Fig. 1. Schematic of the broadband metallic valley TPCs. (a) Geometrical structure of the metallic valley TPCs; the domain wall is marked by black dashed lines. (b) The rhombic primitive cell consists of two equivalent/nonequivalent metallic rods standing on a metallic plate. (c) Calculated band structures of the rhomboidal unit cell for the cases of Δ=0  mm (red dashed lines) and Δ=Δh  (2  mm)+Δr  (2.4  mm) (blue solid lines). Black solid lines represent the dispersion of EM waves in the air (air light line), and the inset shows the first Brillouin zone. (d) Projected band dispersions of the ribbon-shaped supercell consisting of 10 unit cells on each side of the domain wall. The red solid lines and black lines indicate the edge states and projected bulk bands, respectively. The inset enlarges the area enclosed by the green rectangle around the upper bulk modes. (e) Simulated transmission for topologically protected edge states (red curve) and bulk states (blue curve) along the straight domain wall. (f) Simulated Ez field distribution of the topological edge states at 5.53 GHz.

Fig. 2. Design of a seven-order topological rainbow. (a) Tuning the frequency range of edge states and photonic band gap with the height of rods. The solid line denotes the relationship between the frequency f at the Dirac point and the rod height h. (b) Design of the topological rainbow with the excitation source put to the left side. (c) Surface band dispersions for the seven values of rod height h from 12.6 to 11.4 mm, calculated with ribbon-shaped supercells. (d) Transmission spectra of the seven metallic valley PCs with the rod heights h from 12.6 to 11.4 mm. The seven colored areas represent the different frequency ranges of surface waves corresponding to channels of f1 to f7.

Fig. 3. Experimental characterization of the seven-order topological rainbow. (a) Schematic of the topological rainbow assembled from seven pieces of PCs. (b) Experimentally measured transmission spectra (in dB) of the topological rainbow when the source is placed at port S1. (c) Measured Ez field distributions of the edge states in the topological rainbow at 5.17, 5.25, 5.30, 5.36, 5.45, 5.53, and 5.61 GHz.

Fig. 4. Experimental characterization of the asymmetrical channels of the topological rainbow. (a) Channels of the topological rainbow when the excitation source is put to the PCVII on the right side. (b) Experimentally measured transmission spectra (in dB) of the topological rainbow when the source is placed at PCVII. (c) Measured Ez field distributions of the edge states in the topological rainbow at 5.17, 5.25, 5.30, 5.36, 5.45, 5.53, and 5.61 GHz.

Fig. 5. Asymmetric propagating topological frequency router. (a) Schematic of the topological frequency division device that is composed of four TPCs designed with rods of different height h. Red, green, cyan, and blue lines indicate the domain walls corresponding to the PCI, PCII, PCIII, and PCIV, respectively. (b), (c) Stimulated transmission spectra (in dB) when the source is placed at ports 1 and 3, respectively. The transmission spectra at ports 1, 2, 3, and 4 are shown with the red, blue, cyan, and green lines. (d)–(i) Stimulated Ez field distributions at asymmetric frequency channels Ch1, Ch3, and Ch2.

Fig. 6. Dual-channel topological frequency router. (a) Schematic of the dual-channel topological frequency router that is composed of four TPCs designed with rods of different height h. Red, green, and blue lines indicate the domain walls corresponding to the PCI, PCII, and PCIII, respectively. (b) Experimentally measured transmission spectra (in dB) at ports 2 and 3. The transmission spectra at port 2 (S21) are shown with the blue line, while port 3 (S31) is shown with the green line. Shadowed region represents the working frequency range. (c) Measured Ez field distributions at the two frequency channels that are coupled out through Ch1 and Ch2.

Fig. 7. Simulation results for topological rainbow. (a) Schematic of the piece-attached topological rainbow. (b) Simulated transmission spectra (in dB) when the source is placed at port S1, and the scattering parameters are shown for each port. (c) Simulated Ez field distributions of the edge states for each channel, and the prohibited propagation can be seen in the topological rainbow.

Fig. 8. Simulation results for the oppositely propagating channels of the topological rainbow. (a) Configuration for exciting opposite transmitting channels of the topological rainbow, where the point source is put to the right side. (b) Simulated transmission spectra (in dB) when the source is placed at port S8. (c) Simulated Ez field distributions of the edge states in each channel.

Fig. 9. Design of the asymmetric propagating topological frequency router with the excitation source put to different sides to excite the edge states of PCI to PCIV.

Fig. 10. Topological frequency router with the excitation source at the top and bottom sides. (a), (c) Stimulated transmission spectra (in dB) when the source is placed at ports 2 and 4, respectively. The transmission spectra at ports 1, 2, 3, and 4 are shown with the red, blue, cyan, and green lines. (b), (d) Stimulated Ez field distributions at 5.17 and 5.80 GHz, respectively.

Fig. 11. Simulation results for dual-channel topological frequency router. (a) Schematic of the dual-channel topological frequency router that is composed of four metallic valley TPCs. Red, green, and blue lines indicate the domain walls corresponding to the PCI, PCII, and PCIII, respectively. (b) Stimulated transmission spectra (in dB) from port 1 to ports 2 and 3. (c) Stimulated Ez field distributions for the two channels.

Fig. 12. Photograph of the seven-order topological rainbow assembled from seven pieces of metallic valley TPCs.

Fig. 13. Photograph of the dual-channel topological frequency router that is composed of four metallic valley TPCs designed with rods of different height h.

Fig. 14. Photograph of the measurement setup.