Polarization singularity splitting and recovery via geometric moments under tight focusing

Symmetry breaking in practical optical systems can split high-order polarization singularities, thereby restricting further application. We introduce a normalized geometric first-moment factor γ to quantify the pupil-plane symmetry breaking caused by arbitrary geometric binary amplitude obstructions in an incident azimuthally polarized beam. The Richards-Wolf vectorial diffraction simulations indicate that the separation of the C-point pair is governed by γ and scales linearly within the perturbative regime, completely independent of the obstruction shape. When the net first moment is balanced to zero (γ = 0) via complementary obstructions, the split C-point pair recombines to recover the original V-point. These findings establish the geometric first moment as a compact descriptor for polarization-singularity dynamics in tightly focused fields, providing a straightforward and robust strategy for controlling polarization singularities in complex optical systems.