## 4 Conclusions  
In summary, a multidimensional multiplexing mode-locked laser based on a dual-ring integrative structure is proposed for tri-comb generation in this paper. To our knowledge, this is the first attempt to introduce sub-ring multiplexing into a multidimensional multiplexing mode-locked laser for simultaneous multi-comb generation. Combining sub-ring multiplexing and wavelength multiplexing, the proposed dual-ring integrated laser can generate a tri-comb through two modes. In the first mode, sub-ring 1 produced a dual-comb of 1569.1 nm and 1556.1 nm center wavelengths. The corresponding bandwidths of the optical spectrum were 1.4 nm and 3.3 nm, respectively. The relative intensity of pulse trains and optical spectrum peaks were used to identify the generated combs. Based on the definition of dispersion, a rough analysis of the relationship between wavelength and equivalent refractive index was conducted to distinguish the repetition rates of 47.00198 MHz and 47.00436 MHz for the dual-comb above. Furthermore, the small peaks in the RF spectrum of the dual-comb were confirmed to be generated by extra interference among the RF signals. The single comb from sub-ring 2 showed a 1557.3-nm center wavelength, 4.0-nm optical spectrum bandwidth and a 49.59724-MHz repetition rate. The minimum and maximum repetition rate differences of the generated tri-combs were 2.38 kHz and 2.59526 MHz, respectively. In comparison, the other tri-comb generation mode was realized by sub-ring 2 producing a dual-comb. In this case, the minimum and maximum repetition rate differences of the generated tri-comb were able to be changed to 2.74 kHz and 2.59720 MHz. This was achieved by switching the operation mode of the same dual-ring integrated mode-locked laser only. We believe the proposed new dimension of sub-ring multiplexing and the dual-ring integrated mode-locked laser could meet the needs of multi-comb generation with large-range repetition rate differences. Moreover, the repetition rate differences of generated combs can be freely adjusted by changing the resonance length of sub-rings with the proposed new dimension of sub-ring multiplexing. The obtained results indicate that our method can offer a powerful scheme for future multi-comb generation and its application in multi-heterodyne detection-based laser ranging and spectroscopy.  
Figure 5: The output characteristics of the generated tri-comb with single-comb and dual-comb outputs from sub-ring 1 and sub-ring 2, respectively. (**a**) Pulse trains, (**b**) optical spectrum, (**c**) RF spectrum of sub-ring 1 and (**d**) RF spectrum of sub-ring 2. IV designates the pulse train from sub-ring 1, whereas V and VI designate the pulse trains from sub-ring 2. R-1, sub-ring 1. R-2, sub-ring 2.  
**Author Contributions:** Conceptualization, R.Y.; methodology, R.Y. and H.S.; validation, H.S., H.L. and J.W.; formal analysis, R.Y. and H.S.; investigation, J.X.; resources, H.S. and H.L.; data curation, H.S. and H.L.; writing--original draft preparation, R.Y. and H.S.; writing--review and editing, H.Y. and H.F.; visualization, J.W.; supervision, J.T.; project administration, P.H.; funding acquisition, R.Y. and P.H. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the National Major Science and Technology Projects of China (2017 ZX02101006-005), National Natural Science Foundation of China (NSFC) (51605120, 51675138, 61675058), China Postdoctoral Science Foundation (2018T110290), and the Fundamental Research Funds for the Central Universities (HIT. NSRIF. 2020103).  
**Conflicts of Interest:** The authors declare no conflict of interest.