Echoes at Upper Hybrid Resonance. II. Theory for a Cold Nonuniform Plasma

Abstract

The Lagrangian description of a one-dimensional slab of cold nonuniform plasma in a uniform magnetic field is used to calculate the properties of temporal echoes generated by the application of two electric field pulses. Under weak excitation, the electron plasma acts as an ensemble of uncoupled nonlinear oscillators with amplitude dependent eigenfrequencies, which are the local upper hybrid frequencies [ωc2 + ωp2(x)]1/2 in the linear approximation. The decay of the single-pulse response by phase mixing of the oscillators and the action of their nonlinear restoring forces yields, in response to two pulses, a train of echoes in the voltage across the slab. The amplitude of the first echo is studied in detail as a function of magnetic field, electron density, excitation power, and time separation of the excitation pulses. This model successfully accounts for the major features of experimental observations of echoes from afterglow plasmas in a uniform magnetic field. The theoretical echo is strongest in a narrow band of frequencies near the maximum upper hybrid frequency of the plasma slab and shows complex saturation properties in general agreement with experiment.