Photonic Chern insulator through homogenization of an array of particles

We propose a route towards creating a metamaterial that behaves as a photonic Chern insulator, through homogenization of an array of gyromagnetic cylinders. We show that such an array can exhibit nontrivial topological effects, including topologically nontrivial band gaps and one-way edge states, when it can be homogenized to an effective medium model that has the Berry curvature strongly peaked at the wave vector $k=0$. The nontrivial band topology depends only on the parameters of the cylinders and the cylinders’ density, and can be realized in a wide variety of different lattices, including periodic, quasiperiodic, and random lattices. Our system provides a platform to explore the interplay between disorder and topology and also opens a route towards the synthesis of topological metamaterials based on the self-assembly approach.

Figure 1

The dispersion relation for the system as described by Eq. (1) with ${\gamma }_{\varepsilon }={\gamma }_{\mu }=3$. ${\omega }_0$ is a reference frequency, $c$ is the speed of light in vacuum, and $k$ is the in-plane wave vector. (a) ${\omega }_E=1.01{\omega }_0$, ${\omega }_H=0.99{\omega }_0$, ${\mu }_k=0$. (b) ${\omega }_E={\omega }_H={\omega }_0$ and ${\mu }_k=0$. (c) ${\omega }_E={\omega }_H={\omega }_0$ and ${\mu }_k=0.03$. (d) Red and blue represent the Berry flux $\beta$ of the highest and lowest bands in (c), respectively. The Berry flux of the middle band in (c) is exactly zero.

Figure 2

(a) A system consisting of a random array of cylindrical particles (solid blue disks). Each particle is described by its electric and magnetic dipole responses to external electromagnetic fields. There are in total 25 cylinders with the same radius in (a). The cylinders are randomly distributed with equal probability while keeping the minimal distances between cylinders to be larger than $0.8a$, where $1/a^2$ represents the density of the cylinder. (b) A sketch showing the geometry used in deriving the effective parameters ${\varepsilon }_e$ and ${\overleftrightarrow{\mu }}_e$, which provide an effective medium model for the array in (a). The light blue regions are filled uniformly with material having such effective parameters.

Figure 3

(a) The band structure of cylinders in a square lattice. (b) Projected band structure (gray area) and corresponding surface states with the supercell in (c). This supercell is terminated by a PMC boundary on the left and a PEC boundary on the right. Here, red and blue curves represent the surface states at the PEC and PMC boundaries, respectively. The Chern numbers for the band and the gap Chern numbers in (b) are labeled in red and cyan, respectively. The electric field amplitude of the surface state marked by a red star ($k_{y}a/\pi =-0.2$) in (b) is also shown in (c), where red and blue colors represent maximum and zero field amplitudes, respectively. The cylinder has a radius of $r_c=0.1735a$, where $a$ represents the lattice constant of the square lattice. The relative permittivity of the cylinders is ${\epsilon }_c=20$. $\kappa =0$ in (a) and $\kappa =0.08$ in (b) and (c).

Figure 4

(a) The projected bulk band structure (gray regions) and the band structure of surface states (red and blue curves) of a strip geometry consists of five of the supercells, each of which is shown in Fig. 2. The upper and lower boundaries of the strip are terminated by PECs and the strip is periodic along the horizontal direction. The blue and red curves represent the surface states on the upper boundary and the lower boundary, respectively. (b) One-way edge modes supported by a finite random lattice of cylinders. Here, the red star marks the position of the source with an operating frequency at $\omega a/\left(2\pi c\right)=0.547$. The rectangles on the lower end of the figure represent a perfectly matched layer (PML) region that absorbs the waves. All the other outer boundaries in (b) are PEC. The radius of the cylinder and the relative permittivity are $r_c=0.1735a$ and 20, respectively, where $1/a^2$ is the density of the cylinders, and $\kappa =0.4$.