Reconfigurable traveling-wave radiation in valley topological photonic crystals

Abstract

Valley photonic topological insulators have recently attracted much attention, in which the valley degree of freedom provides a promising solution to manipulate light waves. Currently, most studies of valley photonic topological insulators focus on designing valley-dependent transport behavior, but few studies on its radiation properties. In developing functional communication devices for practical applications, studying traveling-wave radiation and its reconfigurable properties in valley photonic topological insulators deserve more attention. In this paper, by adding nematic liquid crystals with tunable refractive index into waveguide channels of valley topological photonic crystals, we propose a reconfigurable traveling-wave radiation system that can dynamically manipulate radiation beams and their coverage regions. Via tuning dispersion of valley-locked waveguide modes controlled by the phase states of liquid crystals, we demonstrate that radiation beams have some unique tunable capabilities in the THz regime, such as single-beam, dual-beam, and multi-beam reconfigurabilities. Moreover, leveraging the idea of digitally encoding waveguide channels, we provide a solution for dynamically steerable traveling-wave radiation in the valley topological photonic platform. The proposed configurations provide more freedom to manipulate traveling-wave radiation and open a pathway for developing reconfigurable traveling-wave antennas in THz multi-link wireless communication system.

Graphical abstract

Keywords

valley topological photonic crystal / reconfigurable travelling-wave radiation / liquid crystals

1 Introduction

Inspired by valleytronics in condensed matter physics, valley photonic topological insulators have emerged as a prominent research field in classical electromagnetic (EM) systems owing to their fascinating topological properties provided by valley degrees of freedom [1–8]. Generally, the striking property of valley photonic topological insulators is that the valley-locked edge states exhibit topologically protected features with robust EM transmission against the local system perturbation. These properties have revealed excellent application potential for developing topological devices, such as on-chip waveguides [9], lasers [10], multichannel intersections [11], and tunable routers [12, 13]. Significantly, the valley-locked edge states are mainly concentrated around the interface (domain walls) between two valley photonic crystals (PhCs) with opposite valley Chern numbers, where the interface is usually one-dimensional and lacks width-designable flexibility. Unlike one-dimensional interfaces, recent studies demonstrate that valley-locked waveguides in heterostructures [14, 15] can design waveguide channel widths as needed, making them more flexible to bridge with conventional waveguides and to achieve high-flux EM transport. Meanwhile, these heterostructures supporting valley-locked waveguide modes are more suitable for high-energy-capacity topological photonic device applications, including energy concentrators [14], topological high-flux waveguides [16, 17], and splitters [18, 19]. Although relevant studies for valley-locked waveguide modes have shown their extraordinary transmission features, their radiation properties have received little attention so far. With the increasing demand for large capacity and versatile components in THz wireless communication systems, it is crucial for the flexible and variable characteristics of robust signal transmission, beam steering, and large space radiation coverage in the multi-link wireless communication with multiple users or devices. Therefore, studying the traveling-wave radiation characteristics in valley topological photonic crystals deserves more attention for real-world applications.

Some previous works based on valley photonic topological insulators have been demonstrated to achieve topological-wave directional radiations [20–22], but achieving steerable radiation beams remains difficult. Recently, traveling-wave radiations in photonic topological insulators have shown encouraging advances [23–27], which exhibit manipulating radiation beam capability. However, it is quite challenging to dynamically reconfigure the radiation beam (i.g., tuning the beam number, beam direction, or space coverage region at a fixed frequency) in wireless communication. Thus, it is of great significance to explore reconfigurable traveling-wave radiation with flexible steerable beam feature in valley topological insulators for designing advanced multifunctional wireless communication components.

In this study, taking advantage of both the refractive index tunability of the nematic liquid crystals (LCs) and the robust propagation property of valley-locked topological edge states, we propose a reconfigurable traveling-wave radiation system in valley topological photonic crystal to achieve dynamically steerable radiation beams. The proposed traveling-wave radiation systems are constructed by embedding LCs into waveguide channels of valley topological photonic crystal, whose tunability is attributed to the dispersion feature of valley-locked waveguide modes controlled by the phase states of the LCs. More importantly, this scheme not only realizes the direction of dynamically steering radiation beam but also achieves multifunctional manipulation (e.g., radiation beam number and radiation coverage region) in dual-beam and multi-beam radiation systems, providing greater flexibility for regulating the radiation beams. Besides, to implement the reconfigurable traveling-wave radiation, only the waveguide channel needs to be tuned in the proposed valley topological photonic platform, which makes implementation of the control circuit much easier. The proposed configuration provides a versatile platform for manipulating radiation beams and enhances the potential for advanced THz wireless communication system applications.

2 Results and discussion

2.1 Design of reconfigurable traveling-wave radiation system

Figure 1(a) shows the proposed reconfigurable traveling-wave radiation system in valley topological photonic crystal, mainly consisting of a dielectric cylinder array and waveguide channels with LCs. The orange cylinders arranged in a honeycomb lattice represent a dielectric cylinder array in the bulk domain; the purple and blue cylinders represent the cladding layer (outside) and liquid crystals (inside) in the internal and external waveguide channels. The whole cylinder array with the height h = 0.11 mm is covered by the top and bottom metal plates to mimic the two-dimensional valley topological photonic system. The proposed configuration system enables us to implement reconfigurable traveling-wave radiation controlled by phase states of LCs in several waveguide channels. Specifically, by applying different coding sequences to programmable waveguide channels 1−8 using a field programmable gate array (FPGA), we can dynamically reconfigure the dispersion of each waveguide channel to realize a flexible manipulation of traveling-wave radiation in the valley topological photonic system, as shown in Fig. 1(b). Therefore, the proposed reconfigurable traveling-wave radiation system could be beneficial for developing functional THz wireless communication devices, such as simultaneously building wireless links with multiple users or devices and providing flexibly controlled beam coverage.

Figure 1(c) gives the detailed configurations of the unit cells in both waveguide channels 1−8 and the bulk domain. The programmable channels 1−4 and 5−8 are constructed by two types of unit cells containing LCs (blue), cladding layer (purple), and dielectric cylinders (orange). The unit cell in the bulk domain has six dielectric cylinders with relative permittivity {\epsilon }_d=11.2 and loss tangent tan\delta = 0.0022, in which the lattice constant a=0.35 mm. Here, the Rogers3010 material can be used to fabricate the real valley PhCs. The diameter of cylinders in each unit cell are d_1=0.44a, d_2=0.28a, d_3=0.33a (cladding layer), d_4=0.30a (LCs), d_5=0.49a, d_6=0.48a, and d_7=0.32a, respectively. The cladding layer with relative permittivity {\epsilon }_c=16 is used for packaging LCs in cylinder C. In our design, we choose nematic LCs for designing reconfigurable traveling-wave radiation because its refractive index can be tuned by dynamically changing the orientation of LC molecules under the bias voltage in the THz frequency range [28–30]. As depicted in Fig. 1(d), when the bias voltage is switched from V_{\mathrm{o}} to V_{\mathrm{e}}, the orientation of the LC molecules will deflect from the pre-aligned horizontal direction (initial state) to the vertical direction (final state). In this case, the nematic LCs we used has a tunable refractive index from n_{\mathrm{o}}=1.565 (initial state) to n_{\mathrm{e}}=1.948 (final state) near the 0.5 THz in Refs. [31–33]. Here, we encode “0” and “1” as LCs in the refractive index n_{\mathrm{o}}=1.565 and n_{\mathrm{e}}=1.948, respectively. Based on this feature, the tunability of traveling-wave radiation is achieved by the digitally manipulating refractive index of the LCs at two phase states in each channel.

To characterize the topological property of the proposed valley photonic crystal, the valley Hall topological phases around K valley can be explained by an effective Hamiltonian model. According to the \mathit{k}\cdot \mathit{p} theory [3, 34], the effective Hamiltonian near the K valley can be written as

(1){\hat{H}}_K=v_D\left({\sigma }_x\delta k_x+{\sigma }_y\delta k_y+{\mathrm{\Delta }}_p{\sigma }_z\right),

in which v_D is the group velocity, \delta k_i (i = x, y) denotes the displacement of the wavevector to the K point, and {\sigma }_i (i = x, y, z) are the Pauli matrices. v_D{\mathrm{\Delta }}_p is the effective mass term, which is determined by the broken inversion symmetry and is proportional to the bandgap width. Similarly, the effective Hamiltonian near the K{}^{\mathrm{\prime }} valley can also be obtained using the time-reversal operation. Here, the inversion symmetry of the unit cell is broken by adjusting the diameter difference between two cylinders \mathrm{\Delta }d (\mathrm{\Delta }d = d_6-d_7), leading to the formation of PhC1 for \mathrm{\Delta }d>0 and PhC2 for \mathrm{\Delta }d<0. The topological properties for PhCs 1 and 2 can be characterized by solving the Berry curvature {\mathrm{\Omega }}_{\mathrm{K}/{\mathrm{K}}^{\mathrm{\prime }}} in Eq. (2) and the valley Chern number C_{\mathrm{K}/{\mathrm{K}}^{\mathrm{\prime }}} in Eq. (3) around the K/K{}^{\mathrm{\prime }} valley:

(2){\mathrm{\Omega }}_{\mathrm{K}/{\mathrm{K}}^{\prime }}(\delta \mathit{k})=\pm {\displaystyle \frac{{\mathrm{\Delta }}_P}{2(\delta k^2+{\mathrm{\Delta }}_P^2{)}^{3/2}}},

(3)C_{\mathrm{K}/{\mathrm{K}}^{\mathrm{\prime }}}={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{\mathrm{H}\mathrm{B}\mathrm{Z}}{\mathrm{\Omega }}_{\mathrm{K}/{\mathrm{K}}^{\mathrm{\prime }}}(\delta \mathit{k}){\mathrm{d}}^2\delta \mathit{k}=\pm 1/(2\text{sgn}\left({\mathrm{\Delta }}_p\right)).}

In parallel, we also numerically calculate Berry curvatures of the band around K valley based on first principles. More details about the designed valley PhCs are illustrated in Appendix A, including the photonic band structure and topological phase transitions controlled by the diameter difference \mathrm{\Delta }d. These analysis and numerical results demonstrate that the proposed valley PhCs have nontrivial topological properties. Moreover, recent studies have shown that valley topological PhCs with the honeycomb lattice can support valley-polarized chiral edge states. These edge states can be controlled by tuning the on-site boundary potentials [35–38], in which effective boundary matrix {\hat{M}}_{\mathrm{e}\mathrm{f}\mathrm{f}} describing the boundary conditions can be expressed as

(4){\hat{M}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\tau }_z\left({\sigma }_y\sin {\theta }_V+{\sigma }_z\cos {\theta }_V\right),

where {\tau }_z is Pauli matrix and {\theta }_V is a phenomenological parameter determined by on-site boundary potentials. For the zigzag boundary {\theta }_V = \pi/2, the required boundary matrix satisfies {\hat{M}}_{\mathrm{e}\mathrm{f}\mathrm{f}} = {\tau }_z{\sigma }_y. Based on this, we can tune the on-site boundary potentials by changing the phase states of the LCs at PhC boundaries, namely channels 1−8, as illustrated in Fig. 1. Therefore, the above theoretical analysis is the basis for realizing the valley topological system and manipulating the EM modes it supports.

2.2 Tunable valley-locked waveguide modes

To verify the above statements, we numerically calculate the band structures and transmission properties of the valley photonic crystal, as shown in Fig. 2. In this work, we investigate the band diagram using COMSOL Multiphysics in the 2D system and radiation properties using CST Microwave Studio in the 3D system. The computational models for CST and COMSOL simulations were constructed with adaptive meshing. According to the bulk-edge correspondence, valley topological edge states can be supported at the interface between PhC1 and PhC2 with the opposite valley chern numbers. To flexibly bridge with conventional waveguides or devices, we construct an A-type interface by introducing an air channel between PhC1 and PhC2, whose supercell structure, electric fields, Poynting vectors (yellow arrows), and projected band diagram are shown in Fig. 2(a). In numerical calculations, we set two types of boundary conditions in supercell structures, where the left and right boundaries along the x-axis are set as periodic boundary conditions, and the upper and lower boundaries along the y-axis are set as scattering boundary conditions. The metal material is set as the PEC model. From the photonic band structure, we can see that the dispersion of waveguide modes in the bandgap can be controlled when the width of the air channel is tuned by changing the distance g between PhC1 and PhC2. This provides channel width degree of freedom for manipulating guided modes. Meanwhile, it can also be observed that electric fields and Poynting vectors at the K valley (f=0.5 THz denoted by the star) are distributed at the PhC-air boundaries and in the air channel when g=0.98a. Remarkably, the direction of the energy flow confined at the PhC1-air boundary (anti-clockwise) is opposite compared with the PhC2-air boundary (clockwise), which implies the valley-locked feature. Additionally, we also further demonstrate in Appendix A that the waveguide modes in the proposed system have the valley-locked unidirectional propagation property. To explore the tunable property of the valley-locked waveguide modes, we further study the B-type and C-type interfaces by changing on-site boundary potentials at a fixed width g=0.98a, where sectional cylinders along the PhC-air boundary are replaced by LCs and cladding layer. Figures 2(b) and (c) show supercell structures, electric fields, Poynting vectors, and the projected band diagram for B-type and C-type interfaces. Similar to the above investigations in the A-type interface, the distribution of waveguide modes and their valley-locked features can still be well preserved in the B-type and C-type interfaces, as can be seen from electric fields and Poynting vectors at f=0.5 THz (valley points denoted by inverted and regular triangles) in Fig. 2. More significantly, we can find from the photonic band structure that the dispersion of the valley-locked waveguide modes can be tuned in the bandgap when the refractive indexs of LCs is switched between initial state n_{\mathrm{o}}=1.565 and final state n_{\mathrm{e}}=1.948. Specifically, the valley-locked waveguide modes work at the higher frequency band of 0.476−0.525 THz (0.494−0.525 THz) in the B(C)-type interface when LCs are in initial state, while they work at the lower frequency band of 0.462−0.525 THz (0.471−0.520 THz) in the B(C)-type interface when LCs are in final state. Here, the dispersion of the valley-locked waveguide mode can be controlled because the on-site boundary potential can be tuned by changing the phase states of the LCs at PhC-air boundaries in the proposed valley photonic system. Given this, the mechanism of the dispersion regulation in B-type and C-type waveguides has a common origin; the difference in dispersion regulation capabilities stems from the waveguide configuration after introducing LCs. Consequently, the dispersion of the valley-locked waveguide modes supported by the B- or C-type interface can be reconfigured when LCs are switched between initial state and final state, which are consistent with the above theoretical analysis. Furthermore, it is worth noting that the dispersion curves of valley-locked waveguide modes are located in the fast-wave region, which is vital for achieving leaky-wave radiation. Here, the ‘fast-wave region’ denotes the region space where the mode’s phase velocity surpasses the speed of light in vacuum, leading to radiation behavior.

To ensure the effective coupling between classical guided modes and valley-locked waveguide modes, we design a transition structure composed of a tapered dielectric waveguide with a partial metal cover. Design ideas of the transition structure are referred to in our previous work [39]. Figures 2(f)−(h) show the terminal-matched A(B)-type waveguide and corresponding simulated E_z field distribution in the xy plane, transmission, and reflection coefficients. One can see that classic guided modes are smoothly converted into valley-locked waveguide modes by the transition structure when EM waves are input from port 1, as displayed in Figs. 2(g) and (h). It can also be found that the transmission and reflection properties of the system shown in Fig. 2(f) quantitatively demonstrate the smooth transition between two waveguide modes, and the working frequency band for the A(B)-type waveguide is consistent with the band diagram in Figs. 2(a)−(b). Moreover, we also studied the transmission and reflection properties of the C-type waveguide structure under two phase states in Appendix B. These results provide a basis for subsequent research into traveling-wave radiation properties under effective coupling and smooth conversion between the two waveguide modes.

2.3 Traveling-wave radiation and steerable beam features

To generate the traveling-wave radiation, the number of rows n in the PhC2 of the B-type waveguide is gradually decreased from n = 5 to n = 1 along the −y direction. The proposed configuration of the traveling-wave radiation system is plotted in Fig. 3(a), in which the PhC2 with n = 1 row serves as the radiation aperture. In order to study the influence of the number of rows n on the traveling-wave radiation, we calculate the leakage rates under the different number of rows n, and the results are displayed in Fig. 3(b). Based on traveling-wave radiation theory [40], the leakage rate can be expressed as follows:

(5){\displaystyle \frac{\alpha }{k_0}}={\displaystyle \frac{c\ln \left({\displaystyle \frac{1}{1-\eta }}\right)}{4\pi Lf}}.

Here, the leakage rate is considered to be uniform along the radiation aperture. L = 5 mm and \eta = 1−(P_L/P_0) represent radiation aperture and efficiency, where P_0 and P_L are input and output power at the termination. We find that the leakage rate for n=1 is much higher than that in other cases (n = 2 to 5). In contrast, the leakage rate decreases dramatically with the increase of n from 2 to 5. For n=1 case, we simulated the E_z field distribution of traveling-wave radiation on the xy plane at f = 0.495 THz plotted in the lower panel of Fig. 3(a). As expected, the EM wave can propagate forward and radiate outward through the single-row cylinder structure, which confirms the radiation property of valley-locked waveguide modes located in the fast wave region in the band diagram of Fig. 2(b). Additionally, we also studied the radiation efficiency of the traveling-wave radiation structure in Appendix C. To further demonstrate the radiation property of the proposed configuration, the radiation patterns at different frequencies (i.e., f = 0.485, 0.495, and 0.505 THz) corresponding to two phase states in LCs are shown in Fig. 3(c). It is seen that radiation beams have different direction angles at various frequencies when the LC is in a specific phase state, which exhibits traveling-wave radiation with frequency scanning feature. In view of dispersion tunable properties, we further studied the traveling-wave radiation properties in the proposed configuration when the LC is in different phase states. In this case, the traveling-wave radiation patterns are presented in Fig. 3(e). We can find that the radiation beams controlled by the phase states of LCs have the fixed-frequency beam scanning behavior in the proposed configuration. Such fixed-frequency beam scanning properties can be understood as the dispersion curves under different phase states corresponding to different propagation constant k_x via choosing a fixed frequency (e.g., f = 0.49 THz) in the band diagram of Fig. 2(b). Thus, the beam angle \theta = \arccos(k_x/k_0) changes with different k_x at a fixed frequency. More details about this section (e.g., the range of radiation beam angles and coverage controlled by the refractive index tuning range) are shown in Appendix D.

To show the transmission and reflection properties under traveling-wave radiation, we simulated the transmission and reflection coefficients for the whole waveguide in two phase states of LCs, as plotted in Fig. 3(d). Compared with the B-type waveguide without radiation shown in Fig. 2(f), the transmission coefficient (|S_{21}|) is significantly reduced in the waveguide due to the EM energy radiation, and the reflection coefficient (|S_{11}|) still has excellent matching performance in the working frequency band. These results demonstrate the traveling-wave radiation and its tunable properties controlled by the phase states of LCs in the proposed valley topological photonic crystals under the terminal-matched condition. As a comparison, we also demonstrate the waveguide transmission and radiation properties based on the trivial photonic crystal with bandgap-guided functionality, which is highly dependent on the waveguide interface configuration (see more details in Appendix E). Therefore, the proposed traveling-wave radiation configuration with the robust transmission and beam stability properties provides useful guidelines for designing tunable topological EM devices (e.g., reconfigurable traveling-wave antennas).

2.4 Dual-beam reconfigurable traveling-wave radiation system

The above discussions only focus on the single radiation beam and its tunable characteristics in the unidirectional space coverage. In fact, the dual-beam traveling-wave radiation system can simultaneously build wireless links with two users or devices, which have significant applications in THz-integrated devices and communication networks. In this scenario, we study the dual-beam traveling-wave radiation in the dual-waveguide channel composed of PhC1 and PhC2, as shown in Fig. 4(a). Ports 1, 2 and ports 3, 4 are set at the terminals of the channels 1 and 2, respctively. Figure 4(b) shows dual-beam traveling-wave radiation E_z field distribution at f = 0.495 THz corresponding to two final states of LCs in two channels when the EM wave is input from port 1 and port 3. It can be clearly seen that both radiation beams toward the forward direction, with a beam angle of 59° (−59°) in the upper (lower) half-space, achieving the symmetrical bidirectional space coverage. To dynamically control dual-beam traveling-wave radiation, we study the dual-beam reconfigurablity at a fixed frequency (f = 0.495 THz), which contain two types: one is dual-beam coverage region reconfigurablity at fixed phase states and the other is dual-beam radiation direction reconfigurablity at fixed port exciation.

In the E_z field distributions of Figs. 4(c, f, g), by selectively exciting different ports, we demonstrate the reconfigurable properties of dual-beam radiation coverage region when the LC in two waveguides is in final state. It is observed in Figs. 4(c) and (g) that the system can generate a radiation beam with a 59° (121°) angle toward the forward (backward) direction and a radiation beam with a −121° (−59°) angle toward the backward (forward) direction when Ports 1 (2) and 4 (3) are excited simultaneously, revealing the asymmetrical bidirectional space coverage. Similarly, contrary to the beam pointing situation in Fig. 4(b), two radiation beams point in the backward direction and a beam angle of 121° (−121°) in the upper (lower) half-space when ports 2 and 4 are excited simultaneously in Fig. 4(f). These results confirm that the dual-beam coverage regions can be reconfigured in the bidirectional radiation by selectively switching the excitation ports at a fixed frequency in our proposed configuration. More interestingly, the dual-beam traveling-wave radiation under the fixed excitation ports can also be dynamically reconfigured by digitally encoding two phase states of LCs in channels 1 and 2, respectively. The dual-beam radiation patterns and corresponding beam angle at f=0.495 GHz when exciting from different ports are plotted in Figs. 4(d, e, h, i), in which initial and final states of LCs in channels 1 and 2 have four different combining states: I (n_{\mathrm{e}}, n_{\mathrm{e}}), II (n_{\mathrm{e}}, n_{\mathrm{o}}), III (n_{\mathrm{o}}, n_{\mathrm{e}}), and IV (n_{\mathrm{o}}, n_{\mathrm{o}}). For convenience, these four combining states are defined as digital coding sequences “11 (I)”, “10 (II)”, “01 (III)”, and “00 (IV)”, shown in the inset of Fig. 4. When the excitation port is fixed, it can be found that not only two beams can be manipulated simultaneously, but also one beam can be independently controlled in a single channel while the other beam remains unchanged in the other channel. These results indicate the beam-steering capability for dual- and single-beam operation. In the meantime, the corresponding beam angle range in channels at four different excitation cases is shown in the inset. It should be pointed out that the radiation beam direction has flexible and variable characteristics due to the tunable property of LC phase states controlled by bias voltage. Here, we introduce the third (middle) phase state of LCs with refractive index n_{\mathrm{m}\mathrm{i}\mathrm{d}}=1.768 and combine it with the initial and final states. The beam angles radiated from the two channels under different excitation cases are shown in Fig. 4(j), in which the four different color polygons represent the beam angles under four different exciting cases. For example, a blue regular triangle corresponds to a 64° beam angle from channel 1 and a −59° beam angle from channel 2 in combining state (n_{\mathrm{m}\mathrm{i}\mathrm{d}}, n_{\mathrm{e}}) under exciting ports 1 and 3; a square corresponds to a 116° beam angle from channel 1 and a −116° beam angle from channel 2 in combining state (n_{\mathrm{m}\mathrm{i}\mathrm{d}}, n_{\mathrm{m}\mathrm{i}\mathrm{d}}) under exciting ports 2 and 4. This result reveals that the radiation beam direction controlled by the phase state of LCs has manipulation diversification. Moreover, we further discussed radiation patterns in introducing middle-phase states, see Appendix F for more details. From the above results, it can be concluded that the proposed dual-beam traveling-wave radiation system offers a flexible and diversified approach to manipulate two radiation beams in the future integrated communication system with two users or devices.

2.5 Multi-beam reconfigurable traveling-wave radiation system

In addition to the dual-beam radiation with two space coverage regions, multi-beam radiation with large space coverage can establish wireless links with targets or devices in multiple discrete regions simultaneously, which play important roles in THz communication system. Given this, we propose a multi-beam reconfigurable traveling-wave system consisting of B-type and C-type waveguides, as depicted in Fig. 5(a). Here, we define outside traveling-wave radiation channels 1 through 4, inside propagation channels 5 through 8, and six ports 1 through 6. To show the radiation property of the proposed multi-beam configuration, the E_z field distributions at f = 0.48 THz under simultaneous exciting ports 1 and 2 when the LCs in all channels are in final state are displayed in Fig. 5(b). Obviously, the proposed multi-beam traveling-wave radiation system achieves larger radiation coverage, i.e., the four radiated beams are located in four different regions (quadrants). More importantly, the multi-beam radiation direction can also be dynamically tuned using an FPGA control circuit. The radiation beam on channels 1 to 4 can be controlled independently by different coding sequences while maintaining the phase state of the LCs in channels 5 to 8 as the final state. Figure 5(c) shows the multi-beam radiation patterns corresponding to four different coding sequences at f = 0.48 THz, which means that the proposed configuration presents flexible manipulating multi-beam capabilities at a fixed frequency. The isolation between the output beams can be further improved, if required, by increasing the lateral size of the photonic crystal structure to reduce evanescent coupling. The current design prioritizes device compactness while demonstrating the core functionality.

Apart from manipulating multi-beam radiation direction, in some applications, it is desired to dynamically control beam coverage region at a certain frequency. To this end, we further study the reconfigurable properties of radiation coverage regions corresponding to two beams via switching the phase state of LCs in channels 5 to 8, as shown in Figs. 5(d)−(h). From the field distributions and radiation patterns, one can observe that dual-beam radiation coverage regions can be arbitrarily reconfigured in four different regions when controlling four different coding sequences. This interesting radiation-coverage reconfigurable phenomenon can be understood in the following way: the wave propagation along the inside channels was reversibly switched between ‘‘on” and ‘‘off” at a certain frequency by altering the initial state and final state of LCs in channels 5 to 8. This property agrees with the C-type waveguide dispersion in Fig. 2(c). Moreover, the beam angles radiated from channels 1 to 4 corresponding to eight different coding sequences are illustrated in Fig. 5(i), in which the red (orange) border and blue (purple) filled area in the polygon represent the beams from channel 1 (3) and channel 2 (4). For example, a regular triangle with a red border and a blue-filled area corresponds to a 70° beam angle from channel 1 and a −82° beam angle from channel 2 in coding sequence III; a blue (purple) circle or a red (orange) pentagon corresponds to a −70° (−110°) beam angle from channel 2 (4) in coding sequence VII or a 70° (110°) beam angle from channel 1(3) in coding sequence VIII. These features indicate that we can exploit the channels with LCs by FPGA control to construct a multi-beam reconfigurable traveling-wave radiation system that can dynamically modulate the beam radiation direction and coverage region. Additionally, we also demonstrated the reconfigurable traveling-wave radiation in other cases (i.e., tri-beam and single-beam reconfigurability), see more details in Appendix G. The proposed multi-beam reconfigurable radiation system could be promising for intelligent wireless devices with multiple links in THz communications. Although we only present the configuration design and simulation demonstration in this work, the proposed traveling-wave radiation configuration can be implemented with fabrication and experiment in further work. Generally, the valley PhCs were fabricated by employing a nanofabrication process on an integrated silicon photonic platform. Detailed discussion on the experimental feasibility of implementing the proposed configuration can be found in Appendix H.

3 Conclusions

In conclusion, we have proposed and demonstrated the reconfigurable traveling-wave radiation system in valley topological photonic crystal, which is able to dynamically manipulate radiation beams and their coverage regions. The tunable traveling-wave radiation systems are constructed by employing LCs in the waveguide channel of valley topological photonic crystal. Via investigating the dispersion relations of waveguide channels loading LCs, we show that valley-locked waveguide modes can be tuned in the topological bandgap by controlling the phase states of the LCs. These findings would be crucial for achieving functional tunability of radiation beams in THz frequency band, such as single-beam, dual-beam, and multi-beam reconfigurability. In particular, by digitally encoding each waveguide channel, the proposed valley topological photonic platform provides a solution for realizing dynamically reconfigurable traveling wave radiation. Moreover, the proposed configuration is flexible to bridge with the existing photonic waveguides or devices and can also reduce the complexity of the biasing circuit since it does not require controlling each unit cell in the bulk domain. Our work provides a promising way to flexibly manipulate traveling-wave radiation in valley topological photonic platform, which has potential applications for designing reconfigurable intelligent devices in the next-generation THz wireless communication system.

4 Appendix A: Topological phase and transmission properties of the proposed valley photonic crystals

In this part, we show the topological phase transition and transmission properties in the proposed valley photonic crystals. When the two dielectric cylinders have the same diameter (i.e., \mathrm{\Delta }d=0, where d_6=d_7=0.4a), a pair of degenerate Dirac points occur at the K/K{}^{\mathrm{\prime }} valley due to the protection of inversion symmetry, as shown in Fig. A1(a). When breaking the inversion symmetry (e.g., \mathrm{\Delta }d = ± 0.16a, where d_6=0.48a, d_7=0.32a or d_6=0.32a, d_7=0.48a), the degenerate Dirac points are lifted, opening up a photonic bandgap. To reveal the properties of the eigenstates at the K valley, Fig. A1(b) gives the E_z phase profile and Poynting vectors at the K valley corresponding to two bands in the case of \mathrm{\Delta }d = ± 0.16a. One can find that the two eigenstates at the K valley exhibit left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) phase vortices around the center of a cylinder, respectively. This typical vortex distribution displays a remarkable feature of valley PhCs. To demonstrate the topological phase transition, we plot the valley phase diagram in Fig. A1(c), illustrating the evolution of eigenfrequencies for LCP and RCP states at the K valley as a function of \mathrm{\Delta }d. By varying \mathrm{\Delta }d, it is obvious that the sequence of LCP (marked by blue arrows) and RCP (marked by red arrows) states are inverted at the critical point \mathrm{\Delta }d=0, which gives rise to a topological phase transition.

To further confirm the topological properties of the system, we numerically calculate the Berry curvature of the first photonic band for \mathrm{\Delta }d = ± 0.16a near K valley, as shown in the right panel of Fig. A1(b). It can be found that Berry curvatures distributed near K valley for \mathrm{\Delta }d=0.16a are opposite to those \mathrm{\Delta }d=-0.16a, which means that valley PhCs with opposite \mathrm{\Delta }d have distinct topological phases. As a result, two topologically distinct VPCs ensure topological protection of edge states at the interface. To explore the transmission behavior of the proposed configuration, we construct a pair of circularly polarized sources (marked by purple stars) with opposite orbital angular momentum (OAM), which are placed in the middle of the A-type waveguide channel, as shown in Fig. A1(d). Among them, the circularly polarized sources consisting of a four-dipole source array with clockwise (anticlockwise) phase distributions are depicted in the left inset. Simultaneously, we simulated the E_z field distributions at the K valley (f=0.5 THz), as shown in the lower panel of Fig. A1(d). When one source array with clockwise (counterclockwise) OAM is placed near the lower (upper) boundary of PhC2(1) and the other source array with counterclockwise (clockwise) OAM is put near the upper (lower) boundary of PhC1(2), achieving unidirectional propagation for EM modes along the left (right) direction. Such unidirectional transmission behavior validates the valley-locked feature of topological waveguide modes.

5 Appendix B: Terminal-matched C-type waveguide and its tunable propagation feature

Here, we further investigated the transmission properties of the C-type waveguide channel in two phase states. The schematic of the terminal-matched C-type waveguide is shown in Fig. A2(a). To begin with, we calculated the transmission and reflection coefficients of the whole waveguide in initial (n_{\mathrm{o}} = 1.565, upper panel) and final (n_{\mathrm{e}} = 1.948, upper panel) states when EM waves are input from port 1, as shown in Fig. A2(b). We can find that the working band of the valley-locked waveguide modes can be more widely shifted within the bandgap when the refractive indices of the LCs are switched between the initial and final states. Specifically, the valley-locked waveguide modes work at the higher frequency band of 0.494−0.525 THz when the LCs are in the initial state, while they work at the lower frequency band of 0.471−0.520 THz when the LCs are in the final state. Simultaneously, the working frequency band of valley-locked waveguide modes is consistent with the band diagram in Fig. 2(c) of the main text. It is worth noting that the smooth transition between two waveguide modes can be realized by the transition structure even in two different working bands. In order to clearly show tunable transmission and isolation for valley-locked waveguide modes, we show the Ez field distributions at f = 0.48 THz when the LCs are in the two phase states, as displayed in Fig. A2(c). When LCs are in the final state(lower panel), the EM energy is efficiently transmitted to the right port along the C-type waveguide channel. However, when LCs are in the initial state(upper panel), the EM energy transmission is prohibited in the C-type waveguide channel. These results demonstrate the switchable transport feature for valley-locked waveguide modes at a fixed frequency and provide a basis for subsequent research into controllable beam coverage in our proposed configuration.

6 Appendix C: The estimates and analyses of the relevant radiation parameters

To explore the impact of LC losses (e.g., absorption coefficient) in the THz band on radiation efficiency, we further study the radiation efficiency simulation results with and without LC losses as shown in Fig. A3. Here, we incorporate the measured absorption coefficient of a representative nematic LC in the THz regime from the reference [31] of the main text into our simulation model, where the corresponding refractive indices are n_{\mathrm{o}} = 1.565 + i0.0158 and n_{\mathrm{e}} = 1.948 + i0.0134 for the 1825 mixture at 0.5 THz. In comparison to the LCs without losses, we can find from the figure that there is a remarkable decrease in the radiation efficiency when the LCs introduce losses in the initial and final states. Meanwhile, it is worth noting that the reconfigurable traveling-wave radiation functionality is not fundamentally hindered in the proposed configuration, although introducing loss decreases the radiation efficiency.

In our current design, the response speed is fundamentally limited by the response time of the nematic LCs. Since the response time of LC-based reconfigurations is related to the thickness of the LC layer and the drive bias scheme [41–44], the beam switching speed can be demonstrated by optimizing the LC thickness and manipulating the electrode bias scheme in future work. Moreover, we can also provide a value of precision for the beam angle in a given voltage or frequency value. That is, a discrete change in LC phase state (refractive index \mathrm{\Delta }n) results in a beam deflection. In practical applications, the resolution of this control is fundamentally limited by the precision of the applied voltage and the uniformity of the LC alignment, which requires systematic experimental characterization in our subsequent research. From our simulated dispersion diagrams [e.g., Figs. 3(b, c)], the valley-locked waveguide mode exists over a bandwidth of approximately \mathrm{\Delta }f\approx 0.47-0.51 THz around our design frequency. Remarkably, compared with the conventional configuration, the traveling-wave radiation in the proposed topological system offers significant advantages, including immunity to bending distortions as well as stable performance across operating bandwidths and beam directions. Therefore, we can make use of the design parameters or control voltages applied to the liquid crystals to tune the beam direction and coverage, which demonstrates the technological feasibility and superiority of traveling-wave radiations in valley photonic platforms with electrically controlled reconfigurability.

7 Appendix D: Beam direction and coverage manipulated by refractive indices for the LCs

To explain the traveling-wave radiation and its beam angles tuned by the LC refractive index, we further study the traveling-wave radiation properties at a fixed frequency (f = 0.490 THz) in the proposed configuration under different refractive indices (e.g., n_{\mathrm{o}} = 1.565, n_{\mathrm{m}\mathrm{i}\mathrm{d}} = 1.768, and n_{\mathrm{e}} = 1.948). To begin with, we study the evolution of the dispersion relations for valley-locked waveguide modes with respect to the refractive index n (i.e., from n_{\mathrm{o}}, n_{\mathrm{m}\mathrm{i}\mathrm{d}} to n_{\mathrm{e}}), to clearly illustrate how the radiation properties are controlled by refractive indices, as presented in Fig. A4(a). It can be observed that the dispersion curve shifts up or down in the bandgap when tuning the refractive index n. Via choosing a fixed frequency f = 0.49 THz (marked by a horizontal dashed line), we noticed that dispersion curves under different refractive indices correspond to different propagation constants k_x (marked by k_1, k_2, and k_3), respectively. In our work, we focus on the radiation of the fundamental (m = 0) wave located in the fast wave region. According to traveling-wave radiation theory [45–48], the radiation condition and beam direction angle for the fundamental wave can be expressed as follows:

(D1)

(D2)\begin{array}{c}{\theta }_m=\text{arccos}\left(\sqrt{{\epsilon }_g}\right),\end{array}

where \sqrt{{\epsilon }_g} is the equivalent dielectric constant. Furthermore, the equivalent dielectric constant can be written as \sqrt{{\epsilon }_g}={\lambda }_0/{\lambda }_g, in which {\lambda }_g is the guided wavelength in the traveling-wave radiation waveguide, and {\lambda }_0 is the wavelength in free space. From the above equations, we can find that the traveling-wave radiation properties are determined by the equivalent dielectric constant \sqrt{{\epsilon }_g}, so varying \sqrt{{\epsilon }_g} will lead to changing beam radiation angles. To verify this, we solve the equivalent dielectric constant \sqrt{{\epsilon }_g} of the traveling-wave radiation waveguide under different refractive indices, where the guided wavelength {\lambda }_g can be obtained from the simulated electric-force line distribution. In this case, we find \sqrt{{\epsilon }_g} < 1 for three different refractive indices at a fixed frequency f=0.490 THz, which can satisfy the condition of fundamental mode radiation consistent with (D1). Meanwhile, the radiation beam angle can also be obtained according to (D2) for different refractive indices. Specifically, these results are shown in Fig. A4(b), where beam angle {\theta }_c and {\theta }_s present the calculation and simulation results. Based on this, we can determine quantitatively the radiation beam angles and coverage at different refractive indices, which agree well with the simulation results. In addition, the beam coverage corresponding to different quadrants can be controlled by exciting the ports while maintaining a constant tunable refractive index range. Therefore, we change the beam angle and coverage by manipulating the refractive index n (or phase states) of the liquid crystal in the waveguide and the excitation port at the waveguide terminal. Therefore, the above analysis provides a clear physical mechanism to explain traveling-wave radiation and its operating principle.

8 Appendix E: Transmission and radiation properties of EM modes in the trivial and topological systems

In this section, we further investigate the connection between the topological properties (e.g., valley protection, robustness) and the radiation characteristics. To begin with, we study the transmission and radiation properties of the straight and bent channels based on trivial photonic crystals, whose operating frequency band is designed to cover the operating frequency band of the proposed topological PhC system, as shown in Fig. A5(a). To explore the transmission and radiation characteristics, we performed full-wave simulations and compared the transmission coefficients when EM waves were input from port 1. The schematic of transmission and radiation configurations is shown in the upper panel of Figs. A5(b)−(e). In the lower panel of Figs. A5(b) and (d), the simulated E_z field distributions at f=0.470 THz show that the straight channel interface can support EM modes forward transmission and radiation. In contrast, as displayed in the lower panel of Figs. A5(c) and (e), EM modes show significant loss in the bending channel interface, predominantly stemming from strong scattering and reflection at the corners. Meanwhile, the simulated transmission coefficients of four configurations are shown in Fig. A5(f). In particular, compared to the transmission scenario in the straight channel, the entire operating frequency band of EM modes is severely deteriorated under the radiation scenario. Additionally, EM modes in the bending channel experience significant loss, resulting in low transmission and narrow bandwidth. These behaviors indicate that topological protection for EM modes is absent in the trivial system.

For further clarifying the effect of topological protection on the radiation, it will be important to demonstrate EM energy transmission and radiation under the bending interface. Next, we study the transmission and radiation properties of bending channels constructed by B-type and A-type channels in the proposed topological system. Here, the transmission and radiation configurations in three different bending (waveguide, radiation 1, and radiation 2) types based on the proposed valley photonic crystals are shown in the upper panel of Figs. A5(g)−(i), respectively. In order to clearly illustrate the transmission and radiation features, we present the E_z field distributions of EM modes at f=0.485 THz in the lower panel of Figs. A5(g)−(i). It can be seen that the presence of sharp corners cannot hinder the transport of waves in the bending waveguide due to the topological protection properties. More importantly, even in the dual-channel radiation system (II), the radiation characteristics remain unchanged compared with the single-channel radiation system (I). Remarkably, as shown in Fig. A5(j), the operating bandwidths in two radiation cases are almost the same as in the transmission case, indicating that the radiation for the topological wave is unaffected by varying configuration types in the proposed topological waveguide. To quantitatively evaluate the radiation performance for the bending configuration, we plot radiation patterns in three different cases at f=0.485 THz, as shown in Fig. A5(k). The inset shows the radiation beam angles corresponding to three different situations. These results demonstrate that the EM wave radiation exhibits robust (e.g., beam stability) in the topological system with suppressed scattering.

From the above discussion, we can find that the transmission and radiation characteristics of topological waves are not affected by the bending interface and its different types due to the robustness feature in the proposed valley topological system. In this way, the traveling-wave radiation with the robustness properties in the topological system makes it difficult to achieve the same functionality in trivial bandgap-guided structure systems. In a word, the traveling-wave radiation in the proposed topological system has major advantages, such as robust against bending interfaces and stability in operating bandwidth and beam. Therefore, valley PhCs offer three key advantages for advanced photonic devices [49–51]: (i) Robust, backscattering-immune propagation due to valley-dependent topological edge states, enabling high-fidelity routing in compact footprints; (ii) Highly tunable light control through the valley Hall phase transition, allowing flexible engineering of interfaces and propagation pathways; (iii) Simple, scalable fabrication using all-dielectric structures with geometric perturbations, avoiding the need for external fields or complex material designs.

9 Appendix F: Radiation patterns in introducing middle-phase states

To clearly demonstrate the reconfigurable traveling-wave radiation characteristics in introducing middle-phase state cases, we give the radiation patterns at a fixed frequency (f = 0.495 THz) under the four different exciting cases. The dual-beam radiation patterns and corresponding beam angle when exciting from different ports are plotted in Figs. A6(a)−(d), in which phase states of LCs in channels 1 and 2 have five different combining states: (n_{\mathrm{o}}, n_{\mathrm{m}\mathrm{i}\mathrm{d}}), (n_{\mathrm{m}\mathrm{i}\mathrm{d}}, n_{\mathrm{o}}), (n_{\mathrm{e}}, n_{\mathrm{m}\mathrm{i}\mathrm{d}}), (n_{\mathrm{m}\mathrm{i}\mathrm{d}}, n_{\mathrm{e}}), and (n_{\mathrm{m}\mathrm{i}\mathrm{d}}, n_{\mathrm{m}\mathrm{i}\mathrm{d}}). As shown in the radiation pattern of Fig. A6, we can find that either one of the two beams or both beams can be flexibly manipulated by the controllable phase states of LCs when selectively exciting any two ports. These results clearly demonstrate that the beam steering capability has flexible and variable characteristics due to the tunable property of LC phase states.

10 Appendix G: Demonstration of reconfigurable traveling-wave radiation in other cases

To show the reconfigurable traveling-wave radiation characteristics in other cases, we further demonstrate the radiation patterns and E_z field distribution at a fixed frequency (f=0.48 THz) under the different coding sequences. From the radiation pattern shown in Fig. A7(a), we can find that the tri-beam radiation can be modulated by the controllable encoding sequence (IX−XII) when simultaneously exciting ports 1 and 2. Similarly, when port 1 or 2 is selectively excited, the single-beam radiation waves can be tuned by steerable encoding sequence (XIII−XVI), as displayed in Fig. A7(c). Moreover, the beam angles radiated from channels 1 to 4 corresponding to the coding sequences (IX−XVI) are illustrated in Fig. A7(b), in which the red (orange) border and blue (purple) filled area in triangles represent the beams from channel 1 (3) and channel 2 (4). These variations indicate that the radiation beam number and direction can be reconfigured by encoding sequences and exciting ports. To present a clearer radiation feature, we simulated the E_z field distributions for traveling-wave radiation with different coding sequences in Figs. A7(d) and (e). It is obvious that the radiation beam and its coverage region can be dynamically reconfigured at a fixed frequency, in agreement with the above radiation patterns. These results demonstrate that the proposed reconfigurable valley topological photonic platform can flexibly and diversely manipulate traveling-wave radiation.

11 Appendix H: The experimental feasibility of the reconfigurable traveling-wave radiation configuration

In this section, the experimental feasibility of the proposed configuration are discussed. The proposed reconfigurable traveling-wave radiation configuration can be fabricated and experimentally characterized. Here, the patterned dielectric thin disks in the bulk region can be deposited on a SiO₂ substrate with thin-film conductive tapes by a magnetron sputtering system, where the conductive tape is locally integrated on each waveguide channel to actuate the phase transitions of LCs. Meanwhile, the PI film alignment layer is spin-coated onto the position in the vicinity of loading LCs to form the pre-aligned molecular direction of LCs. Next, the cladding layer structure for loading LCs can be fabricated using lithography and reactive-ion etching methods. Then, the nematic LCs were injected into the cladding layer. In parallel, the SiO₂ substrate with thin-film conductive tapes and the metal layer needs to be covered on the other side of the structure to achieve the waves confined well between the two metal layers and to seal the LCs. These fabricated techniques and experiment methods with analogous structures can be successfully realized, such as in Refs. [41, 42, 52–57]. In our design, these multi-section conductive tapes can control LCs more efficiently and faster without requiring a relatively large conductive tape area in the proposed configuration. Moreover, the dynamic tuning is realized through FPGA-controlled voltage on conductive tapes, which generates variable currents to switch reversibly between multiple phase states in each waveguide channel.

To characterize the beam steering capability, the beam deflection capability of the proposed configuration can be measured using THz time-domain spectroscopy (THz-TDS) in Refs. [44, 58, 59]. Specifically, the proposed configuration can be fixed at the measured setup. The vector network analyzer is directly connected to the rectangular waveguide located in the configuration terminal for the measurement of the transmission and reflection coefficients of the whole system. At the same time, the receiver was fixed at the center of the rotation stage. After applying the voltage that corresponds to different coding sequences, the THz beam with different beam angles is emitted by the proposed configuration, and finally collected by the receiver at different angles. Although we did not conduct the experiment in our work, we would like to emphasize that our work leverages topological robustness into the reconfigurable traveling-wave radiation for the first time in valley PhCs, providing an innovative platform for incorporating topological protection into existing communications.

References

[1] J. W. Dong , X. D. Chen , H. Zhu , Y. Wang , and X. Zhang , Valley photonic crystals for control of spin and topology, Nat. Mater. 16(3), 298 (2017)

[2] G. J. Tang , X. T. He , F. L. Shi , J. W. Liu , X. D. Chen , and J. W. Dong , Topological photonic crystals: physics, designs, and applications, Laser Photonics Rev. 16(4), 2100300 (2022)

[3] X. Wu , Y. Meng , J. Tian , Y. Huang , H. Xiang , D. Han , and W. Wen , Direct observation of valley-polarized topological edge states in designer surface plasmon crystals, Nat. Commun. 8(1), 1304 (2017)

[4] Y. Tian , Y. Zhang , Y. Li , and M. R. Belić , Vector valley hall edge solitons in the photonic lattice with type-II Dirac cones, Front. Phys. (Beijing) 17(5), 53503 (2022)

[5] H. Fei , M. Wu , H. Lin , Y. Yang , and L. Xiao , Tunable hexagonal boron nitride topological optical delay line in the visible region, Chin. Opt. Lett. 22(5), 053602 (2024)

[6] Y. Yang , H. Jiang , and Z. H. Hang , Topological valley transport in two-dimensional honeycomb photonic crystals, Sci. Rep. 8(1), 1588 (2018)

[7] L. Ye , Y. Yang , Z. H. Hang , C. Qiu , and Z. Liu , Observation of valley-selective microwave transport in photonic crystals, Appl. Phys. Lett. 111, 251107 (2017)

[8] J. You , Q. Ma , L. Zhang , C. Liu , J. Zhang , S. Liu , and T. Cui , Electromagnetic metamaterials: From classical to quantum, Electromagnetic Science 1(1), 1 (2023)

[9] M. Yan , J. Lu , F. Li , W. Deng , X. Huang , J. Ma , and Z. Liu , On-chip valley topological materials for elastic wave manipulation, Nat. Mater. 17(11), 993 (2018)

[10] Y. Zeng , U. Chattopadhyay , B. Zhu , B. Qiang , J. Li , Y. Jin , L. Li , A. G. Davies , E. H. Linfield , B. Zhang , Y. Chong , and Q. J. Wang , Electrically pumped topological laser with valley edge modes, Nature 578(7794), 246 (2020)

[11] L. Zhang , Y. Yang , M. He , H. X. Wang , Z. Yang , E. Li , F. Gao , B. Zhang , R. Singh , J. H. Jiang , and H. Chen , Valley kink states and topological channel intersections in substrate-integrated photonic circuitry, Laser Photonics Rev. 13(11), 1900159 (2019)

[12] J. Han , F. Liang , Y. Zhao , X. Wang , D. Zhao , and B. Z. Wang , Reconfigurable topological wave routing based on tunable valley kink states and valley-polarized chiral edge states, Opt. Express 32(15), 26819 (2024)

[13] J. W. You , Q. Ma , Z. Lan , Q. Xiao , N. C. Panoiu , and T. J. Cui , Reprogrammable plasmonic topological insulators with ultrafast control, Nat. Commun. 12(1), 5468 (2021)

[14] Q. Chen , L. Zhang , F. Chen , Q. Yan , R. Xi , H. Chen , and Y. Yang , Photonic topological valley-locked waveguides, ACS Photonics 8(5), 1400 (2021)

[15] S. Yan , J. Yang , S. Shi , Z. Zuo , C. Wang , and X. Xu , Transport of a topologically protected photonic waveguide on-chip, Photon. Res. 11(6), 1021 (2023)

[16] H. Shao , Y. Wang , G. Yang , and T. Sang , Topological transport in heterostructure of valley photonic crystals, Opt. Express 31(20), 32393 (2023)

[17] R. Zhou , M. L. Chen , X. Shi , Y. Ren , Z. Yu , Y. Tian , Y. Liu , and H. Lin , Protected transverse electric waves in topological dielectric waveguides, IEEE Trans. Antenn. Propag. 72(2), 2058 (2023)

[18] B. Li , H. Shi , E. Wei , J. Yi , G. Li , A. Zhang , and Z. Xu , Valley topological line-defects for terahertz waveguides and power divider, Opt. Mater. 126, 112152 (2022)

[19] Y. Zhao , F. Liang , J. Han , D. Zhao , and B. Z. Wang , Extended terahertz valley-locked surface waves in designer surface plasmon crystals, J. Phys. D Appl. Phys. 56(18), 185101 (2023)

[20] F. Gao , H. Xue , Z. Yang , K. Lai , Y. Yu , X. Lin , Y. Chong , G. Shvets , and B. Zhang , Topologically protected refraction of robust kink states in valley photonic crystals, Nat. Phys. 14(2), 140 (2018)

[21] Z. Zhang , Y. Tian , Y. Wang , S. Gao , Y. Cheng , X. Liu , and J. Christensen , Directional acoustic antennas based on valley-Hall topological insulators, Adv. Mater. 30(36), 1803229 (2018)

[22] R. Jia , S. Kumar , T. C. Tan , A. Kumar , Y. J. Tan , M. Gupta , P. Szriftgiser , A. Alphones , G. Ducournau , and R. Singh , Valley-conserved topological integrated antenna for 100-Gbps THz 6g wireless, Sci. Adv. 9(44), eadi8500 (2023)

[23] W. Wang , Y. J. Tan , T. C. Tan , A. Kumar , P. Pitchappa , P. Szriftgiser , G. Ducournau , and R. Singh , On-chip topological beamformer for multi-link terahertz 6G to XG wireless, Nature 632(8025), 522 (2024)

[24] Z. Xu , X. Sun , H. Wu , Z. Xiong , X. Zhou , H. Yu , X. Yin , D. F. Sievenpiper , and T. J. Cui , Terminal-matched topo- logical photonic substrate-integrated waveguides and antennas for microwave systems, Adv. Sci. (Weinh.) 11(33), 2404163 (2024)

[25] Q. Shen , Y. You , J. Xu , Y. Shen , X. Deng , Z. Wang , W. Min , L. Shen , and S. Xiao , Mechanically scanned leaky-wave antenna based on a topological one-way waveguide, Front. Phys. (Beijing) 15(3), 33601 (2020)

[26] Z. Xu , M. Wang , S. Fang , H. Liu , Z. Wang , and D. F. Sievenpiper , Broadside radiation from Chern photonic topological insulators, IEEE Trans. Antenn. Propag. 70(3), 2358 (2022)

[27] Y. Zheng , J. Chen , Z. Ji , and Z. Y. Li , Continuously tunable topological negative refraction via a tailorable Bloch wavevector in momentum space, Photon. Res. 13(8), 2159 (2025)

[28] Y. Fu , X. Fu , S. Yang , S. Peng , P. Wang , Y. Liu , J. Yang , J. Wu , and T. J. Cui , Two-dimensional terahertz beam manipulations based on liquid-crystal-assisted programmable metasurface, Appl. Phys. Lett. 123(11), 111703 (2023)

[29] X. He , X. Wang , Y. Zhao , R. Zhuo , and F. Liang , Field programmable topological edge array, Photon. Res. 11(3), 476 (2023)

[30] Y. Zhao , F. Liang , J. Han , X. Wang , D. Zhao , and B. Z. Wang , Tunable topological edge and corner states in an all-dielectric photonic crystal, Opt. Express 30(22), 40515 (2022)

[31] M. Reuter , N. Vieweg , B. Fischer , M. Mikulicz , M. Koch , K. Garbat , R. Dabrowski , and Highly birefringent , low-loss liquid crystals for terahertz applications, APL Mater. 1(1), 012107 (2013)

[32] G. Isić , B. Vasić , D. C. Zografopoulos , R. Beccherelli , and R. Gajić , Electrically tunable critically coupled terahertz metamaterial absorber based on nematic liquid crystals, Phys. Rev. Appl. 3(6), 064007 (2015)

[33] Y. Zhao , F. Liang , X. Wang , D. Zhao , and B. Z. Wang , Tunable and programmable topological valley transport in photonic crystals with liquid crystals, J. Phys. D 55(15), 155102 (2022)

[34] T. Ma and G. Shvets , All-Si valley-Hall photonic topological insulator, New J. Phys. 18(2), 025012 (2016)

[35] X. Xi , J. Ma , S. Wan , C. H. Dong , and X. Sun , Observation of chiral edge states in gapped nanomechanical graphene, Sci. Adv. 7(2), eabe1398 (2021)

[36] R. Xi , Q. Chen , Q. Yan , L. Zhang , F. Chen , Y. Li , H. Chen , and Y. Yang , Topological chiral edge states in deep-subwavelength valley photonic metamaterials, Laser Photonics Rev. 16(11), 2200194 (2022)

[37] J. Han , F. Liang , Y. Zhao , J. Liu , S. Wang , X. Wang , D. Zhao , and B. Z. Wang , Valley kink states and valley- polarized chiral edge states in substrate-integrated topological photonic crystals, Phys. Rev. Appl. 21(1), 014046 (2024)

[38] G. Wei , Z. Liu , H. Wu , L. Wang , S. Wang , and J. Xiao , Boundary configured chiral edge states in valley topological photonic crystal, Opt. Lett. 47(12), 3007 (2022)

[39] J. Han , F. Liang , Y. Zhao , X. Ding , X. Wang , D. Zhao , and B. Z. Wang , Highly efficient conversion from classical guided waves to topological chiral edge states, Chin. Opt. Lett. 22(2), 023902 (2024)

[40] J. L. Volakis, Antenna Engineering Handbook, Vol. 1755, McGraw-Hill, New York, 2007

[41] Z. Shen , S. Zhou , S. Ge , W. Duan , P. Chen , L. Wang , W. Hu , and Y. Lu , Liquid-crystal-integrated meta-device: towards active multifunctional terahertz wave manipulations, Opt. Lett. 43(19), 4695 (2018)

[42] S. Savo , D. Shrekenhamer , and W. J. Padilla , Liquid crystal metamaterial absorber spatial light modulator for THz applications, Adv. Opt. Mater. 2(3), 275 (2014)

[43] J. Shu and Y. Zhang , Tailoring meta-liquid crystal for larger tunability, Electromagnetic Science 1(4), 1 (2023)

[44] X. Fu , L. Shi , J. Yang , Y. Fu , C. Liu , J. W. Wu , F. Yang , L. Bao , and T. J. Cui , Flexible terahertz beam manipulations based on liquid-crystal-integrated programmable metasurfaces, ACS Appl. Mater. Interfaces 14(19), 22287 (2022)

[45] D. R. Jackson , C. Caloz , and T. Itoh , Leaky-wave antennas, Proc. IEEE 100(7), 2194 (2012)

[46] X. Li , J. Wang , Z. Li , Y. Li , M. Chen , Z. Zhang , and Y. Geng , Leaky-wave antenna arrays for flexible dual-beam switching, IEEE Trans. Antenn. Propag. 69(12), 8376 (2021)

[47] J. Han , F. Liang , D. Huang , Y. Zhao , J. Liu , D. Zhao , and B. Z. Wang , Topological traveling-wave radiation based on chiral edge states in photonic Chern insulators, Opt. Lett. 49(21), 6233 (2024)

[48] D. Zheng , K. Wu , and W. Hong , Leaky horns: A new paradigm of developing electromagnetic horns based on leaky-wave structures, Electromagnetic Science 3(1), 0090365 (2025)

[49] J. W. Liu , F. L. Shi , X. T. He , G. J. Tang , W. J. Chen , X. D. Chen , and J. W. Dong , Valley photonic crystals, Adv. Phys. X 6(1), 1905546 (2021)

[50] Y. Wang , W. Zhang , and X. Zhang , Tunable topological valley transport in two-dimensional photonic crystals, New J. Phys. 21(9), 093020 (2019)

[51] Y. Zhang , W. Sui , Z. Zhang , H. Zhang , and B. Yang , Effects of structure parameters on kink states of two- dimensional valley photonic crystals, Comput. Mater. Sci. 230, 112498 (2023)

[52] Y. Yang , Y. Yamagami , X. Yu , P. Pitchappa , J. Webber , B. Zhang , M. Fujita , T. Nagatsuma , and R. Singh , Terahertz topological photonics for on-chip communication, Nat. Photonics 14(7), 446 (2020)

[53] J. Ma , X. Xi , and X. Sun , Topological photonic integrated circuits based on valley kink states, Laser Photonics Rev. 13(12), 1900087 (2019)

[54] X. T. He , E. T. Liang , J. J. Yuan , H. Y. Qiu , X. D. Chen , F. L. Zhao , and J. W. Dong , A silicon-on-insulator slab for topological valley transport, Nat. Commun. 10(1), 872 (2019)

[55] X. Liu , J. Huang , H. Chen , Z. Qian , J. Ma , X. Sun , S. Fan , and Y. Sun , Terahertz topological photonic waveguide switch for on-chip communication, Photon. Res. 10(4), 1090 (2022)

[56] T. Cao , L. Fang , Y. Cao , N. Li , Z. Fan , and Z. Tao , Dynamically reconfigurable topological edge state in phase change photonic crystals, Sci. Bull. (Beijing) 64(12), 814 (2019)

[57] Z. Shen , W. Li , B. Jin , and D. Zhao , A liquid crystal- based multi-bit terahertz reconfigurable intelligent surface, APL Photonics 9(1), 016109 (2024)

[58] C. X. Liu , F. Yang , X. J. Fu , J. W. Wu , L. Zhang , J. Yang , and T. J. Cui , Programmable manipulations of terahertz beams by transmissive digital coding metasurfaces based on liquid crystals, Adv. Opt. Mater. 9(22), 2100932 (2021)

[59] J. Yang , X. Chu , H. Gao , P. Wang , G. Deng , Z. Yin , and H. Lu , Fully electronically phase modulation of millimeter-wave via comb electrodes and liquid crystal, IEEE Antennas Wirel. Propag. Lett. 20(3), 342 (2021)

[60] W. Zhang , Y. Li , and Z. Zhang , A reconfigurable reflect array antenna with an 8 μm-thick layer of liquid crystal, IEEE Trans. Antenn. Propag. 70(4), 2770 (2022)

[61] J. Wu , Z. Shen , S. Ge , B. Chen , Z. Shen , T. Wang , C. Zhang , W. Hu , K. Fan , W. Padilla , Y. Lu , B. Jin , J. Chen , and P. Wu , Liquid crystal programmable metasurface for terahertz beam steering, Appl. Phys. Lett. 116(13), 131104 (2020)

[62] J. Liu , F. Liang , Y. Wang , X. Ni , X. Wang , D. Zhao , and B. Z. Wang , Circularly polarized high gain k-band liquid crystal phased array antenna, IEEE Trans. Antenn. Propag. 72(9), 7341 (2024)

[63] J. Lu , C. Qiu , M. Ke , and Z. Liu , Valley vortex states in sonic crystals, Phys. Rev. Lett. 116, 093901 (2016)

[64] X. D. Chen , F.-L. Zhao , M. Chen , and J. W. Dong , Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topological propagation, Phys. Rev. B 96, 020202 (2017)

RIGHTS & PERMISSIONS

Higher Education Press