Topological properties of coupled resonator array based on accurate band structure

The topological edge states of the coupled-resonator optical waveguide (CROW) system have been widely researched. Without breaking time-reversal symmetry nor needing to construct complicated optical modes as pseudospins, the CROW system offers characteristics that are effective for exploration of photonic topological effects and has potential for applications in nanodevices. We extend the previous method by using specific coupling conditions to calculate the system's actual band structure, thereby providing a new perspective of this system. Two types of topological properties, one with and the other without an artificial magnetic field, are identified based on the band diagrams. Fractal spectra including the Hofstadter butterfly spectrum are obtained, revealing the abundant physics associated with the CROW system. By connecting the band diagrams with the corresponding structural parameters, we propose a triangular lattice array with reduced losses at the corners and propose a scheme with the potential for use as a reconfigurable multipath platform.

Figure 1

(a) CROW array with square lattice unit cell composed of site rings (black) and link rings (gray). (b) Four-port model with basic coupling parameters $t$ and $\kappa$. $\theta$ and ${\theta }^{\prime }$ denote the phase accumulation for the whole link ring and the upper part of the link ring, respectively. (c) Coupling model of unit cell of the array with the field amplitudes marked.

Figure 2

Bulk band and corresponding projected band structure under a specific configuration with different coupling parameters. (a), (b) $t=0.7$, topological trivial state without edge state. (c), (d) $t=\sqrt{2}-1$, band gap close to the Dirac cones. (e), (f) $t=0.2$, topological nontrivial state, where the Dirac cone opens and the edge state appears in the gap.

Figure 3

Topological edge states of strong-coupling region. (a) Distribution of electric-field module value and (c) energy-flow arrow diagram when stimulated from the right port of the bend waveguide. Panels (b) and (d) represent the corresponding diagrams, respectively, when stimulated from the left port.

Figure 4

Projected band diagram of square lattice with four special circuit phase accumulations. The single band varies and splits into subbands, and edge states form in the gaps between subbands.

Figure 5

Topological edge states with circuit phase accumulation $\varphi =\pi /2$. (a) Field distribution of edge state in the lower frequency range with a negative edge band slope. (b) Field distribution of edge state in the higher frequency range with a positive edge band slope.

Figure 6

Color-scaled fractal spectra of (a) the square lattice and (b) the triangular lattice obtained from band-diagram analysis. The optical frequency (horizontal axis) and the circuit phase accumulation (vertical axis) correspond to the energy and the magnetic flux in an electronic system, respectively. The colors represent the number of edge states and are equivalent to the absolute value of the topological invariant in each case. Note that the black regions include both high-order edge states and bulk states.

Figure 7

(a) Coupling model for unit cell of the triangular lattice. Subscripts to the field amplitudes refer to the site position; the dash-dotted box outlines the unit cell. Here the link rings are replaced with thin lines. (b) Scheme for on-chip reconfigurable multipath platform. The wavy line represents a nonlinear nanowire and the orange area is a 2D material that, under the control of a pump source, may vary the coupling parameters. (c) Electric-field distribution of unidirectional edge state in the triangular-lattice CROW system. (d) Simulation results showing the construction of a new path by modulating the refractive index along the path shown in panel (b).

Figure 8

Projected band diagram of triangular lattice with four special circuit phase accumulations. The single band varies and splits into subbands, and edge states form in the gaps between subbands.

Figure 9

(a) Simulation results for transmission spectra of through and drop ports over a large frequency scale. (b) Analytical results using Eq. (A1), where the coupling strength $t$ varies with frequency in a Taylor expansion form.

Figure 10

Analytical 1D coupled ring band structure with (a) $t=0.7$, (b) $t=\sqrt{2}-1$, and (c) $t=0.2$.

Figure 11

Projected band diagrams in topological trivial region with (a) matrix method and (b) simulation method, and the corresponding diagrams in the topological nontrivial region with (c) matrix method and (d) simulation method.

Figure 12

Band diagrams with absorbing boundary condition at upper edge. The circuit phase accumulations are $\pi /2$ and $3\pi /2$, as indicated.