To improve the deposition rate of thin films, a novel oscillating pulse magnetron sputtering technology (OPMS) was developed to substitute the traditional high-power impulse magnetron sputtering (HiPIMS). Meanwhile, the relative density and the mechanical properties were also significantly enhanced by this method. In this study, OPMS was used to prepare the pure Cu film, and the effect of the target current on the mode of copper atoms leaving the target (off-target method) under argon gas atmosphere was also investigated. The results showed that with the increase of the target current, the off-target method of copper atoms was transformed from sputtering to evaporation, the surface cracks’ width of the deposited films gradually decreased, and the lattice constants of the Cu films were close to the bulk materials. Furthermore, the deposition rate of Cu films obviously increased from 19 to 103 nm/min. The crystal structures of Cu films showed a face-centered cubic structure, and the grain size increased from 13 to 18 nm, with the target current increased from 2 to 18 A. Moreover, Cu films deposited at currents of 8 and 13 A exhibited excellent adhesion.