## 3 Simulations and Results
In the series of this group's simulation results, as is shown in Figure 6a, when \(d\) increases from 10 to 30 nm at the interval of 5 nm, the Fano resonance is blue-shifted and the transmittance of the resonance dip increases. When \(d\) = 5 nm, due to rectangular stubs being tangent to the elliptical ring resonator, the Fano resonance cannot be generated and we set the minimum value of \(d\) to be 10. According to this finding, we find that the coupling distance \(d\) between the elliptical ring resonator and the MIM waveguide has a significant influence on the transmittance of the Fano resonance dip. In the following parts, with the aim of getting a low transmittance, the coupling distance \(d\) should be small, so we keep \(d\) = 10 nm constant. In general, the lower transmittance can result in a larger extinction ratio and a smaller FWHM, which is useful for obtaining a high FOM and good sensing resolution. Figure 5: (a) Transmission spectrum of different elliptical long axis \(R_{1}\); (b) fitting line of the wavelength of the Fano resonance dip with the changes of elliptical long axis \(R_{1}\). We also try to change the parameter \(h\), which represents heights of two rectangular stubs. As shown in Figure 6b, when \(h\) changes from 40 to 60 nm, the left dip nearly stays the same, while the right one decreases slightly and the FWHM of the transmission spectrum becomes broader. According to the previous analysis, a smaller FWHM means a higher FOM. We make a line chart to record the FOM of different heights of rectangular stubs depending on the results of the simulation. As vividly shown in Figure 7, when \(h\) increases from 40 to 60 nm at the interval of 5 nm, FOM of the structure decreases from 151.96 to 43.05. However, compared with \(h\) = 60 nm, the sensitivity of \(h\) = 55 nm decreases to 1500 nm/RIU and the transmittance becomes higher at the same time. Therefore, it is necessary to compromise between the sensitivity and FWHM. Finally, we also study the influence of the other parameters \(R_{2},r_{1},r_{2}\) and \(w\) on the properties of the structure. As shown in Figure 8a, when changing the length of the long axis of the inner elliptical resonator \(r_{1}\) from 130 to 170 nm at the interval of 10 nm, we can find that the Lorentzian resonance and Fano resonance are red-shifted. However, the impact of \(r_{1}\) on the Fano resonance is small compared with the Lorentzian resonance. Then, we change the length of the short axis of the inner elliptical resonator \(r_{2}\) from 50 to 90 nm at the interval of 10 nm. The simulation result of different lengths of the  
Figure 6: (a) Transmission spectrum of different coupling distances \(d\); (b) transmission spectrum of different heights of rectangular stubs \(h\). Figure 1:0 The values of FOM when heights of the two rectangular stubs increase from 40 to 60 nm. short axis of the inner elliptical resonator is shown in Figure 8b. It is found that the dip wavelength of the Lorentzian resonance is nearly unchanged, while the Fano resonance is red-shifted. At the same time, effects of \(R_{2}\) and \(w\) on the properties of the structure are also studied, as shown in Figure 8c,d. When the length of the short axis of the outer elliptical resonator becomes larger, the dip wavelength of Fano resonance is blue-shifted, the FWHM gets broader, and the transmittance of the Fano resonance dip becomes lower, which can be explained by the fact that when \(R_{2}\) changes from 130 to 145 nm, the coupling distance between the elliptical ring resonator and MIM waveguide gets smaller and the coupling strength becomes stronger. In addition, when the width of the MIM waveguide gets larger, the dip of the Fano resonance is nearly unchanged. At the same time, in order to ensure that the MIM waveguide only exists the fundamental transverse magnetic (TM\({}_{0}\)), the reasonable value of \(w\) is around 50 nm.