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-Landau equation. The stability of a single-mode solution is analyzed, and connections are established with known instabilities of the GinzburgLandau equation. It is found that the principal mode with the largest gain is always stable and hence there is no Benjamin-Feir instability. However, the Eckhaus ~or phase! instability generally occurs if the frequency of a mode lies outside of a designated range centered on the principal mode. Under certain conditions, the Eckhaus instability can cause a sudden chaotization and spikiness in the radiation field. Analytical criteria and scaling for single-mode operation are given and tested by numerical simulations. A theoretical controversy on single-mode operation of the free-electron laser at the University of California at Santa Barbara is resolved. @S1063-651X~98!16009-9#