Reconfigurable Topological Phases in Next-Nearest-Neighbor Coupled Resonator Lattices

We present a reconfigurable topological photonic system consisting of a 2D lattice of coupled ring resonators, with two sublattices of site rings coupled by link rings, which can be accurately described by a tight-binding model. Unlike previous coupled-ring topological models, the design is translationally invariant, similar to the Haldane model, and the nontrivial topology is a result of next-nearest couplings with nonzero staggered phases. The system exhibits a topological phase transition between trivial and spin Chern insulator phases when the sublattices are frequency detuned. Such topological phase transitions can be easily induced by thermal or electro-optic modulators, or nonlinear cross phase modulation. We use this lattice to design reconfigurable topological waveguides, with potential applications in on-chip photon routing and switching.

Figure 1

(a) Square resonator lattice of site rings (blue, red) with relative detuning $\pm M$, coupled with strength $J$ via off-resonant link rings (black) detuned by $\mathrm{\Delta }\nu$. All rings have intrinsic loss $\kappa$. Dashed line indicates the unit cell. (b) Detailed schematic centered on a single link ring, which mediates both nearest-neighbor and next-nearest-neighbor couplings between site rings, with coupling phases $\pm \varphi /4$ and 0, respectively. (c) Tight binding model for the site ring resonances, consisting of a checkerboard lattice with staggered flux $\pm \varphi =\pm 2\pi \mathrm{\Delta }\nu /\mathrm{FSR}$. (d) Bloch band energies $\delta \nu$ (shaded regions) versus site detuning $M$, calculated from the tight-binding model for anti-resonant site and link rings ($\varphi =\pi$). Red and blue areas indicate the sublattice the modes are localized to in the large $M$ limit. Gap Chern numbers $C$ are indicated and topologically nontrivial link bands occur for $|M|<2J\csc \varphi /2$.

Figure 2

Reconfigurable topological waveguide functionality enabled by active control of site ring detunings. The lattice contains $8\times 8$ unit cells, with two domains where $M=M_1$ and $M=M_2$, respectively, with $\kappa =0.05J$ and $\varphi =\pi$ (i.e., site and link rings detuned by $\mathrm{FSR}/2$) throughout. The couplings at the inputs and outputs are ${\kappa }_{\mathrm{ex}}=2J$. (a) Transmission curves for $M_1=J$ and $M_2=3J$; the positions of the input port (1) and output ports (2,3) are shown in (c), and the band gaps are shaded in yellow. (b) Transmission curves for $M_1=3J$ and $M_2=J$. (c)–(d) Intensity profiles at the midgap frequency $\delta \nu =0$.

Figure 3

Nonlinearity-induced topological transition. (a),(b) Solution of the nonlinear scattering equations for a pump beam at $\nu -{\nu }_L=4.6J$. (a) Dynamics of link ring intensities, illustrating relaxation to a stable steady state within 5 ns. The different colors denote different link rings. (b) Resulting pump intensity profile over the $8\times 8$ lattice inducing a Chern insulator phase at the site resonance. Red arrow indicates the pump input position relative to the probe ports (1,2,3). (c) Transmission spectrum at the site resonance without (blue) and with (red) the pump beam. (d) Midgap probe profile when pump is applied [frequency indicated by dashed line in (c)].