Fig. 2: Coupling-strength-controlled TPTs in Floquet TIs.

a, A three-ring model for Floquet TIs. Using nine parameters in a single unit cell, it fully describes the quasienergy band structure. b, TPTs driven by coupling strength. Transmission spectra as functions of wavelength λ and the parameter θ on coupling strength are shown, where θ is negatively correlated with the coupling strength and the amplitude transmittance of an MZI is cos(θ/2). The boundaries of non-trivial bandgaps in one FSR are indicated by purple dashed lines. Theoretical results are in good agreement with experimental results. Boundary states at bandgaps ❶ disappear with a continuous variation of θ near the critical point at θ = 0.272π, while edge modes in bandgaps ❷ and ❸ exist throughout the entire range of θ variation. The attenuation of light for large θ in experiment is due to resonant enhancement in rings, which increases the effective optical length and thus the loss. c–f, Measured spectra at θ = 0.1π (strong coupling, c) and θ = 0.32π (weak coupling, d) and their respective calculated band structures (e,f). The windows of edge modes are visually enhanced. g–j, Imaged real-space distributions of electromagnetic field under different points marked in spectra in c,d: TPT from topological edge modes (g) to forbidden bandgaps (h) at bandgap ❶, edge modes at bandgaps ❷ in weak-coupling regime (i) and randomly distributed bulk mode from the non-degenerate bulk bands (j). k, A boundary cell in Floquet topological insulators (FTIs) is removed by adjusting its coupling to the ‘bar’ state, which forms a lattice defect. High-transmission topological edge modes bypass the hole and present its robustness against atomic vacancies. Note that on the link ring paths we tapped out −35 dB light using diffractive grating couplers for better imaging of light fields, which results in the appearance of regularly distributed bright spots. Noise at the top right arises from light reflection from the input fibre.