Zak phases of chiral photonic crystals designed via transformation optics

We study a class of inhomogeneous chiral photonic crystals with topologically nontrivial bands and edge modes using the approach of non-coordinate transformation optics. By transforming the chiral photonic crystals into achiral binary photonic crystals, the closed-form solutions to the photonic bands of the chiral photonic crystals are obtained. We show that the topological edge modes of these chiral photonic crystals can be accurately predicted by the Zak phases developed from the topological band theory for achiral binary photonic crystals.

Figure 1

The band structures of the chiral photonics crystals. $\theta \left(a\right)$ is supposed to be ${lim}_{z\to a^{+}}\theta \left(z\right)$. The parameters in the virtual space are given by ${\varepsilon }_{11}^{\prime }=4$ and ${\varepsilon }_{22}^{\prime }={\mu }_{11}^{\prime }={\mu }_{22}^{\prime }=1$, where ${\varepsilon }_{33}^{\prime }$ can be arbitrary due to ${\mathbf{L}}^{\prime }=\mathbf{0}$. Panels (a)–(f) are the band structures of $\theta \left(a\right)$ equal to 0, $0.25\pi , 0.5\pi , 0.75\pi , \pi$, and $1.25\pi$ respectively. Panel (a) can overlap with panel (e), and panel (b) can overlap with panel (d) if either one of them is shifted left or right by 1.

Figure 2

(a) Schematic of the transformed structure when $\theta \left(a\right)=0.5\pi$ is chosen. The permittivity (and permeability) parameters of white and black slabs are given by Eq. (14) when $\lfloor z/a\rfloor$ is odd and even respectively. A center of a white slab is chosen as an origin of the system and the width of a unit cell $\mathrm{\Lambda }$ equals $2a$. (b) The band structure of the transformed anisotropic binary photonic crystal. The parameters in the virtual space are given by ${\varepsilon }_{11}^{\prime }=4$ and ${\varepsilon }_{22}^{\prime }={\mu }_{11}^{\prime }={\mu }_{22}^{\prime }=1$. The Zak phases calculated by $|{\psi }_{x^{\prime \prime }}\rangle$ ($|{\psi }_{y^{\prime \prime }}\rangle$) are labeled near the corresponding isolated band.

Figure 3

(a) The structure of the chiral photonic crystal by choosing $\theta \left(a\right)=0.5\pi$, where $\theta \left(a\right)={lim}_{z\to a^{+}}\theta \left(z\right)$. Two unit cells are regarded as a period and the center of a unit cell is chosen as an origin for fear of losing the generality. Color gradient is used to represent the parameters are gradually changing along the $z$ direction. (b) The band structure of the chiral photonic crystal by considering two unit cells as a period. The parameters in the virtual space are given by ${\varepsilon }_{11}^{\prime }=4$ and ${\varepsilon }_{22}^{\prime }={\mu }_{11}^{\prime }={\mu }_{22}^{\prime }=1$.

Figure 4

Calculation of transmittance through the tested finite system. The system is composed of 20 unit cells of the designed chiral photonic crystal on the left and 10 unit cells of a type II achiral photonic crystal on the right. The parameters of the chiral photonic crystal in the virtual space are given by ${\varepsilon }_{11}^{\prime }=3.23$ and ${\varepsilon }_{22}^{\prime }={\mu }_{11}^{\prime }={\mu }_{22}^{\prime }=1$. In the type II achiral photonic crystal, green and blue slabs are slabs A and B respectively. The parameters of the slabs are given by ${\varepsilon }_A=4.2, {\varepsilon }_B={\mu }_A={\mu }_B=1, d_A=0.76a$, and $d_B=1.24a$. [(b), (c)] The transmission spectra of the system calculated by $|{\psi }_x\rangle$ and $|{\psi }_y\rangle$ modes respectively. The spectra are the same.

Figure 5

[(a), (d)] The transmission spectra calculated by $|{\psi }_x\rangle$ and $|{\psi }_y\rangle$ modes respectively. The system is composed of 10 unit cells of a transformed type I achiral photonic crystal on the left and 10 unit cells of a type II achiral photonic crystal on the right. The parameters of the transformed type I achiral photonic crystal in the virtual space are given by ${\varepsilon }_{11}^{\prime }=3.23$ and ${\varepsilon }_{22}^{\prime }={\mu }_{11}^{\prime }={\mu }_{22}^{\prime }=1$, and the parameters of the type II achiral photonic crystal are given by ${\varepsilon }_A=4.2, {\varepsilon }_B={\mu }_A={\mu }_B=1, d_A=0.76a$, and $d_B=1.24a$. [(b), (e)] The band structures of the transformed type I achiral photonic crystal. [(c), (f)] The band structure of the type II achiral photonic crystal. The band structures calculated by $|{\psi }_x\rangle$ and $|{\psi }_y\rangle$ modes are the same. The magenta and cyan strips represent the gaps with $\zeta >0$ and $\zeta <0$ respectively. The Zak phase of each band are denoted near the band and the number in the brackets are the sum of the Zak phases below the gaps.