The corona bar induced pre–ionization is a crucial preliminary process in the operation of ArF excimer lasers, directly impacting the uniformity and stability of output laser. The ultraviolet corona pre–ionization, as the mainstream method, is tightly coupled with the main discharge process, which complicates analysis. Here, we establish a numerical model of a single pulse discharge incorporating an external circuit to analyze the pre–ionization process and its influence on the breakdown characteristics. (1) By adopting detailed input parameters of photoionization model, we observe uniform and dispersed plasma propagation from the corona bar to the main gap. (2) An artificial boundary condition is proposed to investigate the phenomenological effect of high–energy electrons emission, emphasizing the influence of surface discharge along the cathode. (3) The propagation and breakdown characteristics of the two pre–ionization setup methods, photoionization and background electron density, are compared numerically. This study enhances the understanding of the pre–ionization process in ArF excimer lasers and provides theoretical insights for their optimization and design.