Fig. 1. Schematic illustration of a periodically bent silicon waveguide array with a cosine modulation of the spacing G between adjacent waveguides in the propagation direction z to illustrate the band structure of the waveguide array at different frequencies. The inset magnifies the rectangular waveguide structure with parameters P = 5μm, H = 220nm, and W=500nm.

Fig. 2. (a) Quasi-energies under open-boundary conditions with 40 waveguides where the bandwidth Δ is taken as the energy unit. (b) The momentum space quasi-energy band structure (blue solid lines) of the five chosen frequency replicas and the straight cases with ω/Δ=1/6, ω/Δ=1/3, ω/Δ=1/2, ω/Δ=1, ω/Δ=1.6, and ω/Δ=0. Red dashed lines correspond to the case with no dimerization (δκ=0) and uncoupled Floquet replicas to guide the eye for each Floquet replica. (c) The dynamic evolution of the array system for the 20 waveguides with the six frequencies shown in (b).

Fig. 3. FDTD simulations of E_z evolution patterns after injecting light from the upmost boundary waveguide under different driving conditions with the same length of L = 600μm. (a) The results for the curved (n_Λ = 9) and straight (n_Λ = 0) waveguide arrays with waveguide number N=8. (b) The blue solid and red dashed lines correspond to the propagation of each array, where the black and green lines represent the light propagation direction with the driving frequency (n_Λ = 9 and n_Λ = 0). (c) FDTD simulation of the amplitude profiles for the N=4 waveguide array with n_Λ = 9, G_max = 460μm, and G_min = 260μm. (d) The results for the straight N=4 waveguide array with the adiabatic elimination effect.

Fig. 4. FDTD simulations of E_z evolution patterns when injecting light from the upmost boundary waveguide, with different driving periodic numbers at the same length of L = 600μm in simulation. The black dashed circles in (b) and (c) show that the localized field profile exhibits a periodic oscillation pattern between the two boundary waveguides.