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 [Eur. Phys. J. D 55, 433 (2009)]. Electron density is chosen high enough to ensure that many electrons reside in a volume lambda(R) (3), where lambda(R) 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 gamma(x)=(1-(V(x)/c)(2))(-1/2) derived from the electron velocity component V(x) in normal direction rather than the full gamma-factor of the layer electrons.