Impact of Transforming Interface Geometry on Edge States in Valley Photonic Crystals

We investigate how altering the interface geometry from a zigzag to a glide plane interface between two topologically distinct valley Hall emulating photonic crystals (VPC), profoundly affects edge states. We experimentally observe a transition from gapless to gapped edge states, accompanied by the occurrence of slow light within the Brillouin zone, rather than at its edge. We numerically simulate the propagation and measure the transmittance of the modified edge states through a specially designed valley-conserving defect. The robustness to backscattering gradually decreases, suggesting a disruption of valley-dependent transport. We demonstrate the significance of interface geometry to gapless edge states in a VPC.

Figure 1

Interface geometry and dispersion diagrams. (a) A SEM image of a shifted interface of two distinct VPCs for $t=0.25a$, with $a=500\mathrm{nm}$ the lattice constant of both VPCs. (b) Experimentally retrieved dispersion diagram of photonic modes of the shifted interface, where ${\log }_{10}|\mathcal{F}(\mathbf{E}){|}^2$ is plotted versus $k_x$ and the frequency of excitation. A mode gap arises between the edge states (bottom of the graph) and the bulk modes (top of the graph). A close-up of the measured dispersion curve of the edge states for a shift of $0.25a$ is shown in (c), where $|\mathcal{F}(\mathbf{E}){|}^2$ is plotted in a linear scale. The measured dispersion has an M-shape and is consistent with the simulation result (blue-dashed line). Remarkably, there is a slow light region around $k_x=0.8\pi /a$, which is inside the Brillouin zone rather than at its edge. As a result, the edge state also exhibits both a positive and negative group velocity within half a Brillouin zone. (d) Numerically simulated dispersion curves of edge states for $t=0$ (orange), $0.25a$ (blue), and $0.5a$ (red). The gray regions represent bulk modes, and the dark line corresponds to the light line $\omega =ck$. As $t$ increases from 0 to $0.5a$, the edge states transform from gapless to gapped. (e) The group velocity of edge states. With an increase in $t$, a slow-light edge state with $v_g=0$ transitions from the Brillouin zone edge towards its interior.

Figure 2

Defect and transmittance. (a) The geometry of a valley-conserving L3 defect is introduced into a shifted interface, with three small triangles replaced by three large ones (blue). (b) and (c) Simulated electric field amplitudes $|\mathbf{E}|$ on the zigzag interface ($t=0$) and the shifted interface ($t=0.25a$), respectively, at an excitation frequency of 168 THz. The blue arrow indicates the position of the L3 defect. Light enters the VPC interface from the left side and exits from the right side. A decrease in $|\mathbf{E}|$ at the L3 defect and an interference pattern are observed in (c) but not in (b), indicating a significant contrast in the transmittance of that defect between the two VPC interfaces. The simulated transmittance spectra of the zigzag interface and the shifted interface are presented in (d) and (e), respectively. The gray domain indicates the band gap, while the dark one represents the mode gap. The zigzag interface has a transmittance spectrum almost unaffected by the L3 defect. In contrast, the shifted interface exhibits a significant decrease in its transmittance after introducing the L3 defect.