Based on the nonparaxial vectorial beam propagation theory, the far-field characteristics of Laguerre-Gaussian beams are studied. By using the angular-spectrum diffraction formula and the stationary-phase method, the analytic expression of the vectorial electrical field distribution for a Laguerre-Gaussian beam with an initial linear polarization state after some propagation distances is obtained. The electric field components along different directions and the spatial distribution of the equiphase surface are numerically calculated for a Laguerre-Gaussian beam which propagates along the z axis and is initially polarized in the x direction. The results show that, after some propagation distances, the electric field component which is parallel to the propagation direction appears and its intensity distribution shows a double-peak pattern. The value of this component is much lesser than that of the perpendicular component. With the increase of the topological charge, this parallel component increases and the light spot divergence degree enhances. This beam has a helical equiphase surface.