Abstract: Based on vectorial Debye theory, the focusing properties of linearly polarized power-exponent-phase vortex beam focused by a high numerical aperture objective is studied. The influence of the topological charge and power index on the focal intensity and the phase of x-component is investigated in detail. The result shows that different types of focal intensity distribution, such as dark-core shape focal beam with beam size of 2 wavelength, can be generated by controlling the topological charge and power index. In particular, it is found that the phase singularity of power-exponent phase vortex beam has a deviation from the geometrical focus. The potential applications of this study may include particle manipulation.