Key points are not available for this paper at this time.
A simple derivation in the Coulomb gauge of the nonlinear Schr\"odinger equation describing propagation of powerful ultrashort circularly polarized laser pulses in underdense cold inhomogeneous plasmas is presented. Numerical solutions are given for the two-dimensional axisymmetric case for both initially homogeneous plasmas and static preformed plasma columns. These solutions account for (i) diffraction, (ii) refraction arising from variations in the refractive index due to the spatial profile of the electron density distribution, (iii) the relativistic electronic mass shift, and (iv) the charge displacement resulting from the transverse ponderomotive force. The most important spatial modes of propagation corresponding to (1) purely relativistic focusing and (2) the combined action of both the relativistic and charge-displacement mechanisms are described. The latter leads to the formation of stable confined modes of propagation having paraxially localized regions of high intensity and corresponding paraxially situated cavitated channels in the electron density. It is further demonstrated that the dynamical solutions of the propagation tend asymptotically to the lowest eigenmodes of the governing nonlinear Schr\"odinger equation. Finally, the calculations illustrate the dynamics of the propagation and show that the relativistic mechanism promotes the initial concentration of the radiative energy and that the subsequent charge displacement stabilizes this confinement and produces waveguidelike channels.
Borisov et al. (Wed,) studied this question.