Active lower order mode damping for the four rod LHC crab cavity

The high luminosity upgrade planned for the LHC requires crab cavities to rotate bunches into alignment at the interaction points. They compensate for a crossing angle near to 500~$\mu$Rad. It is anticipated that four crab cavities in succession will be utilized to achieve this rotation either side of each IP in a local crossing scheme. A crab cavity operates in a dipole mode but always has an accelerating mode that may be above or below the frequency of the operating mode. Crab cavities are given couplers to ensure that unwanted acceleration modes are strongly damped however employing standard practice these unwanted modes will always have some level of excitation. Where this excitation has a random phase it might promote bunch growth and limit beam lifetime. This paper sets out a method for active control of the phase and amplitude of the unwanted lowest accelerating mode in the crab cavities. The paper investigates the level of suppression that can be achieved as a function cavity quality factor and proximity to resonance