Analysis of Damping Requirements for Dipole Wake-Fields in RF Crab Cavities

Several current particle-antiparticle collider designs require the beams to collide at a small angle. In order to make the luminosity comparable to that of beams colliding head-on, transverse deflecting cavities known as crab cavities will be used. Additional modes, other than the required "crabbing" mode are excited by the beam and if these are excited resonantly then an appreciable dilution of the beam emittance will occur. We provide a single- mode analysis of the damping requirements on the dipole modes which will minimize the resulting angular kicks to the beams, by deriving an explicit solution for the frequency offset of the mode from a bunch harmonic for maximum deflection of bunches, using a novel analytical technique.