## 3 Results
### Resonant EM Fields at the FL Enhancement  
Figure 5 shows a set of resonant EM-field distributions that correspond to the peak of the enhancement factor in Figure 3b. Figure 5a shows the numerically calculated \(R\) spectrum at the normal incidence, which is also represented in Figure 4a, though it is overlapped with the \(R\) spectrum at 5\({}^{\circ}\). The FL-enhancing wavelength corresponds to the red arrow (639.3 nm) in the high-\(R\) band, which was observed during the experiment (Figure 3a,b). The \(|\)E\(|\) and \(|\)H\(|\) distributions in an \(xz\)-section view are respectively shown in Figure 5c,d at the wavelength indicated by the red arrow in Figure 5a. The \(xz\)-section was taken at the center of the Si rod. The resonantly enhanced electric fields mainly exist at the outer surface of the Si rod, whereas the resonant magnetic fields are dominantly localized in the Si rod, taking the maximum value of 11.1, and indicating significant field enhancement inside the Si rod. The magnetic-field distribution exhibits  
Figure 4: (**a**) Numerically calculated \(R\) spectra for Si-rod-array metasurface of Si-rod height 200 nm and \(P\) = 330 nm. (**b**) Measured \(R\) spectra for the metasurface in Figure 2a,c. (**c**) Numerically calculated \(R\) spectra for Si-rod-array metasurface of Si-rod height 200 nm and \(P\) = 400 nm. (**d**) Measured \(R\) spectra for the metasurface in Figure 2b. a multipole feature. Thus, these EM-field distributions indicate that the resonant mode is of a higher magnetic resonance in the Si rod. Figure 6 shows the resonant EM-field distributions contributing to the FL enhancement of the metasurface of SOI-thickness 200 nm and \(P\) = 400 nm. The experimental data are shown in Figure 3c,d. Figure 6a shows, again, the numerically calculated \(R\) spectrum at the normal incidence, which has appeared in Figure 4c. The red arrows indicate the wavelengths corresponding to the peaks of the FL-intensity enhancement factor in Figure 3d. In Figure 6b,c, the \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) distributions are shown in an \(xz\)-section view, respectively. They are excited at the \(R\)-dip wavelength (701.1 nm), which is indicated by the red arrow with label b,c  
Figure 5: (**a**) Numerically calculated \(R\) spectrum for the metasurface of a SOI of 200 nm thickness and \(P\) = 330 nm. Incidence was assumed to be normal. (**b,c**) \(xz\)-section-view \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) distributions, respectively, at a highly FL-enhancing wavelength (639.3 nm), indicated by the red arrow in (**a**). Incident \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) were set to 1.0. Figure 6: (**a**) Numerically calculated \(R\) spectrum for the metasurface of SOI thickness 200 nm and \(P\) = 400 nm. Incidence was assumed to be normal. (**b,c**) \(xz\)-section views of \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) distributions, respectively, at the wavelength (701.1 nm), indicated by the arrow with the label b,c in (**a**). (**d,e**) \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) distributions, respectively, at the wavelength (576.3 nm), indicated by the arrow with the label d,e in (**a**). Incident \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) were set to 1.0.  
in Figure 6a. The \(xz\) section was taken at the center of the Si rod. From the EM-field distributions, it is clear that the incident light efficiently propagates in the Si rod. The magnetic field dominantly exists inside the Si rod, and the electric field is strongly localized at the sidewall of the Si rod. The Si rod works as a waveguide at this wavelength. The EM-field distributions also indicate that they form a 1-wavelength guided mode; consequently, impedance matching is satisfied at the top and bottom interfaces of the Si rod, resulting in the low \(R\) (i.e., \(R\) dip).