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. ( 1998 Elsevier Science B.V. All rights reserved. The possibility that a free-electron laser (FEL) can produce a powerful and coherent optical beam of a single frequency is potentially of great interest for many applications. There is compelling experimental evidence from the FEL at the University of California at Santa Barbara (UCSB) that such a possibility is realizable with long-pulse electron beams [1]. Although there is some theoretical controversy [2,3] as to whether the FEL at UCSB actually attained a single-mode state, there is unambiguous observational evidence that the optical beam in the experiment exhibited a clear tendency to operate on a very narrow bandwidth. Future experiments using very long electron pulses, such as that at the Center for Research in Electro-Optics and Lasers (CREOL) of the University of Central *Corresponding author. Florida, are expected to provide definitive evidence of single-mode operation. In this paper, we report some recent theoretical developments [4] on single-mode operation of an FEL using the Ginzburg—Landau equation (GLE) which has been proposed as a model for the nonlinear evolution of the radiation field [5,6]. The Ginzburg—Landau model was originally motivated by observations of optical “spiking” in several experiments [7—9]. (For a brief review of related theory and experiment in the early 1990s on optical spikes, the reader is referred to Ref. [6]). A qualitative physical mechanism attributing the generation of spikes to the growth of sidebands was outlined by Warren and coworkers [7]. A necessary condition for the excitation of the sideband instability [10] is slippage between the optical and electron beams. However, the experiment at Columbia University [11] did observe spikes (with characteristic width 0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 8 9 0 0 2 ( 9 7 ) 0 1 3 6 3 6 similar to that of a solitary-wave solution [5] of the GLE) apparently without the excitation of the sideband instability. Single-mode operation of the FEL can be realized in experiments with a very long-pulse electron beam when effects involving slippage of the beam can be neglected. It thus represents a good test for the GLE, which has been shown to represent the nonlinear dynamics of the radiation field with reasonable accuracy when the effects of slippage can be neglected [4]. One of the advantages of the GLE is that it has a rich mathematical literature, with antecedents in hydrodynamics [12—14], that can be brought to bear on the issue of stability of singlemode operation. We show by analysis and numerical simulation that the Ginzburg—Landau model provides useful insight on how single-mode operation evolves out of nonlinear interaction of multiple modes. In particular, it yields specific analytical conditions on when single-mode operation is stable, when instability occurs, and the impulsive nonlinear growth of the instability to yield a chaotic and spiky optical beam. We also give quantitative information that may help resolve the controversy as to whether single-mode operation was realised in the UCSB FEL [1]. We refer to Refs. [4,5] for the derivation of the GLE