A plasmonic lens with variant periods was investigated for optical behavior at near-field by means of numerical computational method. To study influence of incident light on different polarization modes, we considered linear polarization, circular polarization, elliptical polarization, radial polarization (RP), and azimuthally polarization in our computational analyses. A finite difference and time domain algorithm is employed in the numerical study. Our computational numerical calculation results demonstrate that focusing performance for the plasmonic lens illuminated under radial polarization is best in comparison to that of the illumination with the other four polarization states. The plasmonic lens with RP illumination can realize superfocusing with ultra-long depth of focus. It is possible to be used as an optical probe or a type of plasmonic lens for imaging with high resolution in the near future.