Calculating Bpm Coefficients with Green’s Reciprocation Theorem

For a highly relativistic charged beam, the Lorentz contraction compresses the electromagnetic field of the beam into the 2-D transverse plane. This results in the induced currents on the beam chamber wall having the same longitudinal intensity modulation as the charged beam. When the wavelength of the beam intensity modulation is large compared to the dimensions of the button electrodes, which are used as beam position monitors (BPMs), the calculation of the induced currents on the buttons may be simplified as a 2-D electrostatic problem. For four-button BPMs, vertical and horizontal signals are monitored from the differences in the induced charges between the top and bottom, and right and left buttons, respectively. In this Note, the coefficients of four-button BPMs are calculated using Green’s reciprocation theorem, which shows that finding induced charges on the buttons due to a charge at a beam position is equivalent to finding induced potential at a beam position due to given potentials on the buttons. In the case of finite element modeling, using the theorem significantly simplifies the calculation of BPM coefficients: the induced potential method needs only three sets of calculations for the sum, vertical, and horizontal signals, compared to one calculation for each beam position for the induced charge method. For the case of analytical expressions, two examples are given: the calculations for optimized button configurations on a small-gap chamber and the circular chamber after conformal transformations of the chamber geometry for the both cases. The BPM coefficients for the analytical results are expressed in simple formulae, which agree with the results of numerical integrals and infinite series obtained from the induced charge method.