Three Dimensional Photonic Dirac Points in Metamaterials

Topological semimetals, representing a new topological phase that lacks a full band gap in bulk states and exhibiting nontrivial topological orders, recently have been extended to photonic systems, predominantly in photonic crystals and to a lesser extent metamaterials. Photonic crystal realizations of Dirac degeneracies are protected by various space symmetries, where Bloch modes span the spin and orbital subspaces. Here, we theoretically show that Dirac points can also be realized in effective media through the intrinsic degrees of freedom in electromagnetism under electromagnetic duality. A pair of spin-polarized Fermi-arc-like surface states is observed at the interface between air and the Dirac metamaterials. Furthermore, eigenreflection fields show the decoupling process from a Dirac point to two Weyl points. We also find the topological correlation between a Dirac point and vortex or vector beams in classical photonics. The experimental feasibility of our scheme is demonstrated by designing a realistic metamaterial structure. The theoretical proposal of the photonic Dirac point lays the foundation for unveiling the connection between intrinsic physics and global topology in electromagnetism.

Figure 1

Band structure of bulk states with $\alpha =\varsigma =3$. (a) Effective bulk band structure on the $k_y\text{−}{k}_z$ plane with the Dirac points marked by blue spheres. (b) The dispersion relation along the $k_y$ and $k_z$ direction, respectively, while (c) gives the dispersion along the $k_y$ direction with $k_z$ fixed at the Dirac point. In both (b) and (c), the Dirac points are marked with the blue points. (d) Equifrequency contour at a frequency below the Dirac points, which indicates the hyperbolic property of the material corresponds to the Dirac cone.

Figure 2

Reflection spectrum around a Dirac point. (a) Configuration of the reflection calculation and Dirac points (two blue dots) in momentum space. One plane wave is incident $({\overrightarrow{k}}_i)$ from the backgroud material $x>0$ and totally reflected $({\overrightarrow{k}}_r)$ back at the interface as indicated. (b) and (c) show the reflected $E$ field with respect to right- or left-circular polarization incidence, respectively. (d) and (e) are similar to (b) and (c) with the incidences of two orthogonal linear polarizations. The incident polarization state is illustrated on the bottom of each panel.

Figure 3

Bulk and surface states of the effective medium. (a) 3D band structures of bulk and surface states of the effective medium. The spin-up (spin-down) surface state between two Dirac points and the vacuum is indicated by the magenta (cyan) surface. The blue and red lines highlight the photonic Fermi arcs at Dirac points. Equifrequency contours at two different frequencies could be seen in (b) and (c) corresponding to the frequency at the Dirac point and below it, respectively. The red and blue lines in (b) and (c) represent the spin-up and spin-down topological surface states, while the black lines represent the effective material and vacuum bulk states.

Figure 4

Surface states indicated by power flow simulated in 3D by CST time domain. The (a) spin-up and (b) spin-down surface states propagate helically along the $z$ direction clockwise and anticlockwise, respectively, on the surface of cuboid effective material which is capsulated by a vacuum.

Figure 5

Dirac points with a realistic metamaterial structure. (a) The metamaterial structure with a top view of one unit cell composed with $C_4$ symmetry distributed helix pairs and the front view of a pair of mirrored helixes with opposite rotation directions with parameters $R=20\mathrm{mm}$, $p_x=p_y=P=90\mathrm{mm}$, $p_z=50\mathrm{mm}$, $g=25\mathrm{mm}$, and $h=20\mathrm{mm}$. (b) and (c) are the simulated band structure of the metamaterial along the $k_z$ and $k_y$ direction, respectively. (d) Surface state dispersion at the Dirac point frequency (0.695 GHz). Frequency and wavevector are in units of GHz and degree, respectively.