Dispersive effects on the main-wave modulational instability in free-electron lasers.

Abstract

The main-wave one-dimensional modulational instability due to the dispersive terms of the wave equation in a free-electron laser is introduced and analyzed. We derive the appropriate dispersion relation and compare its associated growth rate with one due to wave-particle energy exchange alone, as obtained by Davidson and Wurtele under the deep trapping assumption [Phys. Fluids 30, 557 (1987)]. It is found that, depending basically on the relation between some characteristic pa­ rameters, the modulational instability may be governed by wave dispersion. We also discuss the effects of waveguides on these instabilities and the behavior of the unstable modes as a function of the mentioned characteristic parameters. It is known that the initial parametric instability in free-electron lasers (FEL) saturates when particles begin to be trapped in the ponderomotive wells formed by the beatiny of the main electromagnetic wave and the wiggler field. 1• As was discussed by Davidson and Wurtele,2 one essential point is to understand the stability of such a state because, in general, the space-time variation of the main signal may affect its monochromaticity degrading FEL efficiency (of course, we are not talking about a proper tapering of FEL's parameters, which may enhance its gain 3 - 5 ). In the present work, we reconsider the problem of the main-wave stability in the sense defined by Davidson and Wurtele.2 In other words, given an initial steady-state wave, we wish to know if slow perturbations on the phase and amplitude of this wave are unstable functions of time. An important difference, however, will be introduced. Davidson and Wurtele2 discarded ali the slow second derivatives in Maxwell's equations for the main wave based on the reasonable (but not always correct) supposi­ tion that their effect is smaller than the one due to the first derivatives. In their case, the modulational instabili­ ty (MI) was originated by energy exchange between wave and particles. In our treatment we will keep these second derivatives. Putting in another way, we may say that our interest will be the analysis of the instabilities that come out as a combined result of these second derivatives and the already mentioned wave-particle energy exchange. The resultant instability will be called modulational insta­ bility due to kinetic effects (MIK) when the second derivatives are not important. On the opposite limit, when they are decisive in determining the signal of the perturbation squared frequency, the instability will be re­ ferred as due to dispersive effects (MID). Modulational instability due to dispersive effects has been discussed by many authors in many contexts, 6• 7 and here we intend to show under which conditions it may dominate the insta­ bility processes. By definition, two electromagnetic waves A and Aw are present in our system. They are described by the fol­ lowing functions: