Kinetic theory of cross-modulation in a weakly ionized plasma

Abstract

Cross-modulation in plasma is an electromagnetic wave interaction in which the modulation of one ‘disturbing’ wave is imposed nonlinearly on the transport properties of the medium, and thence onto a second, ‘wanted’ wave propagating linearly through it. This analysis is restricted to weakly ionized plasma with allowance for ambient magnetic field, as in the lower ionosphere. A kinetic description is used, based on the Boltzmann equation for the electrons, with electron-molecule collisions described by Boltzmann's collision integral. Because of the small mass ratio this simplifies to a differential form. The perturbation of the electron velocity distribution functionf(v, t) due to the disturbing wave is calculated up to terms quadratic in wave amplitude, which are the lowest order to show the effect. The part of the term quadratic in wave amplitude at zero times the fundamental frequency, and isotropic in velocity space, which represents the perturbation in electron energy distributions, is selectively enhanced by an inverse factor of mass ratio, since the excess energy imparted by the wave to the electrons is transferred collisionally to the molecules at a rate inversely proportional to mass ratio. Modulation of the wave induces modulation of the electron energy distribution. A more general expansion scheme, in velocity-space spherical harmonics, is also presented. To calculate the dispersion relation for the second, ‘wanted’ wave, the linear part of the disturbing wave analysis is adapted, and the amplitude of the wanted wave is given in the WKB approximation as a phase integral of the refractive index along the ray path; this contains moments of the electron energy distribution and is modulated. The predictions of older semi-empirical theories, that the effect is enhanced when the fundamental frequency of the disturbing wave is close to the electron gyrofrequency, and that the second harmonic of the modulation is also imposed on the wanted wave, are confirmed. The wanted wave is predominantly amplitude-modulated, and only amplitude modulation of the disturbing wave is picked up; phase modulation is not transferred. There is no cross-modulation if the collision frequency is independent of collision speed, when contributions from all parts of velocity space cancel.