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 (S.Y. Cai and A. Bhattacharjee, Phys. Rev. A 43 (1991) 6934). 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) (L.R. Elias, G. Ramian, J. Hu, A. Amir, Phys. Rev. Lett. 57 (1986) 424), 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 (Phys. Fluids B 1 (1989) 1097). 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 pinj and cavity loss, and determine the phase instability boundary in the a0−pinj plane. The probability of realizing a single mode starting with random initial conditions is calculated and compared with spectral measurements from the UCSB FEL.