Wave Steering by Relaying Interface States in a Valley-Hall-Derived Photonic Superlattice

Topological notions in physics have become a powerful perspective that leads to the discoveries of interface states. In this work, we present a scheme to steer waves by leveraging the properties of valley interface states (VISs) in a valley photonic crystal (VPC). Due to the chiral characteristics, the VISs deterministically possess either positive or negative dispersion relation, which can cause an obliquely incident wave to undergo a lateral shift. By stacking multiple VPC interfaces, the VIS-induced lateral shifts can relay in the transmitted wave towards the far side of incidence. By selectively engaging one type of VISs, the resultant outgoing wave appears to have undergone negative refraction. While this refraction can be flipped to positive by selectively exciting the other type of VISs with positive dispersion. This finding is verified in microwave experiments. Our scheme opens application scenarios for topological systems in wave manipulations.

Figure 1

(a) Schematic of a VPC strip containing two types of interface labeled by red and blue dashed lines. The topological protection of the VISs is shown in (b)–(f). (b) The projected band structure as a function of radius contrast shows VISs with different chirality, the red (blue) dots indicate the interface state where the corresponding field distribution is plotted in (c). (d) The projected band structure as a function of layer number. (e), (f) The interface state behaves well localized even when the system is only composed of two layers of VPCs with radiation boundary condition. The right inset shows the field distributions.

Figure 2

Schematic drawing of a plane wave incident on a system with a type-I (a) and type-II (d) interface. Both systems have only two unit cells. (b), (e) The transmission spectra as functions of incidence angle $\theta$ and normalized frequency $fa/c$. The color represents the transmission amplitude. (c), (f) The VIS dispersions retrieved from the transmission spectra (red and blue solid curves) show excellent agreement with the band structures calculated by FEM (red and blue dots).

Figure 3

Schematic and simulation result of a two-layer system supporting transmission extreme where the transmitted wave performs as negative (a) or positive (b) refracted, depending on the type of the excited VIS. The length of the system along the interface direction is $60a$. The white arrows indicate the direction of energy flow. (c), (d) Schematic and simulation result of a multiple-layer system supporting negative (c) and positive (d) refraction. (e) The stacking procedure produces a new PC with a larger unit cell, where its band structure is plotted. The red and blue band results from the interface state shown in (a) and (b), respectively.

Figure 4

Experimental demonstration of negative and positive refraction by relaying VISs. (a), (d) Simulated field distributions of VPCs with four type-I (a) and type-II (d) interfaces. Dashed boxes depict the measured region in experiments. The incident beam is generated by a standard X-band waveguide along with absorber materials. (b) The photo of the fabricated VPC superlattice consisting of alumina ceramic cylinders. (c), (e) Measured field distributions of negative refraction through VPC with four type-I interfaces (c) and positive refraction through VPC with four type-II interfaces (e).

Figure 5

Setups for the different simulations in the main text. The corresponding figure labels are shown at the bottom of the corresponding setup.

Figure 6

(a) Propagation of the VIS on the type-I interface. (b) Propagation of the VIS on the type-II interface. (c) Propagation of the VIS on the type-I interface with three dielectric cylinders of the VPCs removed. (d) Propagation of the VIS on the type-II interface with four dielectric cylinders of the VPCs removed.

Figure 7

Interface states that appear in square lattices can also realize negative refraction. (a) The system is a square lattice of dielectric cylinders with relative permittivity ${\epsilon }_r=12.97$. The two structures produce the same band structure shown in (b) with two bulk band gaps. The radius of the cylinder is $0.12a$, the nearest distance of the cylinders is $2a/7$. (c) Two kinds of interface states exist in the two bulk band gaps. (d) The interface state with negative dispersion can induce negative refraction.