Topology-Controlled Photonic Cavity Based on the Near-Conservation of the Valley Degree of Freedom

We demonstrate a novel path to localizing topologically nontrivial photonic edge modes along their propagation direction. Our approach is based on the near-conservation of the photonic valley degree of freedom associated with valley-polarized edge states. When the edge state is reflected from a judiciously oriented mirror, its optical energy is localized at the mirror surface because of an extended time delay required for valley index flipping. The degree of energy localization at the resulting topology-controlled photonic cavity is determined by the valley-flipping time, which is in turn controlled by the geometry of the mirror. Intuitive analytic descriptions of the “leaky” and closed topology-controlled photonic cavities are presented, and two specific designs—one for the microwave and the other for the optical spectral ranges—are proposed.

Figure 1

(a) A “leaky” TCPC based on the near-conservation of the valley DOF in a mirror-terminated photonic structure. TREK states are supported by a domain wall (red horizontal line) between two VPCs with opposite $C_v$. (b) Top and (c) side views and geometry definitions of the unit cell. (d) Energy from an excited TREK state is localized at the cavity for $\omega =0.75(2\pi c/a_0)$. Color: time-averaged energy density. Dimensions: $l=0.12a_0$, $w=0.06a_0$, $d=0.2a_0$, $g=0.03a_0$, and $h=0.94a_0$. The sketches in (a),(b),(c) are not to scale.

Figure 2

Dispersion bands of the TE-polarized TREK state (red line) and trivial surface modes (black and blue lines). The wave number $k$ is along the domain wall for the TREK state and along the PEC mirror for the trivial modes. Black lines: type $a$ trivial mode, propagating downward [direction $-\widehat{x}/2-\sqrt{3}\widehat{y}/2$ in Fig. 1] along the mirror. Blue line: type $b$ trivial mode, propagating upward (opposite to type $a$) along the mirror. Only the TREK state can propagate inside the no-surface-mode band gap. Structure parameters: same as in Fig. 1.

Figure 3

(a) A VPC-based TCPC terminated by PEC reflectors on both ends. (b) The corresponding one-port-two-cavity model of the TCPC.

Figure 4

Examples of the time-averaged energy distributions of the eigenmodes of the TCPC. (a) With zigzag terminations, energy is mostly localized in the two cavities. (b) With armchair terminations, energy is distributed in the waveguide. Horizontal cavity length is $L=13a_0$ for both cases.

Figure 5

(a),(b) Symmetric and antisymmetric eigenmodes data of the TCPC with (a) zigzag terminations and (b) armchair terminations. The circles are the numerical results from COMSOL; the lines are the analytic results from the coupled mode theory; the data points labeled by stars are related to the field profiles shown in Fig. 4. (c),(d) The ratios between the amounts of energy stored in the two cavities and the topological waveguide for (c) symmetric modes and (d) antisymmetric modes. Green lines represent ${\gamma }_{\mathrm{zig}}{a}_0/(2\pi c)$; magenta lines represent ${\gamma }_{\mathrm{arm}}{a}_0/(2\pi c)$. Only the solutions close to the center of the no-surface-mode band gap, i.e., ${\omega }_0-\mathrm{\Delta }\omega <\omega <{\omega }_0+\mathrm{\Delta }\omega$, are plotted, where ${\omega }_0=0.75(2\pi c/a_0)$ and $\mathrm{\Delta }\omega =0.01{\omega }_0$. The gray regions represent the combinations of $\gamma$ and $\alpha$ corresponding to frequencies outside of that range, determined by Eq. (4).

Figure 6

An all-dielectric TCPC. Energy from an excited TREK state is localized at the cavity. The VPC region is formed by triangular Si rods; the trivial PhC region is a Si slab with arrayed holes. Color: time-averaged energy density. Frequency corresponds to $\omega =0.43(2\pi c/a_0)$. $d_1=0.64a_0$, $d_2=0.09a_0$, $d=0.98a_0$, and ${ϵ}_{\mathrm{Si}}=13$.