## 2 Waveguide Design
Finally, we chose \(g\) = 0.3 \(\upmu\)m for a relatively flattened dispersion profile and an appropriate mode coupling distance. To summarize, the geometric parametersof the coupled waveguides are chosen as \(H\) = 0.85 \(\upmu\)m, \(d\) = 0.8 \(\upmu\)m, \(\theta\) = 80\({}^{\circ}\), \(w_{1}\) = 0.9 \(\upmu\)m, \(w_{2}\) = 2.8 \(\upmu\)m, and \(g\) = 0.3 \(\upmu\)m.  
Figure 2: (**a**) The calculated effective refractive indices of the symmetric mode and the anti-symmetric mode of the dual-coupled waveguide, compared with the initial uncoupled modes separated in each core, which are the fundamental TE\({}_{00}\) mode in the narrow waveguide and the TE\({}_{10}\) mode in the wide waveguide. The mode coupling occurs at around 3.8 \(\upmu\)m. Insets: the normalized electric field distribution of the supermodes at 1.55 and 3.6 \(\upmu\)m respectively. The dispersion curves of the anti-symmetric mode in the dual-coupled waveguide as a function of the wavelength for (**b**) varied \(d\), fixed \(\theta\) = 80\({}^{\circ}\), and \(g\) = 300 nm, (**c**) varied \(\theta\), fixed \(d\) = 800 nm, and \(g\) = 300 nm, (**d**) varied \(g\), fixed \(d\) = 800 nm, and \(\theta\) = 80\({}^{\circ}\). (**e**) The dispersion curves of the anti-symmetric mode in Si\({}_{3}\)N\({}_{4}\)-coupled waveguides, the uncoupled mode of a selected single-core waveguide with \(H\) = 0.85 \(\upmu\)m, \(d\) = 0.8 \(\upmu\)m, \(\theta\) = 80\({}^{\circ}\), \(w\) = 1.15 \(\upmu\)m, and the symmetric and anti-symmetric modes in LN dual-coupled waveguide. (**f**) The TOD profiles of the symmetric and anti-symmetric modes of the dual-coupled LNOI waveguide. (**g**) The calculated integrated dispersion of the anti-symmetric mode in Si\({}_{3}\)N\({}_{4}\)-coupled waveguides, the uncoupled mode of a selected single-core waveguide and the anti-symmetric mode in the LN dual-coupled waveguide. The latter two show a similar phase-matching wavelength at around 3.6 \(\upmu\)m.  
To better understand the advantage of our structure, we plot the dispersion profiles of the anti-symmetric mode of Si\({}_{3}\)N\({}_{4}\)-coupled waveguides in [11], an uncoupled mode in a single-core waveguide on LNOI, and the symmetric and anti-symmetric modes of our structure together to have a clear comparison (Figure 2e). The width of the single-core waveguide on LNOI was selected to have a similar phase-matching wavelength as that of the anti-symmetric mode of our structure. As we can see in Figure 2e, the dispersion profiles of the uncoupled mode in a single-core waveguide on LNOI and the symmetric mode of our structure have a similar shape, which has two ZDWs with anomalous dispersion between them and normal dispersion on the longer wavelength side. By comparison, for the anti-symmetric mode of Si\({}_{3}\)N\({}_{4}\) and LN dual-coupled waveguides, similar dispersion profiles with four ZDWs due to the mode coupling in the MIR region can be obtained. Because of the larger refractive index of LN, the ZDW at shorter wavelengths redshifts compared to that of the Si\({}_{3}\)N\({}_{4}\) waveguide. When referring to the MIR region between 2 and 3.25 \(\upmu\)m, the dispersion curve yielded by our structure via LNOI waveguide has a more flattened profile than that of the Si\({}_{3}\)N\({}_{4}\) waveguide, which is preferred to the broadband and flat MIR-DW generation. Figure 2f shows the TOD curves of the symmetric and anti-symmetric modes of the proposed dual-coupled waveguide. We can see that the TODs of the symmetric and the anti-symmetric modes have the opposite sign at the pumping wavelength of 1.55 \(\upmu\)m. The former is positive while the latter is negative. We prefer the latter because a negative TOD means a negative dispersion slope of GVD, which is helpful for enhancing the DW generation on the longer wavelength side (MIR). When high-order dispersion is not negligible, the soliton pump will transfer energy to the DW, which is generated in the normal dispersion regime at the phase-matching wavelength, during the soliton-fission based SCG.