INTERACTION OF A WEAKLY NONLINEAR LASER-PULSE WITH A PLASMA

Based on a one-dimensional model, a perturbation expansion is carried out to solve the equations describing a weakly nonlinear laser pulse in a plasma in which the electrons are treated relativistically and the plasma frequency is much less than the laser frequency. To lowest order, the expansion yields two coupled equations for the vector and scalar potentials. For a pulse which is long compared with a plasma wavelength, the coupled equations reduce to the nonlinear Schrodinger equation with well-known soliton solutions. An initial pulse of hyperbolic-secant shape which is short compared with a plasma wavelength broadens and acquires a characteristic asymmetric shape with a steep trailing edge and a much broader, gently sloping front portion, and has a frequency and wave-number shift which vary from a positive value at the front to a negative value at.the rear of the pulse. The peak and rear part of a short pulse are strongly influenced by nonlinear effects, whereas the front is governed primarily by linear dispersion. The average pulse frequency continually decreases as energy is lost to the plasma wake. The wake-field phase velocity is shown to be approximately equal to the velocity of the pulse peak.