Variational principle approach to short-pulse laser-plasma interactions in three dimensions

Abstract

An approach for describing the evolution of short-pulse lasers propagating through underdense plasmas is presented. This approach is based upon the use of a variational principle. The starting point is an action integral of the form S[a,a(*),straight phi]=integrald(4)x L[a,a(*),straight phi, partial differential(&mgr;)a, partial differential(&mgr;)a(*), partial differential(&mgr;)straight phi] whose Euler-Lagrange equations recover the well-known weakly nonlinear coupled equations for the envelope of the laser's vector potential a, its complex conjugate a(*), and the plasma wave wakes' (real) potential straight phi. Substituting appropriate trial functions for a, a(*), and straight phi into the action and carrying out the integrald(2)x( perpendicular) integration provides a reduced action integral. Approximate equations of motion for the trial-function parameters (e. g., amplitudes, spot sizes, phases, centroid positions, and radii of curvature), valid to the degree of accuracy of the trial functions, can then be generated by treating the parameters as a new set of dependent variables and varying the action with respect to them. Using this approach, fully three-dimensional, nonlinear envelope equations are derived in the absence of dispersive terms. The stability of these equations is analyzed, and the growth rates for hosing and symmetric spot-size self-modulation, in the short-wavelength regime (k approximately omega(p)/c) are recovered. In addition, hosing and spot-size self-modulational instabilities for longer wavelength perturbations (k<<omega(p)/c), and an asymmetric spot-size self-modulational instability are found to occur. The relationships between the variational principle formalism, the source-dependent-expansion (SDE), and moment methods are presented. The importance of nonlinear effects is also briefly discussed, and possible directions for future work are given.