  The generation of collimated, high brilliance $\gamma$-ray beams from a
structured plasma channel target is studied by means of 2D PIC simulations.
Simulation results reveal an optimum laser pulse pulse duration of
$20\,\text{fs}$, for generating $\gamma$-photon beams of brilliances up to
$10^{20}\,\text{s}^{-1}\text{mm}^{-1}\text{mrad}^{-2}\,(0.1\,\%\text{BW})^{-1}$
and photon energies well above $200$ MeV in the interaction of an ultra-intense
laser (incident laser power $ P_L \geq 5$ PW) with a high-Z carbon structured
plasma target. These results are aimed at employing the upcoming laser
facilities with multi-petawatt (PW) laser powers to study the laser-driven
nonlinear quantum electrodynamics processes in an all-optical laboratory setup.
