Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material

We derive in the present work topological photonic states purely based on conventional dielectric material by deforming a honeycomb lattice of cylinders into a triangular lattice of cylinder hexagons. The photonic topology is associated with a pseudo-time-reversal (TR) symmetry constituted by the TR symmetry supported in general by Maxwell equations and the $C_6$ crystal symmetry upon design, which renders the Kramers doubling in the present photonic system. It is shown explicitly for the transverse magnetic mode that the role of pseudospin is played by the angular momentum of the wave function of the out-of-plane electric field. We solve Maxwell equations and demonstrate the new photonic topology by revealing pseudospin-resolved Berry curvatures of photonic bands and helical edge states characterized by Poynting vectors.

Figure 1

Schematic plot of a triangular photonic crystal of “artificial atoms” composed by six cylinders of dielectric material. Dark gray (Red) dashed rhombus and hexagon are primitive cells of honeycomb and triangular lattices. The solid black hexagon labels an artificial atom, while the dashed black one marks the interstitial region among artificial atoms. ${\overrightarrow{a}}_1$ and ${\overrightarrow{a}}_2$ are unit vectors with length $a_0$ as the lattice constant. Right panel: enlarged view of a hexagonal cluster with $R$ the length of the hexagon edge and $d$ the diameter of cylinders. ${ϵ}_d$ and ${ϵ}_A$ are dielectric constants of cylinders and surrounding environment.

Figure 2

(a) Electric fields $E_z$ of the $p_x(p_y)$ and $d_{xy}(d_{x^2-y^2})$ photonic orbitals hosted by the artificial atom at the $\mathrm{\Gamma }$ point. (b) Magnetic fields associated with $E_z$ fields with wave functions of positive and negative angular momenta $p_{\pm }=(p_x\pm i{p}_y)/\sqrt{2}$ and $d_{\pm }=(d_{x^2-y^2}\pm i{d}_{xy})/\sqrt{2}$. The angular momentum of the wave function of the $E_z$ field constitutes the pseudospin in the present photonic crystal.

Figure 3

Dispersion relations of the TM mode for the 2D photonic crystals with ${ϵ}_d=11.7$, ${ϵ}_A=1$, and $d=2R/3$ for (a) $a_0/R=3.125$ (Inset: Brillouin zone of triangular lattice), (b) $a_0/R=3$, and (c) $a_0/R=2.9$. Blue and red are for $d_{\pm }$ and $p_{\pm }$ bands, respectively, and rainbow for hybridization between them. The case of $a_0/R=3$ corresponds exactly to the honeycomb lattice of individual cylinders.

Figure 4

Real-space distributions of the time-averaged Poynting vector associated with the pseudospin-down state at the $\mathrm{\Gamma }$ point below the photonic gap: (a) $a_0/R=3.125$ in the trivial regime and (b) $a_0/R=2.9$ in the topological regime. Other parameters are taken same as those in Fig. 3.

Figure 5

(a) Dispersion relation of a ribbon-shaped 2D topological photonic crystal, which is infinite in one direction and of 45 and 6 artificial atoms for the topological and trivial regions respectively in the other direction. Right panel: enlarged view of (a) around the band gap. Dark gray (Red) curves are for topological edge states. (b) Real-space distributions of $E_z$ fields at points A and B indicated in the right panel of (a). Right panels: time-averaged Poynting vectors $\overrightarrow{S}$ over a period. (c) 3D photonic crystal of height $h$ with two horizontal gold plates placed at two ends symmetrically. (d) Distribution of energy-density of $E_z$ field $u_{E_z}(\mathbf{r})=ϵ(\mathbf{r})|E_z(\mathbf{r}){|}^2/2$ in the 3D topological photonic crystal in (c) stimulated by a linearly polarized source. (e) Leftward and (f) rightward unidirectional energy propagation stimulated by source $S_{+}$ and $S_{-}$, which injects $E_z$ field with wave function of positive and negative, respectively, angular momentum in the region denoted by gray (green) solid frame in (d). The lattice constant and diameter of cylinder are kept same in the whole space $a_0=1\mu \mathrm{m}$ and $d=0.24\mu \mathrm{m}$, while the edge length of hexagon is $R=0.345a_0$ ($a_0/R=2.9$) and $R=0.32a_0$ ($a_0/R=3.125$) in topological and trivial regions, and the frequency of all sources is $\omega =135.6\mathrm{THz}$ within the topological band gap. In the 3D system the height of cylinder is $h=1\mu \mathrm{m}$. Other parameters are same as those in Fig. 3.