### Abstract

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 a_{0} as functions of the detuning parameter p_{inj} and cavity loss, and determine the phase instability boundary in the a_{0} - 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.

Original language | English (US) |
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Pages (from-to) | 88-93 |

Number of pages | 6 |

Journal | Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment |

Volume | 429 |

Issue number | 1 |

DOIs | |

State | Published - Jun 1999 |

Externally published | Yes |

Event | Proceedings of the 1998 20th International Free Electron Laser Conference and 5th FEL User Workshop - Williamsburg, VA, USA Duration: Aug 16 1998 → Aug 21 1998 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics
- Instrumentation

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## Cite this

*Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment*,

*429*(1), 88-93. https://doi.org/10.1016/S0168-9002(99)00077-7