In the paper, relativistic self-focusing in the interaction of laser beam and plasma with periodical density ripple has been studied by the applied WKB approximation and higher-order paraxial theory. The result shows that under the influence of relativistic nonlinear effect, the dielectric function shows the fierce oscillational variation with similar periodicity, which then leads to the intense relativistic beam self-focusing along the propagation distance, such self-focusing also presents similar periodic variation. Besides, in the plasma with periodical density ripple, the initial density and the density ripple amplitude have obvious influence on self-focusing. When the two factors increase, then there will be more strength self-focusing. Choosing the appropriate initial density and the periodic density parameter is benefit to the formation of the more stable self-focusing.

This article presents a preliminary study of the longitudinal self-compression of ultra-intense Gaussian laser pulse in a magnetized plasma, when relativistic nonlinearity is active. This study has been carried out in 1D geometry under a nonlinear Schrodinger equation and higher-order paraxial (nonparaxial) approximation. The nonlinear differential equations for self-compression and self-focusing have been derived and solved by the analytical and numerical methods. The dielectric function and the eikonal have been expanded up to the fourth power of r (radial distance). The effect of initial parameters, namely incident laser intensity, magnetic field, and initial pulse duration on the compression of a relativistic Gaussian laser pulse have been explored. The results are compared with paraxial-ray approximation. It is found that the compression of pulse and pulse intensity of the compressed pulse is significantly enhanced in the nonparaxial region. It is observed that the compression of the high-intensity laser pulse depends on the intensity of laser beam (a0), magnetic field (ωc), and initial pulse width (τ0). The preliminary results show that the pulse is more compressed by increasing the values of a0, ωc, and τ0.

A nonparaxial investigation for propagation characteristics of q-Gaussian laser beam in rippled density plasma is studied by considering the relativistic nonlinearity. The field distribution in the medium is expressed in terms of q parameter and beam width parameter f. Nonlinear parabolic partial differential equation governing the evolution of complex envelope in slowly varying approximation is solved in a modulated density profile. Analytical theory of self-focusing including higher order terms in the expansion of dielectric function up to fourth order is developed and the variation of beam width parameter f with the distance of propagation for different parameters is studied. One may note that increased value of density ripple, laser intensity and depth of modulation, increases self-focusing whereas a lower value of q shows strong self-focusing. A comparative study between paraxial and nonparaxial study has also conducted. This study is useful for research in high energy density physics.

In the present work, excitation of nonlinear ion acoustic wave (IAW) in collisionless plasma by laser beam having null intensity at the center is examined considering relativistic nonlinearity. The differential equation for beam-width parameter is determined considering relativistic nonlinearity using the paraxial and Wentzel–Kramers–Brillouin approximations by the parabolic equation method. The propagation features of the IAW are found to be modified due to the nonlinearity present in the system. The hollow Gaussian beam (HGB) gets nonlinearly coupled with the seed IAW, results in excitation of nonlinear IAW. The interaction of nonlinear IAW with pump beam demonstrated stimulated Brillouin scattering (SBS) of HGB. It is found that the power of IAW and power of SBS is affected with the order of HGB. The power of IAW and backscattered power of SBS is determined analytically and numerically for various orders of HGB. It is found that the power of IAW and the backscattering is diminished for higher order of HGB.

The present paper proposes a model for the generation of Terahertz (THz) radiation by self-focused hollow Gaussian beam (HGB) in collisionless magnetized rippled density plasma. At high intensities, the change in the electron mass occurs due to relativistic effect, introducing a nonlinearity in the plasma leading to the self-focusing of the HGB. The nonlinear interaction of this highly intense self-focused HGB with the electron plasma wave in the rippled density plasma, satisfying proper phase matching conditions, results in the resonant excitation of THz radiations at the beat frequency. We have studied the dependence of generated THz radiations on the order of the HGB as well as on the static background magnetic field. The results show that the intensity of the generated radiations is highly sensitive to both of these parameters. For the current scheme the power of the generated THz waves comes out to be of the order of Gigawatts.

Effect of relativistic nonlinearity on stimulated Raman scattering (SRS) of laser beam propagating carrying null intensity in center [hollow Gaussian beam (HGB)] is studied in collisionless plasma. The construction of the equations is done employing the fluid theory which is developed with partial differential equation and Maxwell's equations. The phenomenon of SRS is shown along with the excitation of seed plasma wave considering relativistic nonlinearity. The power of plasma wave is observed for higher order of HGB. The Raman back reflectivity is studied numerically for various orders of hollow Gaussian laser beam (HGLB) and the numerical analysis shows that these parameters play vital role on reflectivity characteristics. It is observed that the Raman back reflectivity is less for the higher order of HGLB.

The combined effects of filamentation and magnetic field on the third-harmonic generation of electromagnetic beams have been investigated considering extended paraxial rays in magneto plasma. The analysis is done using eikonal method in which eikonal and other relevant quantities are extended up to fourth power of r. The time scale of laser beam is chosen such that the relativistic mass variation of electron becomes dominated source of nonlinearity in refractive index. The expression for coupling between ultra-intense laser beam and electron plasma wave due to relativistic nonlinearity has been deduced. Interaction of the seed plasma wave with the incident filamented laser beam excites the plasma wave and generates third harmonics. The expressions for plasma wave power and third-harmonic power have been derived. The effect of the magnetic field on the power of plasma wave and the third-harmonic power has been carried out. The role of magnetic field has been found to decrease the power of plasma wave and so as the power of third harmonic. Our results can be helpful for various laser plasma diagnostics experiments in which magnetic field present externally or generated spontaneously in the high-power laser–plasma interaction.

This paper presents a model to study the stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) simultaneously at relativistic laser power. At high intensity, the relativistic mass correction for the plasma electrons becomes significant and the plasma refractive index gets modified which leads to the relativistic self-focusing of the pump beam. This filamentation process affects the scattering processes (SRS and SBS) and at the same time the pump filamentation process also gets modified in the presence of the coexisting SRS and SBS due to the pump depletion. We have also demonstrated that the pump depletion and relativistic filamentation affects the back-reflectivity of scattered beams (SRS and SBS) significantly, for the coexistence case.

This paper presents an investigation of a self-consistent, theoretical model, which explains the ring formation in a super-Gaussian laser beam propagating in plasma with transverse magnetic field, characterized by relativistic nonlinearity. Higher order terms (up to r4) in the expansion of the dielectric function and the eikonal have been taken into account. The condition for the formation of a dark and bright ring has been used to study focusing/defocusing of the beam. It is seen that inclusion of higher order terms does significantly affect the dependence of beam width parameter on the distance of propagation. Self-focusing of super-Gaussian beam is studied at various values of super-Gaussian coefficient, m and magnetic field. Further, we have studied some distinct features of critical power curves for varying values of magnetic field.

This article presents the cross-focusing of two high power dark hollow Gaussian beams in plasma when relativistic nonlinearity is operative. A paraxial like approach has been used in the present analysis. In this study, the non-linear dielectric function has been expanded in terms of radial distance from the maximum of the irradiance, rather than from the axis, as is the case of Gaussian beams. The nature of propagation of a hollow Gaussian beam propagating in plasmas has been studied under the influence of relativistic non-linearity. The effect on the order (n) of hollow Gaussian beam on the cross-focusing of two beams has been explored in relativistic non-linearity in this publication.

In the present investigation, the excitation of an upper hybrid wave (UHW) in a hot collisionless magneto-plasma by a relativistic laser beam propagating perpendicular to the static magnetic field and having its electric vector polarized along the direction of the static magnetic field (ordinary mode) is presented. Due to nonuniform intensity distribution of pump laser, the background electron concentration is modified. The amplitude of the UHW, which depends on the background electron concentration, is thus nonlinearly coupled with the laser beam. The effect of nonlinear coupling between the pump laser and UHW is studied. The effect of the relativistic electron mass nonlinearity and the relativistic self-focusing of the pump laser on the excitation of the UHW have been incorporated. The dynamics of the excitation of the UHW in different power domains of the laser beam is accordingly modified. It has been seen that the effect of changing the strength of the static magnetic field on the nonlinear coupling and the dynamics of the excitation of the UHW is significant. The focusing behavior of the UHW may find its relevance in the heating of plasmas near the upper hybrid resonance.