Steady state self-focusing, self-modulation of laser beam in an inhomogeneous plasma

Laboratory as well as PIC simulation experiments reveal strong flow of energetic electrons co-moving with laser beam in laser plasma interaction. Equation governing the evolution of complex envelope in slowly varying envelope approximation is nonlinear parabolic equation. Variational approach is used to solve this problem and a Lagrangian for the problem is set up. Assuming a trial Gaussian profile, authors solve the reduced Lagrangian problem for beam width and curvature. Two scale lengths for inhomogenity along the direction of propagation, one for nonlinearity and other for diffraction management are introduced. Self-focusing, self-modulation as well as self-trapping of the laser-electron-beam plasma system is studied under variety of parameters.