2.1.

Broadband Spoof SPP Valley TPCs

The proposed spoof SPP valley TPCs with topological edge states and engineered PBGs consist of aluminum rods of different heights and radii on top of a metal surface, as shown in Fig. 2(a). Due to the periodic metallic structures,⁵⁵ spoof SPP topological edge states are supported to propagate on the metallic valley TPCs, where the energy is localized at the metallic surface and exponentially decays in both the vertical and transverse directions, and thus the spoof SPP edge states cannot radiate into free space and are self-confined. This means that the proposed design here does not need a covering lid and possesses the extra degree of freedom of modifying the height of the rods to tune the working frequency and broaden the working bandwidth, which does not affect the size or layout of the structure. In addition, compared with other domain walls, the robustness and bandwidth of the edge states along the zigzag-shaped domain wall are wider.⁵⁶ Figure 2(a) shows that our proposed spoof SPP valley TPC is composed of valley photonic crystals (VPCs) with inverted symmetry along the zigzag domain wall. Each unit cell of the VPCs contains two rods of different heights and radii with a lattice constant $a=14\mathrm{mm}$. The height and radius of rod 1 are $h_1=h_0+\mathrm{\Delta }h=13.1\mathrm{mm}$ and $r_1=r_0+\mathrm{\Delta }r=3.15\mathrm{mm}$, respectively, and those of rod 2 are $h_2=h_0-\mathrm{\Delta }h=12.1\mathrm{mm}$ and $r_2=r_0-\mathrm{\Delta }r=1.05\mathrm{mm}$, respectively. When these two rods have a uniform height and radius (that is, $h_0=h_1=h_2=12.6\mathrm{mm}$ and $r_0=r_1=r_2=0.15a=2.1\mathrm{mm}$), the VPCs exhibit $C_{6\mathrm{v}}$ symmetry that leads to a Dirac dispersion near the $K(K^{\prime })$ points in the simulated band diagram at 5.00 GHz [blue dotted lines in Fig. 2(b)]. Note that the valleys are located well below the light line in the air [black lines in Fig. 2(b)]; thus, spoof SPP modes in the neighborhood of the Dirac points can be supported on the VPCs. Then upon breaking the inversion symmetry by simultaneously turning on the height difference $\mathrm{\Delta }h$ and radius difference $\mathrm{\Delta }r$, the Dirac degeneracies are removed and a complete PBG ($4.69\mathrm{GHz}<f_1<5.39\mathrm{GHz}$) is opened [red solid lines in Fig. 2(b)].

Fig. 2

Broadband spoof SPP valley TPCs. (a) Schematic illustration of the designed TPC structure composed of VPCs with inverted symmetry, where the rhomboidal unit cell consists of inequivalent metallic rods with different radii and heights on top of a metal surface. The green line denotes the zigzag-shaped domain wall. (b) Band diagram of the VPCs with and without inversion symmetry, where the blue, red, and black lines represent the dispersions of the VPCs with inversion symmetry, without inversion symmetry, and the light line in air, respectively. The light-yellow region corresponds to the common PBG of the VPCs. Left inset: first Brillouin zone of the VPCs. (c) Simulated eigenmode profiles at $K$ and $K^{\prime }$ valleys for the first and second bands of the VPCs in the x-y plane. The color scale shows the magnitude of the $z$-oriented electric field $E_z$, while the black cones represent the Poynting power flows. (d) Projected dispersion of the valley topological edge states (red lines) for the zigzag-shaped domain wall. The light red, blue, and gray shadow areas represent the edge states and PBGs (PBG1 and PBG2).

The corresponding distributions of the $z$-oriented electric field profiles and Poynting power flow of the modes at the $K$ ($K^{\prime }$) valleys after the perturbation are illustrated in Fig. 2(c). There are four modes that exhibit either left-handed circular polarization or right-handed circular polarization. In addition, within the same band, the polarizations of these modes are opposite at $K$ and $K^{\prime }$ valleys. It is known that VPCs feature nonzero Berry curvatures accumulated around different valleys. We find that the valley numbers $C_{K(K^{\prime })}={\int }_{BZ(K(K^{\prime }))}{\mathrm{d}}^2\delta k[{\nabla }_{\delta k}\times A(\delta k)]/2\pi$ are half-integers,⁵⁷ $C_K=+1/2$ ($C_{K^{\prime }}=-1/2$) for the first band, and $C_{K^{\prime }}=+1/2$ ($C_K=-1/2$) for the second band. Thus, the difference between the valley Chern numbers at $K$ ($K^{\prime }$) points across the zigzag-shaped domain wall in the valley TPCs formed by combining two VPCs with opposite half-integer valley Chern numbers is $\mathrm{\Delta }C=C_K-C_{K^{\prime }}=\pm 1$. According to the bulk-boundary correspondence, it is expected that valley topological edge states will emerge at the domain wall in the valley TPCs within the original PBG of the VPCs.

Then, we numerically calculate the band diagram projection for a $20\times 1$ supercell; the results are shown in Fig. 2(d), where a broadband topological edge state (red-shaded region) is found to be localized at the domain wall, whose propagating directions are locked to the $K$ and $K^{\prime }$ valleys. Interestingly, it should be noted that the edge mode bands are detached from the bulk bands (black lines): a special PBG1 (light blue region) is found. PBG2 also appears above the upper bulk band [see Fig. 2(d)], and according to the band diagram of the unit cell and the dispersion relation of the supercell, it can be seen that it has no relationship with the perturbation $\mathrm{\Delta }$, and it is always present (see section 1 in the Supplementary Material). Note that the upper bulk bands are flat and cover a very narrow frequency range, so they are difficult to excite, functioning like a bandgap in the experiments. Thus, the valley edge states are separate from the bulk bands on both sides, which can be effectively viewed as being sandwiched by two bandgaps (PBG1 and PBG2). This is a unique feature that has not been observed previously in any other topological materials.

To further confirm the mechanism of the emergence of the special PBG1 between the edge and bulk states and the quantitative relationship between the bandwidth of the edge states and the degree of symmetry breaking, we systematically investigate the relationship between the perturbation $\mathrm{\Delta }$ ($\mathrm{\Delta }h$ and $\mathrm{\Delta }r$) and the edge states and PBGs, including the common PBG in the VPCs, the special PBG1, and the edge states in the valley TPCs, as shown in Figs. 3(a) and 3(b). It can be seen that for the common PBG in the VPCs, its bandwidth generally increases monotonically with the perturbation $\mathrm{\Delta }$; for the edge states in the TPCs, the bandwidth also increases monotonically with $\mathrm{\Delta }$. However, the rate of increase of the edge state bandwidth decreases with $\mathrm{\Delta }$ when the perturbation $\mathrm{\Delta }$ is larger than a value (called the threshold value $T$). For the special PBG1 in the TPCs, when $\mathrm{\Delta }<T$, the bandwidths of the VPC-PBG and TPC-edge states vary synchronously with the perturbations and are equal, which is to say that there is no special TPC-PBG1 in this case. When $\mathrm{\Delta }>T$, as the growth rate of the VPC-PBG is larger than that of the TPC-edge states, the special TPC-PBG1 emerges, and its bandwidth also increases monotonically with $\mathrm{\Delta }$. The concept of the threshold value $T$ is derived from the PBG theory,⁵⁸ which states that there is no PBG until the dielectric contrast is increased to some threshold value. Above this threshold, the bandgap opens up and its width usually increases monotonically with the dielectric contrast. Since the dielectric contrast in the valley TPCs is governed by the perturbation $\mathrm{\Delta }$ (see section 2 in the Supplementary Material), we can conclude that the PBG in the valley TPCs is influenced by the perturbation $\mathrm{\Delta }$ as well. The threshold values of opening the same PBG in different ways are different: the threshold values of changing the height difference $\mathrm{\Delta }h$ and radius difference $\mathrm{\Delta }r$ to open PBG1 are $T_h=0.58\mathrm{mm}$ and $T_r=0.82\mathrm{mm}$, respectively. The results presented in Figs. 3(a) and 3(b) illustrate that the valley TPCs with a large enough perturbation $\mathrm{\Delta }$ not only have broadband edge states but also have a special PBG between the edge and bulk states. It is worth mentioning that in this work we achieve large enough perturbation $\mathrm{\Delta }$ by simultaneously adjusting the height difference $\mathrm{\Delta }h$ and radius difference $\mathrm{\Delta }r$ of the different rods that constitute the proposed valley TPCs.

Fig. 3

Valley topological edge states and PBGs. (a) and (b) Evolution of the working bandwidths of the VPC-PBG, TPC-edge states, and TPC-PBG1 with respect to the perturbations of height difference $\mathrm{\Delta }h$ and radius difference $\mathrm{\Delta }r$, respectively. The threshold values of changing the height difference $\mathrm{\Delta }h$ and radius difference $\mathrm{\Delta }r$ to open PBG1 are $T_h=0.58\mathrm{mm}$ and $T_r=0.82\mathrm{mm}$, respectively. (c) Simulated transmission spectrum of the straight spoof SPP valley TPC waveguide. The light red, blue, and gray shadow regions correspond to the edge states, PBG1, and PBG2, respectively. (d) Simulated $|E_z|$ field distributions of the valley topological edge states in the x-y, x-z, and y-z planes at 5.20 GHz.

Then, the transmission spectrum of the spoof SPP valley TPCs is simulated and displayed in Fig. 3(c). In the simulation, a dipole source is placed at the left end of the domain wall to excite the edge states, and a probe at the right end of the domain wall is used to measure the transmission spectrum. The simulated transmission band of the edge states and PBG1 in Fig. 3(c) matches well with the frequency range of the calculated dispersion for the ribbon-shaped supercell in Fig. 2(d). We can also observe a PBG1 between 4.60 and 4.94 GHz, the edge states between 4.94 and 5.40 GHz, and the ever-present PBG2. The simulated $|E_z|$ field distributions of the valley topological edge states in the x-y, x-z, and y-z planes at 5.20 GHz are shown in Fig. 3(d), demonstrating that the edge states are evanescently decaying and well confined at the domain wall in both the vertical and transverse directions. It can be seen that the proposed valley TPCs support the propagating spoof SPP topological edge states.

2.2.

Multifunctional FSDT Topological Photonic Device

The proposed spoof SPP valley TPCs have the advantages that not only the edge mode bands are detached from the bulk bands and a special PBG emerges but also the working frequency is easily tuned by modifying the height $h$ of the rods, which does not affect the size or layout of the structure. Taking advantage of this exotic feature, we proceed to design a broadband multifunctional FSDT topological photonic device, which has an intriguing feature: novel frequency multiplexing of different frequency edge states transporting in different edge channels can be achieved. Figure 4(a) shows the design strategy of the FSDT device, which consists of two spoof SPP valley TPC waveguides with different heights $h$ (TPC I and TPC II). The dispersion relations of the edge states for TPC I and TPC II are marked by the blue and red lines in Figs. 4(c) and 4(d), respectively. Clearly, by carefully optimizing the structural parameters $h$ of TPC I and TPC II, the frequency ranges of the edge states and PBGs can both be adjusted to the desired state, as shown in Figs. 4(c) and 4(d) (for more details on TPC I and TPC II, see section 3 in the Supplementary Material). Note that, for this FSDT topological photonic device, the armchair domain wall between TPC I and TPC II along the $y$ axis cannot support topological interface modes because the valley Chern number across the armchair domain wall is zero (see section 4 in the Supplementary Material).

Fig. 4

Concept of the broadband FSDT topological photonic device. (a) Design strategy of the FSDT device composed of two valley TPCs with different heights $h$, TPC I (blue area), and TPC II (red area). The light-red, yellow, and blue regions represent the LF, IF, and HF waves, respectively. The arrows indicate the propagation directions of the edge states. (b) Schematic of the FSDT device shows versatility when the source is located at different locations. The black dashed line denotes the position of the zigzag-shaped domain wall. (c) and (d) Dispersion curves of the ribbon-shaped supercells of TPC I and TPC II, respectively. The blue and red lines are dispersion curves for the edge states of TPC I and TPC II, respectively. The frequency ranges of the valley topological edge states of TPC I and TPC II are HF + IF (light blue + yellow regions) and IF + LF (light yellow + red regions), respectively. The black dotted lines are in the bulk bands and the PBGs are indicated by the gray regions.

The composite device could achieve the following functions through the combined action of valley edge states and PBGs in two valley TPCs, as shown in Fig. 4(b). The spoof SPP topological edge states in the high-frequency (HF) regime propagate only along the domain wall of TPC I [light blue region in Fig. 4(a)]; in the intermediate-frequency (IF) regime, bidirectional transmission along the domain walls of TPC I and TPC II can be achieved [light yellow region in Fig. 4(a)]; and the low-frequency (LF) waves transmit only along the domain wall of TPC II [light red region in Fig. 4(a)]. Therefore, the device has the functions of WDM and ADM when the excitation source is located at different locations. For example, when the source is located at the center between TPC I and TPC II or in the middle of TPC I and the middle of TPC II, the WDM and ADM functions can be achieved, respectively. Note that in previous valley TPCs, the topological edge states and bulk states are continuous and cannot be used to realize the FSDT device, as demonstrated in section 5 in the Supplementary Material.

To demonstrate the capability of the broadband FSDT topological photonic device based on the proposed valley TPC structures, we have fabricated a finite-sized sample consisting of $20\times 16$ unit cells, as shown in Fig. 5(a). It is composed of aluminum rods of different heights and radii and a metallic substrate, which is fabricated with the traditional metal machining technique. The substrate is 10 mm thick and has corresponding 5-mm air holes, and the aluminum rods are assembled on the metallic substrate. In the experiment, the transmission spectra and field distributions are directly measured by a scanning near-field microwave microscopy system without an extra metallic cover due to the fact that the proposed valley TPCs support the propagation of the spoof SPP topological edge states. A coaxial monopole antenna is placed in the drilled holes of the metallic substrate at ports S2, S3, and S4 to excite the spoof SPP topological edge states, whereas another probe antenna is fixed on a scanning support to measure the transmission spectra and scan the near-field electric field distributions.

Fig. 5

Experimental observation of the broadband FSDT device. (a) Photograph of the experimental sample of the topological photonic device, which is composed of TPC I and TPC II. The blue and red lines represent the positions of the domain walls of TPC I and TPC II, respectively. (b)–(d) Experimentally recorded transmission spectra (in dB) at ports S1 and S5 when the source is placed at ports S3, S2, and S4, respectively. The transmission spectrum corresponding to port 1 is shown as the solid blue line, while that for port 5 is shown as the solid red line. The insets on the right in each figure are the experimental $E_z$ field distributions of the edge states at 4.88, 5.13, and 5.44 GHz (from top to bottom).

When the device is excited from the center port S3 between TPC I and TPC II, the measured transmission spectra and $E_z$ electric field distributions indicate that the WDM function is realized in the experiments, as shown in Fig. 5(b). It can be seen that the IF and HF EM waves within 5.06 to 5.70 GHz propagate to the left along the domain wall of TPC I, and the LF and IF EM waves within 4.72 to 5.38 GHz transmit to the right along the domain wall of TPC II. The inset on the right describes the experimentally measured $E_z$ electric field distributions at 4.88, 5.13, and 5.44 GHz. The edge states at 5.44 GHz (HF region) propagate unidirectionally to the left along the domain wall of TPC I, the edge states at 5.13 GHz (IF region) propagate bidirectionally along the domain walls of TPC I and TPC II, and the edge states at 4.88 GHz (LF region) transmit one-way to the right along the domain wall of TPC II. The robustness of the FSDT device is shown in section 6 in the Supplementary Material.

When the source is located at ports 2 and 4, the measured transmission spectra and $E_z$ electric field distributions are depicted in Figs. 5(c) and 5(d), respectively, illustrating an ADM function. When the excitation source is placed at port 2, the detection receiver at ports 1 and 5 can receive IF and HF waves within 5.09 to 5.61 GHz and IF waves within 5.09 to 5.34 GHz, respectively. When the excitation source is placed at port 4, the detection receiver at ports 1 and 5 can receive IF waves within 5.09 to 5.34 GHz and LF and IF waves within 4.83 to 5.34 GHz, respectively. Overall, the experimental measurements of the transmission spectra and $E_z$ electric field distributions are in good agreement with the simulations. However, the experimental equipment, sample fabrication, and environmental conditions cause a slight difference in the frequencies between the experimental and simulated results (for more simulation details, see section 7 in the Supplementary Material).

4.1.

Sample Fabrications

Our FSDT sample was fabricated with the traditional metal-machining technique. It is composed of aluminum rods of different heights and radii and a metallic substrate. The substrate is 10 mm thick and has corresponding 5-mm air holes, and the aluminum rods are assembled on the metallic substrate.

4.2.

Experimental Characterization

A scanning near-field microwave microscopy system was employed to measure the transmission spectra and near-field electric field distributions. The system comprises a vector network analyzer (Keysight E5063A) and a three-dimensional scanning platform. Two coaxial probes act as the emitting antenna and probe antenna, respectively. The emitting antenna is placed in the drilled holes of the metallic substrate at ports 2, 3, and 4 to generate the valley edge states, whereas the probe antenna is fixed on a scanning support to measure the transmission spectra and scan the near-field electric field distributions.

4.3.

Numerical Simulations

The dispersion relations of the spoof SPP valley TPCs are numerically obtained in the frequency domain with the commercial finite-element method-based software COMSOL Multiphysics. The projected band diagrams are evaluated using a supercell containing 10 unit cells on either side of the domain wall, and periodic boundary conditions are imposed on the left and right sides. The simulations of the transmission spectra and field distributions are performed with the time domain solver of CST Microwave Studios. Open boundary conditions are applied in all directions, and the materials of the valley TPCs are set as perfect electric conductors.