Wake Field Simulation of a Beam Position Monitor

Pickup/Kicker Electrodes are used in particle accelerators for detecting the offset of the beam. The obtained signal is then amplified and applied within the same turn to an identical electrode. This results in an electric field which causes a change in the momentum of the particles and finally a change in their position.

Pick-up/Kicker electrodes can be simulated in two different ways:

1) The electrode can be excited and the impact of the electric field on the moving charged particles evaluated. This is equivalent to operating the electrode as a kicker and can be simulated with CST MICROWAVE STUDIO® (CST MWS).

2) The charged particle beam is used as the excitation. The electric field belonging to the particle beam couples to the electrode which leads to a measurable output voltage. This operating condition would be the pick-up case.

In case there are no excited hollow waveguide modes, these approaches are connected by the reciprocity theorem in antenna theory.

The pick-up/kicker electrode shown in figure 1 is a quarter wavelength electrode similar to the one used in the ESR at GSI. In the simulation only half of the structure is modeled. The opposite electrode is taken into account via a symmetry condition. Port 1 and Port 2 shown in figure 2 are on the outside of the beam tube which is hidden for viewing purposes. The blue/red line in the figure represents the axis of the beam.

The beam is of Gaussian shape. The corresponding characteristics such as deviation, total charge and velocity are denoted in figure 2. The beam does not have to be ultrarelativistic as also shown in figure 2.

The frequency response of the quarter wavelength electrode can be obtained by normalizing the output voltages of the beam spectrum. This normalized output is shown in figure 3. As predicted theoretically [1] the frequency response shows a sinusoidal behaviour.  Furthermore, the results of a pick-up and kicker simulation should agree if a suppression of hollow waveguide modes is ensured. This is nicely shown on the right hand side of figure 3. The deviation of the two curves towards higher frequencies corresponds to the hollow waveguide mode with the cut off frequency f_c. The existence of this waveguide mode violates the validity of the reciprocity theorem meaning that the two operating cases can no longer be converted into each other.

In figure 4 the absolute value of the electric field corresponding to the bunch is shown. In the vicinity of the electrodes the coupling of the electric field of the bunch to the electrodes can be seen. The electric field obtained at the electrodes is then recorded and absorbed by the waveguide ports 1 and 2 as shown in figure 1.

This article shows how CST PS can be used for pick-up simulations. The Kicker simulation can be performed with CST MWS. Under the assumption of non-existing hollow waveguide modes the simulation results of both cases can be converted into each other. The advantage of using CST PS is that the pick-up simulation does not have to obey this unrealistic assumption as in most accelerators.

References:

[1] P. Raabe: "Feldtheoretische Analyse von Detektoren und Ablenkeinheiten für die stochastische Strahlkühlung von Teilchenstrahlen", VDI Fortschrittsberichte, Reihe 21, Nr. 128, 1993.