## 2 Waveguide Design  
We chose 0.85 \(\upmu\)m thick LN layer on an \(x\)-cut LNOI wafer to design the dual-coupled waveguide. Figure 1a shows the 3D schematic diagram and the cross-section (inset) of the dual-coupled ridge waveguide. The two parallel ridge waveguides are spaced with a certain gap. The whole structure is upper-clad with silicon dioxide (SiO\({}_{2}\)) up to a 4 \(\upmu\)m height, which is illustrated in a translucent color. As illustrated in the inset, the \(c\) axis indicates the optical axis of the LN crystal which is aligned with the \(x\)-axis of the structure. The light propagates along the \(y\)-direction of the LN (coincident with the \(y\)-axis of the structure) resulting in an \(x\)-cut assembly. The top view of the structure is shown in Figure 1b. The top widths of the narrower and the wider waveguide are denoted by \(w_{1}\) and \(w_{2}\), respectively. The gap between the two parallel waveguides is defined as \(g\), and the propagation length along the \(y\)-axis is defined as \(L\). The height of the thin-film LN is fixed to \(H\) = 0.85 \(\upmu\)m. The etch depth and sidewall angle are denoted by \(d\) and \(\theta\), respectively. When the two parallel ridge waveguides approach each other gradually, the interaction between the optical fields inside the two waveguides will affect the propagation constants of them due to the refractive index perturbation. Consequently, the group velocity dispersion (GVD) is changed when the two waveguides get close enough that the phase-matching condition of the two modes is satisfied and mode coupling occurs between them. According to supermode theory, mode coupling results in a pair of supermodes, namely, the symmetric and the anti-symmetric modes. The former always produces the normal GVD, while the latter always produces the anomalous GVD. Mode coupling in this work is engineered for the MIR region. By leveraging the abrupt sign-changing of the dispersion that is produced by the anti-symmetric mode on the long wavelength side, the material dispersion of the waveguide is compensated for. In this way, a relative flattened dispersion profile with four ZDWs can be achieved, which is difficult for conventional single-core waveguides. Based on the above principle, Figure 2a illustrates the design of the dual-coupled waveguide. The geometries of the cross-sections of the two waveguides were selected to make the mode coupling occur in the MIR region by satisfying the phase-matching condition. The propagation constants of the two modes from each waveguide should be matched in the MIR region, namely, the effective refractive index should be the same, as \(\beta=n_{\mathrm{eff}}\omega/c\), where \(\omega\) indicates the angular frequency of the light and \(c\) indicates the speed of light in vacuum. We simulate the eff  
Figure 1: **(a)** The 3D schematic diagram of the dual-coupled waveguide on \(x\)-cut LNOI. Inset: the cross-section view of the dual-coupled waveguide. The \(c\)-axis is the optic axis of the thin film LN and is parallel to the \(x\)-axis. **(b)** The 2D top view of the dual-coupled waveguide. of the uncoupled and coupled modes by using the commercially available finite-element method (COMSOL, www.comsol.com, accessed on 6 June 2022). The Sellmeier equations of the material refractive index of LN and SiO\({}_{2}\) are chosen from [26; 27], respectively. In Figure 2a, we can see the two uncoupled modes, and the effective indices of the TE\({}_{00}\) mode in the narrow waveguide and the TE\({}_{10}\) mode in the wide waveguide have an intersection at around 3.8 \(\upmu\)m for the top widths of \(w_{1}\) = 0.9 \(\upmu\)m and \(w_{2}\) = 2.8 \(\upmu\)m respectively with \(d\) = 0.8 \(\upmu\)m and \(\theta\) = 80\({}^{\circ}\). When the two waveguides approach each other and \(g\) = 0.3 \(\upmu\)m, the effective refractive indices and the mode profiles of the supermodes need to be recalculated.