The reflectivity of relativistic ultra-thin electron layers

TL;DR

This paper derives an analytical model for the coherent reflectivity of relativistic, ultra-thin electron layers under oblique incidence, revealing how the Doppler shift depends on the normal velocity component.

Contribution

It introduces a two-fold Lorentz transformation approach to analytically describe the reflectivity of dense, relativistic electron layers with oblique incident light.

Findings

Reflectivity depends on the normal velocity component, not the total gamma factor.

Back-scattered light is directed close to the layer normal.

Doppler shift is governed by gamma_x, related to V_x.

Abstract

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. 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.