The reflectivity of relativistic ultra-thin electron layers

The coherent reflectivity of a dense, relativistic, ultra-thin electron layer is derived analytically for an obliquely incident probe beam. Results are obtained by two-fold Lorentz transformation. For the analytical treatment, a plane uniform electron layer is considered. All electrons move with uniform velocity under an angle to the normal direction of the plane; such electron motion corresponds to laser acceleration by direct action of the laser fields, as it is described in a companion paper [European Physical Journal D 55, 433 (2009)]. Electron density is chosen high enough to ensure that many electrons reside in a volume λR3, where λR is the wavelength of the reflected light in the rest frame of the layer. Under these conditions, the probe light is back-scattered coherently and is directed close to the layer normal rather than the direction of electron velocity. An important consequence is that the Doppler shift is governed by γx equals (1-(Vx/c)2)-1/2 derived from the electron velocity component Vx in normal direction rather than the full γ-factor of the layer electrons. (authors)