3. Incoherent Undulator Radiation

The spontaneous synchrotron radiation produced by individual electrons executing undulatory trajectories in a magnetostatic field is incoherent and is the radiation mechanism used in synchrotron light sources. The magnetostatic field in these devices are formally identical to those employed in free-electron lasers but are commonly referred to as undulators rather than wigglers. The reason for this is that electron synchrotrons produce high-energy electron beams which permit the use of extremely long-period undulations. The use of long-period undulations makes possible the production of relatively large-amplitude magnetostatic fields which are required to ensure the production of a relatively high radiation intensity from this incoherent mechanism. In contrast, since the free-electron laser relies on a coherent emission process, the wiggler magnets employed can be of shorter periods and lower amplitudes. However, incoherent synchrotron radiation is produced in free-electron lasers as well. In this chapter, we present a derivation of the spontaneous undulator radiation emitted as individual electrons propagate through the wiggler. However, it should be remarked that this is only one part of the process. The spontaneously emitted photons can stimulate the emission of more photons or be reabsorbed. The complete physics must include both the spontaneous and stimulated emission mechanisms. Aspects of the stimulated emission including both the linear instability and nonlinear saturation of that instability are presented in succeeding chapters.