Threshold behavior of the three-wave parametric-decay interaction in bounded inhomogeneous plasmas

The three-wave parametric-decay absolute instability in a bounded inhomogeneous plasma, as a function of the plasma length L, the basic gain length L0, and the wavenumber mismatch gradient κ is examined numerically and analytically. The stability boundary in the (Λ,α) plane, with Λ=L/L0 and α=κ L20, is found to be composed of a seesaw structure of branches merging at common values of α and of the product wave frequency shift. As Λ increases, the stability boundary degenerates into the α=8/π line, the plasma tends to perform the wave conversion in a zone of width 4/κ L0, and the best-matched point shifts to within a distance of 2/κ L0 from the boundary of pump incidence. Finally, a novel integral representation of the wave-coupling problem is formulated which allows straightforward application of asymptotic methods under long plasma and/or strong pump conditions.