Composite polarization singularities in superimposed Laguerre-Gaussian beams beyond the paraxial approximation.

The closed-form expression for the free-space propagation of superimposed Laguerre-Gaussian beams beyond the paraxial approximation is derived, and the composite polarization singularities formed by the transverse and longitudinal electric-field components are studied in detail. It is shown that there exist composite C-points and L-lines in vector nonparaxial fields. By suitably varying a control parameter, such as the off-axis distance, relative phase, or amplitude ratio, the motion, creation, and annihilation of composite C-points may appear, and in the process the sum of topological charge remains unchanged. The shift, deformation, combination, and disappearance of composite L-lines may take place. The topological relationship holds true. The results are compared with the previous work.