## 3 Results and Discussions  
With the proposed dual-ring integrated laser, the generation of a tri-comb requires the generation of a dual-comb from one of the sub-rings. Owning to the single direction propagation property of the circulators, direction multiplexing cannot be achieved in the sub-rings of our setup. Considering that polarization multiplexing cannot be achieved with the proposed all single-mode fiber laser either, the dimension of wavelength multiplexing was added to generate a dual-comb from one sub-ring. By coupling a 400-mW pump laser from each WDM into the EDF, tri-comb generation was stably achieved. The reasons a total pump power as high as 800 mW was required are as follows. Firstly, the applied EDF fiber was high-concentration doped. The peak core absorption is 110 dB/m at 1530 nm. It is reasonable to believe the absorption rate is higher than other EDF fibers at 980 nm as well. Secondly, the realization of dual-wavelength mode-locking usually requires a higher pump power. It also requires a precise adjustment of cavity loss to change the gain profile of the laser [28], which is achieved by the cage modules in the proposed laser. However, these essential cage modules introduce approximate 40% power loss for each sub-ring in the fiber-space-fiber coupling process, even under the optimized collimation state and the minimized intracavity loss with wave plates. Moreover, either the collimation or the wave plates should always be adjusted away from the optimized state to maintain the power balance between the dual-rings [29]. According to the discussion above, there are two modes of tri-comb generation with the proposed multidimensional multiplexing mode-locked laser. In the first mode, there are two combs of different wavelengths generated from R-1. At the same time, R-2 outputs only one comb simultaneously. Figure 2 illustrates the pulse train and spectral characteristics of the tri-comb generated through this mode. In our setup, a \(1\times 2\) fiber coupler with a 50:50 coupling ratio is reverse applied to combine the tri-comb from different output ports of R-1 and R-2 into a single fiber for the following pulse train and optical spectrum detection. In Figure 2a, the pulse trains from the dual-ring laser of both R-1 and R-2 are detected with one photodetector. Three stable trains of pulses can be observed with different time intervals and optical intensities, which are marked with labels of I, II and III, respectively. It is clear that the spacing between pulse trains I and II is relatively constant, compared to the spacing between pulse trains II and III. This phenomenon is caused by the intentionally introduced difference in resonance length for the sub-rings. A more detailed discussion of the repetition rates will be performed in the following paragraphs. According to this effect, we can identify that pulse trains I and II are generated from R-1 and pulse train III is generated from R-2. For the optical intensities, despite the peak amplitudes changing over time, the relative intensity among the three pulse trains is still clear enough to distinguish them. More importantly, this relative intensity can be applied to identify the pulse trains from the peaks of the detected optical spectrum. In Figure 2b, the red solid line and blue solid line stand for the optical spectrum from the separated outputs of R-1 and R-2, respectively. For the output spectrum of R-1, two obvious spectral peaks can be found with different peak intensities. With the help of the relative intensity shown in Figure 2a, the average power of comb I can be estimated to be higher than that of comb II. Thus, the stronger peak of 1569.1 nm center wavelength and 1.4 nm bandwidth in Figure 2b corresponds to pulse train I. Centered on a 1556.1 nm wavelength, the weaker peak of 3.3 nm bandwidth in Figure 2b is the optical spectrum of pulse train II, depicted in Figure 2a. A noise peak with an unstable intensity and shape can be observed at approximately 1546.5 nm in Figure 2b as well. We attribute this to the excess energy in R-1.