k-space theory of laser-plasma instabilities in inhomogeneous plasmas

Any three-wave processes governed by the full-wave coupling equations could be Fourier analyzed. The linear theory with linear inhomogeneity leads to the simplest two first-order coupling equations in the k space. Theoretical analysis of the coupling equations relies on the Schrodinger equation and its eigenvalue. We obtained the general eigenvalue formula for different instabilities. By numerical calculations it shows that the eigenvalue precisely predicts the growth and phase shift of the instabilities. For laser-plasma instabilities, we applied it to stimulated Raman scattering, stimulated Brillouin scattering, two-plasmon decay, and Langmuir decay instability, etc., and discussed the growth dependence on different scattering geometries. Compared to real space theories of three-wave processes which often reduce to first-order equations via slow envelope approximation, the k-space theory is more complete and precise, and only requires the hypothesis of linear inhomogeneity. However, such k-space theory is only valid in the linear theory, otherwise convolution is involved. The extension of k-space theory to the nonlinear three-wave coupling is intriguing, but is still under way.