Fig. 1. (a)–(c) Transmission spectra of the S7 TM system with the structural parameter σ=512,12,712, respectively. The central positions of the first three orders of TGs are labeled by dotted lines. The red dots and blue dots indicate two types of singularities. (d) The refractive index distribution (blue line) and |E(z)| (black line) of the PhC at singularity with frequency f=5×1014  Hz and the same parameters as in (b). (e) and (f) Refractive index distribution (blue lines) and |E(z)| (black lines) of the TM system at two types of singularities with frequencies f1=5×1014  Hz and f2=2.5×1014  Hz, respectively.

Fig. 2. (a) Transmission map of the S5 TM system in the parameter space {σ, f}. The blue (red) area indicates a gap (band). (b) The reflection phase map of the S5 TM system. The first-type singularities are located at the singular lines (white lines), which are characterized by the π-phase-jump. The black lines are the mirrored lines. Both of them form the basic structure of the FTBG feature. (c), (d) and (e), (f) Local maps of transmission and reflection phase around the first-type and second-type singularities, respectively. A series of PVPs are indicated by the white circles in (d). (g) Transmission distribution error graph of the S5 TM system after 500 times is calculated when the disorder intensity is 0.1.

Fig. 3. (a) Local transmission map near the second TG of the S7 TM system in the parameter space. The blue (red) area indicates a gap (band). A (B) is the lower (upper) gap-edge state marked by a yellow dot (triangle) when σ=0.4. C (D) is the lower (upper) gap-edge state marked by a green dot (triangle) when σ=0.6. (b) The |E(z)| of state A (B) is indicated by the blue (red) lines. (c) The |E(z)| of state C (D) is indicated by the blue (red) lines. The black lines show the refractive index distribution, s and L means the total length of the system.

Fig. 4. (a) Structure composed of two S7 TM systems which are spliced together. The structural parameters of the left and right TM systems satisfy σl=35 and σr=25, respectively. The black dashed line indicates the interface, and the blue (yellow) areas indicate A-kind (B-kind) layers. (b) The transmission spectrum of the structure shown in (a). (c) The |E(z)| of the topologically protected edge-state with a frequency satisfying f=2.5×1014  Hz. (d) Edge-states in the space of the reflection phase ϕ and frequency f {ϕ, f} when the S5 TM system with σ=0.4 is connected to a reflector with an adjustable phase.

Fig. 5. (a) and (b) Maps of transmission and the reflection phase at the initial position of the S8 TM system in the parameter space. (c) Transmission spectrum near the edge-state corresponding to the structure composed of two S8 TM systems with σl=0.5458 and σr=0.6333. (d) The |E(z)| of the topologically protected edge-state with frequency f=4.156907×1014  Hz.

Fig. 6. (a) and (b) Maps of transmission and the reflection phase at the initial position of the S5 TM system under the periodic boundary conditions in the parameter space. A–H represent the lower and upper gap-edge states for σ=0.125, 0.4, 0.6, 0.875, respectively. (c)–(j) The |E(z)| of the gap-edge states marked in (a). L is the total length of the S5 TM super-cell.