Energy Losses of an Electron Bunch Moving along the Axis of a Circular Waveguide with Periodically Perturbed Wall

The field radiated by an electron bunch moving along the axis of a circular periodic waveguide is obtained from Maxwell´s equations using a perturbation method. The guide shape is defined by its radius a(z) = ao {I + ¿s(z)} where ao denotes the mean guide radius, s(z) the guide geometry and ¿ a small perturbation parameter. Electromagnetic field is calculated up to second order in ¿ . The total energy loss suffered by the bunch and the potential acting on individual particles are related to the following parameters : charge, shape and width of the bunch, energy, guide geometry, by comparatively simple formulas. The influence of these parameters is calculated for a gaussian bunch and a rounded iris waveguide. This example clarifies under what conditions energy losses are virtually independent of energy and shows the importance of the bunch width.