## 3 Results and Discussions
A fine adjustment of the wave plates or the space collimation in cage modules should remove this noise peak. However, the stability of the dual-wavelength comb generation in R-1 was affected in our experiments. We will do more research on this to further improve the results in the near future. For the output spectrum of R-2, depicted by a blue solid line, the only peak is centered on 1557.3 nm with a bandwidth of 4.0 nm. Figure 3 illustrates the radio frequency (RF) characteristics of the generated tri-comb with dual-comb and single-comb outputs from R-1 and R-2, respectively. The outputs from R-1 and R-2 were individually photodetected and monitored by use of a spectrum analyzer with a resolution bandwidth (RBW) of 180 Hz. In Figure 3a, two strong peaks of 47.00198 MHz and 47.00436 MHz can be observed from R-1 with a signal-to-noise ratio (SNR) of over 40 dB. In comparison, there is only one peak of 49.59724 MHz from R-2 shown in Figure 3b. Considering the measurement error of each segment fiber, the experimental result of repetition rates from R-1 and R-2 fit very well with the theoretical analysis presented in Section 2. Figure 2: The pulse train and spectral characteristics of the generated tri-comb with dual-comb and single-comb output from sub-ring 1 and sub-ring 2, respectively. (**a**) Pulse trains, (**b**) optical spectrum. I and II stand for the pulse trains from sub-ring 1, whereas III stands for the pulse train from sub-ring 2. R-1, sub-ring 1. R-2, sub-ring 2. In Figure 3a, there is a slight difference of 2.38 kHz between the two repetition rates of comb I and II from the same R-1. We attribute this phenomenon to the dispersion of different center wavelengths. A brief theoretical analysis can be conducted as follows. According to the definition of repetition rate, the ratio of dual-comb repetition rates from R-1 can be calculated as  
\[\frac{f_{\textit{r}\textit{p}\Pi}}{f_{\textit{r}\textit{p}\Pi\Pi}}=\frac{\frac{c }{n_{\textit{g}\Pi}}}{\frac{c}{n_{\textit{g}\Pi}}L}=\frac{n_{\textit{g}\Pi}}{n_ {\textit{g}\Pi}}, \tag{2}\]  
where \(f_{\textit{r}\textit{p}\Pi}\) and \(f_{\textit{r}\textit{p}\Pi}\) stand for the repetition rates of pulse trains I and II, respectively; \(L\) stands for the equivalent geometrical length of R-1; and \(n_{\textit{g}\Pi}\) and \(n_{\textit{g}\Pi}\) represent the equivalent group refractive index for pulse trains I and II, respectively. As the real R-1 is composed of several different types of fibers and some free space, the equivalent geometrical length and equivalent group refractive index are applied here to accomplish the rough analysis. Based on this approximation, an inverse relation between the repetition rate and the equivalent group refractive index is revealed in Equation (2). As discussed in Section 2, both sub-rings operate in the anomalous-dispersion regime, where the group velocity dispersion parameter of \(\beta_{2}\) is a negative number. According to the definition of \(\beta_{2}\) in [34], the relationship between equivalent group refractive index \(n_{g}\) and optical frequency \(\omega\) can be expressed as  
\[\beta_{2}=\frac{1}{c}\frac{\mathrm{d}n_{g}}{\mathrm{d}\omega}<0. \tag{3}\]  
According to Equations (2) and (3), an inverse relationship between the repetition rate and center wavelength of generated combs can be derived, considering the inverse relationship between wavelength and optical frequency. Therefore, the left stronger peak of 47.00198 MHz repetition rate in Figure 3a belongs to  
to the desired inter-mode interference of combs I and II, the extra interference of a different order of comb modes leads to the side peaks of spectrum shown in Figure 4b. As the number of comb modes needed to generate these extra interferences is much less than the modes for the repetition rate, the SNR of these extra peaks is usually less than 20 dB. In the other tri-comb generation mode of the proposed multidimensional multiplexing mode-locked laser, only one comb is generated from R-1, while the R-2 outputs a dual-comb of two wavelengths simultaneously.