Relativistic increase of critical electron density

Original quasineutrality of a plasma at rest is heavily perturbed when the electrons are induced to oscillate relativistically by a superintense laser beam. This represents one of the major difficulties when studying the propagation of intense linearly polarized electromagnetic waves in plasmas. Particular attention has to be dedicated to the effective relativistic increase of the critical density and the density of maximum laser energy deposition for applications, e.g. fast ignition studies. When the electron particle density at rest n0 is set into motion of relativistic velocity ve in the lab frame it transforms into ne = γen0, γe Lorentz factor [1]. The relativistic transformation of the particle density is a consequence of the well-known Lorentz contraction. In connection with magnetic fields, in order to avoid contradictions, its influence in Poisson’s equation must be taken into account at arbitrary low speeds, when transforming from one reference system into another. Simple example: The magnetic field of a parallel monoenergetic electron beam vanishes in the co-moving reference system and the former Lorentz force on an electron appears now as an additional equivalent electrostatic repulsion resulting from Poisson’s law. Owing to their inertia the ions can generally be considered immobile with density n0 (charge is taken as Z = 1 for simplicity). As a result a strong longitudinal electric field is induced in presence of any electron motion which tries to restore the original quasineutrality. For this reason transforming electron densities just formally may produce erroneous results. Relativistic self-focussing has been observed experimentally. For circular polarization it has been shown that the refractive index of a fully ionized plasma is given by [2],