## 3 Results and Discussion
The dual hump shape is more apparent for MIR-DWs generated in the two normal dispersion regions. In the time domain, because of the lower \(N_{\text{soliton}}\), the compressed high order soliton ejects fewer fundamental sub-solitons compared to the case of \(\tau_{\text{FWHM}}=75\) fs. The DW pulses follow the same rules as previously mentioned with respect to the different group velocity. Here, we choose the propagation length of 0.8 mm and 0.86 mm to study the instantaneous spectral and temporal profiles of the MIR-DW generation. As we can see from Figure 4b, there are dual hump-shaped MIR-DWs centered at 2.45 and 3.6 \(\upmu\)m, respectively. The flatness of the SCG in the MIR region is reduced, of which the variation is 7.44 dB from 1.92 to 3.55 \(\upmu\)m. In regard to the bandwidth (at \(-10\) dB level) of the MIR-DW, it is about 881 nm of the MIR-DW between 2 and 3 \(\upmu\)m. Moreover, the conversion efficiency of the MIR-DW is 17.50%. As for the output spectrum with the propagation length of 0.86 mm in Figure 4c, the phenomenon of the dual hump-shaped MIR-DWs generation is more obvious. The locations of the dual hump-shaped MIR-DWs are around 2.5 and 3.6 \(\upmu\)m. In addition, the conversion efficiency of the MIR-DW reaches as high as 22.53%. The \(-\)10 dB level of bandwidth is around 454 nm of the MIR-DW located between 2 and 3 \(\upmu\)m. Figure 4e,f show that the normalized temporal pulse envelopes at the dual-coupled waveguide output also present the soliton fission effect that splits into two components. For the above situations, the MIR-DW is much stronger than the NIR one because the dispersion characteristics at 1.55 \(\upmu\)m pump wavelength satisfy \(\beta_{2}<0\) and \(\beta_{3}<0\). Therefore, the pump energy is more inclined to be transferred to the DW at longer wavelength side. In order to show the advantages of the proposed structure, we compare the performance of SCG and DW generation between the single-core, \(x\)-cut LNOI waveguide (dispersion in Figure 2e) and the proposed dual-coupled LNOI waveguide. Instantaneous spectra at two different propagation lengths are considered with the same pumping conditions, as shown in Figure 5. It can be seen from Figure 5a that when the propagation length is 0.9 mm, the instantaneous spectrum of the dual-coupled waveguide is apparently broadened with the relatively high degree of flatness. In contrast, for the case of the single-core waveguide, the soliton fission length has not been reached. The spectrum broadening determined by the SPM effect is still weak. We further chose a longer propagation length of 1.8 mm to compare the performance of the two structures. Figure 5b shows that there are  
Figure 3: **(a)** Spectral evolution along the propagation length with the full-width at half maxima \(\tau_{\text{FWHM}}=75\) fs. **(b)** The spectra of the pump and the output at \(L=0.9\) mm. **(c)** The spectra of the pump and the output at \(L=0.95\) mm. **(d)** Temporal evolution along the propagation length with the full-width at half maxima \(\tau_{\text{FWHM}}=75\) fs. **(e)** The normalized temporal pulse envelopes at input and output at \(L=0.9\) mm. **(f)** The normalized temporal pulse envelopes at input and output at \(L=0.95\) mm. intense dual hump-shaped MIR-DWs generated at 2-3 \(\upmu\)m and 3-4.5 \(\upmu\)m, respectively, for the dual-coupled waveguide. As for the single-core waveguide, the output spectrum has only one MIR-DW located at 3-4.5 \(\upmu\)m. The intensity of the MIR-DW is much lower than that obtained in the dual-coupled waveguide. Even if we only focus on the MIR-DW generated at 3-4.5 \(\upmu\)m, the conversion efficiency is about 1.48% for the single-core waveguide and 9.20% for the dual-coupled waveguide. Table 1 shows the comparison of the spectral bandwidth, flatness, and conversion efficiency with some previous works. We note that the listed LNOI single-core waveguide is \(z\)-cut [22], and not the same type used in Figure 2e.