GINZBURG-LANDAU MODEL FOR MODE COMPETITION AND SINGLE-MODE OPERATION OF A FREE-ELECTRON LASER

Mode competition of a free-electron laser is modeled by the Ginzburg-L andau equation. The stability of a single-mode solution is analyzed, a nd connections are established with known instabilities of the Ginzbur g-Landau equation. It is found that the principal mode with the larges t gain is always stable and hence there is no Benjamin-Feir instabilit y. However, the Eckhaus (or phase) instability generally occurs if the frequency of a mode lies outside of a designated range centered on th e principal mode. Under certain conditions, the Eckhaus instability ca n cause a sudden chaotization and spikiness in the radiation field. An alytical criteria and scaling for single-mode operation are given and tested by numerical simulations. A theoretical controversy on single-m ode operation of the free-electron laser at the University of Californ ia at Santa Barbara is resolved.