## 1 Introduction  
The application of dual-combs with slightly different repetition rates has enabled a revolutionary new technique for the field of detection and metrology, e.g., laser ranging [1; 2; 3; 4], velocity measurement [5] and precision spectroscopy [6; 7; 8]. By using the dual-comb-based multi-heterodyne detection technique, temporal and spectral information can be extracted with a high resolution and high update-rate simultaneously. Based on the advantages of dual-comb applications, a third comb has been introduced to expand the measurement range in laser ranging [9]. A basic requirement of this tri-comb laser source for multi-heterodyne detection is the different scale of differences among the repetition rates of the tri-comb. In the application of large-range, precision and rapid laser ranging, the desired differences of repetition rates should be variable from kHz to MHz to enable large range coarse measurements and short-range precision measurements [9]. However, the existing dual-comb generation schemes can hardly fulfill the requirements for the tri-comb technique, as has been discussed in a previous review paper [10]. One conventional dual-comb generation scheme relies on the combination of two independent fully-stabilized or free-running comb sources [1; 3; 4; 5; 6; 7; 8]. However, the obvious disadvantage of this scheme is the system's complexity and expensiveness. Another approach to dual-comb generation is based on the multiplexing of parameters of a single-cavity laser. The natural good coherence of generated combs and the simple system structure of this method are very attractive even for multi-comb generation. Based on the existing reports, the multiplexing parameters for dual-comb generation include the propagation direction [11; 12; 13; 14; 15; 16; 17; 18], polarization [19; 20; 21], wavelength [22; 23; 24; 25] and pulse shape [18; 26; 27]. Limited by the single-ring laser cavity, however, the application of any single multiplexing can hardly generate more than two combs. Although tri-combs and even quad-combs have been reportedly generated by wavelength multiplexing [18], the instability and difficulty of their realization prohibits a broader application of the scheme. In 2019, the research group of Prof. [PERSON] proposed a method of multi-comb generation using multidimensional multiplexing [28]. By combining the multiplexing of polarization and wavelength, a tri-comb and a quad-comb were generated in high quality. This is an innovative direction for multi-comb generation. The only drawback is that the repetition rates of generated combs are still close to each other. Thus, the multidimensional multiplexing method does not conclusively solve the problems with existing multiplexing schemes. It is still worthwhile to study methods for the generation of multi-combs with large-range and variable repetition rate differences. Based on our previous studies of the dual-ring mode-locked laser [29], we have found that it is an effective and simple way to achieve variable repetition rates. Similar dual-ring lasers were reported in [26; 27] to generate different types of pulses, e.g., dissipative and conventional solitons or vector and scalar pulses. However, the lack of mode-locking evolution analysis has prohibited its application as a multiplexing parameter for simultaneous multi-comb generation. In this paper, a multidimensional multiplexing mode-locked laser based on a dual-ring integrative structure is proposed for tri-comb generation. To our knowledge, this is the first attempt to employ sub-ring multiplexing in multidimensional multiplexing lasers to generate multi-combs. To prevent the requirement of high-cost equipment for polarization-maintaining fiber splicing, the whole laser is composed only of single-mode fibers and free space. Wavelength multiplexing is applied as the other dimension for tri-comb generation. Two modes of tri-comb generation can be realized with the proposed laser.