Abstract
Confining photons in a finite volume is highly desirable in modern photonic devices, such as waveguides, lasers and cavities. Decades ago, this motivated the study and application of photonic crystals, which have a photonic bandgap that forbids light propagation in all directions1,2,3. Recently, inspired by the discoveries of topological insulators4,5, the confinement of photons with topological protection has been demonstrated in two-dimensional (2D) photonic structures known as photonic topological insulators6,7,8, with promising applications in topological lasers9,10 and robust optical delay lines11. However, a fully three-dimensional (3D) topological photonic bandgap has not been achieved. Here we experimentally demonstrate a 3D photonic topological insulator with an extremely wide (more than 25 per cent bandwidth) 3D topological bandgap. The composite material (metallic patterns on printed circuit boards) consists of split-ring resonators (classical electromagnetic artificial atoms) with strong magneto-electric coupling and behaves like a ‘weak’ topological insulator (that is, with an even number of surface Dirac cones), or a stack of 2D quantum spin Hall insulators. Using direct field measurements, we map out both the gapped bulk band structure and the Dirac-like dispersion of the photonic surface states, and demonstrate robust photonic propagation along a non-planar surface. Our work extends the family of 3D topological insulators from fermions to bosons and paves the way for applications in topological photonic cavities, circuits and lasers in 3D geometries.
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Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
We thank Q. Yan at Zhejiang University, L. Lu at the Chinese Academy of Sciences and J. C. W. Song at Nanyang Technological University for discussions. The work at Zhejiang University was sponsored by the National Natural Science Foundation of China under grant numbers 61625502, 61574127, 61601408, 61775193 and 11704332, the ZJNSF under grant number LY17F010008, the Top-Notch Young Talents Program of China, the Fundamental Research Funds for the Central Universities and the Innovation Joint Research Center for Cyber-Physical-Society System. Y.C. and B.Z. acknowledge the support of Singapore Ministry of Education under grant numbers MOE2015-T2-1-070, MOE2015-T2-2-008, MOE2016-T3-1-006 and Tier 1 RG174/16 (S). Y.Y. and R.S. acknowledge the support of the Singapore Ministry of Education under grant number MOE2015-T2-2-103.
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Nature thanks J. Bravo-Abad and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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Contributions
Y.Y. initiated the original idea. Y.Y., B.Z. and H.C. designed the experiment. Y.Y., Z.G., M.H. and L.Z. fabricated samples. Y.Y. and Z.G. carried out the measurement and analysed data. Y.Y., H.X., L.Z. and Z.Y. performed simulations. Y.Y., H.X., B.Z., M.H., Z.Y., H.C. and Y.C. provided the theoretical explanations. R.S. assisted in part of the experiment. Y.Y., Y.C., B.Z. and H.C. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Bandstructure evolutions.
a, Hexagonal unit cell and band diagram of the 3D photonic crystal. The green and yellow dots indicate the 3D Dirac points. The plots on the right show the 2D projection of the 3D Dirac cones in the vicinity of the K and K′ points. b–g, Hexagonal unit cell and bandstructures for different l1. Here, a is the lattice constant in the x–y plane and pz is the periodicity in the z direction; the ratio of ttop to l1 remains unchanged, while l1 is gradually compressed to zero. The blue regions represent the first (primary) bandgaps.
Extended Data Fig. 2 Modal analysis.
a, Modal analysis for 3D Dirac points. Current distributions of four degenerate modes near the 3D Dirac point marked by the green dot in Extended Data Fig. 1a. The dashed arrows represent the current directions, and e+ (e−) and m+ (m−) represent the even (odd) transverse-electric and transverse-magnetic modes, respectively. Here, the even and odd modes are classified by the mirror plane indicated by the red dotted lines. b, Modal analysis for the 3D photonic topological insulator with a small perturbation. The first row shows the current distributions of the hybrid modes at the lower bands (green dot in Extended Data Fig. 1b) and upper bands (yellow dot in Extended Data Fig. 1b) near the K valley. The dashed arrows represent the current directions. The previous four degenerate modes hybridize pairwise and split into the lower and upper bands, respectively. The second and third rows show the cross-sectional polarization configurations of electric (red) and magnetic (black) fields, near the K and K′ valleys. The phase difference between electric and magnetic dipole components is 0 or π. c, Modal analysis for the wide-gap 3D photonic topological insulator. The first row shows the current distributions of the lower-band modes (green dot in Extended Data Fig. 1g) near the K valley. The dashed arrows represent the current directions. The second and third rows show the cross-sectional polarization configurations of electric (red) and magnetic (black) fields, near the K and K′ valleys. The phase difference between electric and magnetic dipole components is 0 or π.
Extended Data Fig. 3 Spin–momentum locking of topological surface states.
a, Schematic of the domain wall. The openings of the SRRs in the left and right of the domains are opposite. b, Spin–momentum locking at an isofrequency (4.8-GHz) contour of the surface Dirac cone. On the left is a dispersion diagram of the surface Dirac cone. The black ring indicates the isofrequency contour and the red arrows illustrate the spin–momentum locking at 4.8 GHz. On the right are schematic and numerical results of the polarization configurations of electric (red) and magnetic (black) fields inside the cross-sections of the SRRs (triangles), at eight points marked on the isofrequency contour by green dots and numbers. The phase difference between the electric and magnetic components, Δϕ, varies from 0 to 2π along the contour. The black ring and red dot represent the isofrequency contour and surface Dirac point, respectively.
Extended Data Fig. 4 Simulated field distributions near a sharply twisted 2D domain wall.
a, Schematic of the twisted domain wall: the red and green triangles are SRRs oriented upwards and downwards, respectively. b–f, Distributions of the electric field intensity across the domain wall, with different values of kz.
Extended Data Fig. 5 Measured field distributions and the corresponding dispersion.
a, Measured electric field distributions in the sample of a straight domain wall in the x–y plane at different heights of z at 4.7 GHz. The black dashed line represents the position of the domain wall; the green dashed line denotes the position of the cross-section shown in b. b, Field distributions and normalized electric energy density in the y–z plane with x = 150 mm. The dots and dashed line represent experimental data and exponential fitting, respectively. c, Surface dispersion in momentum space, obtained by Fourier transforming the real-space complex field distributions on the domain wall.
Supplementary information
Supplementary Information
This file contains Supplementary Text sections S1–S5, Supplementary Figures 1–3 and Supplementary References. The Supplementary Text includes the derivation of the effective Hamiltonian at K (K′) valley of 3D photonic TI with k·p theory, Berry curvatures, a brief introduction of bi-anisotropy, theoretical derivation of spin–momentum locking and details of the twisted domain wall measurement
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Yang, Y., Gao, Z., Xue, H. et al. Realization of a three-dimensional photonic topological insulator. Nature 565, 622–626 (2019). https://doi.org/10.1038/s41586-018-0829-0
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DOI: https://doi.org/10.1038/s41586-018-0829-0
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