the simultaneous effect of the temperature and density gradient on the relativistic self-focusing of the gaussian laser beam in an under-dens plasma

In this paper, the interaction of gaussian laser beam with underdense plasma by taking the weakly relativistic ponderomotive nonlinearity has been investigated. for this purpose, the effect of the linear plasma electron temperature and upward exponential electron density profile on the laser propagation has been studied. the nonlinear second-order differential equation of the dimensionless beam-width parameter, f, on the distance of propagation, h, is derived by following wkb and paraxial approximation and solved numerically for the several initial electron temperatures. it is found that, the electron temperature ramp combined with upward ramp density profile would be caused stronger self-focusing where the beam width oscilates with less amplitude and smaller spot size. this could lead to further penetration of the laser beam inside the plasma by reducing the effects of diffraction.