Multiple Brillouin Zone Winding of Topological Chiral Edge States for Slow Light Applications

Photonic Chern insulators are known for their topological chiral edge states (CESs), whose absolute existence is determined by the bulk band topology, but concrete dispersion can be engineered to exhibit various properties. For example, the previous theory suggested that the edge dispersion can wind many times around the Brillouin zone to slow down light, which can potentially overcome fundamental limitations in conventional slow-light devices: narrow bandwidth and keen sensitivity to fabrication imperfection. Here, we report the first experimental demonstration of this idea, achieved by coupling CESs with resonance-induced nearly flat bands. We show that the backscattering-immune hybridized CESs are significantly slowed down over a relatively broad bandwidth. Our work thus paves an avenue to broadband topological slow-light devices.

Figure 1

Experimental setup and schematic. (a),(b) Photograph and diagram of the experimental setup. The lattice constant $a=14.6\mathrm{mm}$. The red arrow in (a) shows the position of the source antenna. The dashed box in (b) indicates the unit cell of the PCI. The gray circles in (b) represent YIG cylinders with a diameter $d=4.4\mathrm{mm}$. The yellow parts represent copper. Here, $g$, $D$, and $\mathrm{\Lambda }$ denote the gap width, the cavity depth, and the cavity width, respectively. (c) Second and third bands of the PCI under an external magnetic field. The gray rectangle indicates the topological band gap. The Chern number of each band is labeled. The first band not shown here is trivial. The inset shows the BZ. (d) CMT model. (e) Calculated dispersions in the decoupling (i.e., $\kappa =0$, the dashed curves) and coupling (i.e., $\kappa \ne 0$, the solid curves) regimes. The blue and red dashed curves represent the NFBs and the CESs, respectively. The detailed parameters can be found in [27].

Figure 2

Coupling between the CESs and the NFBs. (a)–(f) Measured (color) and simulated edge dispersions (black dashed curves) for different $g$ varying from $\delta$ to $6\delta$. In all cases, $D=a$. The blue dashed curves denote the NFBs. The gray curves represent projected bulk states. The color bar represents the momentum-space energy intensity of the measured electric field. The cyan solid curves in (b) represent the fitted dispersion with CMT. (g) Simulated eigenmode profiles ($E_z$) of the edge states for the cases without (top panel) and with coupling (bottom panel). The frequency of each mode can be found in (a) and (d).

Figure 3

Experimental observation of the broadband topological slow light through higher BZ winding. (a)–(c) Measured (color) and simulated (black dashed curves) edge dispersions for $D=2a$ (a), $3a$ (b), and $4a$ (c), respectively. The solid gray curves represent projected bulk states. (d) Simulated eigenmode profiles ($E_z$) of edge states without (left panel) and with (right panel) coupling. The frequency of each mode in the coupling (decoupling) regime can be found in (c) [Fig. S4(a)]. (e),(f) Retrieved group velocities (e) and average group indices (f) from the measured and simulated dispersions of the hybridized CESs for different values of $D$.

Figure 4

Demonstration of the robustness of the broadband topological slow light. (a),(b) Measured transmission (a) and simulated $E_z$ field and energy flux distributions (b) when four YIG pillars are removed. The gray circles in (b) represent removed YIG. (c),(d) Measured transmission (c) and simulated $E_z$ field and energy flux distributions (d) when the edge is blocked by a large PEC obstacle. (e),(f) Measured transmissions (e) and simulated $E_z$ field and energy flux distributions (f) when the hybridized CESs pass through an interface between two regions with different group velocities. The left (right) region has $D=a$ ($D=2a$). The $S_{21}$ ($S_{12}$) parameter is measured by placing the source at port 1 (port 2) and the detector at port 2 (port 1), as indicated by black arrows.