## 4 Conclusions  
In summary, we designed and dispersion-engineered a structure of the dual-coupled waveguide on \(x\)-cut LNOI to achieve highly efficient and flat MIR-DW generation. We studied the mode characteristics and the dispersion engineering of the supermodes of this dual-coupled waveguide. By varying the geometric parameters of the proposed waveguide, i.e., the etch depth, the sidewall angle, and the gap between the dual-coupled waveguide, the influences on the dispersion profiles were revealed. As the dispersion profile of our structure is relatively flat, it effectively generates the MIR-DW having high efficiency and a high degree of flatness. A high spectral flatness of 6.31 dB is achieved in the wavelength range of 1.92-3.55 \(\upmu\)m by reasonable selection of the propagation length. The conversion efficiency of the MIR-DW is as high as 19.31% when the propagation length is longer. This conversion efficiency is relatively high and has the potential to achieve the MIR-DW of the highest conversion efficiency with pumping of 1.55 \(\upmu\)m sources in the near-infrared region. We note that the SHG effect, which exists in the LNOI platform, was not considered in the study. The frequency conversion by the SHG effect does grab energy from the pump source. Consequently, the conversion efficiency of MIR-DW generation is affected and reduced. However, since the conversion efficiency of the SHG effect is typically low, it will not significantly degrade the conversion efficiency of the MIR-DW. We further studied the influence of the pulse width and coupled pulse energy variations on the MIR-DW generation. We found that the dual hump-shaped MIR-DW generation is more obvious when the pulse width becomes shorter and the conversion efficiency of the MIR-DW is higher if we choose an appropriate propagation length. Although the values of the conversion efficiency oscillate when the coupled pulse energy is varied, it is almost always above 10% and has a peak value of 19.31% at 190 pJ. Moreover, we can obtain the spectral flatness as high as 6.31 dB when the coupled pulse energy is 190 pJ. Our results confirm that thin-film LNOI is a promising platform for efficient, broadband, and flat MIR-DW generation. In principle, mode coupling can be engineered at arbitrary wavelength regions on arbitrary material platforms. The results presented in this study can benefit a wide variety of technologies and applications.  
Conceptualization, J.J. and Z.K.; methodology, J.J. and Z.K.; software, J.J.; validation, Z.K.; writing--original draft preparation, J.J.; writing--review and editing, Z.K., Q.H. and S.H.; visualization, J.J.; supervision, Z.K. and S.H. All authors have read and agreed to the published version of the manuscript. This research was funded in part by the National Natural Science Foundation of China under Grant 62075188 and Grant 11621101, in part by the Zhejiang Provincial Natural Science Foundation of China under Grant LY21F050007, in part by Special Development Fund of Shanghai Zhangjiang Science City, in part by Key Research and Development Program of Zhejiang Province under Grant 2022C03051, in part by the Fundamental Research Funds for the Central Universities under Grant 2019 FZA5002, and in part by the Ningbo Science and Technology Project under Grant 2018B10093. We thank [PERSON] for the helpful discussion and comments on this manuscript. The authors declare no conflict of interest.