Introduction The non-linear interaction of highly powerful laser beam with plasma and phenomena corresponding to self-focusing of an intense laser beam has become a matter of significant interest from last three decades because of various wide ranging applications such as laser-driven acceleration

Introduction
The non-linear interaction of highly powerful laser beam with plasma and phenomena corresponding to self-focusing of an intense laser beam has become a matter of significant interest from last three decades because of various wide ranging applications such as laser-driven acceleration, x-ray lasers, harmonic generation etc. 1-5. For the success of these applications, high intensity laser beam should propagate with high precession to directionality over long distance. When laser beam interacts with a nonlinear medium like plasma, it provides an oscillatory velocity to the electron and hence, changes the dielectric constant related to plasma 6. The process of self-focusing of laser beam in the plasma has been studied by various researchers 7-9 and is important for the above mentioned applications. Describing the plasma as a nonlinear medium, the process of self-focusing of electromagnetic beams was initiated and the effective dielectric constant was supposed to consisting effect of the electric vector amplitude 10. Later on expression for the nonlinear dielectric constant was proposed and considered in laser plasma interactions 11-13 and it resulted in to three distinct regions viz steady divergence, self-focusing and oscillatory divergence. Thereafter, the phenomenon of self-focusing of a laser beam in plasma has extended over to the fields of magnetoplasma 14-17, ramped density plasma 17-24, rippled density plasma 25-30.

Wang and Zhou 31 considered the ponderomotive nonlinearity and found that the plasma density is excelled radially from the central axis and results in plasma cavitation at lower laser temperature. Liu and Tripathi 32 proposed a non-paraxial theory and expanded the eikonal in powers of for the self-focusing of electromagnetic beams. They found that because of strong self-convergence of marginal rays, the ring shaped laser intensity profiles are formed within a very few Rayleigh lengths. Later, Amrita and Sharma 33 used the same theory in studying the thermal self-focusing of the laser beyond near axis approximation in collisional plasma. They reported that the extent of self-focusing increases with enhancement in laser power. Further, the temperature dependence of plasma ionization and recombination processes influences the propagation dynamics of laser beam. Due to increase in recombination and ionization, the electron heating gets reduced which in turn reduces the focusing effect 34.However, for hot magnetized plasmas increase in the strength of wiggler magnetic field leads the laser spot size to decrease and vice versa for increase in temperature. With the result the self-focusing becomes more focused for right-hand polarization and vice versa 35.Furthermore, decrease in plasma temperature and increase in oblique magnetic field increases the quality of self-focusing for right-hand polarization and increases defocusing for left-handed polarization 36.Again, Wani et al. 37 reported that the quality of self-focusing is improved in semiconductor quantum plasma (ScQp). The HchG beam gives freedom to decentered parameter which changes the self-focusing nature of laser beam significantly.

Now, keeping in view the ongoing development of high intense laser beams, current work is aimed to examine combined effect of the plasma density transition and higher order axial electron temperature on the self-focusing / defocusing of laser beam propagating in plasma. The effect of optimized parameters pertaining to laser and plasma is seen on self-focusing of the laser beam in plasma under density ramp. Results so obtained reveal that the higher order axial electron temperature is highly useful for self-focusing than the research of neglecting its effect. The paper is organized as: in section 2, electron temperature and nonlinear dielectric constant is presented. In section 3, the basic formulation and the differential equation defining the behavior of beam width parameter is given. Section 4 pertains to results and discussions. Finally, in section 5, the conclusion is added.