Fig. 1. Schematic of triangular lattice in different forms. (a) A-lattice. The primitive cell consisting of one dielectric rod with the permittivity of 11.7, a₁ and a₂ are unit vectors with the length of a as the lattice constant, and d = 0.32a is the rod diameter. (b) B-lattice. The rods are the same as those in A-lattice. a₁^′ and a₂^′ are the unit vectors with length a/2. The solid hexagonals are the unit cell including one rod and six semi-rods in the unit cell. (c) C-lattice. The lattice is the same as A-lattice, but the primitive cell consists of one dielectric rod with the diameter d = 0.578a. (d) The first Brillouin zone with the unit vectors a₁ and a₂.

Fig. 2. Eigenfrequency bands of (a) C-lattice and (b) B-lattice. (c) The p-orbit and d-orbit reversal process with the value of r. It is clear that the d-orbit position has a sharp drop, and it occurs because the dashed unit cell is tangent to rods. The gray region is the common bandgap.

Fig. 3. (a) Edge state dispersion curve from the two lattices. The left curve branch and the right curve branch have the negative and positive group velocities, respectively. (b) The energy flow vectors around the edge corresponding to the edge states P and Q in (a).

Fig. 4. Edge state transport simulations. (a) The straight edge state with anticlockwise source. (b) The straight edge state with clockwise source. (c) The topological edge state with a sharp bend. (d) The ordinary edge state with a sharp bend.

Fig. 5. Model designs of nested topological loops. (a) One layer with clockwise spin source. (b) Two layers with anticlockwise spin source. The dashed arrows denote all the potential photonic paths. The white and gray regions are B-lattice (topologically nontrivial) and C-lattice (topologically trivial), respectively. The red and blue arrows denote the light flows along the loops A and B, respectively.

Fig. 6. Light flows from the clockwise spin source in a one-layer topological loop with normalized angular frequencies: (a) 0.53(2πc/a), (b) 0.5034(2πc/a), (c) 0.5348(2πc/a), and (d) 0.5504(2πc/a). The side length of loop is 9a.

Fig. 7. One-dimensional fields along the line through the source and the common side in Fig. 6(b) with the resonance loop and in Fig. 6(d) with the non-resonance loop.

Fig. 8. Light flows from the anticlockwise spin source in two-layer topological loops with normalized angular frequencies: (a) 0.53(2πc/a), (b) 0.5034(2πc/a), (c) 0.5227(2πc/a), and (d) 0.4991(2πc/a). The outer loop has the side length of 15a.