## 2 Structural Model and Analysis Methods  
A schematic diagram of the presented MIM bus waveguide coupled with two rectangular baffles and a CCDRR is displayed in Figure 1. The width \(w\) of the MIM waveguide, two rectangular baffles, and two annulus cavities remained invariable at 50 nm to ensure that the waveguide only supports the transverse magnetic field (TM\({}_{0}\)) mode. \(g\) represents the coupling gap between the bus waveguide and the CCDRR. The heights of the two rectangle baffles and the separation between them are signified as \(h\) and \(H\), respectively. \(R_{1}\) and \(r_{1}\), \(R_{2}\) and \(r_{2}\) express the external and internal radii of the outer ring and inner ring, respectively. \(d\) is defined as the width of the baffles connecting two rings in the concentric ring, which is fixed at 40 nm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **Sensitivity (nm/RIU)** & **FOM** \\ \hline This work & 2260 & 56.5 \\ [PERSON] et al. [22] & 1500 & 65.2 \\ [PERSON] et al. [23] & 1125 & 75 \\ [PERSON] et al. [24] & 1550 & 43.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of various plasmonic sensors. The white part and green part in Figure 1 represent air and sliver, respectively. The relative permittivity \(\varepsilon_{d}\) of air is 1. Based on the Debye\(-\)[PERSON] dispersion model [31], the description of the relative dielectric constant of Ag is as follows:  
\[\varepsilon(\omega)\!=\varepsilon_{\infty}+\frac{\varepsilon_{S}-\varepsilon_{ \infty}}{1+i\omega\tau}+\frac{\sigma}{i\omega\varepsilon_{0}} \tag{1}\]  
where \(\varepsilon_{\infty}=3.8344\) is the boundless frequency dielectric constant and \(\varepsilon_{s}\!\!=-9530.5\) is the static dielectric constant. The relaxation time and the conductivity of Ag are regarded as \(\tau=\)7.35 \(\times\) 10\({}^{-15}\) s and \(\sigma=\)1.1486 \(\times\) 10\({}^{7}\) S/m, respectively. The formula of the TM\({}_{0}\) mode of the MIM waveguide is as follows [32]:  
\[\text{tan}h(k\omega)\!=-\frac{2k\eta\varepsilon_{c}}{k^{2}\!+\!p^{2}\alpha_{c }^{2}} \tag{2}\]  
where \(k=2\pi/\lambda\) expresses the wave vector in the waveguide, and in free-space, \(k\) is taken as \(k_{0}\!\!=2\pi/\lambda_{0}\); \(\alpha_{c}=\sqrt{k_{0}^{2}(\varepsilon_{in}-\varepsilon_{m})+k^{2}}\) and \(p=\varepsilon_{in}/\varepsilon_{m}\); among them, \(\varepsilon_{in}\) and \(\varepsilon_{m}\) are the permittivity of the insulator and metal, respectively. By analyzing the shifts of the Fano resonance wavelength, the sensing performance of the proposed structure in the waveguide coupled system was investigated. The transmission wavelengths and the effective refractive index's real part in the MIM waveguide can be expressed on the foundation of the standing wave theory as follows [33; 34]:  
\[\lambda_{m}=\frac{2 Re\Big{(}n_{\textit{eff}}\Big{)}L}{m-\frac{\Psi_{r}}{\pi}} \ (m=1,2,\ldots) \tag{3}\]  
\[Re\Big{(}n_{\textit{eff}}\Big{)}=\sqrt{\varepsilon_{m}+\left(\frac{k}{K_{0}} \right)^{2}} \tag{4}\]  
where \(L\) indicates the circumference of the ring cavity; \(\psi_{r}\) signifies the phase shift caused by the reflection of SPPs at the metal\(-\)insulator boundary surface; and \(m\) is a positive integral number, i.e., the number of antinodes of SPPs. The characteristics of the sensor can be evaluated by two important parameters, namely, sensitivity (\(S\)) and _FOM_, which are expressed by the following equation [35]:  
\[S= \frac{\Delta\lambda}{\Delta n} \tag{5}\]  
Figure 1: Two-dimensional schematic diagram of a metal\(-\)insulator\(-\)metal (MIM) waveguide with twin symmetrical rectangle baffles coupled with a connected concentric double rings resonator (CCDRR). \[FOM=\frac{S}{FWHM} \tag{6}\]  
where \(\Delta\lambda\) and \(\Delta n\) are the variation of resonant wavelength and refractive index, respectively. In the next part of the paper, a simulation was run using COMSOL Multiphysics 5.4a.