Polarization dynamics on optical axis

In most tight focusing systems with the incident wave of unique polarization (including the radial and azimuthal polarization), the polarization along the optical (central) axis (the propagation direction) is found to be uniform. In this paper, we will show that when there is an off-axis vortex in the incident field, the polarization state varies along the central optical axis and the polarization singularities, 'C-points', 'L-points' both can exist on axis. The polarization state is demonstrated to be mainly dependent on two parameters, the initial position of off-axis vortex and the Gouy phase of the axial field. Due to the special property of the Gouy phase, we find that moving the off-axis vortex can vary the polarization without changing its handedness. Especially, it is quite interesting to find that one direction of the linear polarization at the geometrical focus exactly corresponds to one position of the off-axis vortex.