Coherent Emission: Linear Theory

In order to give rise to stimulated emission, it is necessary for the electron beam to respond in a collective manner to the radiation field and to form coherent bunches. This can occur when a light wave traverses an undulatory magnetic field such as a wiggler because the spatial variations of the wiggler and the electromagnetic wave combine to produce a beat wave, which is essentially an interference pattern. It is the interaction between the electrons and this beat wave which gives rise to the stimulated emission in free-electron lasers. In the case of a magnetostatic wiggler, this beat wave has the same frequency as the light wave, but its wavenumber is the sum of the wavenumbers of the electromagnetic and wiggler fields. As a result, the phase velocity of the beat wave is less than that of the electromagnetic wave and it is called a ponderomotive wave. Since the ponderomotive wave propagates at less than the speed of light in vacuo it can be in synchronism with electrons that are limited by that velocity. Our purpose in this chapter is to give a detailed discussion of the free-electron laser as a linear gain medium as well as to provide a comprehensive derivation of the relevant formulae for the gain in various configurations in both the idealized one-dimensional and the realistic three-dimensional limits. To this end, we derive the expressions for the gain in both the low and high gain regimes. The low gain regime is relevant to short wavelength free-electron laser oscillators driven by high energy but low current electron beams. In contrast, the results in the high (exponential) gain regime are usually described in terms of a dispersion equation and are appropriate to free-electron laser amplifiers and SASE driven by intense relativistic electron beams.