Edge states in self-complementary checkerboard photonic crystals: Zak phase, surface impedance, and experimental verification

Edge states of photonic crystals have attracted much attention for the potential applications, such as high transmission waveguide bends, spin-dependent splitters, and one-way photonic circuits. Here, we theoretically discuss and experimentally observe the deterministic edge states in checkerboard photonic crystals. Due to the self-complementarity of checkerboard photonic crystals, a common band gap is structurally protected between two photonic crystals with different unit cells. Deterministic edge states are found inside the common band gap by exploiting the Zak phase analysis and surface impedance calculation. These edge states are also confirmed by a microwave experiment.

Figure 1

(a) The schematic of checkerboard PC which consists of two interlaced square lattices of dielectric rods with ${\varepsilon }_{\mathrm{diel}}$ (black patches) and air rods (white patches). The lattice constant of PC is $a$, and the side length of rods is $b=a/\sqrt{2}$. Due to the self-complementarity of checkerboard PC, the unit cell can be chosen to be centered at dielectric rod (blue dashed square) or air rod (red dashed square). (b), (c) The schematics of (b) PC1, which is constructed by periodically repeating the dielectric rod centered unit cell, and (c) PC2, which is constructed by periodically repeating the air rod centered unit cell. (d) Band structure for the transverse magnetic modes of the checkerboard PC with ${\varepsilon }_{\mathrm{diel}}=9$. The inset shows the first Brillouin zone. A complete band gap ranging from $0.245c/a$ to $0.254c/a$ is highlighted by a green rectangle. This band gap is shared by PC1 and PC2 due to the self-complementarity of checkerboard PC.

Figure 2

Zak phase analysis. (a) Band structure, eigen-fields and Zak phase for PC1 whose unit cell is centered at dielectric rod. The schematic of the unit cell is shown in the right inset. The coordinate origin is located at the center of the left boundary of the unit cell. The middle inset shows the electric fields of two eigen-states at high symmetry A point $(k_x=-\pi /a)$ and B point $(k_x=0)$. As $|{\tilde{E}}_{\mathrm{A}}(x=0)|=0$ and $|{\tilde{E}}_{\mathrm{B}}(x=0)|\ne 0$, the Zak phase is $\pi$ which is given along with the reduced 1D band structure for $k_y=0.25\pi /a$. (b) Band structure, eigen-fields, and Zak phase for PC2 whose unit cell is centered at air rod. Due to different field distributions of eigen-states at A point comparing to those of PC1, the Zak phase for PC2 changes to be 0. (c) Projected band structures for the photonic boundary between the semi-infinite PC1 and PC2. The green line represents edge states dispersion, and the inset shows the schematic of photonic boundary. (d) The amplitude of electric fields of one representative edge state with $f=0.223c/a$ at $k_y=0.25\pi /a$ [marked by a green star in (c)]. Fields are localized near the boundary.

Figure 3

Surface impedance calculation. The schemetics of (a) the semi-infinite PC1 and (c) the semi-infinite PC2 for the transmission simulation. (b) The value of $\mathrm{Im}\left(Z\right)$ for PC1 (i.e., $\mathrm{Im}\left(Z_1\right)$ and marked in blue) and $\mathrm{Im}\left(Z\right)$ for PC2 (i.e., $\mathrm{Im}\left(Z_2\right)$ and marked in red) as a function of frequency. The tangential component of incident wave-vector is $k_y=0.25\pi /a$. One edge state exists at $f=0.223c/a$ at which the zero-impedance condition $\mathrm{Im}\left(Z_1\right)+\mathrm{Im}\left(Z_2\right)=0$ is fulfilled (green star). (d) Distribution of surface impedance $\mathrm{Im}(Z_1(\omega ,k_y))$ for PC1. (e) Distribution of surface impedance $\mathrm{Im}(Z_2(\omega ,k_y))$ for PC2. (f) Distribution of $\mathrm{Im}\left(Z_1\right)+\mathrm{Im}\left(Z_2\right)$. For each $k_y, \mathrm{Im}\left(Z_1\right)+\mathrm{Im}\left(Z_2\right)$ drops monotonically from $+\infty$ to $-\infty$ with the increasing frequency, and goes across the value of zero. A white-dashed line outlines the positions $(\omega ,k_y)$, where $\mathrm{Im}\left(Z_1\right)+\mathrm{Im}\left(Z_2\right)=0$ is fulfilled.

Figure 4

Microwave experimental verification. (a) Photo of experimental sample with PC1 on the left and PC2 on the right. The dielectric rod is alumina rod with ${\varepsilon }_{\mathrm{diel}}=8.2$, and the lattice constant of PC is $a=10\mathrm{mm}$. A pink dashed rectangle outlines the scanned region in the experiment, and a green arrow marks the monopole source. (b) Calculated projected bands (black) and edge states (green) dispersions for the boundary shown in (a). (c) Measured dispersions (bright color) for the deterministic edge states, compared with the numerical data (green line). (d) The simulated and (e) measured $E_z$ fields of edge states at the frequency of 7.54 GHz.