## 4 Simulations and Discussions
### Simulation of the Sensing Area for RI Measurement  
Utilizing the higher-order modes produced by the MSC, we designed an RI sensing area of this sensor based on SPR, as shown in Figure 7.  
In this sensing area, most cladding of the fiber has been removed, and the exposed core is coated by a 40 nm thick silver film [13]. The thickness of the 40 nm silver film is selected according to multiple simulations of structures with different thicknesses. Such a structure can excite SPR between the silver film and the evanescent wave. Ignoring the discontinuity of the interface between the sensing region and the optical fiber, we conduct a simplification of the 3D model shown in Figure 7a; thus, we simulate it in a 2D form, as shown in Figure 7b, since the mode field is same in the direction of propagation. In our model (COMSOL Multiphysics), the radius of the core is 6 um and that of the cladding is 18 um. We select Frequency Domain (ewfd) in Electromagnetic Waves module in which the scattered wave type is set as the plane wave in the Scattering Boundary Condition and the Transition Boundary Condition is set to Ag film. Additionally, the physics-controlled mesh is used in the Mesh, as depicted in Figure 7c. The loss of SPR is given by [14]  
\[Loss=8.686\frac{2\pi}{\lambda}\Im n_{neff}\left(\mathrm{dB/m}\right) \tag{6}\]  
where \(n_{neff}\) is the RI of the LP\({}_{\mathrm{nm}}\) mode. As shown in Figure 8, the loss of the LP\({}_{\mathrm{nm}}\) mode is related to the polarization because of the noncircumferential symmetry of the sensing area of this sensor, as shown in Figure 7. If all the cladding of the sensing area is replaced by silver film, the loss of the LP\({}_{\mathrm{nm}}\) mode is assumed to be independent of the polarization, and its distribution is marked by the boxes in Figure 8.  
Among the modes shown in Figure 8, the LP\({}_{\mathrm{01y}}\), LP\({}_{\mathrm{11y2}}\), and LP\({}_{\mathrm{21x1}}\) mode are identified as the LP\({}_{\mathrm{01}}\), LP\({}_{\mathrm{11}}\), and LP\({}_{\mathrm{21}}\) modes due to their loss curves with the largest variation. Under such conditions, where these polarization direction modes are adopted, the sensing structure shows high loss and are extremely sensitive in solutions with different RI. To verify this view, the response curves of the three modes to RI are shown in Figure 9.  
Figure 7: Schematic diagram of the sensing area of this sensor with material settings. (**a**) 3D mode. (**b**) Cross section of the sensing area. (**c**) Mesh of the sensing area. The intensity modulation method [15] is adopted to demodulate the losses of the above modes (the LP\({}_{01}\), LP\({}_{11}\), and LP\({}_{21}\) mode), ranging from 430 nm to 580 nm. This result is defined as \(\mathbf{I_{loss}}\). The fitting results of \(I_{loss}\) are shown in Figure 10, and the sensitivity defined as the loss over RI of each order mode is shown in Table 2. Actually, in the sensing area, the LP\({}_{11}\) mode and the LP\({}_{21}\) mode are used separately to excite SPR to improve the sensitivity of this sensor. The sensitivity of the fundamental mode is calculated only for the comparison among modes. It illustrates clearly that the sensitivity of the sensing area of this sensor for each mode exhibits high linearity. Besides linearity, the value of sensitivity of the LP\({}_{11}\) mode and the LP\({}_{21}\) mode is increased by approximately 330% and 360%, respectively. These values are attributed to the controlled energy distribution of the mode in the sensing area. Therefore, this means that using higher-order modes can improve the sensitivity of the SPR sensor. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{LP_{01}}\) & \(\mathbf{LP_{11}}\) & \(\mathbf{LP_{21}}\) \\ \hline Sensitivity & 1.95 & 8.45 & 9.03 \\ Normalization & 1.00 & 4.32 & 4.62 \\ of Sensitivity & 0.98 & 0.98 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The sensitivity of the LP\({}_{\mathrm{nm}}\) mode by intensity modulation (\(\times 10^{-3}\) dB/RIU).