GENERAL AND APPLIED PHYSICS Combined Effect of Chirp and Exponential Density Ramp on Relativistic Self-focusing of Hermite-Cosine-Gaussian Laser in Collisionless Cold Quantum Plasma Vishal Thakur 1 & Niti Kant 1 Received: 16 October 2018 / # Sociedade Brasileira de Física 2018 Abstract This manuscript presents the dependence of exponential plasma density ramp on relativistic self-focusing of the Hermite-cosine- Gaussian (HcosG) chirped pulse laser in a collisionless cold quantum plasma. Self-focusing along with self-trapping of HcosG chirped pulse laser is studied for different values of laser intensity, decentered parameters, and plasma density. Numerical analysis reveals that these parameters play a key role in achieving the earlier and stronger self-focusing in cold quantum plasma. A comparative study of self-focusing of the HcosG chirped pulse laser with and without exponential density ramp profile is carried out. Exponential ramp profile is found to be more effective in achieving the stronger self-focusing of the HcosG chirped pulse laser. Combined effect of chirp and exponential plasma density ramp on relativistic self-focusing of HcosG laser in a collisionless cold quantum plasma is analyzed, and results reveal the early and strong self-focusing. Keywords Hermite-cosine-Gaussian laser . Self-focusing . Collisionless cold quantum plasma . Exponential plasma density ramp 1 Introduction In the past few decades, interaction of short pulse laser with plasma has been a very attractive area of research due to nu- merous applications, like inertial confinement fusion [1–3], laser electron acceleration [4, 5], and harmonic generation [6]. Whenever laser beam propagates deep into the plasma, the electrons are expelled away from the central axis due to ponderomotive force which in turn modifies the refractive index of the medium leading to self-focusing of the laser beam. Three types of the self-focusing mechanisms were ob- served in plasma due to interaction with laser beam, namely, thermal, ponderomotive, and relativistic self-focusing. Siegrist [7] considered the collective effect of relativistic and ponderomotive forces and reported self-focusing during laser plasma interaction. Later on, Singh and Walia [8] reported self-channeling and relativistic self-focusing of the Gaussian laser beam by following moment theory to analyze the differ- ential equation of beam width parameter, which is non-linear in nature, and then employing Runge-Kutta method to solve it numerically. Further, by considering ponderomotive non-lin- earity, self-focusing of the Hermite-cosh-Gaussian laser beams in magnetoplasma under the effect of decentered pa- rameter was theoretically stated [9]. The importance of density transition and the decentered parameter in the enhancement of self-focusing of the laser beam was analyzed [ 10]. Further, it was extended to Hermite-cosh-Gaussian laser beams, and it was observed that the effects of plasma density ramp and magnetic field become more effective in order to have stronger self-focusing [11]. Also, the sensitiveness of decentered parameter in process of self-focusing was noticed [12]. They observed that by consid- ering the optimized value of decentered parameter, one can have strong self-focusing. Kant et al. [13] described in detail that in absence of the density ramp because of the dominance of diffraction effect, the laser pulse defocuses and on the other hand the phenomenon of self-focusing becomes stronger with the involvement of tangential plasma density ramp. Furthermore, strong self-focusing effects in plasma with quan- tum effects were observed in comparison to that of classical relativistic one [14]. Kant and Wani [15] stated that by opti- mizing laser and plasma parameters, stronger self-focusing was observed under the combined effect of linear absorption and plasma density transition. It was observed that absorption * Niti Kant nitikant@yahoo.com 1 Department of Physics, Lovely Professional University, G.T. Road, Phagwara, Punjab 144411, India Brazilian Journal of Physics https://doi.org/10.1007/s13538-018-00624-7