Realization of a three-dimensional photonic higher-order topological insulator

SJ_Zhang
May. 5, 2025

Abstract

The discovery of photonic higher-order topological insulators (HOTIs) has expanded our understanding of band topology, offering robust lower-dimensional boundary states for photonic devices. However, realizing three-dimensional (3D) photonic HOTIs remains challenging due to the vectorial and leaky nature of electromagnetic waves. Here, we present the experimental realization of a 3D Wannier-type photonic HOTI using a tight-binding-like metal-cage photonic crystal, whose band structures align with a 3D tight-binding model via confined Mie resonances. Microwave near-field measurements reveal coexisting topological surface, hinge, and corner states in a single 3D photonic HOTI, consistent with theoretical predictions. Remarkably, these states are robust and self-guided even within the light cone continuum, functioning without ancillary cladding. This work paves the way for multi-dimensional manipulation of electromagnetic waves on 3D cladding-free photonic bandgap materials, enabling practical applications in 3D topological integrated photonic devices.

Introduction

Recently, the discovery of higher-order band topology^1,2,3 has revolutionized the study of topological matters with unconventional bulk-boundary correspondence that enables lower-dimensional topological boundary states in at least two dimensions lower than the bulk, in contrast to the conventional first-order topological matters whose topological boundary states only live in just one dimension lower than the bulk. For example, the first-order (higher-order) topological phases host topological edge (corner) states in a two-dimensional (2D) system and topological surface (hinge or corner) states in a three-dimensional (3D) system. To date, extensive studies of various higher-order topological phases have been carried out in condensed matter physics^4,5, photonics^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26}, acoustics^{27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42}, mechanics^43,44, electric circuits^45,46, and even thermal diffusions^47,48.

Generally, from the dimensional perspective, higher-order topological phases can be classified into two major classes: 2D^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,27,28,29,30,43,44,45,47,48} and 3D^{20,21,22,23,24,25,31,32,33,34,35,36,37,38,39,40,41,42,46,49}. Compared with the 2D higher-order topological phases whose topological boundary states are limited to first-order one-dimensional (1D) edge states and second-order zero-dimensional (0D) corner states, 3D higher-order topological phases can host first-order 2D surface states, second-order 1D hinge states, and third-order 0D corner states, significantly benefiting multi-dimensional wave manipulation and increasing device integration density. More specifically, there are two different types of 3D HOTIs: 3D HOTIs derived from the generalized Su-Schrieffer-Heeger (SSH) model without quantized multipole moments which are termed Wannier-type 3D HOTIs^31,32,33, and 3D HOTIs with quantized multipole moments which are dubbed as octupole HOTIs^34,35. Compared with the octupole HOTIs, the Wannier-type 3D HOTIs don’t require negative nearest-neighbor couplings and have been experimentally realized in acoustic crystals^31,32,33 and electric circuits⁴⁶ by directly mimicking a 3D SSH model. However, due to the vectorial nature of electromagnetic waves and the lack of mirror reflection symmetry in the vertical direction, the eigenmodes in 3D photonic crystals cannot be simplified as scalar transverse electric (TE) or transverse magnetic (TM) modes. This results in complex band dispersions, making it challenging to describe 3D photonic systems using a simple tight-binding model^50,51. Consequently, in contrast to the extensive experimental realizations of 3D HOTIs in acoustics and electric circuits, their photonic counterparts have been severely lagged, with only a handful of experimental demonstrations of higher-order Dirac²⁰ or Weyl²¹ semimetals with topological hinge states in 3D photonic crystals. However, due to the absence of complete 3D photonic bandgaps in photonic higher-order Dirac or Weyl semimetals, the topological hinge states only exist in a limited range of k_z and could be scattered into the bulk when encountering obstacles, which inevitably jeopardizes their efficient and robust transport. A natural question arises as to whether there exist 3D photonic HOTIs with complete 3D photonic bandgaps and hosting first-order topological surface states, second-order topological hinge states, and third-order topological corner states in the whole Brillouin zone (BZ). More recently, several 3D photonic HOTIs have been theoretically proposed in tight-binding-like photonic crystals based on the confined Mie resonance^22,23,24. However, to date, 3D photonic HOTIs have never been experimentally realized in any photonic system.

Here, we report on the experimental realization of a Wannier-type 3D photonic HOTI in a tight-binding-like photonic crystal, which can be regarded as a photonic realization of the celebrated 3D SSH model. The tight-binding-like photonic crystals comprise coupled dielectric rods embedded with metallic pillars, forming confined Mie resonances and supporting exponentially decayed modes. These exponentially decayed modes function as artificial atomic orbitals, resembling tight-binding models with nearest-neighbor couplings and exhibiting nearly identical band structures. By direct microwave real-space visualization and momentum-space spectroscopy measurements, we experimentally observe coexisting self-guided topological surface states on 2D surfaces, topological hinge states on 1D hinges, and topological corner states confined to 0D corners in a single 3D tight-binding-like photonic crystal without extra claddings, manifesting a dimensional hierarchy of topological boundary states due to the third-order band topology. Furthermore, we experimentally demonstrate that the topological surface, hinge, and corner states are robust against defects. Our work provides a versatile platform to design 3D topological photonic crystals by directly mimicking the tight-binding models, which may inspire the future design of 3D topological integrated photonic devices and chips.

Results

Design of a Wannier-type 3D photonic HOTIs

We begin with a 3D SSH model whose unit cell contains eight sites (blue spheres) coupled with nearest-neighbor intercell (t_v and t_zv) and intracell (t_w and t_zw) couplings, as shown in Fig. 1a. When the intercell couplings t_v (t_zv) are larger than the intracell couplings t_w (t_zw), the 3D SSH model exhibits a Wannier-type third-order topological insulating phase with complete 3D bandgaps and coexisting topological surface, hinge, and corner states (see Note 1 in Supplementary Materials)^31,32,33,34. Moreover, the nontrivial higher-order band topology of the 3D SSH model can be characterized by quantized Wannier centers (see Note 2 in Supplementary Materials) and the dimensional hierarchy of the topological states (see Note 3 in Supplementary Materials). To implement the 3D SSH model in a photonic system, we adopt a 3D tight-binding-like metal-cage photonic crystal (MCPC)²² whose unit cell is shown in Fig. 1b, where eight dielectric rods (blue color) are placed between three parallel perforated metallic plates (golden color) serving as the eight sites and each dielectric rod is surrounded by four metallic pillars (golden color) to confine the slowly decaying Mie resonances’ states, leading to the vectorial electromagnetic waves in 3D photonic crystals being simplified to the scalar-wave-like ones (only TM-like modes) and the 3D MCPC exhibits almost the same scalar-wave-like band structures as those of the 3D tight-binding models (see Notes 4 in Supplementary Materials). In the context of MCPC, the planar (vertical) intracell and intercell couplings can be modulated by the radii R₁ and R₂ of the metallic rods and the distance R between the dielectric rods and the central metallic rods (the radius r₁ of the air holes and the thickness h₁ and h₂ of the metallic plates). Since all nearest-neighbor couplings can be modulated independently and flexibly by tuning the geometrical parameters of the MCPCs, a third-order 3D MCPC can be obtained by setting R₁ = 1 mm, R₂ = R₃ = 2 mm, h₁ = 0.5 mm, h₂ = 3 mm, r = 1.6 mm, r₁ = 1.5 mm, d = 1.9 mm, l₁ = 5 mm, l₂ = 4.5 mm, a = 15 mm, a_z = 10 mm, respectively. The simulated bulk band structure of the third-order 3D MCPC in the first BZ (Fig. 1c) is shown in Fig. 1d, which has three complete photonic bandgaps (floral white regions) and bears a close resemblance to that of the 3D SSH model (see Note 5 in Supplementary Materials). To further unveil the third-order band topology of the 3D MCPC, we design a finite 3D MCPC with 3 × 3 × 3 unit cells and calculate its eigenstates spectrum, as shown in Fig. 1e, we use symbols of different colors and shapes to distinguish topological states and their eigenmode distributions. Although the topological surface, hinge, and corner states exhibit a downward frequency shift compared to those of the tight-binding model due to a slight chiral symmetry breaking of the bulk band of the MCPC (see Note 5 in Supplementary Materials), these topological states still coexist in a single 3D MCPC with a third-order band topology. The simulated E_z field distributions of the topological surface, hinge, and corner eigenstates are shown in Fig. 1f. It can be seen that for the topological surface, hinge, and corner states, their E_z fields are mainly localized at the 2D surfaces, 1D hinges, and 0D corners of the 3D MCPC, respectively, matching well with the tight-binding results.

Fig. 1: Design of a 3D photonic HOTI.

a Schematic of the 3D SSH model, the blue spheres represent the eight sites, and the pink (yellow) rods represent the vertical (planar) intracell and intercell nearest-neighbor couplings. b Unit cell of the 3D MCPC. The golden parts represent the perforated metallic parallel plates and metallic pillars, and the blue parts represent the dielectric rods. The lattice constants in the x-y plane and z-direction are a = 15 mm and a_z = 10 mm, respectively. The other geometrical parameters are R₁ = 1 mm, R₂ = R₃ = 2 mm, h₁ = 0.5 mm, h₂ = 3 mm, r = 1.6 mm, r₁ = 1.5 mm, d = 1.9 mm, l₁ = 5 mm, l₂ = 4.5 mm, respectively. c 3D BZ of the MCPC. d Simulated bulk band structure of the 3D MCPC along high-symmetry lines, the floral white regions represent the three photonic bandgaps. e Simulated eigenstate spectrum of a finite 3D MCPC with 3 × 3 × 3 unit cells. The bulk, surface, hinge, and corner states are represented by grey, orange, blue, and green triangles and circles, respectively. For the surface and hinge states, circles (triangles) indicate these states located at the x-z or y-z plane (x-y plane). f Simulated field distributions of the eigenmodes corresponding to the surface, hinge, and corner states labeled in e, respectively.

Experimental observation of the first-order photonic topological surface states

Now we start experimentally demonstrating the first-order topological surface states in the 3D MCPC. The fabricated experimental sample is shown in Fig. 2a, which consists of 20 unit cells along the x and z directions and 4 unit cells in the y direction. To show the detailed configuration of each layer, the compositional layers are glided for clarity, as shown in Fig. 2b, where the perforated copper plates (golden color) with air holes are adopted to induce vertical interlayer and intralayer couplings, and the perforated air foams (white color) are used to fix the metallic and dielectric rods. Note that all six surfaces of the 3D MCPC can support topological surface states and here we only focus on the front (010) surface (parallel to the x-z plane). Figure 2d presents the calculated topological surface state dispersions (orange lines) along high-symmetry lines of the projected 2D surface BZ (Fig. 2c) and the light cone is indicated by the cyan curve. It can be seen that the first-order topological surface states exist within all three photonic bandgaps and are completely separated from the bulk bands (grey lines). More interestingly, the topological surface states exhibit a well-defined dispersion even within the light cone continuum, indicating the leakage of topological surface states to the surrounding air is omittable in the whole surface BZ. This counterintuitive phenomenon stems from the strong field localization of confined Mie resonances, which concentrate the electric fields around the dielectric rods of the 3D MCPCs with minimal surface leakage. Meanwhile, the wave functions of the topological surface states decay evanescently and retain the vectorial nature of surface electromagnetic waves. This unique property gives rise to the field texture of topological surface states resembling a Néel-type skyrmion⁵² (see Note 6 in Supplementary Materials). This characteristic allows us to observe stable topological surface states across the entire surface BZ, even within the light cone continuum. This is nontrivial since conventional photonic topological surface states either need ancillary cladding to prevent the leakage^53,54,55 or generally have a large radiative loss inside the light cone making them unobservable⁵⁶. We also calculate topological surface states’ quality factor (Q factors) in the third bandgap (see Note 6 in Supplementary Materials). The high Q factors indicate that the photonic topological surface states of the MCPCs are self-guided without extra cladding and well-localized even in the light cone continuum. Moreover, with additional surface modifications, the Q factor of topological surface states can be further increased (See Note 7 in Supplementary Materials). We then perform experiments to measure the transmission spectra of the topological surface (orange color) and bulk (dark grey) states by placing a source antenna (cyan stars in Fig. 2f–g) at the center of the front (010) surface and inserting a probe antenna into the surface or bulk, respectively, as shown in Fig. 2e. Three photonic bandgaps (floral white color) can be observed as the bulk transmission spectrum exhibits three broad dips in the frequency ranges of 14.2–15.2, 15.3–16.8, and 17–18 GHz, respectively. Within the photonic bandgaps, the surface state transmission spectrum exhibits three sharp peaks, indicating the existence of topological surface states on the 3D MCPC surfaces. To directly observe the field distributions of the topological surface states, we employ a probe antenna to measure the electric field distributions of the topological surface states across the entire structure (see Methods). The measured electric field distribution of the topological surface state at 14.6 GHz is shown in Fig. 2f, which agrees well with the simulation results shown in Fig. 2g. The field is predominantly localized on the sample’s surface, signifying the existence of topological surface states. By Fourier-transforming the measured complex electric field distributions from real space to reciprocal space, we obtain the measured surface state dispersion along the high-symmetry lines \(\bar{\Gamma }-\bar{{{{\rm{A}}}}}-\bar{{{{\rm{H}}}}}-\bar{{{{\rm{K}}}}}-\bar{\Gamma }\) in the projected 2D BZ, as shown in Fig. 2h, the measured results (color map) demonstrate excellent consistency with the simulated surface state dispersion (white dashed line).

Fig. 2: Observation of the first-order photonic topological surface states.

a Photograph of the fabricated experimental sample comprising 4 unit cells in the y direction and 20 unit cells in the x and z directions. b The compositional layers are glided for photographing. Each unit cell (green square) comprises two perforated copper plate layers and two air foam layers inserted by metallic pillars (red circles) and dielectric rods (blue circles). c Schematic view of the projected 2D surface BZ. d Simulated surface (orange lines) and bulk (grey lines) state dispersions along high-symmetry lines. The cyan curve represents the light cone. e Measured transmission spectra of the surface (orange color) and bulk (grey color) states. The light beige regions represent the three photonic bandgaps. Measured (f) and simulated (g) electric field distributions of the topological surface states at 14.6 GHz. The cyan star represents the point source. h Measured (color map) and simulated (white dashed line) surface state dispersions along the high symmetry line.

Experimental observation of the second-order photonic topological hinge states

Next, we experimentally demonstrate the second-order photonic topological hinge states originating from the higher-order band topology of the 3D MCPC. Note that all twelve hinges support topological hinge states. Here we only focus on one hinge between (100) and (010) surfaces and simulate its eigenmode dispersion relationship along the k_z direction, as shown in Fig. 3a, the surface states, hinge states, bulk states, and light cone are indicated by the orange, blue, grey and cyan lines, respectively. Similar to the topological surface states, the simulated eigenmode field distribution of the hinge state (Fig. 3b) marked by a black triangle in Fig. 3a shows that the topological hinge state can be well confined at the hinges even if its dispersion is located within the light cone. To excite the topological hinge states, a source antenna (cyan stars in Fig. 3d, e) is placed at the center of the hinge. Subsequently, we use another probe antenna to measure the transmission spectra of the hinge (blue color) and surface (orange color) states, as shown in Fig. 3c, in which the hinge state transmission spectrum exhibits two peaks within the two transmission dips (bandgaps) of the surface states. Figure 3d shows the measured electric field distribution of the topological hinge state at 14.9 GHz, matching well with the simulation result shown in Fig. 3e. The field is predominantly localized on the hinge, indicating the presence of topological hinge states. The measured hinge dispersion (color map) along the k_z direction is obtained through Fourier transformation on the measured hinge states’ complex electric field distribution, as shown in Fig. 3f, which shows excellent agreement with the simulation results (white dashed line).

Fig. 3: Observation of the second-order photonic topological hinge states.

a Simulated bulk (grey lines), surface (orange lines), and hinge (blue lines) state dispersions along the k_z direction. The cyan curve represents the light cone. b Simulated electric field distribution of the hinge states corresponding to the black triangle in a. c Measured transmission spectra of the hinge (blue color) and surface (orange color) states. Measured (d) and simulated (e) electric field distributions of the topological hinge states at 14.9 GHz with the source antenna (cyan star) placed at the middle of the hinge. f Measured (color map) and simulated (white dashed line) hinge state dispersions along the k_z direction.

Experimental observation of the third-order photonic topological corner states

We then experimentally characterize the third-order photonic topological corner states in the 3D MCPC. Note that all eight corners of the 3D photonic HOTI can support corner states, and here we only focus on one single corner. We begin by measuring the transmission spectra of the surface (orange color), hinge (blue color), and corner (green color) states under the excitation of a point source (cyan stars in Fig. 4b, c) placed near the corner. As shown in Fig. 4a, the corner measurement exhibits a transmission peak at about 15.8 GHz (white dashed line), which agrees well with the corresponding eigenfrequency ranges of the calculated corner, hinge, and surface eigenstates shown in Fig. 1e. We also plot the measured and simulated electric field distributions of the corner states at 15.8 GHz in Fig. 4b and Fig. 4c, respectively. It can be seen that the electric field distributions are mainly concentrated at the corner, revealing the tightly localized characteristic of the topological corner states.

Fig. 4: Observation of the third-order photonic topological corner states.

a Measured transmission spectra of the surface (orange color), hinge (blue color), and corner (green color) states. Measured (b) and simulated (c) electric field distributions of the topological corner states with the source antenna (cyan star) placed near the corner.

Experimental demonstration of the robustness of topological surface, hinge, and corner states

Finally, we experimentally demonstrate the robustness of the topological boundary states by introducing defects (removing some dielectric rods) at the surface, hinge, and corner, respectively, as shown in the insets of Fig. 5a–c (red circles) for the experimental measurements and Fig. 5d–f (white dashed cylinder) for the numerical simulations. The measured (Fig. 5a–c) and simulated (Fig. 5d–f) electric field distributions of the topological surface, hinge, and corner states with defects are almost the same as those without defects (Figs. 2–4) except for a sudden decrease in the electric fields at the defect location due to the absence of the dielectric rods, verifying the robustness of the topological surface, hinge, and corner states against defects.

Fig. 5: Experimental demonstration of the robustness of topological surface, hinge, and corner states.

Cutaway views of the measured electric field distributions of the topological surface (a), hinge (b), and corner (c) states with defects, respectively. The insets display photographs of the defects introduced by removing dielectric rods (red circles). Simulated electric field distributions of the topological surface (d), hinge (e), and corner (f) states with defects, respectively. The insets show the defects introduced by removing dielectric rods (white dotted cylinders). The cyan star represents the point source.

Discussion

In conclusion, we have experimentally realized a Wannier-type 3D photonic HOTIs with a dimensional hierarchy of coexisting first-order 2D surface states, second-order 1D hinge states, and third-order 0D corner states in a single 3D MCPC. Moreover, we experimentally demonstrate that all-order photonic topological boundary states are defect-immune and self-guided without any ancillary cladding, making them more suitable for practical applications. The ability to integrate 2D surface states, 1D hinge states, and 0D corner states in a single photonic structure may yield potential applications for multi-dimensional photon steering in 3D photonic devices. Since our work provides a versatile platform for directly mimicking 3D tight-binding models in 3D photonic crystals, we envision more experimental studies on realizing topological lattice defects such as dislocation^57,58,59 disclination⁶⁰ and Dirac vortex^61,62 in 3D photonic crystals. Along these lines, 3D photonic topological insulators and HOTIs operating at telecom or visible frequencies are on the horizon^63,64. Besides, 3D artificial gauge fields induced by lattice site deformation⁶⁵ or 3D non-Hermitian photonics²⁵ induced by lattice sites’ gain and loss are also ready to be explored based on our platform.

Methods

Numerical simulations

All numerical results presented in this work are simulated by the RF module of COMSOL Multiphysics. The bulk band structure is calculated using a unit cell with periodic boundary conditions in all directions. The perforated copper plates and metallic pillars are modeled as perfect electric conductors (PEC). The surface state dispersion is calculated by adopting a 1 × 4 × 1 supercell and applying periodic boundary conditions along the x and z directions, and open boundary conditions along the y direction. The hinge state dispersion along the z direction is calculated using a 6 × 6 × 1 supercell and applying periodic boundary conditions along the z direction, and open boundary conditions along the x and y directions. In the eigenstate and full-wave simulations of a finite 3D photonic crystal, all six boundaries are set as open boundary conditions.

Sample fabrication

The MCPCs are assembled layer by layer and comprise perforated metallic plates, dielectric rods, metal pillars, and perforated air foams. The dielectric rods sintered from alumina have a relative permittivity of 9. The metal pillars are fabricated using surface wire cutting and have a geometric tolerance of about 0.05 mm. The copper plates are fabricated by depositing a 0.035 mm-thick copper layer onto a Teflon woven-glass fabric laminate substrate. We adopt perforated air foam (ROHACELL 31 HF with a relative permittivity of 1.04 and a loss tangent of 0.0025) to fix those metallic pillars and dielectric rods.

Experimental setups

In the experimental measurement, the amplitude and phase of the electric fields are measured by a vector network analyzer (Keysight E5080), connected to two electric dipole antennas with a diameter of 1.2 mm, which serves as the source and probe, respectively. The air holes with a radius of r₁ = 1.5 mm (see Fig. 1b and Fig. 2a, b) are carefully designed in MCPCs not only to insert the probe antenna to measure the electric fields in the sample but also to obtain suitable vertical interlayer couplings. The measured transmission spectra of the topological surface, hinge, and corner states in Figs. 2e, 3c, and 4a are obtained using the experimental setup schematically shown in Figure. S14. To measure the transmission spectra of the surface and bulk states in Fig. 2e, a source antenna (cyan star) is positioned at the center of the front surface and a probe antenna is inserted into the surface (orange dot) or bulk (gray dot) through the air hole, located an equal distance away from the source (see Fig. S14a). To measure the transmission spectra of topological hinge and surface states in Fig. 3c, a source antenna (cyan star) is placed at the center of one hinge and a probe antenna is inserted into the same hinge (blue dot) and surface (orange dot) positioned five unit cells away from the source antenna (see Fig. S14b). To measure the transmission spectra of the topological corner, hinge, and surface states in Fig. 4a, a source antenna (cyan star) is positioned near the corner and a probe antenna is placed at the corner (green dot) one unit cell away from the source, hinge (blue dot), and surface (orange dot) five unit cells away from the source (see Fig. S14c).

To measure the electric field distributions of the topological surface, hinge, and corner states, we insert the probe antenna into 320 air holes one by one and scan automatically along the z-direction with a 1 mm step length (200 steps). By scanning all the air holes, we can obtain 64000 points of measured data (320 air holes multiply 200 steps) to map the electric field distributions of the topological bulk and boundary states of the 3D photonic HOTIs. Note that since the probe antenna cannot be inserted into the same air hole as the source antenna, the measured data of the air hole with the source antenna has been replaced with measured data from the adjacent air hole. We then perform Fourier transformation to the measured complex electric field distributions at each frequency to obtain the projected band structures of the topological surface and hinge states.

Data availability

All data are available in the manuscript, the supplementary materials, or have been deposited in the Zenodo database. https://doi.org/10.5281/zenodo.14999044

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