Photonic Higher‐Order Topological Insulator with Enlarged Non‐Trivial Bandgaps

The emergence of higher‐order topological insulators (HOTIs) has greatly expanded the family of topological materials. While the co‐dimensional bulk‐boundary correspondence is observed in platforms, such as acoustics and photonics, realizing three‐dimensional (3D) photonic HOTIs is relatively challenging due to the complex properties of electromagnetic waves such as polarizations, scattering, and refractive index. In this paper, a photonic HOTI with a simple multilayer structure that supports higher‐order hinge states is proposed. By inserting a central metallic pillar in the unit cell, the 3D bandgap can be well extended, enabling pure and distinguishable surface and hinge modes. The lattice is reconfigurable and flexible, allowing for hinge and surface waves to be generated by controlling the geometrical length of sub‐lattices. The idea of distinguished higher‐order hinge modes is also extended to enlarged higher‐orbital bandgaps. Furthermore, by introducing a central disclination in this photonic model, the one‐dimensional (1D) vertical disclination mode is obtained which is not seen in existing photonic HOTIs. The findings open the door for a high‐performance topological optical apparatus that features efficient one‐way light propagation and energy concentration.


Introduction
The realization of robust unidirectional wave propagation featuring low loss and high speed is a critical goal in modern physics, as it is also essential for fabricating advanced high-performance devices.The development of topological insulators (TIs) in recent decades has inspired tremendous research to find materials with exotic non-trivial phases. [1,2]When a system evolves from a normal insulator (NI) to a TI, the energy bands undergo a topological phase inversion characterized by distinct topological invariants. [3,4]Typically, a d-dimensional (dD) TI supports (d-1) D boundary states, and the system observes the bulk-boundary correspondence.[7] On the other hand, the thriving higher-order topological insulators (HOTIs) are injecting new vitality into this DOI: 10.1002/lpor.20230038410][11][12][13][14][15] Distinguished from the firstorder TIs, HOTIs can exhibit (d-n) D lower-dimensional gapless topological boundary states (n > 1).For instance, a two-dimensional (2D) HOTI hosts zerodimensional (0D) corner modes and a 3D HOTI supports 1D hinge states.[18][19][20] Meanwhile, the concept of HOTIs has also been extended to topological semimetals (TSMs), which have gained constant attention.In TSMs, the gapless point or nodal line, loop degeneracy arises in momentum space, and they are characterized by open surface arcs linked by nodal points like Dirac and Weyl points.[23][24][25][26][27][28] Remarkably, photonic crystals (PhCs) are particularly promising for realizing HOTIs, as electromagnetic (EM) waves contain rich physical information.Meanwhile, the longer wavelength of EM waves is easier to engineer in metamaterials.31][32][33][34][35] The majority of the existing photonic HOTIs are constituted in 2D configurations with specific geometrical arrays, [35][36][37] splitring resonator (SRR), or bar-like metallic lattice. [38,39]The geometrical limitation forbids the photonic HOTIs from pacing into higher dimensions.Recently, a PhC with coupled layers has been proposed, which is composed of dielectric rods covered with drilled metallic plates. [27,40]However, such constructions fail to provide flexible coupling adjustment because the dielectric rods must be finely tuned to form required topological modes.Also, the resulting bandgaps are too narrow to generate well-separated hinge and bulk states.Therefore, there is a need to find a 3D photonic HOTI that has enlarged bandgaps capable of supporting robust hinge modes.
In this paper, we propose a 3D photonic HOTI composed of a hexagonal-shaped PhC with a sandwiched plate-pillar configuration.By adjusting the lengths of the dielectric unit branches, the inter-and intra-cell couplings can be modulated, and interlayer wave couplings can be enabled by the drilled metallic plates.The bandgap is enlarged by inserting a metallic rod in the middle of the lattice, which weakens the next-nearest coupling and strengthens the nearest orbital bonding.The model follows C n -rotational symmetry, and the 2D invariant  (6) is used to determine the HOTI phase and NI phase. [41,42]The simulation results demonstrate that our model supports strongly confined hinge wave transmissions.Further, we examine the HOTI phase within the disentangled higher-orbital bandgap and obtain well-distinguished hinge and surface modes.More inspiringly, we demonstrate a defect-introduced finite-size plate based on our dielectric-metallic PhC and find the 1D vertical disclination state along the central hollow path.The unconventional bulkdisclination correspondence is extended from the 0D cavity to the 1D route.

The Triple Bifurcated PhC and Topological Invariants
We present a 2D honeycomb lattice as depicted in Figure 1a, with lattice constants a = 20 mm and h = 9 mm in the vertical direction.The individual dielectric pillars (yellow color) have three branches that interact with the nearest sites.The branch width is d = 2 mm, and the lengths of the branch pillar are denoted by w 1 and w 0 .Our modified triple bifurcated pillars allow for an expanded or shrunken lattice by selecting pillars with different branches to enhance or recede the intra-cell coupling t 0 and the inter-cell hopping t 1 .To normalize the hopping terms, we set the distance between two adjacent pillars as a/3.The dielectric unit consists of an isotropic silicon material with a relative permittivity of  r = 8.5 and a relative permeability of  r = 1.The remaining part of the lattice is filled with air (blue color) with  r = 1 and  r = 1, as illustrated in Figure 1a.Based on these configurations, the wave function of our crystal is written as  = [ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 , ϕ 5 , ϕ 6 ] if we neglect the next nearest coupling effects.The master Hamiltonian of our 2D PhC is written as where )a , and k x = (1, 0) and k y = (0, 1) are the 2D primitive lattice k-space vectors.By adjusting the length of the pillar branches, we obtain distinct topological and trivial phases.As seen in Figure 1b, the phase transition point occurs at t 1 /t 0 = 1, corresponding to the equal length of w 1 and w 0 .There are two irreducible representations of E 1 and E 2 under the C 6 -symmetry at the Γ point, as pictured in Figure 1b, which represents the p-type and d-type orbitals.Figure 1c describes the field distribution of the p-type and d-type bands.The band inversion happens after the degeneracy process, meaning the pseudo-time-reversal symmetry is allowed in our system.[44] Our model is the typical 2D topological crystalline insulators (TCI) configuration with C 6 -symmetry, and the TCI (NI) hosts when t 1 /t 0 > 1 (t 1 /t 0 < 1), and the phase switch also results in the band inversion.
The system with a C n -rotational symmetry follows the con- where rn is the n-fold rotation operator, and R n is the n-fold rotation matrix of the momentum vector. [43,44]The C 6 -symmetric system processes both twofold and threefold rotational symmetry operators with  process is presented in the Supporting Information.We can track the topological invariants at high-symmetry points (HSPs) of the 2D Brillouin zone (BZ) with these symmetries.A hexagonal lattice hosts rotation invariant points Γ, K, and M, where M, M′, and M″ are twofold HSPs, K and K′ are threefold HSPs, and Γ is a sixfold HSP that also being invariant under C 2 and C 3 operations.][40][41][42] The rotation eigenvalues at the HSPs Π (n) and Γ are defined via invariants [Π p (n) ) is the number of occupied bands at Π (n) and Γ with eigenvalue ) are marked by C n point groups irreducible representations. [42]In degeneracy states, Kramer's theory guarantees that the rotation eigenvalues should be a conjugate pair.Therefore, the rotation invariants have [K 1 (3) ] = [K 2 (3) ] and [M 1 (2) ] = [M 2 (2) ] under C 6 symmetry.For our model, the invariant index is defined by  (6) = ([M 1 (2) ], [K 1 (3) ]).A band transition process occurs means a non-zero  (6) .Figure 1d displays the E z Bloch functions phase profiles of occupied bands at the Γ, Μ, and Κ points under TCI (w 1 = 3 mm and w 0 = 2 mm) and NI (w 1 = 2 mm and w 0 = 3 mm) phases.The rotation eigenvalues with complex conjugate pairs correspond to the opposite phase windings. [41]As shown in Figure 1d, in a TCI phase, the phase (Arg (E z )) profiles of bands 2 and 3 at the Γ point is opposite.Hence, the representations for bands 2 and 3 are labeled as E 1 .Following this rule, we can determine irreducible representations of other HSPs.Phase profiles of bands 2 and 3 at the Γ point in a NI phase indicate Γ 2 (1) = 1 and Γ 2 (2) = 2, meaning bands 2 and 3 have the same C 2 eigenvalues.Therefore, we can obtain the  (6) = (−2,0) and (0, 0) under the TCI and NI phases, respectively.

Unidirectional Wave Propagation against Defects
After figuring out the band topology properties under different phases, we turn to test the performance of the one-way wave transmission with different paths and geometrical defects.We calculate the edge state dispersions of the supercell with periodic conditions along the x-direction and perfect electrical conductor (PEC) conditions in the y-direction, as seen in the inset of Figure 2a.We find a smooth edge state (marked by hollow blue circles) within the band gap of the bulk states in Figure 2a.Then the wave propagation at the interface between topological and trivial areas is studied.As shown in Figure 2b, the topological (trivial) regions are composed of PhCs with t 1 /t 0 = 1.5 (t 1 /t 0 = 0.67) with a Z-shaped interface.The circularly polarized point sources S ± = H 0 e it (x ± iy) are imposed to excite the EM wave.The upper left (right) panel of Figure 2b shows robust wave transmission properties.Also, we give a right circular polarized point source to see the interface wave traveling with defects.It is a witness in the lower panel of Figure 2c that our 2D PhC is immune to the system defects, and the EM wave propagates along the interface without back-scattering, the transmission performance of the Z-shaped interface and defects is also delivered in the Supporting Information.Since our 2D PhC is a C n -symmetric TCI with a non-zero  (6) , the quantized fractional corner charge Q (6) corner = e/4[M 1 (2) ] + e/6[K 1 (3) ] mod e = e/2, [41,42] and our model manifests corner modes at the six corners of the sample.We construct a small finite-sized 2D system with a side length of four unit cells (PEC boundary conditions on six sides), as given in the inset of Figure 2c.By computing the eigenfrequencies of the corner spectra in our sample, we found the solutions in a bulk-corneredge hierarchy order, where the upper and lower corner solutions are located between the bulk and edge states.Figure 2d presents the |E z | field distribution of upper and lower corners at 8.59 and 9.36 GHz, respectively, and corner modes strongly pine at the corners of our established meta-sample.

Constructions of Photonic HOTIs
To create a 3D photonic HOTI, we use a minimal tight-binding model (TBM) shown in Figure 3a that features interlayer hopping terms.Unlike the 2D PhCs, where the periodicity is only in the xy plane, the inter-layer term t z (k z ) (represented by the cyancolored rods) acts as a periodic function in the k z axis throughout the 3D BZ.[23] To ensure a TCI phase throughout the 3D BZ, we impose the condition of t 1 > t 0 (k z ) for −/h < k z < /h, where h is the vertical lattice constant of the 3D PhC.Since our model retains the C 6 symmetry, we can use the index  (6) to measure the band topology at varying k z .We find that the  (6) is non-zero for −/h < k z < /h using the same calculation method as in a 2D TCI scheme, indicating the presence of a complete HOTI phase in our k z -dependent model, see the Supporting Information.
To build a 3D photonic unit cell that always holds a HOTI phase, we adopt an expanded in-plane lattice with w 1 /w 0 > 1.The k z -dependent term is realized by designing a drilled metallic plate (gray colored) covering the dielectric pillars at both ends, as described in Figure 3b.The projected in-plane drilled holes are located between the nearest pillars to enable the inter-layer coupling of neighboring sites at adjacent layers.The distance between the centers of the two nearest drilled holes is l 1 = 5.7 mm, the diameter of the drilled holes is d 1 = 4.8 mm, and the width of the metallic plates is d = 0.6 mm.Therefore, the master Hamiltonian of our 3D photonic HOTI is written as where A(k z ) = A + 2t z cos(k z h). Figure 1c gives a numerical calculation of the bulk dispersion throughout the 3D BZ along the path of Γ-K-M-Γ-A-H-L-A.The full 3D HOTI band marked by a gray region tells a non-trivial band topology throughout the reciprocal space with  (6) = (−2, 0).It is noticed that the bandgap size is relatively small, with a value of Δ/ c = 5.6%, making hinge and bulk modes indistinguishable.Researchers have suggested using the epsilon-nearzero effect in magneto-optical PhCs to expand the bandgap. [46]owever, this scheme has only been studied in a 2D system, and the 3D scenario requires more complex PhC modulations.Our structure is considered a 1D A-B-A type PhC if the entire lattice is regarded as a homogeneous medium region.The drilled PEC plates covering and supporting the dielectric pillars act as the air region.An ideal model for achieving a large bandgap is the periodic intersection of rod and air layers, where the sub-air layers occupy the adjacent upper and lower rod plates. [47]This is also the key to generating higher-order hinge states in the z-direction.
For our 3D PhCs, we can insert a metallic pillar in the middle of the honeycomb lattice, as illustrated in Figure 3d-f.This operation builds a blocking wall to reduce the next-nearest coupling and strengthen the nearest hopping.Furthermore, the central rod confines the air hole size in the unit cell, which helps to formulate a stable atom-bond structure.We define the side length ratio of the metallic pillar and the PEC plate as s r .The insets of Figures 3d, 3e, and 3f show the corresponding photonic bandgap with s r = 0.1, 0.2, and 0.3, respectively.As the s r increases, Δ/ c increases as well, reaching 17.9% when s r = 0.3.The value of s r is related to the size of the pillars, and the bandgap can be further expanded as the central air region is large enough to accommodate the metallic pillar.

Robust Topological Hinge and Surface States
Now we focus on the hinge states with modified large bandgap settings under different meta-samples.As our model hosts the HOTI phase throughout the k z path in the 3D BZ, we construct a layered photonic PhC array to examine robust hinge waves.Figure 4a shows the building of our single-layer metalattice with the top PEC plate deleted for clear visualization.The entire structure has PEC conditions at the side surfaces, while the periodic boundary condition is given in the k z -direction.We perform a full-wave simulation using the finite-element analysis method.Figure 4d displays distinguishable surface and hinge bands with relatively wide slots.Two hinge bands are distributed among the upper (ranging from 9.75 to 10 GHz) and lower (ranging from 8.75 to 9 GHz) band slots between the bulk-surface spectra.Figure 4c shows the eigenmode solutions of the metasample with k z = 0.5 /h, where the hierarchy of bulk-surfacehinge modes matches well with the bulk dispersions.Figure 4b presents |E z | field intensity profiles of the hinge (upper panel) and surface (lower panel) states at different frequencies at k z = 0.1, 0.4, and 0.9 /h, respectively.Our electrical field mode simulations are consistent with the computation results of the hexagonal sample's eigenmodes.
Next, we conduct a multilayer sample to detect the confined hinge and surface states.Figure 5a shows the detailed structure of a 10-layer piling hexagonal meta-cell sample, with each layer containing 37 unit cells.All six side surfaces are set to be PEC boundaries.To excite the EM waves, point sources S 1 and S 2 (marked by the solid blue circle) are placed near the hinge and the side surface of the sample, respectively.Two detectors receive |E z | field distributions of the hinge (marked by the solid green circle) and surface (marked by the solid purple circle) states.The intensity profiles of |E z | at each layer are derived by performing full-wave simulations.Figure 5b shows the slices of layers 1 (L1), 4 (L4), 7 (L7), and 10 (L10) of the hinge and surface states at the excitation frequency range of ≈8.3-10.5 GHz.We find that the photonic hinge and surface modes are well-localized at each layer without bulk scattering.Figure 5c,d presents the |E z | for the surface and hinge states, respectively.There are three peak values at 8.9, 9.7, and 9.9 GHz for the hinge modes, and two peaks at 9.45 and 9.6 GHz for the surface states, which matches the results given in Figure 4.The performance of hinge and surface wave propagation in a parallelogram-shaped sample is analyzed and discussed in the Supporting Information.
In addition, we examine the hinge wave transmission against structural disorders.As shown in Figure 6a, we build an 8-layer hexagonal sample and introduce two types of disorders: defects and displacements.The lattice disorder is located at the central part of the hinge at layer 4 (counting from the top layer).The defect lattice has only four pillars, and the displacement lattice was set by a central axis rotation of  = 10°for the six rods.We place an excitation source (blue solid circle) below the lattice disorder, and two detectors are put at the top and the bottom of the sample (green and brown solid circles).The hinge wave spectrum ranging from ≈8.6 to 10.3 GHz under two types of disorders is presented in Figure 6b-e.As shown in Figures 6b and c, the |E z | profile at the top slice reaches a peak value of 8.9 GHz in both defect and displacement disorders, respectively.Similarly, for the bottom slice transmission spectrum in Figures 6d and  e, the backscattering hinge wave remains qualitatively at 8.9 and 9.7 GHz, respectively.The insets of Figure 6b-e plot |E z | field distribution slices at the top and bottom layer under the two disorders at the peak values of the spectrum, where the backscattering hinge waves are strongly localized.All these results tell that the system remains qualitative vertical hinge wave transmission despite geometrical disorders.

Higher-Orbital Bandgaps with Distinguished HOTI Phases
Generally, PhCs with periodic drilled configurations or dielectric rods have different band spectrums at a lower frequency in comparison with their theoretical solutions in TBMs.The practical structures of PhCs give transmission solutions for each frequency, and the lowest energy band is linear around the center of 2D or 3D BZ.Although the Mie resonance's state of each isolated scatter (dielectric rod) acts as the atomic orbitals in the electronic system, they suffer from a slow decay, leading to the divergence in matrix elements of the TBMs. [48]In this way, the lowest band behaves like a plane wave, and the band structure will not observe the chiral symmetry.In addition, the TBM is characterized by s-, p-, d-, f-and higher-orbital bands originating from the linear combination of atomic orbitals approach.This idea can be extended to the EM wave system. [48]However, the higher-orbit bands are mixed, offering limited explorations on the practical applications in photonic systems.
Recent investigations reveal that confined Mie resonance PhCs with metallic rods inserted between dielectric pillars make the Mie resonance's state decay in an exponential order, and the photonic band diagram remains consistent with the TBM. [49]This method also produces disentangled higher-orbital bands in the 3D band structures.Our framework adopts the central metallic pillar to maintain both the zero-energy property in the lower bands and enlarged bandgaps.As shown in Figure 7b, we calculate the first 18 bands of the PhC without the metallic rods, where the first bandgap is quite narrow and the chiral symmetry of the lowest bands is broken.By inserting a metallic rod (s r = 0.4), we find two enlarged complete 3D bandgaps in Figure 7c.To compute the distinguished higher-order topological phases in the higher-orbital bandgap, we build an 8-layer finite-sized rhombuslike PhC plane in Figure 7a with the same lattice parameters given in Figure 4a and the size of the metallic rod s r = 0.4.Two sources are located on the bottom of the hinge and the side of the sample (marked by solid blue circles) to excite the hinge and surface waves.The hinge detector (solid green circle) and the surface detector (solid purple circle) are placed on the top layer to receive transmission waves.Figure 7d gives the hinge spectrum of the sample at the lower bandgap, while Figure 7e,f plots the corresponding |E z | profiles of surface and hinge states.The inset view of Figures 7e and f describe the surface and hinge wave field distribution of the top layer at 9.6 and 10.1 GHz, respectively.For the higher bandgap, we also compute the hinge and surface mode dispersions, as pictured in Figure 7g.The surface and hinge wave profiles and field intensity slices are given in Figure 7h,i, which match well with the eigenmode diagrams.

Bulk-Disclination Correspondence in Defect-Introduced Photonic HOTI
Now we give a further study on the bulk-disclination correspondence in our flexible photonic HOTI.A 2D C n -symmetric lattice is divided into n sectors, and each sector has an angle of 2/n from the rotation axis, as shown in Figure 8a.To produce a disclination, we insert or remove 1/n part of the finite size plate, [41,42,[50][51][52] and a 0D disclination will emerge in the central part.][46] The sample always supports corner modes, since removing one sector does not disturb the rotational invariant property of the remaining (n-1) sectors.Figure 8b gives Wannier center distributions of the C (n-1) -symmetric plate for the HOTI and the NI phases, respectively.We mark the Wannier center locations in each lattice (orange dots).The corner modes disappear due to the fractional charge distribution in the NI phase.For HOTI phases, both the corner and disclination modes exist since the Wannier centers are pinned on six sides of the lattices.In our 3D HOTI, the corner modes are replaced by the hinge state, and the 0D disclination cavity will be extended to the 1D disclination tunnel.The fivefold disclination-introduced non-trivial PhC plate has a k z -dependent disclination charge of ][52] Figure 8c displays the construction of monolayer plate deformed from the original C 6 -symmetric PhC array in Figure 4a with the central disclination.The top metallic plate is partly removed for a clear visualization.As analyzed in previous sections, our photonic HOTI manifests a second-order phase through the k z -axis, and the band spectrum of the C 5 -symmetric plate is given in Figure 9c, where the well-distinguished disclination modes spanning −/h < k z < /h and the hinge and surface arcs persists within the enlarged bandgap.Figure 9d depicts the eigenmode solutions of the monolayer C 5 -symmetric sample at k z = 0.3/h, and the five eigenstates of the disclination state are located around 9 GHz.
Then we perform a simulation on the field distribution to find the existence of surface, hinge, and disclination states.We construct an 8-layer multilayer sample with disclinations in the central part in Figure 9a.The green, blue, and red solid circles represent wave detectors placed near the photonic metamaterials.The excitation source (solid orange circles) is put near the top corner.All surfaces of the sample and the central disclination path are set as PEC conditions.Figure 9b plots the calculated surface, hinge, and disclination mode field distribution slices at layers 3 (upper panel) and 7 (lower panel), and the insets of Figure 9d give the corresponding wave intensity profiles of the surface (red region), hinge (red region), and disclination (green regions) states.The surface states reach peak values of around 9.2 and 9.9 GHz, which is consistent with the surface eigenmode distributions.
Similarly, the hinge and disclination states have their peak values at around 9 and 10 GHz, respectively.The wave distribution slices in Figure 9b tell confined surface wave transmission and hinge and disclination modes propagation in the vertical direction.

Conclusions
In conclusion, we have designed a 3D photonic HOTI with a nontrivial large bandgap.By introducing a central metallic pillar, we were able to achieve a peak width of bandgap Δ/ c of 17.9%.The band representation theory justifies the different topological phases, which in turn enables the construction of well-performed one-way transmission with imperfections.Our piling PhC configuration can be constructed without changing the location of dielectric rods, making it easier to tune the topological state compared to existing photonic HOTIs.Two types of multilayer samples are built to examine the transmission of higher-order hinge and topological surface states.Our simulation results demonstrate the strongly localized hinge wave propagation against geometrical disorders.We investigate the higher-order topological phases under enlarged higher-orbital bandgaps and find wellseparated hinge and surface modes.By presenting a cut-glue process, the model is capable of supporting 1D disclination wave transport in a disclination-introduced photonic plate, revealing the coexistence of higher-order hinge modes and disclination states.Our model offers a promising approach to fabricating optical devices for high-efficiency and low-loss wave propagation and energy concentration, as well as multimode waveguides in broadband modulation at electromagnetic frequencies.

Figure 1 .
Figure 1.The schematic graph of the 2D PhC and band topology properties.a) Left panel: in-plane view of the PhC.Right panel: 3D view of the unit cell.b) Energy bands of PhCs with t 1 /t 0 = 1.5 (left panel) and t 1 /t 0 = 0.67 (right panel).Insets: Locations of Wannier centers in the TCI and NI phase.c) Topological phase diagram of the PhC as a function of t 1 /t 0 .d) Phase profiles of the E z Bloch function at the HSPs for the TCI (t 1 /t 0 = 1.5) and NI (t 1 /t 0 = 0.67) PhCs.

Figure 2 .
Figure 2. Edge and corner states in our framework.a) Edge state dispersions of the supercell.Inset: The 3D view of the supercell.b) Simulations of EM wave propagation in PhCs with upper NI and lower TCI phases in different paths.c) The corner spectra of our finite-sized sample.Inset: The 3D view of the meta-cell sample.d) Upper panel: distributed corner modes at 8.59 GHz.Lower panel: Corner modes at 9.36 GHz.

Figure 3 .
Figure 3. Design of our 3D photonic HOTI and the corresponding band structure.a) The schematic diagram of the tight-binding model.b) The design of our sandwich-typed 3D PhC.c) Simulated band spectrum of the photonic HOTI.Inset: The 3D BZ of our unit cell.d-f) Photonic bandgaps with bandwidths of Δ/ c = 10.1%, 13.5%, and 17.9% with s r = 0.1, 0.2, and 0.3, respectively.

Figure 4 .
Figure 4. Constructions of the hexagonal photonic PhC array.a) The PhC hexagonal array with an enlarged bandgap framework.b) The simulated electrical field |E z | for the hinge and surface states at k z = 0.1, 0.4, and 0.9 /h, respectively.Upper panel: corner modes at 9.90, 8.85, and 9.78 GHz from left to right.Lower panel: surface modes at 9.68, 9.61, and 9.40 GHz from left to right.c) Solution numbers of the PhC hexagonal array at k z = 0.5 /h.d) The bulk spectrum of our sample.

Figure 5 .
Figure 5. Simulations on multilayer meta-cell hexagonal sample and corresponding |E z | field distributions.a) The architecture of our 10-layer hexagonal PhC arrays.b) |E z | field distribution slices for hinge and surface states at layers 1, 4, 7, and 10, respectively.c,d) Simulated EM wave spectra for the surface and hinge states, respectively.

Figure 6 .
Figure 6.Robustness of the system against disorders.a) Left panel: The architecture of our 8-layer hexagonal PhC structure.Right panel: Two types of disorders.b,c) Simulated transmission spectrum for the hinge states under defect disorders at the top and bottom layer, respectively.The inset views are |E z | field distribution slices at the top and bottom.d,e) Simulated transmission spectrum of backscattering hinge waves by introducing displacement disorders at the top and bottom layers.The inset views give the |E z | field profiles at the top and bottom displacements, respectively.

Figure 7 .
Figure 7. Topological states at lower and higher bandgaps with our PhCs.a) The structure of our 8-layer rhombus plates.b,c) Full 3D bandgaps of the PhCs in the absence of a metallic rod and with a metallic rod of s r = 0.4, respectively.d) The bulk spectrum of the multilayer sample of the lower bandgap with a metallic rod (s r = 0.4).e,f) Simulated transmission spectrum of surface and hinge waves of the lower bandgap.g) The eigenstate dispersion of the sample of the higher bandgap with a metallic rod (s r = 0.4).h,i) Simulated transmission spectrum of surface and hinge waves of the higher bandgap.The inset views of (e), (f), (h), and (i) are |E z | field profiles.

Figure 8 .
Figure 8. Design of our photonic HOTI finite size disclination sample.a) The cut-and-glue process of producing a C 6 -symmetric Ph Carry with a central disclination from the monolayer C 6 -symmetric PhC array.b) The Wannier center distributions of HOTI and NI phases in a disclination-introduced finite-sized sample.c) The construction of a monolayer disclination-introduced photonic plate.

Figure 9 .
Figure 9. Simulations of the topological phase and disclination state transmission in a multilayer architecture.a) The piling photonic sample.b) Field intensity distributions of surface, hinge, and disclination states at layers 3 and 7, respectively.c) The band dispersions of the disclination-introduced photonic plate.d) The eigenmode solutions of the finite-sized sample at k z = 0.3 /h.The insets give the surface, hinge, and disclination wave intensity profiles of the multilayer sample.

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