ABSTRACT
Ultra-short and dense relativistic electron bunches accelerated in laser wakefields promise an elegant route towards ultra-short, high peak brilliance X-ray sources. By wiggling free electron bunches in magnetic undulators, plasma fields or laser fields, photon energies from 100 eV up to multi-MeV are accessible. After summarizing the basics of the universal synchrotron spectrum, we give a short derivation of the coherence condition for undulator radiation and discuss the spectral characteristics for weak and strong deflection. We exploit similarities of electron motion in electromagnetic waves to give modified expressions for Thomson backscattering radiation and consider the influence of the energy distribution of the electrons and their divergence on the emitted spectrum and photon number. In the context of a wakefield accelerator, the plasma-focusing forces lead to the emission of copious amounts of betatron radiation during the acceleration process. We modify the undulator formalism to describe the betatron emission process, and give a short review of recent experimental findings. Finally, in the appendix we introduce the numerical calculation of electron trajectories for arbitrary wiggling fields, and the emitted radiation from these trajectories, along with some key code examples. These may be used by interested readers as a first step for own numerical simulations.
