupole) and DTL (drift-tube-linac) accelerators fall into this category. When testing a BPM with a thin wire excited with either pulses or high-frequency sinusoidal currents, the EM wave represents the principal (TEM) mode in a coaxial transmission line, which is equivalent to a highly relativistic (β = 1) beam. Thus wire measurements are not suitable for simulating slow particle beams in high bandwidth diagnostic devices that couple to the image currents in the beam tube wall. Attempts to load the tin wire either capacitively or inductively to slow the EM wave down have met with limited success. In general, the equations used to represent the 2-D response of cylindrical-geometry BPMs to charged-particle beams make several assumptions: (1) the BPM electrodes are flush with and grounded to the surface of the conducting beam tube; (2) the beam is a line source (pencil beam); (3) the longitudinal extent of the EM field of a beam particle at the beam tube wall is zero, corresponding to a highly relativistic beam. The purpose of this paper is to make some quantitative estimates of the corrections to the conventional approximations when a BPM is used to measure the position of low velocity (low-β) beams