Topological photonics [1–4] is an emergent field at the intersection of optical study and topological physics research, which has actively enlightened the wisdom on the novel control of electromagnetic (EM) waves. Optical waves engineered within a variety of topological phases [5–9] exhibit fascinating properties such as backscattering immunity and defect insensitivity, which are highly favored in scenarios demanding the reduction of crosstalk or scattering losses. For example, by employing an external magnetic field to gyrotropic materials, the quantum Hall effect can be imitated in photonic systems to provide unidirectional propagation channels [10–14]. More convenient ways of achieving topological transport are the photonic quantum spin Hall effect [15–19] or valley Hall effect [20–25], which require no time-reversal symmetry breaking. Following such recipes, topological photonics has served as a great support to the development of practical optical devices in the past few years [9,26–29]. Within these available approaches, valley topological photonic crystals (TPCs) [30–33] are particularly attractive for device application due to various advantages such as the easy design, robust topological transport, large band gap, and compact size. A variety of novel devices have been proposed based on valley TPCs, such as reflectionless waveguides [34,35], valley-conserved topological antennas [36], topological filters [37], topological cavities [38,39], valley-dependent quantum photonic circuits [40], topological power splitters [41,42], robust all-optical logic gates [43], and topological lasers [44]. Note that these topological photonic devices based on valley TPCs suffer from propagation losses due to Anderson localization induced by fabrication disorders at telecommunication wavelengths. However, at lower frequencies (microwave and terahertz regimes), Anderson localization is negligible as the length scale of such disorders is much smaller than the operating wavelength [45,46].

On the other hand, frequency multiplexing devices [47–50] of rainbow trapping and frequency division are another family of essential components in optical communication systems that allow for a large capacity of information transmission. In traditional solutions with metamaterials [47,51,52], plasmonic waveguides [50,53], nonlinear optics [54], or photonic crystals [55], the wave propagations are prone to defects, disorders, and bending losses. The idea of using topological materials, however, can provide a feasible plan to overcome these shortages, where novel photonic devices such as topological rainbows [56–60] and topological wavelength division multiplexers (WDMs) [61–64] have been presented. Among the existing demonstrations, the frequency channel isolation and division are usually induced by the topological slow light mode, where the wide range of wavevector components leads to spatial localization of topological edge mode. However, the resulting static spatial local mode or finely tuned flat band is usually weak against perturbations or fabrication inaccuracies, and the localization size can hardly be ideal to avoid the coupling between different channels. Meanwhile, the excitation or out-coupling of edge modes in such topological frequency multiplexing devices is very difficult due to the zero group velocity of the slow light mode and flat edge band, which makes such frequency multiplexing topological devices unsuitable for practical use.

In this paper, we present an ideal solution to frequency multiplexing by proposing and experimentally demonstrating a new type of asymmetric propagating topological rainbow and frequency router based on floating topological edge mode. The multiple channels in the designed frequency multiplexing topological devices can be directly pumped by one single-point source placed at the end of the topological photonic crystal. These channels can then be easily coupled out to different ports. The principle of channel separation is accomplished by directly truncating the band gap on a topological edge band, which is attributed to the novel feature of the floating edge band being sandwiched between the top and bottom band gaps in our design. To be specific, through geometrical parameters tuning, it can be arranged that the frequency of the edge states in one TPC slab corresponds to the band gap of the second TPC slab, so that the wave propagation of this frequency channel finds prohibition at the interface between these two slabs. By successively implementing such design with gradually tuned TPC slabs, an edge band covering a wide range of frequencies can be truncated into multiple channels with each of them stopping at different spatial positions, and a propagating topological rainbow is thus designed. Meanwhile, proper domain walls are designed and attached to the prohibited positions of each frequency channel in the topological rainbow to design a topological frequency router for coupling the energy out. More importantly, the upper and lower frequency gaps of the frequency multiplexing topological devices will each introduce truncation to provide different channels with opposite directions of propagation. The topological rainbow and frequency router are thus here first-time empowered with asymmetrical transportation.

Here, we adopt the design of metallic valley TPCs [65] that consist of aluminum rods standing on a board in a triangular lattice, as depicted in Fig. 1(a). Such TPCs support the propagation of spoof surface plasmon polaritons (SPPs) due to the periodic structures [66]. The spoof SPP modes along the z-direction are self-confined. Compared to the traditional realization of two-dimensional systems that require a pair of parallel metallic plates for polarization selection and energy confinement [14,16,22,67], the proposed design here does not need a covering lid and thus possesses an extra degree of freedom in tuning as the height of rods, which makes the metallic crystals better candidates for both implementing topological transport and device performance optimization.

A rhomboid unit cell with the lattice constant of a=14mm is shown in Fig. 1(b) (top panel), which consists of two metallic rods with a radius of r=2.4mm. When the sizes of the rods are the same, the photonic crystal possesses inversion symmetry, and the Dirac band dispersion emerges near the K (K′) valley of the Brillouin zone and is shown in Fig. 1(c) as red dashed lines. A perturbation Δ can then be introduced into the size of sublattice rods to break the inversion symmetry. By implementing different heights Δh or radii Δr to the rods in Fig. 1(b) (bottom panel), the degeneracy at the Dirac point is lifted and a complete photonic band gap is formed at the K and K′ valleys. By simultaneously tuning the height difference h1−h2 and the radius difference r1−r2 of metallic rods, a large band gap of 21% (normalized to central frequency) is achieved in Fig. 1(c).

By implementing a reversed set of parameters (height and radius) in the sublattice rods from the upper and lower regions in Fig. 1(a), a domain wall can be formed as indicated with the dashed line. The mirror arrangement of rods across the domain wall constrains the valley Chern indices (in the lower two bands) as opposite for the two valley TPCs, where a contrasting integer valley Chern number ±1 is formed and is opposite for K and K′ valleys. Topological edge modes are then formed and localized near the domain wall to provide topological transport.

Contrary to traditional valley TPCs, it can be noticed in Fig. 1(d) that the edge mode bands in the metallic valley TPCs are detached from the bulk band projections and become floating in band gaps, which is an important feature that enables the design of frequency multiplexing topological devices such as a topological rainbow and frequency router, and asymmetric transportation channels as will be discussed later. Note that different from the usual indication of band gaps as the full region between two bulk bands, we here use band gaps to specify the void region between the edge band and bulk bands. A little part of the edge mode is located above the air light line and can scatter into the background; thus only the spoof SPP edge modes outside the light cone are used in our design. We then calculate the transmission spectra of these TPCs waveguides, as displayed in Fig. 1(e), where apparently the edge modes provide broadband propagation along the domain wall in the valley TPCs. The simulated field pattern of the valley edge states at 5.53 GHz is also displayed in Fig. 1(f), verifying that the edge modes are evanescently decayed along both the vertical (z) and transverse (±y) directions.

In this metallic valley TPC, the freedom in the rod height can be utilized to manipulate the frequency of edge states and the band gaps. We first define an average rod height h=(h1+h2)/2, illustrated in Fig. 1(b), for convenience. And as shown in Fig. 2(a), while the average height is continuously tuned from h=11 to 14 mm, the frequency range of the edge mode decreases from 5.61–6.34 GHz to 4.63–5.06 GHz (see Appendix A for more details), where we note that the higher bounds of the edge modes always merge with the upper bound of the first band gap as the flat bulk bands indicate with the dashed red line. Although only the lower bounds of the edge state bands are detached from the bulk band projection, it should be noticed that a second band gap appears above the flat bulk band in Figs. 1(d) and 2(a), and the flat bulk bands cover a very narrow frequency range so that the edge state bands can be effectively seen as being sandwiched between two band gaps.

Taking advantage of the exotic feature of the floating topological edge band in the metallic valley TPCs, we proceed to design a broadband propagating topological rainbow by selecting a series of discrete parameters from Fig. 2(a) (see Appendix B for details). The proposed design of a seven-order topological rainbow is presented in Fig. 2(b), where the edge mode frequency ranges (together with the band gap) are tuned as gradually increasing for each of the seven photonic crystals from left to right (noted as PCs, including PCI to PCVII). The frequency range of edge modes in each PC is indicated by the arrowed line in black and the band gaps are indicated with red lines. It can be seen that there is a mismatch in the frequency range between two neighborhood PCs. Each of such differing frequency ranges is noted as f1 to f7 as shown in Fig. 2(b), which serve as the multiplexing channels in the designed topological rainbow.

To understand the frequency channel separation, we first take the channel f1 as an example. When the EM waves in the frequency range of f1 are excited from the left side of PCI in Fig. 2(b), the topological edge mode will propagate to the right coming across a band gap by reaching PCII (and no reflection due to topological protection), given the fact that the edge bands and band gaps in PCII are tuned to relatively higher frequencies. This will lead to prohibited propagation of channel f1 at PCII. Then if channel f2 is excited from the left side of PCI in Fig. 2(b), the propagation will be supported by both PCI and PCII, so the prohibited propagation will be at the interface between PCII and PCIII. Similarly, each channel of f1 to f7 will stop at different spatial positions of the TPCs, i.e., the interface between two adjacent PCs. This presents the desired frequency channel separation. Importantly, all the frequency channels can be easily excited with a single point source buried in the PCI at the domain wall, since the edge states in the PCI already cover the full frequency range from channel f1 to f7. We then realized the propagating topological rainbow with easy point source excitation. In Figs. 2(c) and 2(d), we show the calculated band dispersion and transmission spectra for each of the seven PCs in the topological rainbow.

To experimentally verify the proposed seven-order topological rainbow, we fabricated a finite-sized sample consisting of 37×16 unit cells, where the number of unit cells in each block of PCs is 5×16, as shown in Fig. 3(a). In the experiment, a source antenna is put at the PCI and inserted through the drilled hole in a metallic substrate to excite EM waves, and the signals at different spatial positions are measured by a probe antenna. We experimentally measured the transmission spectra as shown in Fig. 3(b), where the frequency ranges of edge states in different PCs of the topological rainbow are verified. By using microwave near-field scanning microscopy, we also directly mapped the electric field distributions along the domain walls and experimentally observed the prohibited propagation of EM waves for each rainbow frequency channel from f1 to f7 in Fig. 3(c). It can be seen that the EM waves in each channel stop at distinct spatial positions, exhibiting the topological rainbow (see Appendix C for simulation results).

The seven-order topological rainbow can also be excited in a reversed way, where the source is put to the right-hand side of the TPCs, as shown in Fig. 4(a). Interestingly, in such a configuration, the whole spectra of edge states in PCVII (instead of PCI) will be excited, and the mismatch between adjacent PCs (from right to left) due to the upper frequency band gap provides new channels of f8 to f13 for topological transport as indicated with different colors in Fig. 4(a). These channels also each find prohibited transmission at the interface between adjacent PCs, similar to the case in Fig. 2(b). Such new channels support the transmission from right to left as opposite to the channels of f1 to f6, making the topological rainbow device asymmetric in transmission. It should be mentioned, however, that the channel f7 supports wave propagation in both directions.

We then experimentally characterized the topological transport with a point source placed in PCVII. The transmission spectra to different ports are shown in Fig. 4(b), and the field distributions for several selected frequencies from channels f7 to f13 are shown in Fig. 4(c), where the rainbow effect is again demonstrated. The simulation results for the opposite channels are put in Appendix D.

The two types of excitation for the topological rainbow provide different working frequencies of 5.10–5.70 and 5.55–6.25 GHz, respectively, which originate from the different frequency ranges of edge states supported in PCI and PCVII. These asymmetric channels further increased the information-carrying capacities of the designed topological rainbow device, which can find application in novel optical communication systems.

To further enhance the results on asymmetric transportation and achieve the frequency division, we can assemble a design of an asymmetric topological frequency router with multiple coupling ports so that the energy in each frequency channel can be coupled out instead of rainbow trapped. The design is shown in Fig. 5(a), which consists of four metallic TPCs (PCI, PCII, PCIII, and PCIV), and proper domain walls are designed and attached to the prohibited propagation positions of each frequency channel in the topological rainbow for coupling the energy out (see Appendix E for detailed parameters of TPCs and the design method of the asymmetric topological frequency router). The frequency multiplexing device consists of 18×18 unit cells, and four ports are designed with ports 1 and 3 being the input ports and ports 2 and 4 serving as output ports for different frequency channels.

The transmission spectra are shown in Figs. 5(b) and 5(c) for the configuration of the source being placed at ports 1 and 3, respectively. Three channels are designed to provide both asymmetrical and symmetrical transportation. From the field distributions in Figs. 5(d) and 5(e), we can see that when a source antenna is put at the left side of port 1 to excite all the frequencies of edge states in PCI, the frequency in Ch1 can propagate to port 2, but such frequency channel is not supported while being excited from port 3. On the other hand, when a source antenna is put at the right side of the device as port 3 to excite the edge state in PCIII, as shown in Figs. 5(f) and 5(g), the frequencies in Ch3 are supported and can propagate to port 4, but this channel is not supported by exciting from port 1. And a symmetrical transportation channel is also designed as Ch2 in Figs. 5(h) and 5(i) that always exists regardless of the excitation source being located on the left or right.

Although the designed device supports asymmetric propagating channels, we should note that the propagation for each channel along the domain walls is reversible, e.g., from port 3 to port 1 for Ch2. We provide more discussion on this in Appendix F.

Due to the size limit of the permitted sample in the experimental setup and the requirement for the output ports to be on the same side, we choose to design and fabricate a dual-channel topological frequency router that consists of 19×15 unit cells shown in Fig. 6(a). For experimental verification, we measured the transmission spectra from port 1 to port 2 and port 3, which are shown in Fig. 6(b), where high transmittance for each of the two working channels can be found (simulation results are shown in Appendix G). It can be seen that a source antenna is put to port 1 to excite all the frequencies of the two channels. While being excited, the edge states of Ch1 and Ch2 will propagate along the domain wall of PCI to the first sharp bend. Ch1 comes across with the forbidden propagation by PCIII and thus turns right entering PCII and subsequently arriving at port 2. The energy in the frequency channel of Ch2, however, can keep propagating with PCIV; it is then coupled out from port 3. Such sharp turns are possible due to the topological propagation of valley Hall edge states. It should also be noted that a part of energy in Ch2 can also make a turn towards port 2; however, such propagations are stopped by PCII. As expected, the EMWs within 5.1–5.4 GHz propagate along the domain walls of PCII (Ch1) to output port 2, and those within 5.4–5.7 GHz transmit along the domain walls of PCIII (Ch2) to port 3. We also measured the field distributions along domain walls for each frequency channel, as shown in Fig. 6(c), and it can be seen that the energy at different frequency channels is coupled out through separate ports. Note that the dual-channel topological frequency router can be composed of three metallic TPCs (PCI, PCII, and PCIII), but we have added PCIV to block energy output on the right side of the router, so that the energy in frequency channel Ch2 can make a sharp turn to port 3. Such results clearly demonstrate the idea of achieving frequency division with the proposed topological frequency router, which provides robust routes to realize multi-channel topological devices for optical communication.

The band dispersion of the topological photonic crystals is numerically calculated with the frequency domain solver of commercial software COMSOL Multiphysics. The projected band diagrams are evaluated using a supercell configuration 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 electric field distributions are performed using the time domain solver of CST Microwave Studios. Open boundary conditions are applied in all directions, and the metals are set as perfect electric conductors.

The TPC samples of the seven-order topological rainbow and dual-channel topological frequency division are fabricated with the traditional metal machining technique. They are composed of aluminum rods of different heights and radii placed on metallic substrates. The substrates are 10 mm thick and contain air holes in radii of 2 mm, and the aluminum rods are assembled to the metallic substrates through mechanical mounting. The fabricated samples are shown in Appendix H.

A scanning near-field microwave system is employed to measure the transmission spectra and near-field electric field distributions. The system comprises a vector network analyzer (Keysight E5063A) and a 3D scanning stage. Two coaxial probes act as the emitting and probe antennas. The emitting antenna is placed in a drilled hole of the metallic substrate to excite the edge states and thus excite states with a full range of moments covering the −1 to 1 (π/a). However, the part within the light cone will scatter to surrounding air and does not contribute to the energy transport, and the other probe antenna is fixed to a module driven by a moving stage to measure the transmission data at discrete spatial positions. A detailed setup can be seen in Appendix H.

In summary, we have proposed and experimentally realized the asymmetric propagating topological rainbow as well as a dual-channel asymmetric topological frequency router. The proposed topological multiplexing topological devices rely on robust topological propagating modes instead of localized modes and can capture and divide the EM signals into multiple frequency channels. These devices are designed based on metallic valley TPCs and spoof SPP modes, which are not only optimized with large band gaps, but can also be efficiently excited with a simple point source. By directly mapping the electric field propagation and measuring the transmission spectra, the spatial channel separation, asymmetric transportation, and out-coupling of different EM wave frequency channels have all been experimentally demonstrated. Our proposed asymmetric frequency multiplexing devices pave the way for designing robust and compact multi-frequency topological photonic devices. Moreover, such devices can be easily applied to terahertz or telecommunication wavelengths through advanced micro-nanofabrication photolithography processes, providing a new approach for the development of topological photonic devices for next-generation terahertz or optical communications.

Acknowledgment. D. W. acknowledges support from the Anniversary fellowship of the University of Southampton.

Author Contributions:J. M., D. W., and C. O. conceived the idea and designed the prototype. J. M. and L. N. performed the theoretical analysis and numerical simulations. J. M. and Y. Y. fabricated the sample and performed the experiments. C. O. led the project and J. M. wrote the manuscript. H. L., Y. L., Q. X., Y. L., L. N., and Y. Y. helped with the manuscript and data analysis, and J. H. and W. Z. supervised the project. All authors contributed to the manuscript.