## 3 Results and Discussion  
The SCG in the proposed dual-coupled LNOI waveguide is governed by the following GNLSE [28]:  
\[ \begin{array}{rl}\frac{\partial\widetilde{A}}{\partial x}&=i[\beta(\omega)- \beta(\omega_{0})-\beta_{1}(\omega_{0})(\omega-\omega_{0})]\widetilde{A}(\text{ z},\omega)-\frac{\alpha(\omega)}{2}\widetilde{A}(\text{z},\omega)\\ &+i\gamma(\omega)(1+\frac{\omega-\omega_{0}}{\omega_{0}})\mathcal{F}\Big{\{}|A (\text{z},t)|^{2}A(\text{z},t)\Big{\}}\\ \\ &\widetilde{A}(\text{z},\omega)=\mathcal{F}\{A(\text{z},t)\}=(\frac{A_{\text{ eff}}(\omega)}{A_{\text{eff}}(\omega_{0})})^{-1/4}\widetilde{\text{E}}(\text{z}, \omega)\\ \\ &\gamma(\omega)=\frac{n_{2}n_{0}\omega_{0}}{cn_{\text{eff}}(\omega)\sqrt{A_{ \text{eff}}(\omega)}A_{\text{eff}}(\omega_{0})}\\ \\ A_{\text{eff}}(\omega)=\frac{(\int\int_{-\infty}^{+\infty}|E(x,\text{z},\omega )|^{2}dxdz)^{2}}{\int\int_{-\infty}^{+\infty}|E(x,\text{z},\omega)|^{4}dxdz} \end{array}  \tag{3}\]  
where \(\widetilde{A}(\text{z},\ \omega)\) is the Fourier transform of \(A(\text{z},\ t)\) given by Equation (4), \(\beta\) is the propagation constant of the selected mode, \(\beta_{1}\) is the first order dispersion parameter, \(\omega_{0}\) is the angular frequency of the pump, \(\alpha\) is the linear propagation loss of the nonlinear waveguide, and \(\gamma\) is the nonlinear coefficient. \(A_{\text{eff}}\) is the effective mode area given by Equation (6), \(\widetilde{\text{E}}(\text{z},\ \omega)\) is the Fourier transform of the electric field. \(n_{2}\) is the nonlinear refractive index, \(n_{0}\) is the linear refractive index, \(c\) is the speed of light in the vacuum, and \(n_{\text{eff}}\) is the effective index of the selected mode. \(E\) (\(x\), \(z\), \(\omega\)) is the frequency-dependent intensity of electric field distribution. We safely neglected the stimulated Raman scattering in the model for two reasons. First, the contribution of the Raman effect typically becomes significant after the soliton fission length [29]. The propagation lengths we considered to evaluate the performance of SCG and MIR-DW generation are just around the soliton fission length, where the Raman contribution is still weak. Second, the Raman effect was neglected in previously reported experimental results that aimed at SCG and Kerr frequency comb generation in LNOI platforms [21; 22; 23] because its contribution is weak. The \(\chi^{(2)}\)-based second harmonic generation (SHG) is also neglected in this study because it makes a substantial contribution to the frequency generation on the shorter wavelength side, not the MIR region. The dispersion parameters considered in our study are up to the 11-th orders in order to fulfill the dispersion characteristics within a broadband spectrum. Here, we chose the anti-symmetric mode of our structure as the selected mode. Because of the weak mode coupling between the anti-symmetric mode and other high-order modes [11], we can assume that there is no mode coupling. We simulate Equation (3) with the fourth-order Runge-Kutta method [30]. We chose the femtosecond pulse centered at 1.55 \(\upmu\)m as the pump source. The pulse envelope is a hyperbolic secant shape as:  
\[A(\text{z}=0,\tau)=\sqrt{P_{0}}\text{sech}\bigg{(}\frac{\tau}{\tau_{0}}\bigg{)} +\sigma_{\text{Quantum}} \tag{4}\]  
where \(\tau_{0}=\tau_{\text{FWHM}}/2\text{ln}(1+2^{1/2})\approx\tau_{\text{FWHM}}/1.7632\) is the pulse width, \(\tau_{\text{FWHM}}\) is the full-width at half maxima, and \(\sigma_{\text{Quantum}}\) is the quantum noise. The full-width at half maxima of the pump \(\tau_{\text{FWHM}}\) is 75 fs and the peak power \(P_{0}\) is 4.5 kW, corresponding to a pulse energy of 190 pJ. Assuming a commercially available repetition rate of 100 MHz for the mode-locked fiber laser at 1.55 \(\upmu\)m, the average power consumption of this work is only 19 mW. The length of the dual-coupled waveguide is 1.2 mm, and the propagation loss was chosen to be 3 dB/m [21]. As shown in Figure 3a,d, the spectral and temporal evolution profiles are presented along the propagation length.