This dissertation explores the inertial confinement fusion using the fast ignition concept (FI) and the cone-in-shell method. Unlike the conventional approach where the same laser is used for compression and ignition, here, two separate laser beams are being used. Here, we will focus on the ignitor beam, generated by the LFEX, the most powerful laser in the world. This beam is a picoseconds pulse reaching an intensity greater than 10^19 W/cm^2 , used to generate relativistic electrons from the cone target to the imploding fuel core, giving it the energy necessary to initiate the thermonuclear reaction. However, picoseconds laser pulses are preceded by a pedestal of few nanoseconds and several orders of magnitude lower intensity than the main picoseconds one, preventing the generation of an ideal hot electron beam. This dissertation will show how a pedestal can affect the relativistic electron generation, and how Plasma Mirrors (PM) can be used in order to improve the beam contrast, thus increasing the energy absorption of the fuel core. We will also see how we setup the PM for the fusion chamber of the Institute of Laser Engineering of the Osaka University.