Eigenmodes and mode competition in a high-gain free-electron laser including alternating-gradient focusing

We solve the eigenvalue problem for a high gain free-electron laser in the “water-bag” model including alternating-gradient focusing by a variational-solution-based (VSB) expansion method. Such VSB expansion method is very efficient for finding the eigenvalue. The results agree with those obtained by numerical simulation quite well. We further discuss the mode degeneracy and mode competition.