Fig. 1. (a) Original supercell structure without dimensional change. (b) Supercell structure modulated according to the original cell size; the size change is dr, according to the displacement formula. (c) Band structure of the original cell size. There are two Dirac points at point K; P1 represents a low-frequency Dirac point with a normalized frequency of about 0.257, and P2 represents a high-frequency Dirac point with a normalized frequency of about 0.537. (d) Band structure of the supercell at point P1. (e) Band structure of the supercell at point P2.

Fig. 2. (a) Schematic diagram of the structure arrangement, phase distribution, and supercell of the dual-band Dirac-vortex cavity. The dashed line represents the cavity radius, the background color represents the topological phase, and the white hexagon shows the structural arrangement. (b) Normalized frequency near low-frequency P1 mode. (c) Electric-field distribution and magnetic field polarization distribution near low-frequency P1 topological mode; the left picture shows the overall electric-field distribution of the model, and the right picture shows the local amplification of the center of the model, where the black arrow represents the direction of magnetic field polarization. (d) Normalized frequency near high-frequency P2 mode. (e) Electric-field distribution and magnetic field polarization distribution in high-frequency P2 topological mode.

Fig. 3. Scale relationship between effective mode diameter and cavity diameter, FSR, far-field divergence angle; mode spectrum and mode distribution of Dirac cavity topological mode. (a) and (b) are low frequency, and (c) and (d) are high frequency.

Fig. 4. Near and far fields and mode frequency of the low-frequency Dirac cavity topology mode at (a) w=1; (b) w=2; (c) w=3. Near and far fields and mode frequency of the high-frequency Dirac cavity topology mode at (d) w=1; (e) w=2; (f) w=3.

Fig. 5. Near- and far-field distribution and mode frequency at (a) w=4 and (b) w=8 in low-frequency Dirac cavity with high winding number. Near- and far-field distribution and mode frequency at (c) w=4 and (d) w=8 in high-frequency Dirac cavity with high winding number. (e) Phase gradient of high winding number and Dirac cavity mode far-field topological number.

Fig. 6. Frequency distribution of high winding number Dirac cavity mode.

Fig. 7. (a) Dual-wavelength emission Dirac cavity 3D model lattice structure band. (b) 3D model mode field spectrum and Q factor. (c) High-frequency mode far-field intensity and linear polarization component intensity. (d) Low-frequency mode far-field intensity and linear polarization component intensity.

Fig. 8. Lattice band and arrangement diagram. (a) Hexagon structure used in the text. (b) Kagomé structure. (c) Circular structure.

Fig. 9. Kagomé structure lattice diagram. Orange represents the dielectric material, white represents the cavity, the dotted line represents the unit cell, and the solid hexagon represents the supercell structure. (a) Before modulation. (b) After modulation. (c) Band structure after modulation; the P1 point bandgap width is about 9%, and the P2 point bandgap width is about 5.6%.

Fig. 10. (a) Dirac cavity arrangement diagram. (b) shows the normalized frequencies of low frequency and high frequency topological cavity modes, the near-field distributions of electric field modes, and the far-field topographies in the Kagomé structure. The black arrow in the near field is the polarization direction of the near-field magnetic field, and the green arrow in the far field is the polarization direction of the far-field electric field.

Fig. 11. Near-field electric field distribution and far-field spot morphology of all cavity modes in Dirac-vortex cavity near (a) low- and (b) high-frequency Dirac points. This includes bimodal postures not given in the text.

Fig. 12. Influence of the overall size and phase continuity of the cavity on the emission of the optical field.