Simulations of field emission of electrons from a custom built electron gun are used to determine the effect that offsets in alignment of the cathode to the grid and the cathode to grid gap have on the trajectories of electrons transmitted through the grid. The simulations are a first step towards understanding how to optimize the electron gun and determine the magnitude of optical aberrations that limit focusing of the emitted electron beam. A finite element method is employed to simulate field emission from the cathode, an array of carbon nanotube pillars. This is done using a three dimensional model that attempts to account for field enhancement due to both the pillar geometry and individual carbon nanotubes. Qualitative comparisons of the configurations tested indicate larger transmissivity through the grid can be achieved by offsetting the pillar array but only at the expense of a skewed elevation angle distribution.