Space and time behavior of backscattering instabilities in the modified decay regime

The space and time behavior of parametric backscattering instabilities is computed analytically in the so-called modified decay regime. The plasma is assumed to be homogeneous and of finite length. The propagation of the pump wave and its finite pulse duration both are taken into account, its depletion is ignored. The parametric growth is solved in terms of fluctuating initial and boundary conditions corresponding to thermal noise at equilibrium. Fluctuating source terms, representing spontaneous emission of waves, are accordingly retained in the coupled mode equations. The initial stage of the instability is investigated in detail; the time from which the time asymptotic concept of absolute or convective instability applies is computed. Approximate expressions for the fluctuations of the waves, that are uniformly valid for any gain factor and any time, are derived.