## 2 Experimental Setup  
The structure of the multidimensional multiplexing mode-locked laser is shown in Figure 1. Although it is an integrated laser, its dual-ring structure enables extra sub-ring multiplexing for multi-comb generation. This dual-ring structure is realized with a shared common-path and two symmetrical sub-paths. In the common-path, a 0.5-m-longerbium-doped fiber (EDF, Liekki ER110-4/125) is bidirectionally pumped by two 980-nm laser diodes via wavelength-division multiplexers (WDMs). The WDMs in our setup transmit and reflect lights of 1550 nm and 980 nm, respectively. The common path acts as the gain medium of the whole dual-ring laser. To ensure the unidirectional propagation of the laser in each sub-path, two circulators are applied to connect the common path and two sub-paths. Accordingly, two sub-rings are formed and labelled as R-1 (red) and R-2 (blue), respectively. Accordingto the previous studies presented in [21], a passive cancellation of the common mode noise can be achieved with this system structure. In each sub-ring, a semiconductor saturable absorber (SESA, Batop SA-1550-25-2 ps) is applied to achieve the self-starting mode-locking. To ensure a free but recoverable switching among the mode-locking states of the dual-ring integrated laser, a cage module is added in each sub-ring to adjust the intracavity loss. The evolution of the mode-locking status of this dual-ring laser has been experimentally explored in our previous works, reported in [29]. Although only dual-comb generation was analyzed, the law of mode-locking evolution for this dual-ring laser is invariable. The simultaneous mode-locking in both sub-rings always requires a balance of power in the dual-ring structure. This can only be achieved by adjusting the intracavity loss of each sub-ring precisely and continuously. Therefore, the existence of these two cage modules is important to some extent for the proposed multidimensional multiplexing mode-locked laser. Although the function of intracavity loss control can be realized by the in-line tunable fiber optical attenuator as well, it is quite hard to record the mode-locking status and to switch among them in the experiments. In comparison, intracavity loss control with the cage module can be realized in two ways. The combination of rotating scaled zero-order wave plates (Thorlabs, WPH05M-1550 and WPQ05M-1550) and a fixed polarizing beam splitter (Thorlabs, PBS104) ensures precise, continuous and periodically repeated loss control. Furthermore, the micrometers for the aiming adjustment of the collimators enable large-range, precise and recoverable loss control. Moreover, the design of such a free space cage module is required for repetition rate stabilization in the near future. The free space length of both the cage modules are adjusted to be approximately 0.2 m.  
In contrast to the existing multi-comb generation method using a single-ring mode-locked fiber laser, the new dimension of the sub-ring enables various resonance lengths for the generation of individual combs. Accordingly, the repetition rate of the generated combs can be changed within a large range. The repetition rate can be estimated with the parameters of ring laser as follows. \[f_{\text{r}rp}=\frac{c}{L_{\text{eqVR}}}=\frac{c}{n_{1}L_{1}+n_{2}L_{2}+\dots+ n_{m}L_{m}}, \tag{1}\]  
where \(L_{\text{eqVR}}\) stands for the equivalent optical length of the ring laser; \(c\) is the speed of light in a vacuum; \(n_{1}\), \(n_{2}\) and \(n_{\text{m}}\) stand for the refractive index of each segment in the ring laser; and \(L_{1}\), \(L_{2}\) and \(L_{\text{m}}\) are the length of each segment in the ring laser. In our setup, R-1 (including the common path and red sub-path shown in Figure 1) is composed of 0.5 m EDF fiber, 3.7 m SMF-28e+ fiber and 0.2 m free space path. If the refractive index of EDF and SMF-28e+ are both estimated as 1.4679 [30],  
Figure 1: The experimental setup of the multidimensional multiplexing mode-locked laser.