Bound States in the Continuum on a Silicon Chip with Dynamic Tuning

Expanding bound states in the continuum (BIC) beyond photonic crystal systems may enable broader applications benefiting from the unique properties of BIC states. We use photonic integrated circuit to realize a Fabry-Pérot BIC on a silicon chip. The devices consist of cascaded ring resonators with tunable resonance frequencies and phase delays. As a result, the BIC state is dynamically tuned with electrical I/O. We analyze the mechanism of the formation of BIC states in this waveguide system and point out the fundamental differences between the BIC states and electromagnetically induced transparency states in integrated photonics. The high transmission protected by the BIC state enables versatile optical filters, which are capable of independent control over switching, peak position, and quality factor. We also demonstrate the scalability of this platform. This integrated silicon photonic platform brings opportunities for practical BIC applications.

Figure 1

(a) A schematic model of the dual-resonator system that demonstrates Fabry-Pérot BIC. A₁ to A₈ are the electric field amplitudes in the bus near the coupling ports, a₁ and a₂ are the amplitudes of the resonance modes at steady state, and ω₁ and ω₂ are the intrinsic resonance frequencies of the two resonators. The intrinsic decay rate is γ_i/2 for both resonators; the coupling rate is γ_c/2 at all the gaps. (b) Schematic of a silicon photonics realization. Two ring resonators have slightly different radii (r₁, r₂) coupling through two bus waveguides. The gaps (g) between rings and buses are of the same size to enforce the same coupling rate γ_c(g)/2. Thermo-optic heaters (pink bars) tune ω₁, ω₂, and Δφ with independent control voltages (V₁, V₂, and V_t). (c) Optical image from the fabricated dual-resonator device. White circles denote the ring resonators, red rectangles are the thermo-optic phase shifters. (d)–(f) Energy-level diagrams of rigorous EIT analog, indirectly coupled EIT, and Fabry-Pérot BIC, respectively. 2Δ is the frequency offset between two resonators, J is the direct coupling rate between two resonators, and ${\gamma }_c({{\gamma }^{\mathrm{\prime }}}_c)$ is the coupling rate between ground state and excited states. (g),(h) Simulation of EIT-like (Fano resonance) and BIC transmission with zero energy offset (Δ = 0) between two resonators. Spectra are plotted as functions of probe beam detuning relative to resonance wavelength (1533 nm).

Figure 2

Transmission spectra of BIC filters. (a) Tuning the phase delay between two rings. The first curve is the on-resonance case with a symmetric transparency window. V_t is the voltage of the phase shift heater between two rings. (b) Tuning the peak position with fixed width. (c) Tuning the peak width (Q-factor) with fixed position. Spectra in (b) and (c) are all on-resonance. Blue curves in (a)–(c) are the same. Black arrows are used as a guide to the peaks of interest in each panel. All the curves are shifted vertically for clear illustration. Black, blue, and red arrows highlight ω_o, ω₁, and ω₂, respectively, as described in Eqs. (4) and (5). Simulation counterparts can be found in Appendix pp2

Figure 3

(a) Optical image of the triple-resonator device. White circles denote the ring resonators, red rectangles are the thermo-optic phase shifters. The left-hand and middle rings are the first loop, the middle and right-hand rings are the second loop. (b) Tuning the phase delay in the first loop. The blue curve is the on-resonance case for both loops with two transmission peaks. (c) Simulation reproduction of (b). (d) Tuning the right-hand peak’s position with fixed width while tuning the left-hand peak’s width (Q-factor) with fixed position. Spectra in (d) are all on-resonance in both loops. Blue curves in (c) and (d) are the same. Black arrows are used as a guide to the peaks of interest in each panel. All the curves are shifted vertically for clear illustration.

Figure 4

(a) Simulation of the single-bus coupling rate (γ_c/2) as a function of gap size using a 15-μm-radius ring and a single bus waveguide (width of 300 nm and resonant wavelength of 1533 nm). Dashed lines indicate frequency detunings from the resonant wavelength. (b) Measured transmission spectra of a single 15-μm-radius ring resonator with double-bus coupling. V_r is the voltage of the thermo-optic heater near the ring resonator. The black arrow shows the same transmission dip under different ring heater voltages.

Figure 5

(a) Schematic of the simulation structure. (b)–(d) Simulation reproduction of Fig. 2. Unless noted beside the curves, the default parameters to the BIC model are: Δ = 61 GHz; γ_c/2 = 667 GHz; Δφ = 0. All the curves are shifted vertically for clear illustration. Black, blue, and red arrows highlight ω_o, ω₁, and ω₂, respectively, as described in Eqs. (4) and (5). The simulation results reproduce the experimental data. The simulations show apparent transmission dips indicating intrinsic resonance wavelengths of the rings. Note that the curvature near the peak is slightly different in the simulations compared with the experiments, which also results in larger Q-factors for the windows. This is due to the additional loss in the fabricated device that causes the transmission to fall off quickly around near-BIC points.