• Generation of high energy beam with LWFA is the most efficient and low in cost as compared with conventional accelerator. • In LWFA, we firstly performed this acceleration with the combination of up ramp and down ramp density region. • We firstly performed the work related with production of high energy beam with self-injected electrons. • We firstly performed the work with ultra – short laser pulse and polygonal plasma density profile for LWFA. High-energy particle beam is used to probe the local or long range structure and properties of materials. Particle accelerators are also used in material science to understand radiation damage, particularly in studies of structural material to be applied in fusion power generator and satellites. Laser Wakefield Acceleration (LWFA) is a promising technique to build compact and powerful particle accelerators for generating multi-GeV electron beams. The efficient coupling of laser energy with plasma is very important to generate high energy electrons, which is the subject to optimum laser and plasma profiles. In this article, we have used high intensity ultra-short laser pulse and a polygon plasma density profile to analyse the behaviour of LWFA in the bubble regime, which corresponds to the evacuation of plasma electrons in the wake of the laser pulse. Many electrons get self-injected in this wakefield and get accelerated behind the laser pulse up to very high energies, producing high energy beam useful for material processing along with their other applications. We have diagnosed some parameters such as electron density variation, wakefield magnitude and energy spectrum of the electrons accelerated in this mechanism.