Monitoring of particle beams at high frequencies

Abstract To calculate the response of non-intercepting beam monitors, we must know the electromagnetic field of a charged particle beam, in cylindrical beam pipes of various cross-sections. For a beam, modulated in intensity, we can approximate the field by modulating the two-dimensional solution for a uniform beam with the varying intensity of the beam. This may not be accurate for beams with high-frequency modulation and low speed. Therefore, the three-dimensional field of a modulated beam is calculated for circular and rectangular cross-sections. With this field, we calculate the response of electrostatic, wall-current, loop and stripline monitors, both for intensity and position measurement. The resulting formulas are valid at all frequencies. They are fairly simple and easy to apply to practical monitors. For the cases that are difficult to solve by the direct method, a very general reciprocity formula is presented.