Ginzburg-Landau model for a long-pulse low-gain free-electron laser oscillator

The Ginzburg-Landau model for the radiation field of a free-electron laser (FEL) was originally derived for a high-gain amplifier. With a view to making precise comparisons with experimental data from the long-pulse FEL oscillator at the University of California at Santa Barbara (UCSB), we have developed a new formulation of the Ginzburg-Landau model starting from the low-gain oscillator equations. We implement a small-amplitude expansion of the radiation field, and derive the coefficients of the Ginzburg-Landau equation by analysis as well as by Mathematica. Stability analysis of the Ginzburg-Landau equation produces results similar to those obtained by T.M. Antonsen and B. Levush. These include the stability of the main mode (no Benjamin-Feir instability), phase-unstable off-centered modes (Eckhaus instability), as well as relaxation to the single mode which occurs much faster in amplitude than in phase. We obtain the saturated radiation amplitude a₀ as functions of the detuning parameter p_inj and cavity loss, and determine the phase instability boundary in the a₀ - p_inj plane. The probability of realizing a single mode starting with random initial conditions is calculated and compared with spectral measurements from the UCSB FEL.