Growth of spike in relativistic Gaussian laser beam in a plasma and its effect on third-harmonic generation

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 δ = (ωp 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, ωp is the plasma frequency, and a 00 is the normalized laser amplitude. For δ > 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.