## 3 Results
### Resonant EM Fields at the FL Enhancement
We also note that the field enhancement is particularly significant among the various resonant modes of the Si rods; the maxima of the electric- and magnetic-field intensity are 557 and 279, respectively, for the incident intensities of 1. In Figure 6d,e, the \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) distributions are shown in the same \(xz\) views to Figure 6b,c, respectively, at the \(R\)-peak wavelength (576.3 nm) indicated by the red arrow with label d,e in Figure 6a. The EM-field distributions also indicate a waveguide mode in the Si rod, which forms a 3/2-wavelength mode. Thus, the resonant mode is a 3/2-wavelength guided mode. This mode does not have impedance matching at the interface of Si and BOX. As a result, the \(R\) cannot be small, and indeed forms a peak at the wavelength (Figure 5a). Figure 7 shows the resonant EM-field distributions that contribute to the FL enhancement in the metasurface of a SOI of 340 nm thickness and \(P\) = 400 nm. The corresponding experimental data are shown in Figure 3e,f. Figure 7a shows the numerically calculated \(R\) spectrum at the normal incidence. The wavelengths indicated by the red arrows correspond to the peaks of the FL enhancement factor in the experiment (Figure 3f). In Figure 7b,c, the \(\left|\mathbf{E}\right|\) and \(\left|\mathbf{H}\right|\) distributions are shown in an \(xz\)-section view, respectively, at the \(R\)-dip wavelength (651.0 nm) indicated by the red arrow with label b,c in Figure 7a. The \(xz\) section was taken at the center of the Si rod. One of the noteworthy features is that the magnetic field is confined in the Si rod, whereas the enhanced electric field is strongly localized at the sidewall of the Si rod. Figure 7: (**a**) Numerically calculated \(R\) spectrum for the metasurface of SOI-thickness 200 nm and \(P\) = 340 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 (651.0 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 (616.1 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. Thus, we can attribute this mode to a higher magnetic mode. Note that it is not simply attributed to a multipole mode. This is partially because the height of the Si rod is fairly high for the effective wavelength inside the Si rod, and there is a degree of freedom to form three-dimensional complex EM modes. In Figure 7d,e, the \(|\,\mathbf{E}|\) and \(|\,\mathbf{H}|\) distributions are shown in an \(xz\)-section view, respectively, at the \(R\)-dip wavelength (616.1 nm) indicated by the red arrow with label d,e in Figure 7a. The section was taken at the center of the Si rod. This resonant EM mode is also ascribed to a higher magnetic mode. The magnetic field forms a complicated distribution, and the electric field is mainly enhanced at the sidewall of Si rod. Summarizing the results in Figures 5-7, we realize that there is a tendency for the large FL-intensity enhancement factors to appear at resonant wavelengths associated with \(R\) peaks or dips at the normal incidence. The largest FL enhancement factor in Figure 3d was found at the 1-wavelength guided resonance in Figure 6b,c. The second largest FL enhancement factor in Figure 3b originates from a higher magnetic resonance in Figure 5. The third largest FL enhancement factor in Figure 3d was attributed to the 3/2-wavelength guided resonance in Figure 6d,e. A common feature of these three modes is that the enhanced electric-field intensity is strongly localized at the outmost surface of the Si rods. Thus, the resonant electric-field enhancement was identified as a key factor for the large FL enhancement factors. Further discussion is presented in the next section.