Vol.:(0123456789) Optical and Quantum Electronics (2019) 51:109 https://doi.org/10.1007/s11082-019-1818-8 1 3 Evolution of the beam‑width parameter of zeroth‑order Bessel–Gaussian beams in collisional plasma with density ripple L. Ouahid 1  · L. Dalil‑Essakali 1  · A. Belafhal 1 Received: 16 October 2018 / Accepted: 16 March 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract In the present paper, we have investigated the self-focusing of zeroth-order Bessel–Gauss- ian beams in collisional plasma with density ripple. Based on Wentzel–Kramers–Brillouin and the paraxial ray approximations, the nonlinear differential equation governing the evo- lution of beam-width parameter is evaluated. The variation of beam-width parameter with propagation distance is examined for different values of ripple wave number and beam parameters. From our study, it is seen that stronger self-focusing is obtained when the den- sity ripple is important. The results corresponding to Gaussian beams are deduced from this work as a particular case. Keywords Zeroth-order Bessel–Gaussian beams · Wentzel–Kramers–Brillouin approximation · Collisional plasma · Density ripple 1 Introduction The study of high laser beams self-focusing in plasma has taken a considerable interest for many researchers both theoretically and experimentally due to their large applications in laser ablation (Hafeez et al. 2008), generation of X-rays (Arora et al. 2014), excitation of an upper hybrid wave (Purohit et al. 2008), inertial confinement fusion (Michel et al. 2015), electron acceleration (Tajima and Dawson 1979), etc. Recently diverse studies have focused on the self-focusing at laser plasma interaction with weakly relativistic and pon- deromotive regime and by considering the impact of linear absorption for Gaussian, cosh- Gaussian and Hermite-cosh-Gaussian laser beams (Navare et al. 2012; Valkunde et al. 2018; Vhanmore et al. 2017; Patil et al. 2015, 2016; Urunkar et al. 2018; Patil and Takale 2016). However, with the progress of short pulse lasers technology, a production of suita- ble density is feasible which can be observed in gas-jet plasma experiment (Lin et al. 2006; Kuo et al. 2007). In this context, several research works have been interesting to the propa- gation of laser beams in plasma under ramp density profile. The self-focusing of Gaussian * A. Belafhal belafhal@gmail.com 1 Laboratory LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Chouaïb Doukkali University, P. B.: 20, 24000 El Jadida, Morocco