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 Ginzburg-Landau 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.