Objective Terahertz (THz) technology has broad application prospects in astronomy, security, biomedicine, broadband wireless communication, and other fields. However, the current THz system is bulky and has limited applications. THz integrated photonics is the key to further development and wide applications of THz technology, among which photonic topological insulator (PTI) is a good integration platform. Topological edge states (TESs) in the PTI bandgap have caught extensive attention. They can realize light transmission only along the interface and have no backscattering, with robustness to disorder and defects. Among PTIs, valley photonic crystal (VPC) constructed based on the photonic quantum valley Hall effect do not need to introduce magnetic fields or pseudospins, but only need to break the spatial inversion symmetry, and TESs will be formed on the edge of two photonic crystals with opposite valley Hall phases. However, once a traditional topological photonic device is designed, its functional characteristics are difficult to change. Manipulating the topological phase and realizing dynamic TES tuning will result in breakthroughs for designing THz photonic crystal chips, which becomes a research hotspot in this field. As a soft material with excellent properties, liquid crystals (LCs) are sensitive to external fields such as light, electricity, magnetism, and heat. Meanwhile, it is an ideal method to realize the dynamic control of THz topological devices by dynamically tuning the refractive index of LCs with an external electric field. Conventional LC- based topology devices adjust the TESs or topological angular states by changing the overall topological properties, and they are difficult to fabricate via experiments. Additionally, the study on THz tunable TESs based on local LCs has not been reported. Methods Different from the entire device filled with LCs, we only fill the hole of the topological interface with the LCs and design a tunable THz VPC. Firstly, a two-dimensional photonic crystal is constructed to break the spatial symmetry by changing the duty cycle of the two air cavities to open the bandgap. Then a VPC is constructed, and the tunable TESs are studied. Meanwhile, we construct a Z- shaped waveguide and add LCs after the first bend, further design a forked wavelength division multiplexer (WDM), and add LCs to the upper branch. Additionally, the TES characteristics with different THz frequencies are analyzed, with the effect of a defect on TES transmission studied finally. The refractive index change of LCs at the VPC interface can tune the TES transmission. This transition of TESs breaks conventional bulk- boundary correspondence, which attributes the existence of TESs in VPCs to bulk topology while disregarding the role of the interface refractive index. Results and Discussions We start with the basic properties of THz VPCs (Fig. 1) and pattern the VPCs on a silicon slab. Each unit cell of these VPCs comprises two inequivalent circle holes, R1=0.25a, R2=0.08a. The Dirac point originally located at K(K') is opened, creating a bandgap. Hz phase distributions of the upper and lower bands of VPC 2 and VPC 3 at the K(K') point have opposite directions. The Poynting vector also exhibits vortex properties of opposite chirality (black arrow) along with topological band inversion. VPC 2 and VPC 3 with bandgaps and band inversion will generate TESs at the interface of their composition. The influence of changing the LC refractive index of the interface on edge states in VPC is demonstrated. The projected energy band of the supercell with beard interfaces is calculated (Fig. 2). The dispersion curve of the edge state shifts down, indicating that some operating frequencies no longer maintain TESs. As the LC refractive index increases, the curve shifts down further, but the TES always maintains a frequency range. When the LC refractive index is adjusted under different applied voltages, TESs can be tuned over a certain frequency range (between blue and black dashed lines). At the boundary, the electric field has a local enhancement effect with its direction along the x direction. Then, an LC tunable Z- shaped topological waveguide is constructed (Fig. 3). When the LC is not filled, the waveguide maintains a high transmittance in the range of 0.90-1.03 THz. After the LC is filled and n=1.8, the upper edge of the transmittance curve shifts to low frequency. The passband range rises as the LC refractive index decreases. No matter what the refractive index of the LC is, the THz transmittance is the same at 0.950 THz, while at 1.005 THz, the THz wave has a very different field distribution. At 0.990 THz, the tunable transmission of the TESs in the Z- shaped waveguide and the electric field diagrams is shown (Fig. 4). Additionally, an LC- tunable wavelength division multiplexer (WDM) is designed (Fig. 5). At 0.990 THz and n=1.5, the THz wave passes through the upper branch, and under n=1.7 the THz wave passes through the lower branch. This is consistent with the THz transmission rule of the Z-waveguide above. When n increases from 1.50 to 1.68, the THz wave transmits mainly from port 2, and the transmittance is about 80 degrees o. As n increases from 1.68 to 1.80, the output of THz waves is mainly from port 3, and the transmittance is close to 100 degrees o. A point defect with no LC added is introduced to the interface and the electric field on the interface at 0.990 THz is shown (Fig. 6). The transmittance spectrum is almost unaffected by the defect. The LC tunable topological photonic devices constructed by pure LCs or dielectric rods have high requirements for LC packaging and manipulation. The VPC structure in our study has sound backscattering immunity and stable mechanical strength. The enhanced THz near- field at the boundary can interact with LCs in the air cavity. Additionally, it is convenient to control the orientation of LCs to change the refractive index by an external electric field, which ensures the TES tunability. We have simplified the LC integration and manipulation methods, which is conducive to follow-up experiments and further research. Conclusions THz integrated photonics is the key to further development and widespread applications of THz technology. VPCs are a good platform for realizing integrated devices and their dynamic control is in high demand. We propose a THz VPC with tunable TESs based on LCs, with a focus on the influence of LCs on the topological transmission characteristics. The topology- protected edge state of the Z- shaped waveguide can be dynamically tuned in the range of 0.98-1.00 THz, while the topological transmission characteristics in the range of 0.90-0.98 THz are unchanged, indicating that the device has sound robustness. Additionally, we construct a THz WDM which shows excellent multiplexing properties and defective immunity. In the future, the design can be further optimized to implement programmable broadband THz topology on- chip devices. Therefore, our study plays a significant role in promoting the wider application of PTIs and THz technology, and the results are of significance for a deep understanding of TESs and the development of THz integrated chips.
