Orbital angular momentum, as a basic physical quantity of electromagnetic waves, has been widely studied since 1992. Recently, the geometric phase metasurface, which is also known as Pancharatnam-Berry (P-B) phase metasurface, has been proposed. Because of its frequency-independent and angle-dependent phase control characteristics, it can generate high-performance and broadband vortex wave. However, the current design of reflective metasurface encounters the following problems: 1) the reflected vortex wave is partly blocked by the feeding antenna; 2) in practical applications, the cross-polarized field will inevitably be induced due to the feed antenna and the reflective metasurface. How to avoid the cross-polarization is still worth further investigating. In this work, an offset-fed vortex wave generator is proposed. It consists of a right-handed circularly polarized Archimedes spiral antenna and a reflective metasurface. Firstly, the offset feeding design is introduced to avoid generating the cross-polarized fields caused by the feeding antenna. A geometric meta-atom of the reflective metasurface is designed at a working frequency of 8.5 GHz. By regularly arranging meta-atoms with different orientation angles, the convergence and phase compensation functions are imparted only to the co-polarization field. The cross-polarized field is intentionally weakened and refracted along other directions. Subsequently, a low cross-polarized vortex wave with an enhancement effect is obtained at the desired observation position. There are three contributions made in this work: 1) a P-B meta-atom is proposed to fabricate the reflective metasurface; 2) the conversion relationship between the co-polarized and cross-polarized field is studied from the initial state to the final state, and the four transformation processes are demonstrated in detail; 3) an offset-fed vortex wave generator is established which allows one to generate high-performance vortex beam with arbitrary OAM mode. The experimental results are in good agreement with those simulation results, proving the proposed method effective and feasible. The proposed design shows its advantages including simple structure, polarization selectivity, and regional field enhancement effect, which has great potential applications in vortex wave communication and OAM-based target detection.