Fig. 1. Fundamental energy band pattern and mass term distribution of unit cell. (a) Geometries of trivial photonic crystal (left side of the solid line) and topological photonic crystal (right side of the solid line), a1 and a2 are unit vectors with length as the lattice constant of a, and the right panel shows a magnified hexagonal unit cell; (b) distribution of z-component of magnetic field (H_z) for dipole mode (left) and quadrupole mode (right); (c) energy band distribution of unit cell when Dirac mass term m<0 (left) (the inset shows Brillouin zone), m=0 (middle), and m> 0 (right); (d) distribution of quality items

Fig. 2. Design and intrinsic mode distribution of optical Dirac cavity. (a) Drawing of designed configurable Dirac cavity, the mass term varies radially from inside to outside with the cavity center as a circle center and it is modulated by performing δ, the unperturbed cells are represented by yellow hexagons, the cells with the largest expansion are represented by red hexagons, and the outermost gray region is the air layer; (b)(c) band structure of topological unit cell at δ=0.053 and magnetic field (H_z) at Γ point carried by px(py) and dxy(dx2-y2) photonic orbitals; (d)(e) energy band and magnetic field (H_z) distribution of trivial unit cells for δ=-0.115; (f) Dirac cavity pattern designed using the above two kinds of unit cells, and the figures 5 and 3 in the cavity are the number of layers of two kinds of unit cells, respectively; (g) mass term distribution of unit cells at different positions on the dashed line in Fig. 2(f), and the domain wall is set to change from non-trivial photonic crystal to trivial one; (h) electric field E distribution of lower-order intrinsic modes obtained from first-principles simulations, and f and Q correspond to intrinsic frequency and quality factor, respectively; (i)(j) mass term distribution after the two unit cell positions are interchanged and their electric field E distributions of low-order eigenmodes

Fig. 3. Distribution of energy bands and eigenmodes of constructed optical Dirac cavity for some unit cells. (a)(b) Energy band of topological unit cell B and magnetic field (H_z) at Γ point carried by p_x(p_y) and d_xy(dx2-y2) photonic orbitals; (c)(d) energy band and magnetic field (H_z) distributions of trivial unit cell; (e) diagram of configurable Dirac cavity design; (f)(h) distribution of mass term inside the cavity; (g)(i) electric field E distributions of lower-order intrinsic modes

Fig. 4. Eigenmodes of optical Dirac cavities with complex mass term distributions. (a)(b) Mass term function of m=tanh(x/2) and corresponding electric field (E) distribution; (c)(d) mass term function of m=sinπx/3 and corresponding electric field (E) distribution

Fig. 5. Modal distributions of Dirac optical cavities, ①‒④ are four different forms of mass term distributions discussed in this paper. Column A is diagrams of cavity structure with spatial variations of mass terms, indicated by color changes of the mass terms, with the introduction of cavities and disorder and marked by black circles in the diagram; column B is the defective optical cavity after inversion of mass terms of counterpart column A. Defectiveness A and defectiveness B are electric field ∣E∣ distributions of far-field eigenmodes of defective optical cavities shown in A and B, respectively; modal distributions of perfect optical cavities after removal of the defects are given by the columns of without defect A and without defect B. (a)‒(f) Mass term distribution ①; (g)‒(l) mass term changes in step as shown in ②; (m)‒(o) mass term satisfies m(x)=tanhx2(x>0); (p)‒(r) mass term satisfies m(x)=sinπx3

Fig. 6. Coupling of optical cavities to spin-polarized boundary states.(a)(c)(e) General design of Dirac cavity with domain wall, and spin orientation and location of excitation source [(a) right circular polarization on the left side, (c) left circular polarization on the right side, and (e) left circular polarization on the left side]; (b)(d)(f) selectively oriented excitation of cavity mode corresponding to the system on the left