## 3 Simulations and Results
As a consequence, the structure has a relatively low transmittance at the dip. Figure 2: Transmission spectrum of the whole system (the black line), without two rectangular stubs (the red line), without the elliptical ring resonator (the blue line), with only one rectangular stub (the green line) when \(R_{1}~{}=~{}300\) nm, \(R_{2}~{}=~{}140\) nm, \(r_{1}~{}=~{}150\) nm, \(r_{2}~{}=~{}70\) nm, \(d~{}=~{}10\) nm, \(l~{}=~{}480\) nm, \(h=60\) nm. The Fano resonance is easily influenced by changing refractive indices of the dielectric due to its asymmetrical and sharp line shape. Therefore, we can make use of the sensitivity of the Fano resonance to monitor the changes of vital parameters. There are two vital parameters for evaluating the sensing performance of the Fano system: the sensitivity S and the FOM, which are expressed as (2) and (3):  
\[\text{S}=\frac{\Delta\lambda}{\Delta n} \tag{2}\]  
\[\text{FOM}=\frac{S}{\textit{FWHM}} \tag{3}\]  
Herein, _FWHM_ means the full width at the half maximum. Figure 4a describes the transmission spectrum of the system when the refractive index changes from 1 to 1.05 at an interval of 0.01. When the refractive index changes, other parameters of the structure remain the same, as shown in Figure 3a. The fitting line shows that the sensitivity is 1550 nm/RIU with \(R_{1}~{}=~{}300\) nm, \(R_{2}~{}=~{}140\) nm, \(r_{1}~{}=~{}150\) nm, \(r_{2}~{}=~{}70\) nm, \(d~{}=~{}10\) nm, \(l~{}=~{}480\) nm, \(h=60\) nm., and its FOM is 43.05. Figure 4: **(a)** Transmission spectrum of different refractive indices; **(b)** the fitting line of the wavelength of the Fano resonance dip with the changes of refractive index. Figure 3: The Hz field distribution of the resonant dip: **(a)** the whole system at \(\lambda=1105\) nm; **(b)** the whole system at \(\lambda=1545.5\) nm; **(c)** without the elliptical ring resonator at \(\lambda=1105\) nm; **(d)** without two rectangular stubs at \(\lambda=1105\) nm. In this part, we mainly study the influence on the transmission spectrum when we change several main parameters of the structure. First of all, when the influence of elliptical long axis \(R_{1}\) is studied, \(R_{1}\) increases from 260 to 300 nm, and other parameters, such as \(R_{2}\leavevmode\  
ext{nm}\) are kept constant. In addition, the proportional relationship between \(R_{1}\leavevmode\
obreak\ \text{and}\leavevmode\
obreak\ r_{1}\) is two at this time. For an MIM waveguide coupling resonator, the resonance wavelength can be calculated by standing wave theory:  
\[\lambda_{m}=\frac{2 Re\big{(}n_{eff}\big{)}L}{m-\psi_{r}/n} \tag{4}\]  
where \(L\) is the length of the cavity, positive integer \(m\) is the number of antinodes of the standing SPP wave, \(\psi_{r}\) is the phase shift of the beam reflected at one end of the cavity, and \(Re\big{(}n_{eff}\big{)}\) is the real part of the effective index of an MIM waveguide. From the results of simulation shown in Figure 5a, we can find that as \(R_{1}\) increases, the Fano resonance is red-shifted, which is consistent with Equation (4). That is, \(\lambda_{m}\) increases when \(L\) increases. Here, the increase of \(L\) is achieved by increasing \(R_{1}\). Meanwhile, when \(R_{1}\leavevmode\
obreak\ \text{increases}\), the transmittance of resonance dip on the left dip increases slightly, while the right one shows an obvious decrease that is desired. From the fitting lines of various \(R_{1}\) in Figure 5b, we can find that with the increase of \(R_{1}\), sensitivities of the structure become better. Theoretically, we can get the greater sensitivity by increasing \(R_{1}\) continuously. However, the increase of \(R_{1}\) is not conducive to the miniaturization of the cavity area. At the same time, in order to achieve a low transmittance, \(R_{1}\) is not less than 290 nm. In addition, we also study the influence of the coupling distance \(d\) on the dip of Fano resonance.