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.

The study proposes a semi-analytical model for the pulse compression of two co-propagating intense laser beams having Gaussian intensity profile in the temporal domain. The high power laser beams create the relativistic nonlinearity during propagation in plasma, which leads to the modification of the refractive index profile. The co-propagating laser beams get self- compressed by virtue of group velocity dispersion and induced nonlinearity. The induced nonlinearity in the plasma broadens the frequency spectrum of the pulse via self-phase modulation, turn to shorter the pulse duration and enhancement of laser beam intensity. The nonlinear Schrodinger equations were set up for co-propagating laser beams in plasmas and have been solved in Matlab by considering paraxial approximation. The propagation characteristics of both laser beams inside plasma are divided into three regions through the critical divider curve, which has been plotted between pulse width τ01 and laser beam power P01. Based on the preferred value of critical parameters, these regions are oscillatory compression, oscillatory broadening, and steady broadening. In findings, it is observed that the compression of the laser beam depends on the combined intensity of both beams, plasma density, and initial pulse width.

The mathematical model for the propagation of intense laser pulse in a plasma having Gaussian profile is investigated. The model has been formulated considering that the relativistic–ponderomotive nonlinearity dominates over other nonlinearities in the plasma. Model equation for self-compression and self-focusing properties of the laser pulse has been set up and solved by both semi-analytical and numerical methods. The result indicates that due to the effect of group velocity dispersion, diffraction of the laser pulse and the nonlinearity of medium, the pulse width parameter as well as beam width parameter of pulse gets focused at a different normalized distance, and hence the normalized intensity is also deferred at those points. Numerical simulation shows an oscillatory behavior of intensity during propagation in the plasma either having minimum beam radius (r 0) or having minimum pulse duration (t 0) depending on the normalized distance.