## 4 Discussion  
To physically understand the sensing properties of the metagrating sample, we calculated its reflection spectra using a Lumerical FDTD Solver (Lumerical Solutions, Vancouver, BC, Canada). In the calculations, refractive indices of different materials are plotted in Figure 5a, where the Cu and Au are described with Drude Model. The plasma frequency and damping frequency of Cu are taken as \(6.38\times 10^{4}\) cm\({}^{-1}\) and \(2.55\times 10^{2}\) cm\({}^{-1}\), and those of Au are \(7.25\times 10^{4}\) cm\({}^{-1}\) and \(2.16\times 10^{2}\) cm\({}^{-1}\), respectively, as given in a previous study [38]. The refractive index of ZnSe is taken as 2.42, as reported in another study [39]. For the ODT, its refractive index is considered as a constant of 1.46 [40]. The thickness of ODT is taken as 3 nm, while the geometric dimensions of other layers are given in Figure 1a. Our simulated wavelength shifts of the four modes in the metagrating are plotted in Figure 4b. Their detailed wavelength shift values are given in Table 3. These simulated results are in general agreement with the measured results in Figure 4a and Table 2. There are small quantitative discrepancies in shift values between theory and experiment, which likely originate from uncertainties in optical parameters of materials used in calculations. In addition, the P2 resonance is unresolved in Table 2 due to insufficient signal-to-noise ratio in the measurement. The different sensing behaviors of LSPRs and PSPs can be understood from their field properties, as shown in Figure 5b-e. The LSPRs L1 and L2 are generally more confined in space than the PSPs P1 and P2, and thus the LSPRs have larger field enhancement. The surface field enhancement factor was obtained by calculating the mode field profile first, and then by normalizing the maximum field at the top surface of the metagrating to the incident light. In particular, surface fields of the L1 and L2 modes are enhanced by a factor of 140, while the P1 and P2 modes are enhanced by only 10. Besides this difference in field enhancement, the P1 and P2 modes, on the other hand, exhibit larger effective sensing area with their fields spreading not only on Au strips but also on the ZnSe surface, which is beneficial for accessing more ODT molecules. Overall, the field enhancement is the dominant factor, which explains the better sensitivity of the LSPRs. It is noted that our comparison on sensing properties of LSPRs and PSPs is made for an ultrathin monolayer \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{\theta}\) (deg)} & \multicolumn{2}{c}{**Mode L1**} & \multicolumn{2}{c}{**Mode P1**} & \multicolumn{2}{c}{**Mode L2**} & \multicolumn{2}{c}{**Mode P2**} \\ \cline{2-9}  & \(\boldsymbol{\lambda}\) (\(\boldsymbol{\mu}\)m) & \(\boldsymbol{\Delta}\) (nm) & \(\boldsymbol{\lambda}\) (\(\boldsymbol{\mu}\)m) & \(\boldsymbol{\Delta}\) (nm) & \(\boldsymbol{\lambda}\) (\(\boldsymbol{\mu}\)m) & \(\boldsymbol{\Delta}\) (nm) & \(\boldsymbol{\lambda}\) (\(\boldsymbol{\mu}\)m) & \(\boldsymbol{\Delta}\) (nm) \\ \hline
30\({}^{\circ}\) & 8.557 & 10 & 6.714 & 2 & 4.356 & 9 & 3.385 & 5 \\
40\({}^{\circ}\) & 8.586 & 9 & 7.341 & 5 & 4.354 & 9 & 3.698 & 4 \\
50\({}^{\circ}\) & 8.64 & 14 & 7.881 & 3 & 4.359 & 9 & - & - \\
60\({}^{\circ}\) & 8.75 & 16 & 8.209 & 5 & 4.383 & 5 & 4.167 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured wavelength shift values of local surface plasmon resonances (LSPRs) and propagating surface plasmon resonances (PSPs) in the metagrating sample. Figure 4: **(a)** Measured and (**b**) simulated wavelength shifts of both local surface plasmon resonances (LSPRs) and propagating surface plasmon resonances (PSPs) in the metagrating sample. analyte. For bulky targets like thick embedding medium used in earlier reports [14], different results might be expected, since both the field enhancement and the spatial overlap with the analyte molecules likely play significant roles.