Second-order photonic topological insulator with corner states

Higher-order topological insulators (HOTIs) which go beyond the description of conventional bulk-boundary correspondence, broaden the understanding of topological insulating phases. Being mainly focused on electronic materials, HOTIs have not yet been found in photonic crystals. Here, we propose a type of two-dimensional second-order photonic crystals with zero-dimensional corner states and one-dimensional boundary states for optical frequencies. All of these states are topologically nontrivial and can be understood based on the theory of topological polarization. Moreover, by tuning the easily fabricated structure of the photonic crystals, different topological phases can be realized straightforwardly. Our study can be generalized to higher dimensions and provides a platform for higher-order photonic topological insulators and semimetals.

Figure 1

(a) The 2D photonic SSH model and the first Brillouin zone. We set $a=1.5\mu \text{m}$, $r=0.12a$, and the relative dielectric constant $\epsilon =12$ for all the cases. The coupling strength is denoted as $t_a$ and $t_b$. (b)–(d) The band inversion induced by changing $l$. $+$ ($-$) indicates the even (odd) parity of the band. (b) $l=0.14a$ (topologically trivial PBG). (c) $l=0.25a$ (band-gap closing). (d) $l=a/2.8$ (topologically nontrivial PBG). We choose the values of $l$ in order to make a maximal overlap between the band gaps in (b) and (d). (e) Photonic eigenmodes of a combined structure where the PC in (d) is inside a box and the PC in (b) is outside the box, as shown in the inset. There are four degenerate states localized at the four corners in the band gap. Here, edge (bulk) modes are denoted by orange (gray) points, whereas the corner modes are represented by blue points.

Figure 2

Projected band structure along the $k_x$ direction for Fig. 1 (isotropic). There are 1D edge states appearing in the first band gap which are topologically protected by bulk polarization as depicted by the solid blue line. The dashed red line labels the frequency of the corner states.

Figure 3

(a) Schematic of bulk-edge-corner correspondence. The nontrivial topology of the bulk (cyan region) leads to the emergence of the edge states, while the polarization of the edge (indicated by the black arrows) results in the formation of the topological corner state (indicated by the shallow yellow region). (b) There are in total four degenerate corner states at the four corners with frequency $66.0\mathrm{THz}$. The electrical fields $E_z$ (a.u. stands for arbitrary units) are strongly localized at the corners (a superposition of the four degenerate corner states is shown here). The blue dashed line indicates the boundary between the two PCs. (c) The zoom-in structure of the corner, where the dashed black lines indicate the border of the unit cells. The topological unit cell has a much larger $l$ ($l=a/2.8$) compared to the three trivial unit cells ($l=0.14a$).

Figure 4

Anisotropic 2D photonic SSH model where $l_x\ne l_y$. The PCs have the same $a$, $r$, and $\epsilon$ as the isotropic 2D photonic SSH model but have deformed lattice structures characterized by $l_x$ and $l_y$. The eigenstates are solved by comsol. For the outer PC, we set $l_x=l_y=0.14a$. For inner PCs, we set (a) $l_x=0.37a$ and $l_y=0.35a$. Nontrivial phase in both $x$ and $y$ directions. Two sets of degenerate edge states (DESs) with different frequencies. For the DESs in the upper (lower) figure, the frequency is $60.2\mathrm{THz}$ ($61.1\mathrm{THz}$). (b) $l_x=0.15a$ and $l_y=0.35a$. The $x$ direction in the topologically trivial phase and the $y$ direction in the topologically nontrivial phase. There are only 1D DESs in the upper and lower sides. The frequency is $64.2\mathrm{THz}$. (c) $l_x=0.35a$ and $l_y=0.15a$. The $x$ direction in the topologically nontrivial phase and the $y$ direction in the topologically trivial phase. There are only 1D DESs in the left and right sides. The frequency of the state in this figure is $65.2\mathrm{THz}$. The solid green circles emphasize the differences in the corner structures and field strengths in (a)–(c). The dashed blue lines label the boundaries of two PCs.

Figure 5

Projected band structures for Fig. 4 along the $k_x$ and $k_y$ direction as shown in (a) and (b), respectively. (a) There are no 1D edge states along the $k_x$ direction. (b) There are 1D edge states in the band gap along the $k_y$ direction as depicted by the solid blue line.

Figure 6

Classification of all kinds of the generalized 2D photonic SSH model with different configurations in the $x$ and $y$ directions. $A\left(B\right)=1$ means the 2D photonic SSH model is in the topologically nontrivial phase along the $x$ ($y$) direction. $A\left(B\right)=0$ means the 2D photonic SSH model is in the topologically trivial phase along the $x$ ($y$) direction. The blue area represents isotropic cases and the yellow area represents anisotropic cases. The dashed lines represent there are no 1D edge states and solid lines represent the existence of 1D edge states. The solid circles mean there are corner states. The edge (or corner) states of the same color have the same frequency, whereas the states with different colors have different frequencies.