## 2 Waveguide Design
It can be seen from Figure 2a that mode coupling occurs at the MIR region and there is no crossing point between the curves of the symmetric and anti-symmetric modes. By comparison, in the visible and near-infrared (NIR) region, the curves of the symmetric and anti-symmetric mode are degenerate to the uncoupled TE\({}_{10}\) mode in the wide waveguide and the uncoupled TE\({}_{00}\) mode in the narrow waveguide, respectively. The mode profiles of the symmetric and anti-symmetric modes at 1.55 \(\upmu\)m (pumping wavelength) and 3.6 \(\upmu\)m in the mode coupling region are also shown in Figure 2a. The symmetric mode evolves from the uncoupled TE\({}_{10}\) mode in the wide waveguide in the NIR, while the anti-symmetric mode evolves from the uncoupled TE\({}_{00}\) mode in the narrow waveguide in the NIR. Hereafter, in this study we focus on the anti-symmetric mode and its dispersion profile. We studied the influence of etch depth, sidewall angle, and gap variation on the dispersion curve. Figure 2b-d show the calculated dispersion as a function of wavelength when \(d\), \(\theta\), and \(g\) vary, respectively. It can be seen in Figure 2b that the variation in the etch depth has a more significant influence on the ZDW at longer wavelengths than the one at shorter wavelengths. For the ZDW at shorter wavelengths, it redshifts when \(d\) increases from 600 to 650 nm, then it blueshifts as \(d\) increases further from 650 to 800 nm. For the ZDW at longer wavelengths, when \(d\) increases from 600 to 800 nm, it redshifts monotonously. Since the MIR-DW is generated in the normal dispersion region where the value of the dispersion is below zero, a larger value of the ZDW at longer wavelengths is preferred in order to generate a stronger MIR-DW at longer wavelengths. Only when \(d\) is equal or greater than 650 nm can the dispersion profile possess the normal dispersion in the NIR region; thus, a four-ZDW profile. As we can see from Figure 2c, the variation in the sidewall angle which can be tuned from 40\({}^{\circ}\) to 80\({}^{\circ}\) according to the advanced fabrication technology [20] has little influence on the ZDW at shorter wavelengths. When \(\theta\) increases from 60\({}^{\circ}\) to 65\({}^{\circ}\), the ZDW at shorter wavelengths blueshifts, then it redshifts to the initial location as \(\theta\) further increases from 65\({}^{\circ}\) to 80\({}^{\circ}\). By comparison, for the ZDW at longer wavelengths, it blueshifts as \(\theta\) increases from 60\({}^{\circ}\) to 80\({}^{\circ}\). When \(\theta\) is equal to or larger than 70\({}^{\circ}\), the dispersion profile is a four-ZDW shape; otherwise, it has two ZDWs only and keeps the anomalous dispersion between them. Since we pump the dual-coupled waveguide with a 1.55 \(\upmu\)m source, it is beneficial to locate the pump at the falling edge of the dispersion profile in the anomalous dispersion region. In this way, we can obtain both negative GVD and the third-order dispersion (TOD) parameters, which is preferred for MIR-DW enhancement. For the reasons described above, we fixed \(\theta\) to be 80\({}^{\circ}\) in this study. Consequently, \(d\) = 0.8 \(\upmu\)m and \(\theta\) = 80\({}^{\circ}\) were chosen to fulfill the desired dispersion profile of the dual-coupled waveguide. Figure 2d shows that the ZDWs at shorter wavelength are almost superimposed when \(g\) varies from 200 to 400 nm. For the ZDW at longer wavelengths, it blueshifts as \(g\) increases from 200 to 400 nm. As for the shape of the dispersion profile, when \(g\) is equal to 200 nm, it becomes a two-ZDW curve. In other cases, it is a four-ZDW profile. A smaller \(g\) does not yield a stronger change in the dispersion profile. Since the mode coupling can occur over a larger spectral bandwidth when \(g\) decreases, the variation in the phase constant of the anti-symmetric mode with frequency dependency is reduced, which results in a reduced depth and spectral range of the normal dispersion region.