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

The Ginzburg}Landau model for the radiation "eld of a free-electron laser (FEL) was originally derived for a high-gain ampli"er (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 "eld, and derive the coe$cients 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), phaseunstable o!-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 a 0 as functions of the detuning parameter p */+ and cavity loss, and determine the phase instability boundary in the a 0 !p */+ plane. The probability of realizing a single mode starting with random initial conditions is calculated and compared with spectral measurements from the UCSB FEL. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr; 42.65.Tg; 52.35.Mw