## 3 Theoretical Analysis  
### The Power of Different Modes in the Cladding  
The sensitivity of the sensor could be improved by using only the higher-order mode to excite SPR in this sensor. To verify this assertion, the powers of the LP\({}_{01}\), LP\({}_{11}\), and LP\({}_{21}\) modes in the cladding are calculated. The electric field \(F_{I}(R)\) of the LP\({}_{\rm{nm}}\) mode (ignoring the phase \(e^{il\phi},l=0,1,\cdots\)) in the step fiber is expressed as [4]  
\[\bigg{[}\frac{\rm{d}}{\rm{d}R^{2}}+\frac{1}{R}\frac{\rm{d}}{\rm{d}R}-\frac{l^ {2}}{R^{2}}+a^{2}\Big{(}k^{2}n_{co}^{2}-\beta^{2}\Big{)}\bigg{]}F_{I}(R)=0\;0 \leq R<1 \tag{1a}\] \[\bigg{[}\frac{\rm{d}}{\rm{d}R^{2}}+\frac{1}{R}\frac{\rm{d}}{\rm{d}R}-\frac{l^ {2}}{R^{2}}-a^{2}\Big{(}\beta^{2}-k^{2}n_{cl}^{2}\Big{)}\bigg{]}F_{I}(R)=0\;1 \leq R<\infty \tag{1b}\]  
where \(R\) is the normalized radius, \(a\) is the radius of the fiber, \(k\) is the wavenumber in the vacuum, \(\beta\) is the propagation constant, \(n_{co}\) is the core index, and \(n_{cl}\) is the RI of the cladding. \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Index** & **Parametric Meaning** & **Index** & **Parametric Meaning** \\ \hline \(F_{I}\) (\(R\)) & The electric field & \(\gamma\) & The efficiency of converting the fundamental mode into a higher order mode of MSC \\ \(R\) & The normalized radius & \(\lambda\) & The wavelength \\ \(a\) & The radius of the fiber & \(n_{eff}\) & The effective RI of the LP\({}_{\rm{nm}}\) mode \\ \(k\) & The wavenumber in the vacuum & \(D\) & The core distance \\ \(\beta\) & the propagation constant & \(L\) & The fusion length of fiber \\ \(n_{co}\) & the RI of the core & \(r_{co}\) & The radius of the core in the simulations \\ \(n_{cl}\) & the RI of the cladding & \(r_{cl}\) & the simulations \\ \(U\) & the core parameter & \(Loss\) & The loss of SPR in the sensing area \\ \(W\) & the cladding parameter & \(n\) & The refractive index in the simulations \\ \(V\) & The normalized frequency & \(I_{\rm{loss}}\) & The total loss of the LP\({}_{\rm{nm}}\) mode, including \\ \(\eta_{LP01-light}\) & The efficiency of a light source producing & \(I_{min}\) & The minimum detected loss \\ \(\eta_{LPnm-cl}\) & The proportion of the power of the LP\({}_{\rm{nm}}\) mode in the cladding to the total power & \(P_{in}\) & The incident power \\ \(\eta\) & The ratio of the power enhancement of the & & \\ higher order mode to the & & \(P_{out}\) & The output power \\ fundamental mode & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter indices related to the SPR sensor. By substituting Equation (1) into the boundary conditions that \(F_{I}(R)\) and \(F_{I}^{\prime}\) continue when \(R=1\), characteristic eigenequations follow these relationships:  
\[U\ J_{n+1}(U)/J_{n}(U)=WK_{n+1}(W)/K_{n}(W) \tag{2a}\] \[U\ J_{n-1}(U)/J_{n}(U)=WK_{n-1}(W)/K_{n}(W)\] (2b) \[V=\sqrt{U^{2}+W^{2}}=k\cdot a\sqrt{n_{co}^{2}-n_{cl}^{2}} \tag{2c}\]  
where \(n\) is the order of mode, \(U\) is the core parameter, \(W\) is the cladding parameter, and \(V\) is the normalized frequency [4]. The bisection method is adopted to solve these equations using some subroutines of Fortran [12] (GNU Fortran 10.3.0, Rev5, built by the MSYS2 project). The proportion of the power of the LP\({}_{\text{nm}}\) mode in the cladding to the total power is expressed as [4]  
\[\eta_{LPnm-cl}=\frac{\int_{1}^{\infty}F_{I}^{2}(R)RdR}{\int_{0}^{\infty}F_{I} ^{2}(R)RdR} \tag{3}\]  
The percentage of cladding power to the total power of the first three modes (the LP\({}_{01}\), LP\({}_{11}\), and LP\({}_{21}\) mode) is illustrated in Figure 2. The value of the LP\({}_{21}\) mode is discontinuous in the inset of Figure 2 due to the defined error range (0.00001) of the bisection method. This indicates that in the cladding, the power of the higher-order modes is higher than that of the fundamental mode. In other words, compared to the fundamental mode, the higher-order modes can effectively improve the power of the evanescent wave for SPR excitement.