The discovery of the quantum Hall effect in electronic systems ushered in a new era in condensed matter physics, where topological phases of matter hold promise for revolutionizing dissipationless transport dynamics [1–6]. However, electronic systems with nontrivial topology are hard to implement and manipulate. Photonic crystals (PCs), the analogues of conventional crystals with the atomic lattice replaced by a periodic medium, provide an excellent means of studying topological physics in a highly controllable manner [7]. As the quantum Hall effect, quantum spin Hall effect, and quantum valley Hall effect have been successfully extended to photonic systems [8–15], topological photonics brings new degrees of freedom to manipulating light. These photonic topological insulators are characterized by gapless edge states with one-dimensional lower real-space distributions than the bulk states, making them suitable for use in linear and nonlinear on-chip photonic devices [16–19]. More recently, higher-order topological insulators (HOTIs) have been introduced in photonic systems, in which the traditional bulk-boundary correspondence has been extended to bulk-edge-corner correspondence [20–24]. In contrast to the conventional topological insulators, the two-dimensional HOTIs can host gapped one-dimensional edge states and zero-dimensional corner states in the gap, characterized by the quantized bulk polarization. This allows for the design of topologically robust cavities and waveguides in the same structure, greatly enhancing the localization and transmission of light in integrated photonics. Since the existence and robustness of topological corner states (TCS) have been adequately demonstrated, the HOTIs cut a conspicuous figure in topological high-quality nanocavities, topological insulator lasers, topologically protected nonlinear imaging, and other potential applications [25–28]. Aiming to future development, it remains valuable to visualize multiple TCS with various electromagnetic characteristics and to enrich the manipulation methods, such as more implementation schemes of various TCS, and pseudospin manipulations [29–32].

In this work, we demonstrate six distinct types of TCS in the filling anomaly C6v-symmetric PCs with various electromagnetic characteristics, in which the specific TCS can be supported by selecting diverse splicing boundaries. For multiple TCS in the same structure, we observe that the TCS overlapping in the spectral and spatial domains inevitably lead to interweaving with each other, which provides the possibility of TCS manipulation with pseudospin sources as well as the directional excitation. According to the simulation and experiment, when we put a chiral source at the center of structural symmetry, we discover that the bonding coupled TCS at lower energy exhibit clear pseudospin dependence, while the anti-bonding coupled TCS do not. More interestingly, we further design a special quadrangular structure with two zigzag and two armchair splicing boundaries to avoid the coupling between adjacent TCS. By adjusting the structure size, our work provides a versatile approach to light manipulation based on TCS, applicable across a broad electromagnetic wave range including visible light, infrared light, and microwaves.

The PCs studied in this work consist of a triangular lattice of hexagonal clusters, each composed of six adjacent dielectric rods, as shown in the insets of Fig. 1(a). The distance between adjacent rods plays a crucial role in determining the topological phase of the PCs. When the intracell distance of the nearest rods is equal to the intercell distance, the band structure of the PCs exhibits four-fold degenerate Dirac cones at the Brillouin zone center Γ, due to the band-folding version of the one in honeycomb lattice [12]. By adjusting the relative value of intracell and intercell distances, the Dirac cone at Γ can be gapped out, leading to a topological phase transition. In our study, the numerical calculations of the eigenmodes and electric field distributions were based on COMSOL Multiphysics. In all simulations, we use 2D PCs to mimic our sample. The geometrical parameters were set as follows: triangular lattice of hexagonal a0, radius of rods r=0.1·a0, and intracell distance of nearest rods d. The relative dielectric constant of rods is set as ε=8.55. The periodic boundary condition and scattering boundary condition were used for corresponding interfaces. As shown in Fig. 1(a), for shrunken configuration, the frequency of p states at the Γ point is smaller than d states, corresponding to the ordinary insulator (OI). Conversely, for the expanded configuration, the frequencies of p states are higher than that of d states, corresponding to the topological insulator (TI). And further details about the band inversion are shown in Appendix A. For the h3c(6) PCs, the topologically insulating phases are characterized by the bulk polarization, which is associated with the Wannier centers. As shown in Fig. 1(b), the Wannier centers for the expanded PCs are at maximal Wyckoff positions with the corner fractional charge Qc=12, indicating a nontrivial second-order topologically insulating phase (see Ref. [33] for more details).

To visualize the existence of TCS with diverse types, we designed several photonic splicing structures combining expanded PCs (d=0.38·a0) and shrunken PCs (d=0.28·a0). First, we numerically simulated the projected band structures of 1D zigzag and armchair splicing boundaries, as shown in Appendix B. At the interface between OI and TI, there exist two types of 1D topological edge states in the bulk bandgap. Due to the spin-momentum locking, the edge states corresponding to zigzag and armchair splicing boundaries can both be selectively excited and propagate unidirectionally along the 1D interfaces. Here, a pseudospin-dependent source is realized by using three-point sources, with each one differing in phase by 23π. It is worth noting that the edge states with zigzag and armchair splicing boundaries exhibit diverse electromagnetic characteristics in both the real and spectral spaces. For the zigzag interface, a bandgap centered at normalized frequency of 0.590 appears in the gapped edge states. Meanwhile, for the armchair interface, there is a bandgap centered at normalized frequency of 0.620 between the edge state and bulk states. Thus, the diversity of topological states can be determined by the splicing boundaries.

Consequently, we have exhaustively explored all possible seamless splicing boundaries to expand the diversity of TCS, where the TI lattice is surrounded by the OI lattice. Through simulations, we have discovered several new types of TCS in various structures as shown in Appendix C. For example, as shown in Fig. 6(a) in the trapezoidal structure with four zigzag splicing boundaries, we have identified two types of TCS. The first TCS, named Z1, presents an electric dipolar distribution at the 120 deg of splicing corner with eigenfrequency f1=0.588. And Z2 presents a unipolar distribution at the 60 deg of splicing corner with eigenfrequency f2=0.597. As shown in Fig. 6(b), in the rhombic structure with four zigzag splicing boundaries, we have found two Z1 and two Z2 types of TCS, which have similar characteristics to those in the trapezoidal structure. We further calculated the eigenvalues of the triangular and hexagonal structures with pure zigzag splicing boundaries as shown in Figs. 6(c) and 6(d). As expected, the supported three Z2 types of TCS and six Z1 types of TCS present the same features as the TCS mentioned before. In short, the diversity of TCS can be determined by the splicing boundary condition, and the zigzag splicing boundaries can only support two types of TCS, namely Z1 and Z2. Similarly, we have discussed the eigenmode of seamless splicing structures with pure armchair boundaries as shown in Fig. 7. There are three extra types of TCS named A1, A2, and A3, which present three different electric quadripolar field distributions at the 60 or 120 deg of splicing corners. And the corresponding eigenfrequencies are 0.612, 0.613, and 0.614, respectively. In addition, we have explored splicing corners that combine zigzag and armchair boundaries as shown in Fig. 8. Interestingly, we found that the 150- and 90-deg splicing corners cannot support TCS, while the 30-deg splicing corner can support a new type of TCS called ZA corner modes with a quadripolar field at a frequency of 0.614.

In a word, our simulations demonstrate that the diverse combinations of zigzag and armchair splicing boundaries can host six distinct types of TCS with unique electromagnetic characteristics, derived from the diverse configurations of Wannier centers at the splicing boundaries. For 60- or 120-deg corners between the same types of zigzag (or armchair) splicing boundaries, the overlapping interfaces of TI lattices and OI lattices are different. And the strongest localized fields always appear at those Wannier centers. Consequently, the electromagnetic characters of different types of corner states are different. As for the splicing corners that contain both zigzag and armchair boundaries, the TCS are hard to support, maybe attributed to the diverse bandgap positions between the two types of one-dimensional splicing boundaries. We have designed a special structure where these six types of TCS can be independently excited, and their corresponding boundary conditions and electromagnetic characteristics are shown in Fig. 10. Our implementation based on all-dielectric materials can operate effectively at any electromagnetic frequency, from visible light to infrared light and microwave. As an example of proof, we performed a microwave experiment with a0=18mm, and the height of Al2O3 pillars is 25 mm. By using the near-field scanning technique, we detected the electric distributions of six corner modes, which fit simulations well as shown in Fig. 1(c). These discoveries of multiple HOTIs have potential applications in multiplexed photonic devices, where different wavelengths of light can be trapped at different spatial positions.

Next, we have studied the collective coupling effects of multiple TCS. As depicted in Fig. 9, there is no coupling between the different types of TCS. Specifically, the spectrum and electric field distributions of Z2 and ZA TCS remain unchanged even though they are integrated into the same structure. However, when multiple same types of TCS (overlapping in the spectral and spatial domains) are integrated into the same structure as shown in Fig. 2, they unavoidably interweave with each other, leading to a division of the spectrum. In the meantime, TCS with a phase difference appear at the equivalent corners of the structure. In the case of two Z2 types of TCS coupling, the degenerate eigenmodes convert to two asymmetric modes (φ1+φ2 and φ1−φ2). As shown in Fig. 2(b), the eigenfrequency of anti-bonding coupled TCS is higher than that of bonding coupled TCS. This is because the Z2 TCS presents a unipolar distribution, and the energy of the coupled system will increase when two synchronous nanocavities with even symmetry come closer together. For anti-bonding coupled TCS, the pseudospin polarization at adjacent splicing corners is consistent. Meanwhile, for bonding coupled TCS, the adjacent nanocavities present a π phase difference, indicating an opposite pseudospin polarization. And the phenomenon is similar when two Z1 types of TCS are designed into the same structure whether it is a trapezoidal or rhombic splicing structure as shown in Fig. 7.

As for the case of three Z2 types of TCS coupling, the effect is slightly different. As shown in Fig. 2(c), the spectrum of coupled TCS is divided into two levels, although there are three coupled eigenstates (φ1+φ2+φ3, φ1+φ2−φ3, φ1−φ2−φ3). This is because two bonding coupled TCS with lower eigenfrequency are two-fold degenerate eigenmodes, which show two synchronous nanocavities and one nanocavity with a π phase difference. And the eigenfrequency of anti-bonding coupled TCS with three synchronous nanocavities is higher, which can also be attributed to the energy increase of the coupled system mentioned before. Similarly, we have discussed the coupling effect between six Z1 types of TCS. As shown in Fig. 2(d), there are six coupled eigenstates. Due to the energy degenerate, the spectrum of the six coupled TCS can be divided into four levels. The lowest-frequency coupled TCS display six end-to-end dipoles, in which the adjacent nanocavities have the same phase of dipole oscillation (indicated as s mode, ∑i=16φi). Meanwhile, for the highest-frequency coupled TCS, the adjacent nanocavities have the opposite phase of dipole oscillation (indicated as f mode, φ1−φ2+φ3−φ4+φ5−φ6). As for the coupled TCS with frequencies in between, there are two two-fold degenerate eigenmodes. According to the simulated electric field distributions, the configuration of TCS shows the characteristics of dipole-like and quadrupole-like patterns depending on the direction of dipole oscillations (indicated as p and d modes). In addition, the coupling effects of multiple TCS in armchair splicing structures are similar, except for the weaker coupling strength and narrower localized fields as shown in Fig. 8. Based on the coupling effects between multiple TCS, we can achieve more manipulation of light.

Furthermore, we have studied the pseudospin-polarized characteristic of the coupled TCS systems. As both simulations and experiments shown in Fig. 3, when we put a chiral source at the center of structural symmetry, we observed an obvious difference about the directional excitation of coupled TCS with different frequency. For bonding coupled TCS with lower frequency as shown in Fig. 3(c), the B corner has a relatively higher local field intensity than the A corner when a left-handed circularly polarized (LCP) source is put at the middle of the 1D boundary between two corners. On the contrary, if a right-handed circularly polarized (RCP) source is used at the same place, the A corner will have a relatively higher local field intensity than the B corner as shown in Fig. 11. Meanwhile, for the anti-bonding coupled TCS with higher frequency, things get different. As shown in Fig. 3(d), the A corner has the same high local field intensity as the B corner whether LCP or RCP is put at the same place. According to the theoretical analysis, the underlying physics is that the bonding coupled TCS at the A corner or B corner are polarized with opposite pseudospin structure, causing a pseudospin-polarized dependence. Meanwhile, for anti-bonding coupled TCS, the synchronous nanocavities share the same pseudospin structure at the A corner and B corner, which leads to a pseudospin-polarized independence. For the chiral source, it can be regarded as a combination of two different directional polarizations with a phase difference. Therefore, the phase difference at different corners affects the selectively excitation by a chiral source. For confirmation of the localization of corner states, we map the out-plane electric field intensity |Ez|2 by the microwave near-field scanning systems, in which the experimental measurement agrees well with the simulated results.

Similarly, the three-corner coupled and six-corner coupled TCS systems also present abundant pseudospin-polarized characteristics as shown in Fig. 11. For the three-corner coupled case, the anti-bonding coupled TCS with higher frequency present three synchronous nanocavities, where the phase of electromagnetic oscillation at A, B, and C corners is the same. In the meantime, the local field intensity at the A corner is equal to that at the B corner and slightly higher than that at the C corner when an LCP source is put at the middle of the 1D boundary between the A and B corners. As for bonding coupled TCS with lower frequency, the phase of electromagnetic oscillation at the A, B, and C corners is different. As a result, the local field intensities at the C, A, and B corners decrease gradually when an LCP source is put at the middle of the 1D boundary between the A and B corners. If an RCP source is put at the same place, the difference is that the local field intensities at the C, B, and A corners decrease gradually as shown in Fig. 3(e). For the six-corner coupled case, the lowest-frequency coupled TCS present different phase of electromagnetic oscillation at six different corners with chiral source excitation. Consequently, the local field intensities at the F, E, A, D, B, and C corners decrease gradually when an LCP source is put at the middle of the 1D boundary between the A and F corners as shown in Fig. 3(f). And the local field intensities at the A, B, F, C, E, and D corners decrease gradually when an RCP source is used. As for other coupled TCS in this system, the directional excitation of specific corner modes based on chiral sources is more complicated. In short, for coupled TCS with phase difference at different corners, the relative local field intensity at different corners can be manipulated by the specific chiral sources. By adjusting the spatial positions of the chiral source, the light manipulation will be more flexible.

Interestingly, we further designed a special splicing structure to avoid the coupling between two same types of TCS. As shown in Fig. 4, two zigzag and two armchair splicing boundaries are grouped together to form a quadrangular structure, in which two ZA types of TCS are designed into the same structure. According to the simulation as shown in Fig. 9, the A corner and B corner are strictly isolated with each other, and no coupling occurs in this system. Through putting a probe at the A corner or B corner, the simulated local field intensities excited by a chiral source at another corner are obtained. As shown in Fig. 4(a), we observed that only the A corner is excited whether an LCP or RCP source is put at the lower right position. This phenomenon can be observed over a broad electromagnetic wave range only if we adjust the structure size. As an example of proof, we also performed a microwave experiment with millimetric structural parameters. The chiral source is realized by using three-point sources, with each one differing in phase by 2π/3. Other parameter settings are consistent with the design mentioned above. As shown in Figs. 4(c) and 4(d), the simulated electric field distribution fits well with measured results. In this case, the uncoupled TCS are pseudospin-polarization independent. On the contrary, only the B corner will be excited and the A corner is no longer able to be excited if the chiral source is put at the higher left position, as shown in Fig. 12. In our opinion, the TCS present obvious path dependence rather than pseudospin dependence, which may contribute to the diverse bandgap positions of zigzag or armchair splicing boundaries between OI and TI lattices.

In summary, we theoretically propose and experimentally demonstrate various types of photonic topological cavities in the same structure. Through selecting different splicing boundaries, only specific TCS with specific electromagnetic characteristics are supported. More importantly, we observe interweaved multiphotons between the multiple same types of TCS, which present spectrum division and directional localization of pseudospin dependence. Through enough computation, we find that coupled TCS with a specific phase difference at different corners can be directional localization based on a specific chiral source. Furthermore, we design a special splicing structure that can avoid the coupling between two adjacent TCS. Our finding greatly enriches the cognition of TCS and paves more way to light manipulation in TCS, which inspires the trap of light in integrated photonics.