On Wakefields With Two-Dimensional Planar Geometry

In order to reach higher acceleration gradients in linear accelerators, it is advantageous to use a higher accelerating RF frequency, which in turn requires smaller accelerating structures. As the structure size becomes smaller, rectangular structures become increasingly interesting because they are easier to construct than cylindrically symmetric ones. [1,2] One drawback of small structures, however, is that the wakefields generated by the beam in such structures tend to be strong. Recently, it has been suggested that one way of ameliorating this problem is to use rectangular structures that are very flat and to use flat beams. [3] In the limiting case of a very flat planar geometry, the problem resembles a purely two-dimensional (2-D) problem, the wakefields of which have been studied in Ref. [4]. In this work we consider the purely 2-D problem that is sketched in Fig. 1. The beam is considered to be infinitely long in the horizontal x-direction; it propagates with the speed of light c in the longitudinal z-direction from z = 1 to z = +1. The beam distribution in the y-z plane is arbitrary. The environment consists of boundaries which are independent of x, but are otherwise unrestricted; for example, in the y-z plane they can be of arbitrary shape, and they can be made of metal, dielectric or plasma material. We do assume, however, that the beam trajectory is entirely in free space and that it nowhere intersects the boundaries. A test charge e in the beam (or trailing the beam) also moving in the z-direction at the speed of light samples the force due to the wakefield generated by the beam. For these conditions, a theorem that we call the “planar wake theorem” was proven in Ref. [4]. The theorem states that the total transverse wake kick received by the test charge is independent of the y-positions of the beam and the test charge, and is also independent of D, the longitudinal separation between the beam and the test charge (see Fig. 1). In addition, the theorem states that the longitudinal wake kick is also independent of y, though it does not say anything about its D-dependence. In this report, in Section II, we rederive the planar wake theorem. In Section III, we add a corollary to the theorem, applicable to the case when, in addition to the above conditions, the boundaries also have up-down symmetry. For this case, we will prove that the transverse wake kick not only is constant, but in fact is equal to zero. The proof consists of a simple application of the planar wake theorem. However, it was Ref. [3] which triggered the present extension to the theorem. Finally, in Section IV, we make additional observations concerning the wakefields in very flat 3-D accelerating structures.