Up to this point we have described the laser–plasma interaction and the generation of the nonlinear force exclusively on the basis of plasma hydrodynamics and Maxwell’s theory. The hydrodynamics was limited to space charge quasi-neutrality according to Schlüter’s assumption for deriving the two-fluid equations [Eqs. (3.67) and (3.69)], which with the requirement of momentum conservation of obliquely incident laser radiation on plasma had to be extended to Eqs. (4.62) and (4.33) or (4.55), where some high-frequency electrical charge effects had to be realized. The essential point for the appearance of the nonlinear force was that there was a phase shift between E and H of the waves in an inhomogeneous medium according to a WKB approximation (Hora et al. 1967), which could be understood from the Rayleigh case of inhomogeneous media (Hora 1957) (see Fig. 3.7).
We now look for another derivation of the nonlinear force based on the single-particle motion of an electron in the laser field in an inhomogeneous plasma, and then we contrast the case of any low-density plasma. We are considering the geometry of Fig. 6.1, where laser radiation from vacuum penetrates into a plasma with a continuously growing electron density and therefore decreasing (real part) of the optical refractive index until the critical density. The motion of a single electron in vacuum is the quiver motion, where the electron follows the electric field E with the vacuum amplitude Ev of the linearly polarized laser, oscillating in the y direction and results in an electron velocity vy. Taking the cross-product of this velocity with the magnetic field results in a Lorentz force such that the electron will perform a longitudinal oscillation with twice the optical frequency resulting in figure-8-like motion (Fig. 6.1, vacuum range). The longitudinal motion is a vy/c effect and therefore a somewhat relativistic process. We shall see that this is essential even at laser intensities many orders of magnitudes less than the relativistic threshold, in fact for the whole subrelativistic range.
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