Fig. 1. LZ model in Harper waveguide lattices. (a) Conceptual map of silicon optical waveguide lattice with $M$ unit cells for demonstrating four-level Harper model. The width of the $n$th waveguide is $W_n$, determining the propagation constant $k_n$ of the fundamental TE waveguide mode, and each waveguide has a height of $h=220\mathrm{nm}$, and length of $l$ corresponding to the modulation period. The $g_{n,(n\mathrm{mod}4)+1}$ is the separation distance between the $n$th and $((n\mathrm{mod}4)+1)$th waveguides, which is closely related to the coupling coefficient $C_{n,(n\mathrm{mod}4)+1}$. (b) Propagation constant of fundamental TE mode $k_n$ versus $W_n$. The propagation constant spectral flow of Harper model with (c) $M=10$ and (d) $M=2$, where ${\beta }_x$ is the evolution parameter normalized to $2\pi$ along the propagation direction. (e) LZ model near ${\beta }_x=0.75\pi$ with different coupling coefficients $c$. All the propagation constant spectra are normalized to the free space wavenumber of light, $k_0=2\pi /\lambda$ ($\lambda =1.55\mu \mathrm{m}$).

Fig. 2. Topological phase and edge states of the Harper waveguide lattice. (a) Silicon eight-waveguide lattice with length $L$, near ${\beta }_x=0.75\pi$. This structure refers to the region bounded by the dashed black frame in Fig. 1(a). (b) The Zak phase of 1D finite chain model corresponding to three different cross sections with ${\beta }_x=0.67\pi$, ${\beta }_x=0.75\pi$, and ${\beta }_x=0.83\pi$. The red and blue dots represent the Zak phase of the first band and fourth band, respectively. The orange and purple rhombus denote the Zak phase of the second band and third band, respectively. (c) The cross-sectional field intensity distributions of $|E{|}^2$ with ${\beta }_x=0.67\pi$, ${\beta }_x=0.75\pi$, and ${\beta }_x=0.83\pi$. The upper (lower) three figures correspond to TES1 (TES2). (d) The Bloch bands of the waveguide lattice shown in Fig. 1(a) in 2D parameter space ${\beta }_x{\beta }_y$ under periodical boundary condition. ${\beta }_y$ is the Bloch vector in the reciprocal space along the $y$ direction. The Chern numbers of four bands are labeled by the white characters.

Fig. 3. Edge-to-edge channel converter with different device lengths. The field intensity distributions of $|E{|}^2$ for edge-to-edge channel conversion of the two TESs in the waveguide lattice in Fig. 2(a). (a) $L=10\mu \mathrm{m}$, (b) $L=20\mu \mathrm{m}$, and (c) $L=100\mu \mathrm{m}$.

Fig. 4. Experimental demonstration. (a) SEM image of the device. The upper (lower) panel is used to test the edge-to-edge channel conversion effect of TES2 (TES1) with $L=100\mu \mathrm{m}$. Sections A and E are the grating couplers for coupling in and out of the waveguide energy, respectively. Section B corresponds to the adiabatic coupler for exciting the TESs, and section D represents output branch waveguides for testing the conversion effect. (b) The partially amplified SEM image for section C associated with Fig. 2(a). (c) The amplified SEM image for the region bounded by the dashed-line rectangle in section D. The simulated field intensity distributions of $|E{|}^2$ with TES2 (d) and TES1 (e). The simulated and experimental power contrast ratio ${\rho }_{2\to 3}$, ${\rho }_{3\to 2}$ versus light wavelength with (f) $L=100\mu \mathrm{m}$ and (g) $L=300\mu \mathrm{m}$. The red circles (lines) and blue circles (lines) represent the estimated ${\rho }_{2\to 3}$ and ${\rho }_{3\to 2}$ from the experiment (simulation), respectively.