  Using two-dimensional (2D) and three-dimensional (3D) kinetic simulations, we
examine the impact of simulation dimensionality on the laser-driven electron
acceleration and the emission of collimated $\gamma$-ray beams from hollow
micro-channel targets. We demonstrate that the dimensionality of the
simulations considerably influences the results of electron acceleration and
photon generation owing to the variation of laser phase velocity in different
geometries. In a 3D simulation with a cylindrical geometry, the acceleration
process of electrons terminates early due to the higher phase velocity of the
propagating laser fields; in contrast, 2D simulations with planar geometry tend
to have prolonged electron acceleration and thus produce much more energetic
electrons. The photon beam generated in the 3D setup is found to be more
diverged accompanied with a lower conversion efficiency. Our work concludes
that the 2D simulation can qualitatively reproduce the features in 3D
simulation, but for quantitative evaluations and reliable predictions to
facilitate experiment designs, 3D modelling is strongly recommended.
