Type-II Dirac Photons at Metasurfaces

Topological characteristics of energy bands, such as Dirac and Weyl nodes, have attracted substantial interest in condensed matter systems as well as in classical wave systems. Among these energy bands, the type-II Dirac point is a nodal degeneracy with tilted conical dispersion, leading to a peculiar crossing dispersion in the constant-energy plane. Such nodal points have recently been found in electronic materials. The analogous topological feature in photonic systems remains a theoretical curiosity, with experimental realization expected to be challenging. Here, we experimentally realize the type-II Dirac point using a planar metasurface architecture, where the band degeneracy point is protected by the underlying mirror symmetry of the metasurface. Gapless edge modes are found and measured at the boundary between the different domains of the symmetry-broken metasurface. Our Letter shows that metasurfaces are simple and practical platforms for realizing electromagnetic type-II Dirac points, and their planar structure is a distinct advantage that facilitates applications in two-dimensional topological photonics.

Figure 1

(a) The sketch of a single unit of the designed metasurface with the following parameters: $D_x=D_y=2.5\mathrm{mm}$, $a=2.3\mathrm{mm}$, $b_1=b_2=0.9\mathrm{mm}$, $c=0.8\mathrm{mm}$, $w_1=0.1\mathrm{mm}$, $w_2=0.2\mathrm{mm}$, $d=2\mathrm{mm}$, $t=18\mu \mathrm{m}$. (b) The top view of the designed sample with the inset showing a photo of part of the fabricated sample.

Figure 2

(a) Electric field maps of the $E_z$ component of the calculated two modes of the single unit of the designed metasurface. The eigenfrequencies of the electric dipole modes with a dipole moment along the $x$ direction ($\mathrm{EDM}x$) and $y$ direction ($\mathrm{EDM}y$) are 8.52 and 16.12 GHz, respectively. (b) The calculated dispersion relations along the $\mathrm{\Gamma }\text{−}X$ direction, with the red and black solid lines denoting $\mathrm{EDM}x$ and $\mathrm{EDM}y$, respectively. The blue dashed line is the light line, and the inset shows the first Brillouin zone and high-symmetrical points. (c) 3D view around the DP (red point); the transparent blue surface is the equal-energy surface at the degenerate frequency, with the IFC cut by the surface plotted at the bottom (a cross shape formed by blue and orange lines). (d) Dispersion relation around the DP along the $\mathrm{\Gamma }\text{−}X$ direction, obtained via simulations (red and black open circles) and the $\mathbf{k}·\mathbf{p}$ method (red and black solid lines).

Figure 3

IFCs in the first Brillouin zone obtained via (a) simulations and (b) experiments. The high-symmetrical points are indicated on the top of the figure. The type-II DPs are seen along the $\mathrm{\Gamma }\text{−}X$ direction at the 10.6 GHz IFC.

Figure 4

The top view of a single unit of (a) type-$A$ Jerusalem cross with $b_1=0.3\mathrm{mm}$, $b_2=1.5\mathrm{mm}$ and type-$B$ Jerusalem cross with $b_1=1.5\mathrm{mm}$, $b_2=0.3\mathrm{mm}$, respectively. (b) The dispersion relation of the first two bands along the $\mathrm{\Gamma }\text{−}\mathrm{X}({\mathrm{X}}^{\prime })$ directions. (c) Electric field maps of the $E_z$ component of the first (band 1) and second (band 2) bands at $k_x=0.5$ ($\pi /D_x$). (d) The Berry curvature for the first band of type-$A$ Jerusalem cross on the first Brillouin zone obtained via comsol.

Figure 5

The top view of the designed (a) $AB$-type stack, (d) $BA$-type stack samples. (Insets) The corresponding enlarged view of the fabricated sample at the domain walls. (b),(e) The project band diagrams along the $x$ direction for the $AB$- and $BA$-type stacks, respectively. The regions shaded in gray denote the bulk states, and the blue dashed lines denote the light lines along the $\mathrm{\Gamma }\text{−}\mathrm{X}({\mathrm{X}}^{\prime })$ directions; the blue solid lines in (b) and the red solid lines in (e) show the TISs, while the black lines in (b) show the NTISs. (c),(f) The projected bands for the $AB$- and $BA$-type stacks obtained from experiment, respectively.

Figure 6

Simulated field distribution of the $E_z$ component and Poynting vector profiles for $AB$- and $BA$-type domain walls at opposite wave vectors on the $x\text{−}y$ plane at a distance of 1 mm above the surface.