Observation of Three-Dimensional Photonic Dirac Points and Spin-Polarized Surface Arcs

Three-dimensional (3D) Dirac points inheriting relativistic effects from high-energy physics appear as gapless excitations in the topological band theory. Hosting fourfold linear dispersion, they play the central role among various topological phases, such as representing the degeneracy of paired Weyl nodes carrying opposite chiralities. While they have been extensively investigated in solid state systems for electrons, 3D Dirac points have not yet been observed in any classical systems. Here, we experimentally demonstrate 3D photonic Dirac points in the microwave region with an elaborately designed metamaterial, where two symmetrically placed Dirac points are stabilized by electromagnetic duality symmetry. Furthermore, spin-polarized surface arcs (counterparts of Fermi arcs in electronic systems) are demonstrated, which opens the gate toward implementing spin-multiplexed topological surface wave propagation. Closely linked to other exotic states through topological phase transitions, our system offers an effective medium platform for topological photonics.

Figure 1

Design of three-dimensional photonic Dirac metamaterial. (a) Schematic of the metamaterial structure. Each unit cell consists of eight helical elements. Along the $z$ direction, there are two sets of helical elements related by mirror symmetry indicated by $M$ (four left-handed and four right-handed helical elements). In the $x\text{−}y$ plane, the unit cell possesses $C_4$ rotation symmetry along $z$ to ensure the degeneracy between the two circularly polarized transverse states. (b) Sample fabricated with printed circuit board technology. The helical elements are etched in the 1-mm-thick hosting material (dielectric constant of 2.2). Between two structured layers there is a 3-mm-thick spacer with the same dielectric constant. The in-plane period (marked as the white dashed squares) is $6.8\times 6.8{\mathrm{mm}}^2$. The period along $z$ is 8 mm. (c) Sample configuration for the experimental measurement. The sample is stacked layer by layer to form a three-dimensional bulk metamaterial. In the measurement, two antennas are used, with one working as the source (magenta) and the other working as probe (cyan). (d) The full wave simulation shows the presence of two fourfold degenerate Dirac points at the same frequency around 7.92 GHz. (e) The simulated dispersion along the $k_z$ directions around the Dirac points. (f) Similar to (e) but along $k_y$.

Figure 2

Observation of three-dimensional photonic Dirac points. (a),(c) Experimentally mapped band projection around the Dirac point along the $k_z$ and $k_y$ direction, respectively. (b),(d) The corresponding simulation results from CST microwave studio. Blue and magenta lines indicate light cone and surface states, respectively.

Figure 3

Spin-polarized photonic surface arcs and surface waves. (a)–(c) Experimentally measured surface state arcs at the frequency 7.8 GHz (below Dirac point), 7.92 GHz (at Dirac point), and 8.1 GHz (above Dirac point), respectively. (d)–(f) The corresponding simulated bulk and surface states, where cyan and magenta curves indicate opposite spinful surface states from CST microwave studio. Black solid lines indicate simulated bulk states. (g) Measured real space surface wave propagation excited by left circular polarization. (h) Spin-polarized surface states in momentum space after Fourier transformation from (g). (i),(j) Similar to (g),(h) but excited by right circular polarization. In each panel, the dashed circle indicates the light cone.

Figure 4

Illustration of the photonic topological phase transition on the metamaterial (effective medium) platform. Adding a bianisotropic term in the $y\text{−}z$ plane transforms the Dirac metamaterial (a) into a photonic topological insulator (b), where red dashed lines indicate gapless interface states across the gap. Breaking either inversion symmetry $P$ or time-reversal symmetry $T$ splits the Dirac points (a) into two transient phases (c) and (d), where each triple degenerate point (TDP) consists of two longitudinal modes and one circularly polarized transverse mode. After shifting the two longitudinal modes (shift $L$) away from each other, (c) and (d) transform into (g) and (h), respectively. The TDPs in (e) and (f) are topologically neutral. By maintaining only the $C2$ rotational symmetry along the $z$ direction, i.e., breaking the $C4$ rotational symmetry in (e), or breaking the degeneracy of the two longitudinal modes in (f), they can transform into nodal lines (i). Hollow square, triangle, and ellipsoid represent fourfold, triple, and double degeneracy, respectively. Colorful (black) symbols indicate chiral (neutral) gapless excitations. The gray- (black)-dashed, red (blue), and cyan (magenta) lines indicate mutually orthogonal: electric- (magnetic)-longitudinal [$E(L)/H(L)$], left (right) circularly polarized [LCP  (RCP)] transverse- and linear-polarized [TE (TM)] transverse modes, respectively. Corresponding hypothetical constitutive relation examples for panels (a)–(i) are given in Table S1.