Evolution of topological singularities below the light line in momentum space

Polarization singularities that exist in momentum space have brought new opportunities in various fields such as enhanced optical nonlinearity, structured laser sources, and light field manipulation. However, previous researches have predominantly focused on the polarization singularities above the light line, because they have no leakage and are referred to bound states in the continuum. Here, by extending the polarization fields to Fourier components of the evanescent field on a dielectric metasurface, polarization singularities of different Fourier orders are discovered below the light line. When continuously changing the geometrical parameters of the metasurface, a Fourier order transition process of the polarization singularity is observed through the bandgap closing at the boundary of the Brillouin zone, which finally leads to the annihilation of two singularities with opposite topological charges below the light line. These findings expand the understanding of polarization singularities in the near-field region and may find applications in light field manipulation and light-matter interaction.