Fig. 1. Schematic of controllable polarization singularity splitting. (a) Schematic of offsetting APB to generate split C-points for displacement sensing. The red line represents the axis of the objective, and the black line represents that of the APB. (b) and (c) are the polarization distribution of the incident APB under co-axis and off-axis conditions, respectively. The red dashed circle represents the objective aperture with radius of R, the red and black dots represent the objective center and V-point in APB, respectively. (d) and (e) are the corresponding polarization distributions at the focal plane. The black dashed circles indicate the position of polarization singularity. In (b)–(e), the short line segments denote the localized linear polarization, the blue and red ellipses denote the left- and right-handed circular polarizations, respectively.

Fig. 2. The Stokes phases (a)–(d) and corresponding polarization ellipses (e)–(h) of the focus field with different offset angles of α=0, π/2, and π/4, respectively. Here, β=0 in (a) and (e), and β=0.3 for others. Background in (e)–(h) is the normalized intensity. The white segment represents linear polarization, the blue ellipse represents left-handed elliptical polarization, and the red ones the right-handed. The NA of the objective is 0.9. The arrow in the yellow circle represents the offset angle.

Fig. 3. Splitting distances with different β. (a)–(c) The distribution of the Stokes phase and (d)–(f) the polarization ellipse states with β=0.1, 0.2, and 0.3, respectively. The NA is kept as 0.9, and background in (d)–(f) is the normalized field intensity. (g) The relationship between the splitting distances of the C-point pair and β under different NAs.

Fig. 4. Local longitudinal SAM density (Sz) of the off-axial scenarios. (a) Spatial distribution of the Sz at the focal plane under various β, offset direction along the x-axis. (b) plots the Sz along the yellow dashed central line in (a). (c) plots the gradient of Sz in (b). Moreover, (b) and (c) share the same x-axis. The wavelength in the simulation is 532 nm, and the NA is 0.9.

Fig. 5. The far-field distribution of the scattered signal of a nanostructure near the C-points. (a) The scattering far-field intensity distribution when nanostructures are located at different positions along the y-axis within the focal field. Here, β=0.3 for all situations. (b) and (c) The corresponding Q-values at various positions when nanostructures sweep along the y-axis, with β=0.3 and β=0.1, respectively.