This article studies the relativistic self-focusing of an intense laser beam in underdense plasma. Due to relativistic self-focusing of the laser beam, a transverse intensity gradient is created, and this can generate a plasma wave at the pump wave frequency. The mechanism of the plasma wave, second-harmonic generation, and the parameters which govern the second-harmonic yield, have been studied in detail. For a typical laser with wavelength (λ=1.053 μm), intensity (1018 W cm−2) and a plasma having density (ωp/ω0=0.25), the second-harmonic intensity was found to be around (1014 W cm−2).

This paper presents the stimulated Raman scattering (SRS) of relativistic laser beam in unmagnetized plasma. The effect of the relativistic electron mass nonlinearity and the relativistic self-focusing of the pump laser on the SRS process is studied. The effect of nonlinear coupling between the pump laser and scattered laser beam has been incorporated. Semianalytical solution for the back reflectivity for SRS has been obtained. For a typical laser wavelength (λ0=1.053μm), intensity (1018Wcm−2), and a plasma having density (ne∕nc=0.08), the back reflectivity comes out to be ∼10%.

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In this work, the problem of relativistic self-focusing of a q-Gaussian laser beam propagating in unmagnetized plasma is studied using a simple heuristic approach. The q-Gaussian profile is analyzed with respect to full width at half maximum (FWHM), intensity distribution, and critical power. A modified version of the Akhmanov method is used to solve the non-linear differential equation for the q-Gaussian beam. The results showed that the FWHM of the q-Gaussian beam substantially depends on the q-factor, and for high values of q, a small part of the beam energy becomes contained within the FWHM. The q-parameter is more effective in the diffraction term than the nonlinear term. The critical power of the q-Gaussian laser beam needed for self-focusing as q→1 becomes extremely high, and in this case, it is very hard to wave-guide such a beam. Moreover, the critical power of self-focusing for q-Gaussian is highly dependent on the value of q-parameter, and as q goes to zero, the critical power needed for self-focusing becomes extremely high. All the derived expressions for the q-Gaussian beam become identical to those for a normal Gaussian beam as q approaches to infinity.

Second harmonic generation of a right circularly polarized Gaussian electromagnetic beam in a magnetized plasma is investigated. The beam causes Ohmic heating of electrons and subsequent redistribution of the plasma, leading to self-defocusing. The radial density gradient, in conjunction with the oscillatory electron velocity, produces density oscillation at the wave frequency. The density oscillation beats with the oscillatory velocity to produce second harmonic current density, giving rise to resonant second harmonic radiation when the wave frequency is one-third of electron cyclotron frequency. The second harmonic field has azimuthal dependence as exp(iθ). The self-defocusing causes a reduction in the efficiency of harmonic generation.