Multiple toroidal dipole bound states in the continuum in dielectric metasurfaces

We investigate bound states in the continuum (BICs) in dielectric metasurfaces consisting of four corner or square patches in the unit cell of a square lattice. Friedrich and Wintgen (FW) BICs and symmetric-protected (SP) BICs, both with extremely large quality factors, appear at the center of Brillouin zone, and are recognized as vortex polarization singularities (<i>V</i> points) in the momentum space and carry topological charges <i>q</i> = ±1. In particular, the FW BICs and SP BICs are represented as electric or magnetic toroidal dipole (TD) modes characterized by electric or magnetic fields circulating around the surface of a hypothetical torus, with magnetic or electric fields looping inside the torus. The symmetry axis of an electric or magnetic TD can be oriented either normal or tangential to the metasurface, leading to a longitudinal or transverse TD. Occasionally, a few pairs of electric or magnetic TD modes are gathered in the unit cell to form multiple TDs. By carefully arranging the dielectric patches and substrates, the underlying metasurfaces support a combination of multiple electric and magnetic TD BICs or quasi-BICs oriented in longitudinal and transverse directions.