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Abstract
Topological photonics has become a new and fascinating area in recent years, which enables electromagnetic waves to propagate with negligible backscattering and excellent robustness even when encountering sharp corners or defects. But the flexible tunability of edge and corner states is challenging once the topological photonic crystals (PhCs) have been fabricated. In this paper, we propose a new all-dielectric PhC with C3 symmetry constructed by hexagonal array of petal-like aperture embedded in silicon background. The proposed configuration has much wider energy gap than its triangular counterpart, and hence is suitable for wideband and high-capacity applications. When the apertures are filled with liquid crystals (LCs), the topologically-protected edge and corner states can be regulated through changing the refractive index of the LCs under different bias voltages. Moreover, the robustness of topological protection of edge and corner states is further demonstrated. This is the first demonstration of LC based tunable valley higher-order photonic topological insulator. The tunability of the proposed topological PhCs may be beneficial for development of tunable optical waveguides, reconfigurable topological microcavities, and other intelligent topological optical/terahertz devices.
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1. Introduction
Topological insulators (TIs) have tremendous potential applications in electronic system due to their fantastic characteristics of insulating in the bulk and conducting on the interfaces. The edge states are topologically protected to transport unidirectionally and robust against disorders and defects. Recently, topological phenomenon has been discovered in the system of the quantum Hall effect (QHE), the quantum spin-Hall effect (QSHE), and the quantum-valley Hall effect (QVHE) [1,2]. In the system of QHE, the time reversal symmetry (TRS) is broken through external magnetic fields. QSHE relies on spin-orbit coupling to cause spin-up and spin-down electronics to propagate in opposite directions owing to TRS. Besides, valley associated with energy extrema at the high symmetry points of K and K′ in the first Brillouin zone is considered as a new degree of freedom (DOF) in the two-dimensional materials, and valleytronics has been developing rapidly [1]. Analogous to electric system, the topological concept was introduced into classical fields, including optics [3–6] and elastics [7–9]. And then photonic topological insulators (PTIs) have been discovered theoretically and experimentally in various platforms, such as photonic crystals (PhCs) [3], bi-anisotropic materials [5], meta-waveguides [10,11], and metasurfaces [12]. According to bulk-edge correspondence [1], topologically protected edge states are strongly confined between trivial and nontrivial PTIs with different topological phases. In order to transform topological phases and realize topologically protected edge states in photonic system, the TRS or space-inversion symmetry need to be broken [2]. Haldane firstly proposed topological PhCs through gyromagnetic material under external magnetic fields and the unidirectional edge states at microwave frequency immune to backscattering were observed experimentally [3,4]. But the requirement of external magnetic fields limits their applications. Under preserving TRS, the concept of QSHE was used to design topological PhCs with bi-anisotropic media. In this system, the spin states were created by two orthogonal linear combinations of the transverse electric (TE) and transverse magnetic (TM) modes [5,6]. Afterwards, all dielectric PTIs were proposed in PhCs with hexagonal crystals, in which the expansion and shrinkage of the hexagonal unit cell lead to band inversion [13,14]. PTIs based on QHE and QSHE are difficult to be constructed. Alternatively, topological valley photonic crystals (VPCs) [15–17] have attracted enormous interests because they are significantly easier to design. Various PTIs based on QVHE have been reported through breaking space-inversion symmetry in hexagonal lattice structure to open the twofold degeneracies at the K or K′ point [15–23]. Meanwhile, various tunable and reconfigurable PTIs have also been proposed through mechanically controlling [24,25], electrically controlling [26–30], temperature controlling [31], optically controlling [32], etc. Tunable topological phase transition was demonstrated in a photonic system composed of coupled ring resonators loaded with thermal heater elements [33]. Among these tuning schemes, liquid crystal (LC) based tunable PTIs [26–29] are quite promising to regulate topological edge states on demand, since the electrically controlling of LCs is quite flexible and efficient. Non-Hermitian control can also be used to realize tunable topological photonics. It’s based on gain and loss and enables controllable bulk topology, topological lasing and light steering [34,35]. However, it needs active semiconductor devices and external pumping.
In the conventional PTIs, only the transmission of topological edge states was paid many attentions. Recently, higher-order topological insulators (HOTIs) have been introduced into photonic and acoustic system [36–45], which are not in accordance with the traditional bulk-edge correspondence. And the lower dimension states protected by the bulk band topology are confined at the corners of the structure. So it can be regarded as a new DOF to manipulate the electromagnetic (EM) waves, which will have great benefits on optical cavity modes and development of new optical devices [40]. The photonic HOTIs are usually implemented on Su-Schrieffer-Heeger (SSH) model and kagome structures [45–47]. A photonic HOTI with kagome lattice supporting edge states and higher-order corner states was reported [47], in which the edge states and two kinds of corner states exist at different frequency bands. It was demonstrated that photonic HOTIs and higher-order topological phase transitions can be obtained in C3 symmetrical PhCs by tuning a single geometrical parameter [48]. Topological edge and corner states were also observed in designer surface plasmon crystals and the corner state frequency can be tuned by modifying the location or diameter of the corner pillars [49]. But once the structure and material parameters are determined, the corner and gapped edge states are difficult to tune. Temperature controlling phase-change materials like Sb2S3 and Sb2Se3 were adopted to implement tunable edge and corner states [50]. But the change of temperature is usually slow and inflexible. Comparatively, electrically controlling is much flexible and efficient. In the last year, a reconfigurable LC based photonic HOTI mimicking quantum spin Hall effect was reported [51]. By varying the refractive index of LCs, the eigenfrequency of the edge and corner states can be tuned. However, the energy gap of the band structure is limited and highly dependent on the refractive index of LCs. As the refractive index of LCs increases, the eigenfrequency of edge states gradually move to lower frequency, and the energy gap becomes narrower and narrower. When the refractive index of LCs reaches 1.69, the relative bandwidth is only 5.3%. This means that the working frequency band of the edge state is quite limited. Large energy gap and bandwidth are highly desired for broadband and high-capacity applications.
In this paper, a broadband valley PhC constructed with petal-like aperture embedded in silicon background is proposed. The configuration has C3 symmetry and its mirror symmetry at K(K′) point can be broken by rotating the petal-like aperture. When the mirror symmetry at K(K′) point is broken, the band degeneracies are lifted, and a topological band gap appears. The topological band gap ranges from frequency of 0.231c/a to 0.291c/a, where a is the lattice constant and c is the speed of light in free space. The band gap of the proposed petal-like aperture configuration is much broader than its counterpart. The broadband topologically protected valley edge states are demonstrated. The edge modes can propagate smoothly along interface with sharp corners. Moreover, tunable VPC waveguide is realized by virtue of nematic LCs filled into the apertures. The proposed configuration supports different topological edge and corner states by changing the bias voltage. Furthermore, a beam splitter with a tunable bandwidth is constructed and valley-locked transport is demonstrated. The tunable corner states are realized through varying the refractive index of LCs filled in the apertures. This electrically controlling approach is much more flexible compared to the other controlling methods. The designed tunable VPCs will have great potential applications in photonic integrated circuits and communication systems. The main contributions of this work are summarized as follows. (1) A new petal-like aperture VPC configuration is proposed and investigated. This structure reveals wider energy gap and broader bandwidth than its triangular counterpart, and hence it’s quite suitable for high-capacity applications. The topologically protected property with backscattering immunity of the proposed structure is demonstrated. (2) Tunable LC filled petal-like aperture PhCs are designed and the tunable topologically protected edge state transports are demonstrated. These tunable PhCs may find potential applications in optical/THz switches and reconfigurable or intelligent photonic/THz integrated circuits with high throughput requirements. (3) A tunable photonic HOTI with valley pseudospin based on the LC filled petal-like aperture configuration is designed and validated. This is the first demonstration of LC based photonic HOTI emulating quantum valley Hall effect. Compared to reported LC based tunable photonic HOTI emulating quantum spin Hall effect, where the scatterers are shrunken and expanded in a limited range when the lattice dimension is fixed, our proposed configuration has more design flexibility by rotating the scatterers within the lattice.
2. Valley photonic crystals
In two-dimensional (2D) hole-type topological PhCs, edge states with both TE [16,21,52,53] and TM [54,55] modes have been studied. In this work, TM mode is considered, since we need to place metallic plates on both the top and bottom surfaces of the silicon based PhCs. The metallic plates are used as electrodes for providing bias voltage on the LCs filled in the PhCs. The perfect electric conductor (PEC) boundary conditions from the metallic plates at z = 0 and z = h will not affect the field distributions of TM mode. The nonzero electromagnetic field components in 2D TM mode are Hx, Hy, and Ez, in which Ez satisfies
Based on the Floquet-Bloch theorem, Ez(r) and ɛ(r) can be expanded as
It’s well known that the VPCs can be achieved by deforming a photonic structure with C6v symmetry, e.g. a honeycomb lattice with a circle air aperture in silicon background, into a structure with C3 symmetry, e.g. a triangular aperture, due to the breakdown of inversion symmetry [54]. However, we will show that breaking the inversion symmetry cannot always ensure the generation of an energy gap. Considering the different configurations in Fig. 1(a), if a unit cell has mirror symmetry at the K or K′ point (more specifically, along the ΓK or ΓK′ direction) in momentum space, we name it C3v symmetry. On the other hand, if the mirror symmetry at the K or K′ point is broken by a perturbation, e.g., by deflecting the scatterer by a certain angle, we name it C3 symmetry. Figure 1(b) shows three kinds of petal-like aperture elements with deflection angle α = 0°, 30°, and −30°, labeled with “C3v”, “VPCA”, and “VPCB”, respectively, both in the direct space and the momentum space. The deflection angle α is measured from the positive x axis. The mirror planes are shown with dashed lines in all the configurations and their corresponding momentum spaces. It can be see that that mirror symmetry at K/K′ point exists for α = 0° and is absent for α = 30° or −30°. We can further conclude that when the deflection angle α = nπ/3, where n is an integer, the mirror symmetry at the K or K′ point exists and the configuration satisfies C3v symmetry. The configuration will be C3 symmetrical for the other deflection angles. We have investigated the electric field profiles of eigenstates and band diagrams for several C3v and C3 symmetrical configurations, based on conventional triangular aperture and our proposed petal-like aperture. The representative results are shown in Figs. 1(c)–1(f). We conclude that if the mirror symmetry at the K or K′ point is preserved in C3v symmetrical configurations, we can always observe the Dirac cone at K(K′) valley. Only when the mirror symmetry at the K or K′ point is broken, the Dirac cone will disappear, and there will be an energy gap in the band structure. In other words, reducing C3v symmetry to C3 symmetry contributes to open an energy gap. For example, when α = nπ/3, the mirror symmetry at the K or K′ point exists, and the structure supports Dirac cone without any energy gap. In contrast, when α = (n + 0.5)π/3, there will be a maximum energy gap in the band structure, as shown in Fig. 1(f) where α= 30° or −30°.
From Eq. (6) we can see that the size of the energy gap mainly depends on the size of the perturbation area (V1 and V2 regions in Fig. 1(a)), the permittivity difference between the aperture filler and background materials, and the electric field profiles of the K valley state as shown in Figs. 1(c) and 1(d). Note that when the deflection angle α = nπ/3, the perturbation area is maximum, but the perturbation strength will be zero according to (6) since the integral in V1 is completely conceled by the integral in V2 for such a deflection angle. Instead, the maximum perturbation strength will be obtained when α = (n + 0.5)π/3. However, the maximum perturbation strength and hence the maximum energy gap also depends on the geometrical configuration of the aperture. For a triangular aperture, there is only one geometrical parameter, i.e. the side length L, that can be designed. In contrast, for a petal-like aperture constituted by three cylindrical holes with the same radius r and center-to-center distance d, embedded in background material of silicon as shown in Fig. 1(a), two geometrical parameters, i.e. r and d, can be optimized to maximize the energy gap. We have compared the maximum energy gap of the triangular and petal-like aperture configurations, and the results are shown in Figs. 1(e) and 1(f). For a fair comparison, we make the triangular and petal-like apertures have their respective maximum aperture area for a fixed lattice size, and the geometrical parameters are derived as follows: a = 5 µm, L = 0.865a, r = 0.25a, d = $\sqrt 3$r. From Fig. 1(e), we can see that the maximum energy gap for the triangular aperture is opened from 0.2071c/a to 0.2195c/a (12.42-13.16 THz), where c is the free space light speed and a is the lattice constant. So the gap/mid-gap ratio is only 5.85%. In contrast, the gap/mid-gap ratio of the petal-like aperture for the same lattice constant and the same background material can reach 22.33%, with energy gap from 0.2317c/a to 0.2900c/a (13.89-17.39 THz).
Fig. 1. Different PhC configurations. (a) Mirror symmetry broken from C3v to C3 and the perturbation area for a comparison between triangular and petal-like aperture elements, where C3v symmetry means it contains mirror symmetry along the ΓK or ΓK′ direction in the momentum space. (b) Three kinds of petal-like aperture VPC elements, i.e. C3v, VPCA, and VPCB, in which the deflection angle α= 0°, 30°, and −30°, respectively, both in the direct space and momentum space. The dashed lines denote the mirror planes. (c) and (d) depict the phase and magnitude profiles of Ez at K and K′ valley states for VPCB configuration, respectively. The clockwise and counter-clockwise red circular arrows in (c) represent the right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) orbital angular momentum, respectively. The arrows in (d) represent the Poynting vector directions. (e) and (f) The band diagrams for triangular and petal-like aperture elements, respectively, with lattice constant a = 5 µm, L = 0.865a, r = 0.25a, d = $\sqrt 3$r, and the relative permittivity of the background material ɛr = 11.7.
Through the band diagrams for VPCA (α= 30°) and VPCB (α= −30°) elements are the same, they have different topological properties. From the electric field profiles shown in Figs. 1(c) and 1(d) we can see that the K and K′ valley states at the same band have opposite chiralities with clockwise and counter-clockwise orbit angular momentums (OAMs). Meanwhile, the K(K′) valley states at the first and second bands also reveal opposite chiralities. Through only the field profiles for VPCB are shown in Fig. 1, we also obtained the field profiles for VPCA, and band inversion was observed. That is to say, the field distribution for the first (second) band of VPCA is exactly the same as that for the second (first) band of VPCB. Besides, the K(K′) valley state of VPCA has the same field distribution as that of the K′(K) valley state of VPCB. The valley Chern number can be employed to represent the topological property of the VPC system. Valley Chern numbers of the first band at K and K′ point are given as
where Ω(k) is the Berry curvature, and the integration is calculated over half of the first Brillouin zone around the high-symmetry point of K or K′. The strict calculation of the Berry curvature and Chern number can be conducted by virtue of the first principle method [56]. One can also estimate the valley Chern numbers according to CK/K′ = 1/2 sgn(ΔP(K/K′)) [15]. Then we can obtain CK = −1/2 and C K′ = 1/2 for VPCA, while CK = 1/2 and C K′ = −1/2 for VPCB. Therefore, VPCA and VPCB reveal different nontrivial topological phases. Moreover, since |ΔCK/K′| = | CVPCAK/K′ - CVPCBK/K′| = 1, there will be topologically protected unidirectional edge states at a domain wall between VPCA domain and VPVB domain, according to the bulk-edge correspondence.To investigate the topological edge states, a zigzag domain wall is constructed in a hexagonal-lattice PhC by virtue of VPCA and VPCB unit cells. In our calculations, as shown in Fig. 2(a), the VPC supercell is composed of 14 VPCA unit cells and 14 VPCB unit cells along the y direction. The periodic boundary conditions are used along the x direction, and perfect electric conductor boundary conditions are used along the y direction. There are two types of domain walls, i.e. AB type and BA type. AB-type (BA-type) domain wall is formed by arranging VPCA (VPCB) elements in the upper domain and VPCB (VPCA) elements in the lower domain. The dispersion diagrams of these two kinds of supercells are shown in Fig. 2(b). As expected, the gapped edge states are observed between the bulk states. As shown in Fig. 2(b), for the two edge modes indicated as a green star and a pink star, they have two different valley-pseudospin states, i.e. Φ+AB with kx = −0.6π/a and Φ-AB with kx = 0.6π/a; While for the two edge modes indicated as a pink triangle and a green triangle, they have two different valley-pseudospin states, i.e. Φ-BA with kx = −0.6π/a and Φ+BA with kx = 0.6π/a. The energy flow directions for Φ+AB (Φ+BA) and Φ-AB (Φ-BA) states have the opposite directions, as shown in Figs. 2(c) and 2(d). The edge state dispersion curves of AB-type and BA-type domain walls intersect each other at the K and K′ valleys. The group velocity of an edge state is related to the slop of the dispersion curve at the corresponding point of the edge state. It can be see that the group velocity at the K(K′) valley for AB-type and BA-type domain walls is opposite. In a word, from the supercell simulated results of the two kinds of domain walls, it can be concluded that the valley-polarized edge states at K or K′ are locked to one-way propagation direction for a certain type of domain wall, which manifests valley chirality of the valley-Hall effects.
Fig. 2. Valley-locked edge states of the AB-type and BA-type supercell. (a) Structure of AB-type and BA-type supercell. (b) The dispersion diagrams of the AB-type and BA-type supercell, where the black dots represent the bulk states, the red and blue dotted lines represent the projected edge states at the AB-type domain wall, and the red and blue solid lines represent the projected edge states at the BA-type domain wall. (c) and (d) The Ez field distributions and Poynting vectors of the edge states, where the red arrows represent the Poynting vector, and Φ+AB (Φ+BA) and Φ-AB (Φ-BA) represent the mode propagates forward and backward along the AB (BA)-type domain wall, respectively.
3. Demonstration of topological edge states
Based on the band structures of the honeycomb photonic crystal with petal-like holes, we design a topological waveguide with domain wall between VPCs with different valley Chern numbers. In order to observe the propagation of topologically protected edge states, we build the domain walls with a straight path and a Z-shaped path and place a z-polarized line electric source on the left side of the interface. The Ez field distributions for the straight interface and the Z-shaped interface are shown in Fig. 3(a) and 3(c), respectively. We can see that the EM wave propagates along the domain wall, even though there are two sharp corners of 120° on the propagation path. The EM wave is not seriously affected by backscattering and propagates smoothly in the frequency range from 0.2314c/a to 0.2625c/a within the complete band gap.
Fig. 3. Simulated Ez field distributions and transmission coefficients. (a), (b), and (c) Simulated Ez field distributions at the frequency of 0.25c/a with straight path, straight path with defect, and Z-shaped bent path, respectively. (d) Simulated transmission coefficients for the paths in (a)-(c). (e) and (f) Simulated Ez field distributions when placing a RCP and LCP chiral source at the center of the straight path, respectively.
Besides, to study the robustness of edge state propagation against defects, we introduce a defect (denoted by black five-pointed star in Fig. 3(b)) into the straight path. The Ez distribution depicted in Fig. 3(b) demonstrates the robustness of valley edge states. We also calculate the transmission coefficients of the edge states along the different domain walls, and the results are shown in Fig. 3(d). Note that the calculated transmission coefficients don’t include the coupling loss from the source to the waveguides, since we place the reference plane S1 apart from the source. From Fig. 3(d) we can find that all the straight, Z-shaped and defected domain walls support near-unity transmissions in the edge state frequency band. The backscattering loss from the sharp bending and defects are negligible. Therefore, these results demonstrate that the valley edge states are topologically protected and robust to sharp corners and defects. Through analyzing the band structures of supercell, we can find that each edge state has a certain valley chirality. Therefore, the edge state has the unique characteristics of unidirectional transmission. To demonstrate the one-wave propagation characteristics, a chiral source with a certain OAM at the working frequency of 0.25c/a is placed at the center of the straight interface (denoted by a black five-pointed star). As shown in Figs. 3(e) and 3(f), if the source is RCP (OAM index L = 1), the EM wave propagates only to the right port; if the source is LCP (OAM index L = −1), the EM wave propagates only to the left port. This validates the unidirectional behavior of the edge states due to the valley-momentum locking.
4. Tunable topological edge states, optical switch, and beam-splitter
If the petal-like aperture are filled with nematic LCs, the proposed VPC will be tunable. Under external bias voltage, the refractive index of the filled LCs will be changed. Because the external voltage will change the orientations of LC molecules. As depicted in Fig. 4(a), without external voltage, the long axes of all the LC molecules will be along the horizontal direction due to the effect of the alignment layer. However, with a larger enough external voltage, the long axes of all the LC molecules will be deflected to the vertical direction. The refractive index of LCs will be changed from 1.45 without external voltage to 1.75 with external voltage [27]. From the band diagrams calculated by COMSOL Multiphysics, we can find that the band gap is shifted when the refractive of index of LCs is changed. But the topological properties at the K and K′ valleys are not modified. It has been demonstrated that the topological phase transition can be realized by rotating the petal-like holes. We design a supercell consisted of 14 VPCA unit cells and 14 VPCB unit cells along the y direction, where the rotation angle α = 30° in VPCA and α = −30° in VPCB. The dispersion diagrams for the designed supercell with LC refractive index being 1.45 and 1.75 are shown in Figs. 4(c) and 4(d), respectively. The frequency of edge states between complete band gap ranges from 0.2267c/a to 0.2563c/a when the voltage is off. But the frequency range changes to from 0.2231c/a to 0.2486c/a when the voltage is on. Therefore, the topological edge states can be manipulated flexibly by the external voltage.
Fig. 4. Tunable VPC and its band structures (a) The configuration of tunable VPC in which LCs are filled in holes. (b) The band diagrams for the different refraction indexes. (c) and (d) Calculated dispersion diagrams for the AB-type interface along the x direction when the refraction index of LCs is 1.45 and 1.75, respectively.
Based on the proposed tunable LC filled petal-like aperture element, many reconfigurable photonic and terahertz (THz) devices can be implemented. Figure 5(a) shows the real-world three-dimension (3D) structure of a general device. Two metallic plates working as electrodes are placed at the top and bottom of the VPC. A bias voltage is applied to the electrodes. Though the thickness of VPC is finite, according to image theory, the top and bottom metallic plates will generate multiple images of the silicon substrate and LC filler, and hence make the VPC equivalent to that with infinite thickness. As an example, we designed a tunable optical switch based on a Z-shaped domain wall composed VPCA and VPCB elements, as shown in Fig. 5(b). We employ a Z-shaped propagation path to validate its immunity to backscattering, and place a source at frequency of 0.25c/a at the left port. For different refraction indexes of LCs filled in holes, the frequency ranges of edge states differ. When the bias voltage is off, the refractive index of LCs is 1.45 and the EM waves can smoothly propagate along the Z-shaped waveguide, as shown in Fig. 6(c). But when the bias voltage is on, the refractive index of LCs is changed into 1.75 and the EM waves can’t propagate along the waveguide, as depicted in Fig. 5(d). This is because the source frequency falls into the edge-state band of the waveguide when the bias voltage is off, while it falls out of the edge-state band when the bias voltage is on.
Fig. 5. Tunable LC based photonic crystal devices. (a)General configuration. (b) An optical switch made from Z-shaped domain wall. (c) and (d) The simulated Ez field distributions of topological edge states at frequency 0.25c/a when the bias voltage is off and on, respectively.
Fig. 6. Reconfigurable beam-splitting waveguide and transmission of topological edge states. (a) Configuration. (b) The Ez field distribution of topological valley-edge states at 0.25c/a when the refraction index of LCs is 1.45. (c) and (d) Transmission coefficients of topological edge states when the refraction index of LCs is 1.45 and 1.75, respectively.
In topological valley photonic crystals, there exists fascinating phenomena of valley-locked edge states. Now we also implement an optical beam splitter with a tunable bandwidth. The harpoon-like beam-splitter waveguide, as depicted in Fig. 6(a), are consisted of four domains, which are configured by VPCA and VPCB unit cells with different topological phases. Firstly, the splitting of the valley interface state is observed when the input port at Port 1 is excited by a z-polarized electric line source. The excited valley edge state is split into two branches and then propagate along the bent paths to the Port 2 and Port 3, instead of going straight to Port 4. This is because the path to Port 4 is BA-type domain wall, while the paths to Port 2 and Port 3 are AB type. The AB and BA type domain walls have different valley chiralities. Due to the valley-momentum locking characteristic of valley-polarized edge states, each type of domain wall only supports one-way edge state transport.
In order to calculate and compare the transmission coefficients from Port 1 to Port 2, Port 3, and Port 4, we place four linear probes at the four reference planes (denoted by S1, S2, S3, and S4) perpendicular to the interface. Then the planar density of power flow at each reference plane is calculated at the operating frequencies and the transmission coefficients, i.e. S21, S31, and S41 are derived. Figures 6(c) and 6(d) show the transmission coefficients of topological edge states when the voltage is off and on, respectively. Some observations and conclusions can be drawn from Figs. 6(c) and 6(d) as follows. 1) There is almost no output wave at Port 4 in the working band of the edge states, which agrees with our expectation. This result manifests that the inter-valley scattering at the intersection of AB type and BA type domain walls is forbidden. 2) The working frequency band of the beam splitter can be tuned. The working frequency of the beam splitter is from 0.2268c/a to 0.2518c/a (13.61-15.11 THz) when the bias voltage of LCs is off. However, the working frequency band becomes from 0.2232c/a to 0.2402c/a (13.39-14.41 THz) when the voltage is on. This will benefit to frequency reconfigurable applications. In determining the working bandwidths herein, we take the mode distribution into consideration in addition to the value of the transmission rate. Specifically, the bandwidth is determined from the frequency range where the transmission rate is not less than 0.707 times (−3 dB) its peak value and pure edge state exists via checking the field distributions. Note that these bandwidths obtained from full-wave simulations of the whole structure have some discrepancy with those obtained from eigenmode simulations of a supercell shown in Fig. 4. 3) The bandwidth of the beam splitter becomes slightly smaller when the refraction index of LCs is changed from 1.45 (voltage off) to 1.75 (voltage on). This result is consistent with the estimation from Eq. (5). When the refraction index of LCs increases from 1.45 to 1.75, the permittivity difference between silicon and LC materials, i.e. Δɛ decreases. Then the perturbation strength and bandwidth of edge states decrease. 4) The splitting ratio of output to Port 3 over output to Port 2 is about 2:1, though their path lengths seem like the same. However, under a careful observation, we find that the path along the upper branch to Port 2 is longer than the path along the lower branch to Port 3 by a lattice constant, i.e. a, at the intersection. This asymmetry is inevitable for the zigzag domain wall and is the cause of the unequal splitting. We note that this phenomenon was also observed in [57]. Moreover, through regulating the asymmetry by several times lattice constant (e.g. 2a or 4a), different splitting ratios were obtained [57]. Inspired by their work, our configuration shown in Fig. 6(a) can also be modified by increasing the asymmetry to implement other splitting ratios.
5. Tunable topological corner states
Generally, the existence of gapped edge states may lead to second-order corner states which don’t propagate but are highly localized at the corner of the domain wall with topological protection. When the holes are filled with air, from the dispersion diagram of AB-type supercell structure, there exist the gapped edge states. The gap between edge states and bulk states ranges from 0.2625c/a to 0.2910c/a, which may lead to the emergence of second-order corner states. In order to demonstrate the existence of corner states, we construct a triangular box-shaped supercell, where triangular VPCA region is surrounded by outer triangular annular VPCB region. When analyzing the eigenstates, perfect matching layer boundaries are imposed around the whole structure.
The calculated eigenfrequencies are shown in Fig. 7(a), where the black dots denote the edge states, while the green dots represent the bulk states. From Fig. 7(a) one can see that there are two pairs of near-degenerated corner states (denoted as red and blue dots). We name them lower frequency corner states (LFCSs) (denoted as red dots) and higher frequency corner states (HFCSs) (denoted as blue dots). The working frequency of LFCSs is near 0.2689c/a, while the frequency of HFCSs is near 0.2860c/a. Figure 7(b) shows the field distributions of edge states and corner states. From Fig. 7(b), we can find that the electric fields are strongly confined at the corners of the triangular interface between VPCA and VPCB regions. There are two kinds of modes, i.e. monopole and dipole modes. The most unique characteristic of topological states is that they are immune to structure defects. In order to demonstrate the robustness of the corner states, a kind of defect is introduced near the corners. As shown in Fig. 7(c), the defect denoted by a black five-point star is implemented by rotating one petal-like aperture. Figure 7(d) shows the eigenfrequencies for the structure with defect. We can find that the corner states still exist with the defect. From the field distributions as shown in Fig. 7(e), it is found that the electric field energy is highly localized near the corners without scattering to the interface or bulk. Consequently, it is demonstrated that the corner states are topologically protected against imperfections. On the other hand, we find that the defects have an effect on the resonant frequency. Though the corner states still exist with the defects, the frequency of the corner states shifts slightly. As shown in Fig. 7(d), the points labeled “(3)” and “(6)” represent the shifted corner states with defects. The electric fields of these shifted corner states are still confined at the corners, as shown in Fig. 7(e).
Fig. 7. Topological edge and corner states in a triangular VPC structure. (a) The dispersion diagram of the VPC. The black and green dots represent the edge states and bulk states, respectively, and the red and blue dots denote corner states with the lower and higher frequencies, respectively. (b) The Ez field distributions for the edge states and three degenerate corner states at lower frequency of 0.2689c/a and higher frequency of 0.2860c/a, respectively. (c) The triangular VPC structure with defects introduced by rotating the petal-like aperture (denoted by a black five-pointed star). (d) The dispersion diagram of the triangular VPC structure corresponding to (c). (e) The Ez filed distributions corresponding to corner states in (d).
Besides, the corner states can be effectively tuned through external voltage applied on the LCs filled in the holes. The band gap is shifted with the changes of the refractive index of LCs. Accordingly, the working frequency of corner states is then dynamically tuned by adjusting the external voltage. As shown in Figs. 8(a) and 8(b), the eigenfrequencies of the structure are calculated corresponding to the LC refractive index being 1.45 and 1.75, respectively. We can see that the corner states still exist, but the working frequency is shifted when LC refractive index is changed. The working frequency of LFCSs changes from 0.2593c/a to 0.2505c/a. HFCSs disappear when the refractive index of LCs is 1.75. In this way, we can simply realize the tunable corner states in the proposed topological structure. Moreover, we find that through controlling the external voltage, the corner states can be changed into edge states. Figure 8(c) shows the Ez field distribution of edge states with frequency of 0.2505c/a when the voltage is off. But when the voltage is on, the edge states can be changed into corner states, as shown in Fig. 8(c).
Fig. 8. Topological edge and corner states in a tunable triangular VPC structure. (a) and (b)The eigenfrequencies of tunable triangular VPC structures when the refraction index of LCs is 1.45 and 1.75, respectively. (c) Edge states near the frequency of 0.2505c/a are turned into the corner states when the applied voltage is turned on.
6. Discussion and conclusion
A comparison between the proposed structure and other LC based tunable topological PhCs reported in literature is listed in Table 1. All the previously reported works employed scatterers with a regular shape, e.g. cylindrical, triangular, or square shape. The energy gap and bandwidth of their edge states are limited. Contrastively, in this work, we introduce a composite petal-like aperture configuration to extend the bandwidth of edge states. Benefiting from the petal-like configuration, our proposed LC filled PhCs support topological edge states with wider bandwidth and broader tunable range than the previous works. We further validate the interconversion between edge states and corner states at the same fixed frequency. This work is also the first demonstration of LC based tunable photonic HOTIs mimicking quantum valley Hall effect. Compared to the reported LC photonic HOTIs mimicking quantum spin Hall effect, where the spin DOF is obtain by shrinking and expanding the scatterers within limited lattice dimension, our proposed configuration has more design flexibility by rotating the scatterers within the lattice.
Table 1. Comparison between different LC based reconfigurable topological PhCs
The valley-Hall PTIs are much easier to realize than their spin-Hall counterparts, since the valley pseudospin can be obtained by breaking inversion symmetry, while the generation of spin-Hall degree of freedom usually involves combinations of TE and TM polarizations.
However, we demonstrate that the topological band gap will not necessarily generate with broken inversion symmetry. The in-plane mirror symmetry at K(K′) point plays a vital role in preserving/removing the Dirac cone. Specifically, only breaking the mirror symmetry at K(K′) point can ensure a topological band gap. For high-capacity communication and information processing applications, broad working bandwidth is highly desired. For this purpose, we propose a new petal-like aperture PhC configuration, which reveals much wider energy gap than its triangular counterpart. Moreover, tunable PTIs can be implemented with LC filled petal-like aperture PhCs, in which wideband edge states, reconfigurable edge states and corner states are observed. Compared to other tunable schemes, LC tuning features much more flexibility, higher efficiency, low cost, low power consumption, and excellent adaptability in the THz and optical frequency range. The voltage needed to drive the LCs depends on the distance between the upper and lower electrodes, i.e. the thickness of LCs along z direction. As reported in [58], a bias voltage of 11 V can fully deflect 8-µm thick LC. With the development of large birefringence LC material, the tunable range of LC PTIs will be enlarged. Compared to other LC based tunable PhCs, this work supports wideband topological protected edge states and interconvertible edge states and corner states at the same frequency. More importantly, this work is the first demonstration of LC based valley higher-order PTI. Though all the simulations are performed in 2D, the overall band diagrams and the physical interpretations for a real-world 3D design will remain unchanged from the 2D situation.
In conclusion, we demonstrate that reducing the VPC symmetry from C3v to C3 by breaking the mirror symmetry at the K(K′) point is the origin of energy gap in band structure. Besides, we propose a new VPC configuration composed of composite petal-like aperture in silicon background, which reveals much broader bandgap than its triangular aperture counterpart and hence may find potential applications in high-capacity communications and information processing. Moreover, tunable VPCs based on LCs filled into the petal-like apertures supporting broadband topological protected 1D edge states and 0D corner states have been demonstrated. The refractive index of LCs in the VPCs changes with the external bias voltage, which will cause the change of light wave transmission intensity in the photonic crystal waveguide. Therefore, the transmitted light wave can have broadband modulation effects. The tunable topological waveguide can be used for optical/THz switches as well as modulators with high on/off ratio. Besides, because of the strong localization of corner states, this work can contribute to development of high-Q corner cavity. More importantly, we find that the edge states and corner states can be interchangeable at the same frequency, which might be further exploited to design novel optical devices.
Funding
National Natural Science Foundation of China (62171082); Natural Science Foundation of Sichuan Province (2022NSFSC0483); Key-Area Research and Development Program of Guangdong Province (2019B010158001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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