Longitudinal Wake Field for an Electron Beam Accelerated through a   Ultra-High Field Gradient
  Electron accelerators with higher and higher longitudinal field gradients are desirable, as they allow for the production of high energy beams by means of compact and cheap setups. The new laser-plasma acceleration technique appears to constitute the more promising breakthrough in this direction, delivering unprecedent field gradients up to TV/m. In this article we give a quantitative description of the impact of longitudinal wake fields on the electron beam. Our paper is based on the solution of Maxwell's equations for the longitudinal field. Our conclusions are valid when the acceleration distance is much smaller than the the overtaking length, that is the length that electrons travel as a light signal from the tail of the bunch overtakes the head of the bunch. This condition is well verified for laser-plasma devices. We calculate a closed expression for the impedance and the wake function that may be evaluated numerically. It is shown that the rate of energy loss in the bunch due to radiative interaction is equal to the energy emitted through coherent radiation in the far-zone. Furthermore, an expression is found for the asymptotic limit of a large distance of the electron beam from the accelerator compared with the overtaking length. Such expression allows us to calculate analytical solutions for a Gaussian transverse and longitudinal bunch shape. Finally, we study the feasibility of Table-Top Free-Electron Lasers in the Vacuum Ultra-Violet (TT-VUV FEL) and X-ray range (TT-XFEL), respectively based on 100 MeV and 1 GeV laser-plasma accelerator drivers. Numerical estimations presented in this paper indicate that the effects of the time-dependent energy change induced by the longitudinal wake pose a serious threat to the operation of these devices. 
Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller and Mikhail   Yurkov