## 3 Results and Discussion
From Table 1, we can see that the bandwidth of the MIR-DW (at \(-20\) dB level) of the proposed dual-coupled LNOI waveguide is about 1630 nm, which is the largest one. Meanwhile, our dual-coupled waveguide shows the highest degree of flatness within the largest spectral spanning in the mid-infrared, benefited from the well-tailored dispersion profile. Consequently, the conversion efficiency of our work shows significant improvement. We note that the comparison is somewhat unfair because our performances are determined by the numerical results rather than an experiment. However, considering the significantly improved level, the proposed waveguide has reasonable potential to yield a high-performance MIR-DW in the form of practically fabricated devices. Figure 4: **(a)** Spectral evolution along the propagation length with the full-width at half maxima \(\tau_{\text{FWHM}}=50\) fs. **(b)** The spectra of the pump and the output at \(L=0.8\) mm. **(c)** The spectra of the pump and the output at \(L=0.8\) mm. **(d)** Temporal evolution along the propagation length with the full-width at half maxima \(\tau_{\text{FWHM}}=50\) fs. **(e)** The normalized temporal pulse envelopes at input and output at \(L=0.8\) mm. **(f)** The normalized temporal pulse envelopes at input and output at \(L=0.86\) mm. We further studied the influence of coupled pulse energy variation on MIR-DW generation, especially the conversion efficiency and the flatness. As the conversion efficiency of the MIR-DW is as high as 19.31% with \(\tau_{\text{FWHM}}\) = 75 fs, \(P_{0}\) = 4.5 kW and \(L\) = 0.95 mm, we just changed the peak power of the pump to study this effect, while the rest of the pump conditions and the propagation length remained constant. Figure 6a shows the output spectra with different coupled pulse energy ranging from 170 to 340 pJ, corresponding to the conversion efficiency of the MIR-DW. The conversion efficiency of the MIR-DW is as high as 19.31% with \(\tau_{\text{FWHM}}\) = 75 fs, \(P_{0}\) = 4.5 kW and \(L\) = 0.95 mm, and the peak power of the pump to study this effect is \(\tau_{\text{FWHM}}\) = 75 fs. The full-width at half maxima of the pump is \(\tau_{\text{FWHM}}\) = 75 fs. The full-width at half maxima of the pump is \(\tau_{\text{FWHM}}\) = 75 fs. The full-width at half maxima of the pump is \(\tau_{\text{FWHM}}\) = 75 fs. The comparison of the spectral bandwidth, flatness, and conversion efficiency with some previous works. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Structure**} & **MIR-DW Bandwidth (at** & \multirow{2}{*}{**Flatness**} & **Conversion Efficiency** & \multirow{2}{*}{**Reference**} \\  & \(-\)**20 dB Level)** & & & \\ \hline \multirow{2}{*}{Si\({}_{3}\)N\({}_{4}\) single-core waveguide} & \multirow{2}{*}{0 nm} & \(\sim\)15 dB & \multirow{2}{*}{\(<\)1\%} & [6] \\  & & (2.9–3.7 μm) & & \\ \multirow{2}{*}{Si\({}_{3}\)N\({}_{4}\) dual-coupled waveguide} & \multirow{2}{*}{\(\sim\)928 nm} & \(\sim\)19 dB & \multirow{2}{*}{\(\sim\)1–5\%} & [11] \\  & & (2.56–3.65 μm) & & \\ \multirow{2}{*}{LNOI single-core waveguide} & \multirow{2}{*}{N/A} & \(\sim\)15 dB & \multirow{2}{*}{N/A} & [22] \\  & & (1.84–217 μm) & & \\ \multirow{2}{*}{LNOI dual-coupled waveguide} & \multirow{2}{*}{\(\sim\)1630 nm} & \(\sim\)6.31 dB & \multirow{2}{*}{18.48\%} & \multirow{2}{*}{This work} \\  & & (1.92–3.55 μm) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of the spectral bandwidth, flatness, and conversion efficiency with some previous works. Figure 5: (**a**) The instantaneous spectral profiles of the initial pump, and the single-core and the dual-coupled waveguides of \(x\)-cut LNOI at \(L\) = 0.9 mm. The full-width at half maxima of the pump is fixed to \(\tau_{\text{FWHM}}\) = 75 fs. (**b**) The instantaneous spectral profiles of the initial pump, and the single-core and the dual-coupled waveguides of \(x\)-cut LNOI at \(L\) = 1.8 mm. The full-width at half maxima of the pump is fixed to \(\tau_{\text{FWHM}}\) = 75 fs. the peak power of the pump varying from 4 to 8 kW.