## 4 Discussion  
Let us discuss the FL enhancement observed in this investigation. Considering the electric-field distribution in Figure 5b, the intensity \(|\,\mathbf{E}|^{2}\) has the maximum value of 50.4 at the bottom of the Si rod, while the intensities at most of the outmost surfaces of the Si rod are estimated to be 10 or less. Therefore, although resonant electric-field enhancement is one of the origins of the large FL-intensity enhancement (-1000), it cannot fully account for the enhancement effect. A similar situation to this study was found for metallic metasurfaces; in this case, the FL-intensity enhancement mechanism was examined by considering all the processes from the excitation to FL emission [48]. In the present dielectric metasurface, such full consideration is also necessary to account for the FL enhancement effect. Concretely, the enhancement factor (EF) is expressed such that  
\[\text{EF}=\frac{N_{\text{enh}}}{N_{0}}\frac{\eta_{\text{enh}}}{\eta_{0}}\frac {\gamma_{\text{R,enh}}(\mathbf{k})}{\gamma_{\text{R,0}}(\mathbf{k})} \tag{1}\]  
where \(N_{\text{enh}}\) and \(N_{0}\) are excited populations in the metasurface and reference configurations, respectively; \(\eta_{\text{R,enh}}\) and \(\eta_{0}\) are quantum yields of the FL molecules in the metasurface and reference configurations, respectively; \(\gamma_{\text{R,enh}}(\mathbf{K})\) and \(\gamma_{\text{R,0}}(\mathbf{K})\) are radiative transition rates dependent on outgoing wavevector \(\mathbf{K}\) in the metasurface and reference configurations, respectively. The transition rates are calculated, in principle, based on quantum mechanical transition probabilities. The subscripts enh and 0 denote the metasurface and reference configurations, respectively. We note that the quantum yield \(\eta\) is represented with radiative and nonradiative transition rates, such as \(\eta=\gamma_{\text{R}}/(\gamma_{\text{R}}+\gamma_{\text{NR}})\), where \(\gamma_{\text{NR}}\) denotes the nonradiative transition rate. The EF in Equation (1) represents a generalized Purcell factor. Note that, when we consider an ideally simplified situation that a single two-level system is excited only once per unit time, the EF reduces into the original Purcell factor of \(\gamma_{\text{R,enh}}/\gamma_{\text{R,0}}\)[60], because of \(N_{\text{enh}}/N_{0}=1\) and \(\eta_{\text{enh}}/\eta_{0}=1\). In comparison with the original Purcell factor, the EF in Equation (1) is substantially elaborated in accordance with the actual experimental configuration. The EFs have been experimentally evaluated in Figure 3b,d,f, whereas it is extremely difficult to experimentally determine the constituent factors at the right-hand side of Equation (1). Therefore, we discuss the EF from the electric-field distributions evaluated in the computations. First, we refer to the excitation efficiency, which is proportional to the excited populations in Equation (1). We compared electric-field intensities at 532.0 nm on the outmost metasurfaces of the Si rods, at which the dispersed FL molecules were located, with those on the reference Si wafer; the electric fields are shown in the Supplementary Materials (Figure S6). As a result, we estimated the intensity ratio of \(\left|\mathbf{E}_{\text{enh}}\right|^{2}/\left|\mathbf{E}_{0}\right|^{2}\) to be 22 on average in the case of 200 nm SOI thickness and \(P=400\) nm (Figure 3c,d), where \(\mathbf{E}_{\text{enh}}\) and \(\mathbf{E}_{0}\) denote electric-field vectors in the metasurface and reference, respectively. Thus, it is impossible to attribute the origin of the large EF up to 1000 only to the factor of \(\left|\mathbf{E}_{\text{enh}}\right|^{2}/\left|\mathbf{E}_{0}\right|^{2}\). Second, definite spectral changes in the FL spectra in Figure 3 indicate that FL-emitting electronic transitions are modified in the metasurfaces; that is, the radiative-transition-rate ratio of \(\gamma_{\text{R,enh}}(\mathbf{K})/\gamma_{\text{R,0}}(\mathbf{K})\) is substantially modulated.