Sideband Instabilities

The growth of sidebands of the primary signal in free-electron lasers can occur after the bulk of the electron beam becomes trapped in the ponderomotive potential formed by the beating of the wiggler and radiation fields. Trapped electrons execute an oscillatory bounce motion in the ponderomotive well, and the waves formed by the beating of this oscillation with the primary signal are referred to as the sidebands. The difficulties imposed by the growth of sidebands are that they can compete with and drain energy form the primary signal as well as considerably broaden the output spectrum. Sideband control, therefore, is an important consideration for free-electron laser configurations that operate in the trapped-particle regime. Examples of such systems include (1) tapered wiggler configurations designed to trap the beam at an early stage of the interaction and then extract a great deal more energy from the beam over an extended interaction length and (2) oscillators which run at sufficiently high power over an extended pulse time that the electron beam becomes trapped upon entry to the wiggler. In order to understand the basic nature of the sideband mechanism, we consider the process within the framework of an idealized one-dimensional model. The purpose of the analysis is to examine the gain of the sideband modes that result from the trapped electron motion under a variety of conditions including the effect of a tapered wiggler field and the inclusion of a fluctuating space-charge field.