## 4 Discussion  
We first investigated the influences of the geometric parameters. Figure 4 shows the simulated reflection spectra when independently adjusting a certain parameter. As seen in Figure 4, the peak exhibited a redshift when the grating height \(h_{2}\), the sub-period \(\Lambda_{2}\), the sub-filling-factor \(f_{2}\), or the structure period \(\Lambda_{1}\) increased. These can be explained based on Equations (2) and (3) as follows. As \(h_{2}\), \(\Lambda_{2}\), or \(f_{2}\) increases, the silicon-to-gold ratio \(p=2f_{2}\Lambda_{2}h_{2}/(\Lambda_{1}h_{1})\) in each unit cell increases. Figure 3: (**a**) Retrieved optical constants for the designed narrowband PA. (**b**) Electric field (map) and displacement current (arrow) distributions. (**c**) Magnetic field distribution. (**d**) Energy dissipation (in W/m\({}^{3}\)). Note that the vertical dashed line in (**a**) and the profiles in (**b–d**) are at \(\lambda=4.071\)\(\upmu\)m. The electromagnetic field and displacement current distributions in (**b,c**) are self-normalized, respectively. The color of the arrows in (**b**) represents the intensity of the displacement current. Accordingly, the effective index of the plasmonic mode \(n_{\text{eff}}\) increases, leading to a redshift of the resonance wavelength. Compared with those in Figure 4a,b, the more significant redshift in Figure 4c is due to the larger increase in \(n_{\text{eff}}\). However, the redshift in Figure 4d is mainly attributed to the increase in \(\Lambda_{1}\). As seen in Figure 4a, a slight increase in the FWHM was observed when \(h_{2}\) increased. This is mainly because the confined magnetic field in the silicon slit and the displacement current in the gold film increased. In Figure 4b, the FWHM also increased slightly with an increased \(\Lambda_{2}\), which mainly resulted from the increase in the silicon slit width (corresponding to the increase in the energy dissipation length). In Figure 4c, the FWHM increased with an increased \(f_{2}\), which is mainly due to the narrower silicon slit at a larger \(f_{2}\). Since the electric field concentrates at the silicon ridges, a narrower silicon slit results in a stronger electric field, thus enlarging the displacement current. Both the increases in displacement current and energy dissipation length can lead to the increase in the energy dissipation rate, thus broadening the FWHM. Compared with those in Figure 4a,b, the FWHM in Figure 4c changed more significantly. The main reason for this is that the electric field at the silicon ridges is sensitive to the distance between the two silicon strips. Therefore, the energy dissipation rate changes dramatically with a different \(f_{2}\). However, in Figure 4d, the FWHM exhibited almost no change with an increased \(\Lambda_{1}\). This is because \(\Lambda_{2}\) was also adjusted to keep \(\Lambda_{2}/\Lambda_{1}\) unchanged. The results suggest that it is convenient to tune the absorption wavelength and bandwidth by adjusting the structure parameters. However, careful attention should be paid to the sub-filling-factor, because the FWHM of the absorption peak is very sensitive to the distance between the two silicon strips. Figure 5 shows the calculated reflection spectra at different angles of incidence, where the incidence was tilted in the _xz_-plane. In Figure 5a, it can be observed that the peak splits at oblique incidence. As the incident angle increases, the two split peaks gradually separate from each other. These phenomena are in accord with Equations (2) and (3). In Figure 5b, we see that there are two narrow peaks with good intensity extinctions (\(\sim\)10 dB) at an incident angle of 2.4 degrees. Furthermore, when the incident angle was 3.8 degrees, there was a narrow absorption peak at \(\sim\)3.8 \(\upmu\)m with high  
intensity extinction (\(\sim\)22 dB). These results indicate that dual-band absorption can be achieved by adjusting the incident angle.