## 3 Results and Physical Understanding  
Figure 2 shows the simulated spectral responses of our structure at normal incidence. The structure parameters were chosen to be \(\Lambda_{1}\) = 3.5 \(\upmu\)m, \(h_{1}\) = 0.1 \(\upmu\)m, \(h_{2}\) = 0.7 \(\upmu\)m, \(\Lambda_{2}\) = 0.7 \(\upmu\)m, and \(f_{2}\) = 0.37. Based on these parameters, we knew that the width of each silicon strip was \(f_{2}\Lambda_{2}\) = 259 nm. As can be observed in Figure 2, the transmission was nearly zero in the investigated spectral region. There was only a single reflection dip across the 3-5 \(\upmu\)m wavelength range. The reflection dip was at a wavelength of 4.071 \(\upmu\)m with an intensity extinction of \(\sim\)21 dB. Thus, based on the proposed structure, a mid-IR narrowband notch filter operating in the reflection mode can be achieved. In comparison with notch filters presented in [4] and [19], our structure had the advantage of a narrower stop band. Compared with those in [17; 26; 27], our filter was simpler in structure.  
The absorption spectrum A(\(\lambda\)) was calculated by \(1-\text{T}(\lambda)-\text{R}(\lambda)\). The backscattered light was included in R(\(\lambda\)). We obtained a single absorption peak at 4.071 \(\upmu\)m, as shown in Figure 2. The peak had a high absorption rate on resonance (\(\sim\)99.2%). The full-width at half-maximum (FWHM) linewidth of the absorption peak was \(\sim\)31 nm, corresponding to a Q-factor of \(\sim\)137. These results indicate that our structure can realize perfect absorption of the TM-polarized light at a wavelength of 4.071 \(\upmu\)m.  
In order to understand the spectral responses illustrated in Figure 2, we retrieved the optical constants for the designed narrowband PA. Figure 3a plots the retrieved \(\varepsilon(\lambda)\) and \(\mu(\lambda)\), which were obtained by the inversion of the transmission and reflection coefficients (\(S\) parameters) [33]. As is seen, either Re(\(\varepsilon\)) or Re(\(\mu\)) was negative in the investigated wavelength range. This indicates that there was no transmission across the wavelength range. Moreover, at a wavelength of 4.071 \(\upmu\)m (dashed line),  
Figure 2: Simulated reflection (R), transmission (T), and absorption (A) spectra at normal incidence.  
both Re(\(\varepsilon\)) and Re(\(\mu\)) crossed zero. This suggests that the reflection at a wavelength of 4.071 \(\upmu\)m was minimized [34]. As a result, at a wavelength of 4.071 \(\upmu\)m, the absorption was maximized.  
To further understand the absorption mechanism, the electric field \(|\mathbf{E}|\) and the displacement current \(\mathbf{D}=\varepsilon\mathbf{E}\) distributions are plotted in Figure 3b; the magnetic field \(|\mathbf{H}|\) distribution is plotted in Figure 3c. In Figure 3b, we can observe that the electric field was concentrated strongly at the silicon ridges. The silicon slit was surrounded by a strong displacement current loop. For this reason, the magnetic field confined in the silicon slit was strong, as shown in Figure 3c. These results indicate that the electromagnetic resonance was excited in our structure. Figure 3d illustrates the simulated energy loss at 4.071 \(\upmu\)m. It is observed that the energy dissipation was attributed to the Ohmic loss in the gold film. Energy loss mainly occurred in the gold film beneath the two silicon strips, because there the displacement current was strong.  
In comparison with the usual dielectric-metal structures in the literature [19], our structure enabled the electromagnetic field on resonance to be away from the gold film. Therefore, the loss of the resonance mode decreased, and the FWHM of the absorption peak was narrowed. In addition, the electromagnetic field outside the silicon strips would be advantageous for refractive index sensing of the surrounding medium. In the following, we will show this.