In this work role of density profiles for the nonlinear propagation of intense laser beam through plasma channel is analyzed. By employing the expression for the dielectric function of different density profile plasma, a differential equation for beamwidth parameter is derived under WKB and paraxial approximation. The laser induces modifications of the dielectric function through nonlinearities. It is found that density profiles play vital role in laser-plasma interaction studies. To have numerical appreciation of the results the propagation equation for plasma is solved using the fourth order Runge-Kutta method for the initial plane wave front of the beam, using boundary conditions. The spot size of the laser beam decreases as the beam penetrates into the plasma and significantly adds self-focusing in plasma. This causes the laser beam to become more focused by reduction of diffraction effect, which is an important phenomenon in inertial confinement fusion and also for the understanding of self-focusing of laser pulses. Numerical computations are presented and discussed in the form of graphs for typical parameters of laser-plasma interaction. 1. Introduction There has been considerable interest in the interaction of laser beam with plasma. Among the nonlinear optical effects, the self-interaction of intense laser beam occupies a significant place. The interaction of intense laser beam with plasma modifies the dielectric function of the medium. Moreover, the dielectric constant of the plasma depends on amplitude of laser beam, which further changes the propagation characteristics of the laser beam. Laser beam propagating in plasma channel can create its own waveguide in which geometric and diffraction divergence are removed and beam is self-focused. Self-focusing of laser beams is an important nonlinear process in laser produced plasma and laser based charged particle accelerators. For ultrafast laser pulses lasting the order of a picoseconds or less, the drift velocity of electrons in a plasma can be comparable to the velocity of light, causing a significant increase in the mass of the electron and consequently in the effective dielectric constant of the plasma. The nonlinearity in the dielectric constant arises on account of relativistic variation of mass for arbitrary magnitude of intensity and the density perturbations. With the recent advancements in laser technology beam power can be a thousand times larger than the critical value. High power lasers are capable of achieving very high electric field and thus plasma can sustain extremely intense
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