Abstract
A paraxial ray formalism is developed to study the evolution of an on axis intensity spike on a Gaussian laser beam in a plasma dominated by relativistic and ponderomotive non-linearities. Ion motion is taken to be frozen. A single beam width parameter characterizes the evolution of the spike. The spike introduces two competing influences: diffraction divergence and self-convergence. The former grows with the reduction in spot size of the spike, while the latter depends on the gradient in non-linear permittivity. Parameter delta = (omega(P)r(00)/c) a(00)/(3.5 r(00)/r(01)) characterizes the relative importance of the two, where r(01) and r(00) are the spike and main beam radii, (p) is the plasma frequency, and a(00) is the normalized laser amplitude. For delta > 1, the intensity ripple causes faster self-focusing of the beam; higher the ripple amplitude stronger the focusing. In the opposite limit, diffraction divergence increases more rapidly, slowing down the self-focusing of the beam. As the beam intensity rises due to self-focusing, it causes stronger generation of the third harmonic.