Three-dimensional photonic Dirac points stabilized by point group symmetry

We discover a pair of stable three-dimensional (3D) Dirac points, a 3D photonic analog of graphene, in all-dielectric photonic crystals using structures commensurate with nanofabrication for visible-frequency photonic applications. The Dirac points carry nontrivial $Z_2$ topology and emerge for a large range of material parameters in hollow cylinder hexagonal photonic crystals. From Kramers theorem and group theory, we find that only the $C_6$ symmetry leads to point group symmetry stabilized Dirac points in 3D all-dielectric photonic crystals. The Dirac points are characterized using $\vec{k}·\vec{P}$ theory for photonic bands in combination with symmetry analysis. Breaking inversion symmetry splits the Dirac points into Weyl points. The physical properties and experimental consequences of Dirac points are also studied. The Dirac points are found to be robust against parameter tuning and weak disorders.

Figure 1

(a) Structure in real-space unit cell of hexagonal PCs. Hollow cylinders and micropillars are made of the same material with isotropic permittivity. (b) Top-down view of hexagonal PC. ${\vec{a}}_1$ and ${\vec{a}}_2$ are the two lattice vectors. $a\equiv |{\vec{a}}_1|=|{\vec{a}}_2|$ is the lattice constant in the $x\text{−}y$ plane. The height of each unit cell is $h=0.6a$. The diameter of each micropillar is $0.1a$ and its height is $0.2a$. The outer and inner radii of the hollow cylinder are $R_{\mathrm{out}}$ and $R_{\mathrm{in}}$, respectively, whereas its height is $0.4a$. (c) First Brillouin zone with a pair of DPs along the $\mathrm{\Gamma }\text{−}A$ line as kinks of the $Z_2$ number vs $k_z$. A DP with $n_D=1$ $\left(-1\right)$ is labeled as a red (blue) point. (d) Dispersion close to a DP.

Figure 2

(a) Photonic band structure of a hexagonal PC with $C_6$ and inversion symmetry (blue curve indicating light line). Zoom in: Band structure along $\mathrm{\Gamma }\text{−}A$ line. The $p$ bands (red) cross the $d$ bands (green) at $(0,0,\pm K_z)$ with $K_z=0.338\frac{2\pi c}{h}$. The gray curves represent the $f$ bands. (b),(c) $E_z$ field profile in the $x\text{−}y$ plane of (b) $d$ bands and (c) $p$ bands at $\mathrm{\Gamma }$ point. A unit cell is depicted by the black dashed curves. Parameters: $R_{\mathrm{out}}/a=0.5$ and $R_{\mathrm{in}}/a=0.4$, and permittivity $\varepsilon =12$.

Figure 3

IS broken hexagonal PC with $C_6$ symmetry. (a) Lateral (left) and top-down (right) views of the structure in a real-space unit cell. (b) Photonic bands along the $\mathrm{\Gamma }\text{−}A$ line with $k_z>0$. Two WPs are found at (0, 0, $K_{z1})$ and (0, 0, $K_{z2})$ with $K_{z1}=0.2\frac{2\pi c}{h}$ and $K_{z2}=0.26\frac{2\pi c}{h}$, respectively. The other two WPs in the $k_z<0$ region are not shown. (c) Depicting the four WPs in the first Brillouin zone. Green (yellow) spheres denote WPs with chirality $-1$ (+1). (d) Chern number $N_c$ for the $p_{+}-d_{+}$ bands below the WPs (blue) and for the $p_{-}-d_{-}$ bands below the WPs (red). (e) Photonic spectrum near the two WPs. Left: for WP at (0, 0, $K_{z1})$; right: for WP at (0, 0, $K_{z2})$.

Figure 4

(a) Angular-momentum distribution on a sphere surrounding the WP at $(0,0,-0.2\frac{2\pi c}{h})$ for an IS broken PC (same parameters as in Fig. 3); calculated for the band below the WP. (b) Measurement setup for backscattering suppression in a PC with DPs. If the small defect at the center of the PC preserves $C_6$ symmetry, the backscattering is suppressed; otherwise, the backscattering considerably reduces transmission. (c),(d) Phase diagrams of hollow-cylinder hexagonal PCs with (c) and without (d) IS for various $R_{\mathrm{out}}$ and $R_{\mathrm{in}}$.

Figure 5

Photonic dispersion in $k_x\text{−}{k}_y$ plane around the $(0,0,k_z)$ point for (a) $k_z<K_z$, (b) $k_z=K_z$, and (c) $k_z>K_z$. There are two nearly degenerate bands above and below the Dirac point. The arrows indicate the orbit angular momentum distribution for the pseudospin up bands. Note that the orbit angular momentum has been subtracted by a universal constant $\frac{3}{2}\hslash$ (i.e., the average orbit angular momentum between the $p_{+}$ and $d_{+})$ state.

Figure 6

Mie resonances of infinitely long hollow cylinder. (a) The scattering cross section as a function of the frequency of the incident plane wave with TM polarization $(E_z$ polarization, $z$ direction is along the cylinder). (b) The scattering cross section vs the frequency of the incident plane wave with TE polarization $(H_z$ polarization). (c)–(e) The electric-field patterns $|E_z|$ at the first, second, and third resonances for the TM polarization, respectively. They are identified as the $p$-, $d$-, and $f$-wave scatterings. (f)–(h) The electric-field $|E_z|$ patterns in the $x\text{−}y$ plane in a unit cell (averaged over the $z$ direction) for the $p,d$, and $f$ states in the hollow cylinder photonic crystal, respectively.

Figure 7

Photonic band structure of hexagonal photonic crystal without micropillars. The height of the unit cell is $0.6a$. The green curve represents the $p$ bands, whereas the red curve is for the $d$ bands. The blue curve denotes the light line. The height of the hollow cylinder is $0.4a$ (there are no micropillars). Other parameters: $R_{\mathrm{out}}/a=0.5,R_{\mathrm{in}}/a=0.4$, and $\varepsilon =12$.

Figure 8

(a) Supercell structure for the calculation of the edge states. In the middle is the photonic crystal with micropillars twisted clockwise (represented by blue circles). In the other regions are the photonic crystal with anticlockwise twisted micropillars (depicted as gray circles). The structure of the two types of photonic crystals are shown above the supercell. For a given supercell we calculate the photonic band structure with a given $k_z$ for various $k_y$ $(k_x=0$ as the system has finite size along the $x$ direction). (b) Chern numbers for the $p_{+}$ and $d_{+}$ bands (green) and for the $p_{-}$ and $d_{-}$ bands (yellow) below the Weyl points as functions of $k_z$ for the photonic crystal with micropillars twisted clockwise. (c) Chern numbers vs $k_z$ for the photonic crystal with anticlockwise twisted micropillars. (d) Total Chern number difference $\mathrm{\Delta }{N}_C=N_C\left(\mathrm{clockwise}\right)-N_C\left(\mathrm{anticlockwise}\right)$ as a function of $k_z$.

Figure 9

(a) The supercell structures calculated. Top: the heterostructure with two types (clockwise and anticlockwise twisted micropillars) of photonic crystals. Bottom: the structure with only one type of photonic crystal (the one with anticlockwise twisted micropillars). (b) The photonic spectrum with $k_z=0$ for various $k_y$. The blue curves give the spectrum for the heterostructure, while the gray curves give the spectrum for the supercell with only one type of photonic crystal [the same color scheme is applied to (c) and (d), as well]. The gray curves are on top of the blue curves for (b)–(d). The projected bulk photonic bands are depicted by the red and black points for the upper and lower band edges, respectively. (c),(d) The photonic spectrum for the heterostructure and uniform structure with $k_z=0.225\frac{2\pi }{h}$ and $k_z=0.3\frac{2\pi }{h}$, respectively. There are four edge states in (c). We label four special points a, b, c, and d, with $k_y=\pm 0.2\frac{2\pi }{a}$. The properties of these four points will be shown in Fig. 10.

Figure 10

Electromagnetic energy density $\int dz\varepsilon \left(\vec{r}\right){\left|\vec{E}\left(\vec{r}\right)\right|}^2$ in the $x\text{−}y$ plane for the a, b, c, and d points in the photonic spectrum in Fig. 9.

Figure 11

Spatial field patterns on the surfaces of a block that contains a unit cell for the $p_{\pm }$ and $d_{\pm }$ bands at $\vec{k}=(0,0,0.1\frac{2\pi }{h})$ in chiral photonic crystal. In each figure, on the three boundary planes (i.e., the $x\text{−}y,y\text{−}z$, and $x\text{−}z$ planes) we plot the energy density $\varepsilon |\vec{E}{|}^2\left(\vec{r}\right)$ by color. The red color represents high energy density, while the blue denote low energy density. The Poynting vector (the time-averaged value $\text{Re}[{\vec{E}}^{*}\times \vec{H}]/2)$ distribution is also plotted in the $x\text{−}y$ plane.

Figure 12

Dependence of the position of the Dirac point $K_z$ on the inner radius $R_{\mathrm{in}}$ of the hollow cylinder. For small $R_{\mathrm{in}}$ the Dirac points are moved to the $\mathrm{\Gamma }$ point and annihilated in pairs.

Figure 13

Splitting of the Dirac points vs simulation resolution. Parameters: $R_{\mathrm{out}}/a=0.5$ and $R_{\mathrm{in}}/a=0.4$, and $\varepsilon =12$.