Parasitic loss of a gaussian bunch in a closed cavity

Excitation of higher-order modes in RF cavities has been under rather extensive study recently. The parasitic energy loss of a particle bunch has been estimated more than once. Our toy model begins with a closed cavity represented by two infinite conducting plates separated by a distance g. Since we are interested only in the ultrarelativistic case, we shall assume that all particles in the bunch travel with the speed of light c. The electromagnetic fields excited by a δ-function bunch has been worked out in a previous note. It is not difficult to show that a δ-function bunch passing through a closed cavity loses an infinite amount of energy to parasitic cavity modes. Besides, the infinity originates from an ''ultraviolet divergence''; i.e., it diverges when summing over eigenmodes with short wavelengths. To overcome this difficulty, one possibility is to introduce a gaussian bunch of finite bunch length so that cavity modes with wavelengths sorter than the bunch length will be sufficiently suppressed to overcome the ultraviolet divergence and the parasitic energy loss will thus become finite. An exact formula is derived for our simplified model in this paper. It is then applied to numerical calculation. 2 refs., 1 fig., 1 tab