# Nanogap Plasmon Resonator: An Analytical Model
## 3 Analytical Model vs. Simulations and Experiments
The results of our quasi-static model extrapolated to \(ka\)\(\sim\)1 were still in qualitative agreement with the computer simulations of \"large\" nanocylinders (see Figure 3). The agreement was better for a smaller gap size \(d\) and a higher plasmon resonance \(q\). Indeed, our quasi-static theory worked better when the spatial scale of the electric field variance \(L_{q}\)\(\sim\)\(L/q\)\(\sim\)\(\sqrt{ad}/q\) was much smaller than the wavelength \(\lambda\). In this case, we believe that all other parts of the plasmonic system can be considered as an external environment by simply renormalizing the applied electric field. The model is valid when the plasmon size \(L_{q}\)\(\sim\)\(\sqrt{ad}/q\) is less than the skin depth \(L_{q}\ll\frac{1}{k\sqrt{|\varepsilon_{m}|}}\). Figures 2 and 3 show that the largest field enhancement was obtained in the red and infrared spectral bands, where the optical loss in the metal was relatively low. On the other hand, the silver-silicon system resonated for \(\lambda<550\) nm when the parameter \(\Re h\) in the Equation (17) became greater than one. Figure 3: Analytical results in the gap between the Ag cylinder and the Ag surface as a function of wavelength and \(d/a\) ratio: (**a**) electric field intensity \(|E/E_{0}|^{2}\) and (**b**) reflectance. Numerical results for the “large” silver cylinder with a radius of \(a=50\) nm and a period of \(D=200\) nm: (**c**) electric field intensity \(|E/E_{0}|^{2}\) and (**d**) reflectance. Moreover, we also compared our analytical model to other theoretical results obtained with a local analogue model (LAM) developed by [PERSON]'s group [62]. A good agreement was obtained between our analytical model and the LAM of [PERSON]'s group for the excitation wavelength of 785 nm (see the red line and red crosses in Figure 2b). For the excitation wavelength of 532 nm, our analytical model was still in relatively good agreement with the LAM (see the green line and green crosses in Figure 2b). Furthermore, we compared our analytical model to the experimental results obtained on the enhancement factors of the SERS effect recorded by [PERSON] group [63]. The nanostructures used for these experiments were arrays of gold dimers on a 50 nm-thick gold film, composed of two disks with a radius of 70 nm, height of 40 nm, and gap of 20 nm. They obtained \(|E/E_{0}|^{2}\) ratios of 1.25 \(\times\) 10\({}^{2}\) and 5.48 \(\times\) 10\({}^{2}\) at the excitation wavelengths of 532 nm and 785 nm, respectively, for a corresponding _d/a_ ratio of 0.29. We observed a good agreement between our analytical model and the experimental results recorded by [PERSON] group (see the red/green lines and the red/green stars in Figure 2b). The small difference can be due to the difference in the incidence angle of the excitation beam between our analytical model (45\({}^{\circ}\) with respect to the surface normal) and the experimental results (normal incidence; the polarization of the excitation light was along the dimer direction). We analytically calculated the electric field intensity \(|E/E_{0}|^{2}\) at the excitation wavelengths of 532 nm and 785 nm in the gap between the Au cylinder and the Au surface for the case considered by [PERSON] group and [PERSON]'s group, where the polarization of the excitation light was along the dimer direction. This configuration corresponded to ours at an incidence angle \(\theta=90^{\circ}\) with respect to the surface normal (see the red and green dashed lines in Figure 2b). In this case, \(|E/E_{0}|^{2}\) was more enhanced and an even better agreement was reached with the above-mentioned results of [PERSON] and [PERSON]. Besides, a comparison between our analytical model and the numerical simulations obtained by [PERSON]'s group was made [16]. The structure employed for these simulations was a disk on a 200 nm-thick gold film, the 40 nm gap between them of which was made by a layer of SiO\({}_{2}\). The radius of the disk was 250 nm, and its height was 40 nm.