Topological Dirac-vortex modes in a three-dimensional photonic topological insulator

Abstract

Recently, topological Dirac-vortex modes in Kekulé-distorted photonic lattices have attracted broad interest and exhibited promising applications in robust photonic devices such as topological cavities, lasers, and fibers. However, due to the vectorial nature of electromagnetic waves that results in complicated band dispersions and fails the tight-binding model predictions, it is challenging to construct three-dimensional (3D) topological photonic structures with Kekulé distortion, and the photonic topological Dirac-vortex modes have thus far been limited to two-dimensional (2D) systems. Here, by directly mapping a 3D Kekulé-distorted tight-binding model in a 3D tight-binding-like photonic crystal exhibiting scalar-wave-like band structures, we theoretically propose and experimentally demonstrate topological Dirac-vortex modes in a 3D photonic topological insulator for the first time. Using microwave near-field measurements, we directly observe robust photonic topological Dirac-vortex modes bound to and propagating along a one-dimensional (1D) Dirac-vortex line defect, matching well with the tight-binding and simulation results. Our work offers an ideal platform to map tight-binding models in 3D topological photonic crystals directly and opens a new avenue for exploiting topological lattice defects to manipulate light in 3D space.

Topological Dirac-vortex modes have attracted broad interest but are thus far limited to two-dimensional photonic systems. Here, authors report the first experimental observation of topological Dirac-vortex modes in a 3D photonic topological insulator.

Introduction

Recently, the interplay between real-space topological lattice defects (TLD)^1,2 and reciprocal-space band topology^3–5 has given rise to many novel topological phenomena and promising applications^6–8, such as cavities^9–17, lasers^18–22, waveguides^23–32, fibers^33,34, and three-dimensional (3D) photonic topological insulators in synthetic dimensions³⁵. In particular, topological Dirac-vortex modes bound to a vortex defect in Kekulé-distorted lattices, which are Jackiw-Rossi zero modes originating from the Dirac equation with mass vortices³⁶, have attracted widespread attention in many areas ranging from high-energy physics³⁷, condensed-matter physics³⁸ to topological physics^{14–22,34,39–45} due to its unique properties such as scalable mode areas, arbitrary mode degeneracies, vector-beam emission, and large free spectral range. However, to date, most previous studies of topological Dirac-vortex modes have been limited to two-dimensional (2D) systems that support zero-dimensional (0D) localized modes.

More recently, one-dimensional (1D) vortex-string chiral modes bound to and propagating along a Dirac-vortex line defect^43,44 and 0D monopole topological modes localized in a 3D Dirac-vortex volume defect⁴⁵ have been experimentally demonstrated in 3D acoustic crystals, extending topological Dirac-vortex modes from 2D to 3D systems. However, in sharp contrast to the studies of topological Dirac-vortex modes in 3D acoustic crystals^43,45 where the couplings and on-site energies can be engineered flexibly to implement the discrete lattice models (tight-binding models) easily and directly, their photonic counterparts have been severely lagged due to the vectorial nature of electromagnetic waves and the inherent challenge of discretely modeling 3D photonic systems^46,47, which usually makes the tight-binding model prediction fail. Therefore, it is still an open question whether topological Dirac-vortex modes can be realized in 3D topological photonic structures (even in theory).

Here, we theoretically propose and experimentally demonstrate topological Dirac-vortex modes in a 3D photonic topological insulator^48–54 by inducing Kekulé distortion in a 3D tight-binding-like metal-cage photonic crystal^55–57 whose bulk band dispersions resemble scalar waves and match well with those of the 3D tight-binding model. Using microwave near-field measurements, we directly observe topological Dirac-vortex modes bound to and propagating along the 1D Dirac-vortex line defect in a 3D photonic topological insulator. Moreover, we experimentally demonstrate that the photonic topological Dirac-vortex modes exhibit robustness against various defects and obstacles, making such modes well-suited for robust electromagnetic wave manipulation in 3D space. Our work not only experimentally extends photonic topological Dirac-vortex modes from 2D to 3D for the first time, but also provides a versatile platform to explore novel physical phenomena and practical applications enabled by TLDs in 3D topological photonic crystals.

Results

Topological Dirac-vortex modes in a three-dimensional Kekulé-distorted honeycomb lattice

We start with topological Dirac-vortex modes in a 2D Kekulé-distorted honeycomb lattice, as schematically illustrated in the upper panel of Fig. 1a, where the topological Dirac-vortex modes (red region) are tightly localized around the 0D vortex core due to the gapped Dirac cones induced by mass terms with phase vortices. The winding number of Dirac vortices equals the number of topological Dirac-vortex modes. Then we vertically stack the 2D Kekulé-distorted honeycomb lattices with uniform interlay couplings to construct a 3D Kekulé-distorted honeycomb lattice which supports 1D topological Dirac-vortex modes (red arrows) bound to and propagating along the 1D vortex line defect, as shown in the lower panel of Fig. 1a. To study this unique 1D topological Dirac-vortex modes, we adopt a 3D tight-binding model whose unit cell consists of two layers of honeycomb lattices with intralayer couplings t_a (green rods), t_b (blue rods), t_c (orange rods), and interlayer couplings t_z (gray rods), respectively, as shown in Fig. 1b (see the Hamiltonian in Supplementary Note 1). The intralayer coupling strengths are $t_a=t_0-\delta t\cos \left(\phi +4\pi /3\right)$, $t_b=t_0-\delta t\cos \left(\phi +2\pi /3\right)$, and $t_c=t_0-\delta t\cos \left(\phi \right)$, where t₀ represents the initial coupling strength, δt and φ represent the amplitude and phase of the Dirac mass, respectively. When all intralayer couplings are equal (t_a = t_b = t_c, δt = 0), the bulk band structure (green lines) of the 3D honeycomb lattice in the first Brillouin zone (BZ) (Fig. 1c) is gapless with an eightfold degenerate double Dirac point at A point, as shown in Fig. 1d. When we introduce periodic Kekulé distortion characterized by different intralayer couplings (t_a ≠ t_b ≠ t_c, δt ≠ 0 and $\phi =\pi /3$), the eightfold degenerate double Dirac point will be broken, resulting in a gapped bulk band structure (gray lines) with a complete 3D topological bandgap (orange region) and topological surface/hinge states (see the Wilson loop and eigenstate calculation in Supplementary Note 2 and Supplementary Note 3). To study the nontrivial topological states supported by a 3D aperiodic Kekulé-distorted honeycomb lattice characterized by a position-dependent phase modulation φ(r), we calculate the dispersion of a finite hexagonal supercell of 3D aperiodic Kekulé-distorted honeycomb lattice with open boundary conditions in the xy plane and periodic boundary conditions along the z direction, as shown in Fig. 1e. The 3D aperiodic Kekulé-distorted honeycomb lattice supports 1D topological Dirac-vortex modes (red line) and topological hinge states (blue line) within the 3D topological bandgap (orange region). The topological hinge states and the topological Dirac-vortex modes can be selectively excited by placing a point source (green star) at the hinge or center of the 3D Kekulé-distorted honeycomb lattice, respectively, as shown in Fig. 1f, g, we can see that the topological hinge states and topological Dirac-vortex modes are tightly localized and propagate along the hinge or vortex line defect, respectively. Since the topological hinge states have been experimentally demonstrated in a 3D photonic higher-order topological insulator⁵⁸, we only focus on the 1D topological Dirac-vortex modes in this work.

Fig. 1. Topological Dirac-vortex modes in a 3D Kekulé-distorted honeycomb lattice.

a Upper panel: 2D Kekulé-distorted honeycomb lattice hosting 0D localized topological Dirac-vortex modes (red region). Lower panel: 3D Kekulé-distorted honeycomb lattice supporting 1D propagating topological Dirac-vortex modes (red arrow). b Unit cell of the 3D honeycomb lattice with intralayer (t_a, t_b, and t_c) and interlayer (t_z) couplings. c 3D Brillouin zone. d Calculated bulk band structures of the 3D honeycomb lattice without (green lines) and with (gray lines) Kekulé modulation, respectively. The orange region represents the 3D topological bandgap. e Calculated dispersions of the topological Dirac-vortex modes (red lines) and hinge states (blue lines), and the gray lines and orange region represent the bulk states and 3D topological bandgap, respectively. Energy distributions of the topological hinge states (f) and Dirac-vortex modes (g), respectively. The green star represents a point source.

Topological Dirac-vortex modes in a 3D photonic topological insulator

Now we design a 3D photonic topological insulator with Kekulé distortion by implementing the 3D tight-binding model (lower panel of Fig. 1a) in a 3D tight-binding-like metal-cage photonic crystal with confined Mie resonance⁵⁰. Figure 2a, b shows the perspective and top views of a unit cell of the 3D tight-binding-like metal-cage photonic crystal, in which the twelve dielectric rods (white rods) serve as sites. Each dielectric rod is embedded with three metallic rods (copper rods) to confine the Mie resonance to satisfy the tight-binding approximation in photonic systems. The perforated metallic plates with air holes are used to introduce interlayer couplings. Remarkably, the embedded metallic rods and perforated metallic plates confine the Mie resonances of the dielectric rods, making the complex vectorial electromagnetic waves in 3D photonic crystals simplified to scalar-wave-like ones and creating chiral symmetric photonic band structures that ideally match those of the tight-binding models with nearest-neighbor couplings. Note that tight-binding models with nearest-neighbor couplings are sufficient to describe the vectorial nature of electromagnetic waves here due to the tightly confined Mie resonances, while the next-nearest-neighbor couplings in 3D Haldane model (stacked 2D Haldane models with alternating magnetic flux directions) are required to describe a 3D photonic Chern (axion) insulator^53,54 with time-reversal symmetry breaking. The Kekulé distortion can be introduced in the 3D tight-binding-like metal-cage photonic crystal by two steps: displacing three dielectric rods (white color) from their original positions toward the center of the unit cell (along the gray dashed lines) with a distance m₀ and then rotating them with an angle φ around the original center of each dielectric rod, as illustrated in Fig. 2b. Similar to the tight-binding model, when m₀ = 0 and φ = π/3, the simulated bulk band structure of the 3D tight-binding-like metal-cage photonic crystal (green lines) exhibits a double Dirac point with eightfold degeneracy at the A point, and the degeneracy will be lifted (gray lines) and open a 3D complete topological photonic bandgap (orange region) when m₀ ≠ 0 and φ = π/3, as shown in Fig. 2c. More significantly, if m₀ ≠ 0, the 3D photonic bandgap persists for arbitrary angle of φ, as shown in Fig. 2d, in which the color represents the angle φ varying from 0 to 2π (see the band inversion in Supplementary Note 4). The distance m₀ and rotating angle φ (air hole radius r) are used to adjust the intralayer (interlayer) couplings of the 3D tight-binding-like metal-cage photonic crystals to implement the tight-binding model. Here, all interlayer couplings are the same (see the correspondence between the tight-binding models with asymmetric interlayer couplings and 3D tight-binding-like metal-cage photonic crystals with different air holes in the Supplementary Note 5).

Fig. 2. Topological Dirac-vortex modes in a 3D photonic topological insulator.

a Schematic of the unit cell of a 3D tight-binding-like metal-cage photonic crystal. The copper parts represent the perforated metallic plates and metallic rods, and the white parts represent the dielectric rods. b Top view of the unit cell of the 3D tight-binding-like metal-cage photonic crystal. The lattice constants in the xy plane and z direction are a = 15 mm and a_z = 11.7 mm, respectively. The other geometrical parameters are R = 5 mm, r₁ = 1.5 mm, r₂ = 2.4 mm, r = 1.5 mm, m₀ = 0.9 mm, h₁ = 1 mm, and h₂ = 4.85 mm, respectively. The Kekulé distortion is introduced by displacing the dielectric rods by a distance m₀ and a rotating angle φ. c Simulated photonic bulk band structures without (green lines) and with (gray lines) Kekulé distortion, respectively. The orange region represents the 3D topological photonic bandgap. d 3D topological photonic bandgaps with different angles φ and fixed displacement distance m₀ = 0.06a. The color represents the angle φ, varying from 0 to 2π. e Schematic of the modulation angle φ of the aperiodic Kekulé distortion with a winding number of +1. f Simulated dispersion of the photonic topological Dirac-vortex modes (red lines) along the k_z direction, the orange region represents the minimum Dirac-vortex bandgap. g Simulated electric field distribution of the photonic topological Dirac-vortex eigenmodes at k_z = 1.5π/a_z marked by the black dot in f. h Simulated electric field distribution of the photonic topological Dirac-vortex modes excited by a point source (green star) at 18.7 GHz.

We then implement aperiodic Kekulé distortion to a 3D tight-binding-like metal-cage photonic crystals to open a varying vortex bandgap by defining a modulation vector $\mathbf{m}=m_0{e}^{jw\phi \left(\mathbf{r}\right)}$ (Dirac mass) with a winding number w = 1 and position-dependent phase φ(r) = arg(r) varying continuously from 0 to 2π, as schematically shown in Fig. 2e. Figure 2f presents the simulated dispersion of the topological Dirac-vortex mode (red lines) along the k_z direction within the minimum vortex bandgap ranging from 18.5 to 19 GHz (orange region). Note that the bandwidth of the topological Dirac-vortex modes can be tuned by changing the interlayer couplings, which are determined by the radius of the air holes and the thickness of the perforated copper plates. Figure 2g shows the E_z field distribution of the topological Dirac-vortex eigenmode marked by a black dot in Fig. 2f, from which we can see that the topological Dirac-vortex mode is tightly localized around the vortex core. It is worth noting that the number of topological Dirac-vortex modes is determined by the Dirac-mass winding number w, and multiple topological Dirac-vortex modes, having nearly identical group and phase velocities, can exist simultaneously with large Dirac-mass winding numbers (see Supplementary Note 6 for details). We employ a point dipole source (green star) to excite the topological Dirac-vortex modes in a 3D Kekulé-distorted photonic topological insulator whose top and bottom boundaries are perfectly matched layers (PML) to eliminate reflection. As shown in Fig. 2h, the topological Dirac-vortex modes are bound to and propagate bidirectionally along the 1D vortex line defect. Note that the spin-locking topological Dirac-vortex modes can propagate unidirectionally if excited with a source array (see Supplementary Note 7).

Experimental observation of topological Dirac-vortex modes in a 3D photonic topological insulator

Next, we experimentally demonstrate the topological Dirac-vortex modes in a 3D tight-binding-like metal-cage photonic crystal. The fabricated experimental sample is shown in Fig. 3a, which consists of forty layers (20 unit cells) of perforated copper plates with air holes and perforated air foams inserted with dielectric and metallic rods. The large and small air holes in the perforated copper plates are used to introduce interlayer couplings, and a probe antenna is inserted into the small air holes to map the topological Dirac-vortex modes. Figure 3b, c shows the top view of a perforated air foam inserted with dielectric and metallic rods. The top and bottom of the sample are wrapped with microwave absorption materials to eliminate the reflection. We first measure the transmission spectra of the topological Dirac-vortex modes (red line) and bulk states (gray line) by placing a point source antenna (green star) at the center of the sample and inserting a probe antenna into the top center (red star) or boundary (gray star) of the sample, as shown in Fig. 3d, in which we can see that the transmission of the topological Dirac-vortex modes (red line) within the bulk bandgap (orange region) is much higher than that of the bulk states (gray line). To directly observe the topological Dirac-vortex modes, we insert a probe antenna into the small air holes one by one to map the E_z field distribution of the topological Dirac-vortex modes at 18.7 GHz, as shown in Fig. 3e. The topological Dirac-vortex modes are bound to and propagate vertically along the 1D vortex line defect, agreeing well with the simulation results shown in Fig. 2h and unambiguously verifying the existence of topological Dirac-vortex modes in the 3D Kekulé-distorted photonic topological insulators. Note that the attenuation of topological Dirac-vortex modes in the experiment (Fig. 3e) is attributed to the fabrication errors and material absorption losses. The unique propagation characteristic of the topological Dirac-vortex modes can also be revealed by the measured electric field distributions as a function of the excitation frequencies and the z coordinates of the probe antenna, as shown in Fig. 3f. For the frequency range of 18.5–19 GHz within the bandgap, the topological Dirac-vortex modes can propagate upward and downward simultaneously, indicating broadband propagation of topological Dirac-vortex modes along the vortex line defect in the bandgap. Meanwhile, the vanished upward and downward transmissions outside the bandgap indicate the absence of topological Dirac-vortex modes along the vortex line defect. Moreover, by Fourier-transforming the complex electric field distributions from real space to reciprocal space, we can obtain the measured dispersion (color map) of the topological Dirac-vortex modes, as shown in Fig. 3g, which matches well with the simulation results (cyan solid lines).

Fig. 3. Experimental observation of topological Dirac-vortex modes in a 3D photonic topological insulator.

a Photograph of the fabricated experimental sample consisting of 40 layers of perforated copper plates and air foams inserted with dielectric and metallic rods. b Photograph of a perforated air foam inserted with metallic (copper part) and dielectric (black circles) rods. c Magnified image of the perforated air foam. d Measured transmission spectra of the bulk (gray line) and topological Dirac-vortex (red line) modes, the orange region represents the 3D topological photonic bandgap. e Measured electric field distribution of the topological Dirac-vortex mode excited by a point source (green star) at 18.7 GHz. The red and gray stars represent the probe antennas. f Measured electric field distributions as a function of the excitation frequencies and the z coordinates of the probe antenna. g Measured (color map) and simulated (cyan solid lines) dispersions of the topological Dirac-vortex modes.

Finally, we examine the robustness of the topological Dirac-vortex modes. We introduce local defects (green dashed square) in the upper center of the vortex line defect by either removing the central six dielectric rods or replacing them with six metallic rods (blue circles), as shown in Fig. 4a, b, respectively. We first measure the transmission spectra of the bulk states without (light gray line) and with (dark gray line) local defects, as shown in Fig. 4c, d, from which we can see they are almost the same and exhibit transmission dips matching well with the simulated photonic bandgap (orange region). We then measure the transmission spectra of the topological Dirac-vortex modes with (blue line) and without (red line) local defects. We can see that they almost overlap with each other, indicating the robustness of the topological Dirac-vortex modes despite the presence of various defects. We repeat the near-field imaging measurements to directly observe the propagation of the topological Dirac-vortex modes in the presence of local defects, as shown in Fig. 4e, f, respectively. The topological Dirac-vortex modes can circumvent the defects (green dashed square) and continue to propagate along the 1D vortex line defect. For comparison, we present the simulated electric distributions of the topological Dirac-vortex modes with defects, as shown in Fig. 4g, h, which agree well with the experimental results. Besides these two local defects, topological Dirac-vortex modes exhibit robustness against more complex defects, such as interlayer coupling perturbation and structural damage (see Supplementary Note 8). More interestingly, alternating interlayer couplings will hybridize the upward and downward spin-locking topological Dirac-vortex modes and generate exotic third-order face-centered modes at the top and bottom surface of the 3D Kekulé-distorted photonic topological insulator⁴⁴ (see Supplementary Note 9). Since topological Dirac-vortex modes are bound to and propagate along a 1D Dirac-vortex line defect inside a 3D photonic crystal with unique properties such as tunable mode areas, arbitrary mode degeneracies, vectorial beam emission, and large free spectral range, they can be used to design topological waveguides, fibers, lasers, and other photonic devices with good performance and specific functions. The main challenge to extend the 1D topological Dirac-vortex modes into the optical frequency range is the metallic absorption loss, since current designs compose metallic structures to realize the confined Mie resonance. To address this challenge, we predict that point defect cavities in 3D all-dielectric photonic crystals can be utilized as the confined Mie resonances to realize the 1D topological Dirac-vortex modes in a 3D all-dielectric photonic topological insulator.

Fig. 4. Robustness of the photonic topological Dirac-vortex modes.

Photographs of the local defects by removing six dielectric rods (a) or replacing them with six metallic rods (b) at the upper center of the vortex line defect. c, d Measured transmission spectra of the topological Dirac vortex modes with (blue lines) and without (red lines) local defects in (a) and (b), respectively. The light (dark) gray line represents the measured transmission spectrum of the bulk states without (with) the local defects. Measured electric field distributions of the topological Dirac-vortex modes with six dielectric rods removed (e) or replaced by six metallic rods (f) at 18.7 GHz, respectively. g, h Simulated electric field distributions of the topological Dirac-vortex modes with six dielectric rods removed (e) or replaced by six metallic rods (f) at 18.7 GHz, respectively. The green star and green dashed square represent the point source and local defects, respectively.

Discussion

In conclusion, by directly emulating a 3D Kekulé-distorted tight-binding model in a 3D tight-binding-like metal-cage photonic crystal, we have theoretically proposed and experimentally demonstrated 1D topological Dirac-vortex modes in a 3D photonic topological insulator for the first time. We also experimentally observed that the photonic topological Dirac-vortex modes are robust against various defects or obstacles, making them suitable for robust manipulation of electromagnetic waves in 3D space. Moreover, we show that the 3D tight-binding-like metal-cage photonic crystals exhibit scalar-wave-like band dispersions resembling those of the tight-binding models, making the experimental realization of 3D photonic topological phases an easy task. We envision that other topological defects, such as dislocation, disclination, and monopole topological modes, can be readily realized in the 3D tight-binding-like metal-cage photonic crystals.

Methods

Numerical simulations

All numerical results presented in this work are simulated using the RF module of COMSOL Multiphysics. The bulk band structures are calculated using a unit cell with periodic boundary conditions in all directions. The perforated copper plates and metallic rods are modeled as perfect electric conductors (PEC) in the simulation. The dispersion of the topological Dirac-vortex modes is calculated by adopting a hexagonal supercell and applying periodic boundary conditions along the z direction and open boundary conditions along the x and y directions. In the full-wave simulations of a finite 3D tight-binding-like photonic crystal, all boundaries are set as open boundary conditions.

Materials and experimental setups

The copper plates are fabricated by depositing a 0.035-mm-thick layer of copper onto a Teflon woven-glass fabric laminate. We use perforated air foam (ROHACELL 31 HF with a relative permittivity of 1.04 and a loss tangent of 0.0025) to fix the metallic and dielectric rods. In the experimental measurements, the amplitude and phase of the electric fields are measured using a vector network analyzer (Keysight E5080) connected by two electric dipole antennas serving as the source and probe, respectively. To excite the topological Dirac-vortex modes, a point source antenna is placed at the center of the sample, and a probe antenna is inserted into the air holes one by one to scan the electric fields.

Author contributions

Z.G. conceived the idea and supervised the project. B.Y., Y.F.Q., Z.Y.W. and X.X. performed the simulations. B.Y., Y.F.Q., Z.Y.W., X.X. and Z.G. designed the experiments. B.Y., Z.G., Z.Y.W., Y.F.Q., X.X., L.Y., Y.M., Z.X.Z., Y.X.Z., M.Q.C. and J.M.C. fabricated samples. B.Y., Y.F.Q. and Z.Y.W. carried out the measurements. B.Y., Y.F.Q., Z.Y.W., X.X. and Z.G. analyzed data. B.Y. drafted and Z.G. revised the manuscript.

Peer review

Peer review information

Nature Communications thanks Junsuk Rho and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All data are available in the manuscript, the supplementary materials, or have been deposited in the Figshare database at 10.6084/m9.figshare.29163452.

Code availability

We use commercial software COMSOL Multiphysics to perform electromagnetic numerical simulations. Requests for computation details can be addressed to the corresponding authors.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-61238-7.