Self-focusing of electormagnetic waves as a result of relativistic electron-mass variation

Relativistic electron-mass variations due to the presence of intense electromagnetic radiation in the plasma cause a nonlinear refractive index. Using a variational principle the latter is obtained up to fourth order in the electric field amplitude and it is shown that nonlinear effects of the second order lead to self-focusing of a beam of radiation. By nonlinear optics considerations, the self- focusing length of an axially symmetric beam is obtained. Including higher- order dispersive effects it is shown that within the thin-beam approximation thecomplex electric field envelope obeys a cubic nonlinear Schrödinger equation with an attractive self-consistent potential. The cylindrically symmetric nonlinear Schrödinger equation predicts collapse of the radiation at the self-focusing distance. The nature of the self-focusing singularity is analysed and it is shown that higher-order nonlinearities saturate the amplitude. Then oscillations of beam radius along the axial direction occur.