Ginzburg–Landau model for a free-electron laser: from single mode to spikes

Single-mode operation of a free-electron laser is modeled by the Ginzburg–Landau equation. The linear stability of a single-mode solution is analyzed, and connections are established with known instabilities of the Ginzburg–Landau equation. It is found that there is no Benjamin–Feir instability and hence, the principal mode with the largest gain is always stable. However, the Eckhaus (or the phase) instability generally exists for a mode with frequency outside a range centered on the principal mode. This gives rise to two distinct possibilities: either there is spontaneous frequency shifting to the stable mode with the largest growth rate and a consequent tendency to approach single-mode operation, or there is a sudden chaotization and spikiness in the radiation field. Analytical criteria and scaling are given and tested by numerical simulations.