## 4 Simulations and Discussions  
### Simulation of MSC  
In our sensor, the wavelength of SPR ranges from 430 nm to 580 nm. Thus, to produce a relatively pure higher-order mode in the visible band and to maximize \(\gamma\), the optimized central wavelength of visible MSC is selected to be 500 nm. Ignoring the influence of external fiber bending on the coupling area, we establish a model of the MSC with Rosft; the area of fusing is shown in Figures 1 and 5. In this model, the radius of two optical fibers is 2.99 um for \(\mathrm{LP_{01}}\) mode, and 5.15 \(\upmu\)m (7.15 \(\upmu\)m) for \(\mathrm{LP_{11}}\) mode (\(\mathrm{LP_{21}}\) mode). The model aims to analyze how the core distance (\(D\)) and the fusion length of fiber (\(L\)) affect \(\gamma\). As shown in Figure 5, the fundamental mode originates from the bottom of the left fiber as the triangle marked in Figure 5a. It should be noted that the incident power equals the total power. In the fusion area, the total power transmits from one fiber to another fiber periodically. In the end, most of the power flows out of the top of the right fiber corresponding to the right fiber shown in Figure 5b, in the form of the \(\mathrm{LP_{11}}\) mode or the \(\mathrm{LP_{21}}\) mode, and a small amount of power, the power of the fundamental mode, exits from the top of the left fiber, corresponding to left fiber, depicted in Figure 5b. Therefore, \(D\) and \(L\) are related to the ratio of the power distribution between the two fibers, which affects \(\gamma\). Thus, optimizing these parameters can improve the \(\gamma\) in the visible band (430-580 nm). Figure 4: The mode effective RI curves for the \(\mathrm{LP_{01}}\), \(\mathrm{LP_{11}}\), and \(\mathrm{LP_{21}}\) modes with the core radius of the fiber \(a\) (um). Using the MOST module, the optimized result is achieved, as shown in Figure 6a. It clearly shows that there are many options for \(D\) and \(L\) to realize \(\gamma>0.95\) at \(\lambda=500\) nm. However, when \(D\) and \(L\) are determined, \(\gamma\) will peak at the central wavelength and decline on both sides, as shown in Figure 6b. Therefore, in order to obtain high \(\gamma\) in the entire visible band, the choice of the parameters for \(D\) and \(L\) is limited. According to our simulation, for the LP\({}_{11}\) mode and the LP\({}_{21}\) mode, \(D\) is \(8.08\), and \(10.61\) um, and \(L\) is \(1253.3\) and \(3700\) um. For example, at the central wavelength, the orange pentagram indicates the parameters (\(D=8.08\) um and \(L=1253.3\) um) of the LP\({}_{01}\)-LP\({}_{11}\) coupler shown in Figure 6(a), and the coupling efficiency \(\gamma\) of this coupler is derived from the power of LP\({}_{11}\) over the power of LP\({}_{01}\)-P\({}_{11}\) in Figure 6b, which is \(95.69\%\). Similarly, \(\gamma\) of the LP\({}_{01}\)-P\({}_{21}\) coupler is \(90.60\%\), which is marked with the black pentagram shown in Figure 6c. As shown in Figure 6b, the efficiency of conversion (\(\gamma\)) is less than 1 and it is relatively low for the LP\({}_{01}\)-LP\({}_{21}\) mode coupler in the visible wavelength range. This is because the evanescent field of the higher-order mode extends significantly, which easily allows the higher-order mode to transition to another fiber and propagate forward as the fundamental mode during the propagation in the fusion area. Figure 5: A model of the visible MSC in the area of fusing. (**a**) is the OXZ plane, and (**b**) is the OXY plane, \(L\) is the fusing length of the fiber, and \(D\) is the core distance between two cores. Figure 6: Simulations of \(\gamma\) at the central wavelength and the power of the MSC in the visible band. (**a**) the coupling efficiency (\(\gamma\)) of the MSC at \(\lambda=500\) nm, with different core distance (\(D\)) and varied coupling length (\(L\)). (**b**) The normalized power under different wavelengths. (**c**) \(\gamma\) of the LP\({}_{01}\)–LP\({}_{21}\) coupler at \(\lambda=500\) nm, with different \(D\) and varied \(L\).