# Nanogap Plasmon Resonator: An Analytical Model
## 3 Analytical Model vs. Simulations and Experiments  
In any real SERS detection experiment, a layer of molecules to be investigated is placed in the gaps formed by metallic particles distributed on a metal mirror. In this work, numerical simulations (COMSOL) were employed to reproduce the periodic grating of the metallic nanocylinders, whose dimensions were, respectively, 10 nm and 50 nm for the radius \(a\) of the nanocylinder for each grating and 200 nm for the periodicity \(D\) for both gratings. The arrays of nanocylinders were deposited on a metallic surface, then illuminated by \(p\)-polarized light at a 45\({}^{\circ}\) angle of incidence comparable to the work of [PERSON] group [10]. To obtain the reflectance and electric field measurements, we employed periodic boundary conditions and periodic ports. The 2D model was set up for an unit cell of the grating, flanked by Floquet boundary conditions describing the periodicity. This means that the solution on one side of the unit is equal to the solution on the other side multiplied by a complex-valued phase factor. The phase shift between the boundaries was evaluated from the perpendicular component of the wave vector. Port conditions were used to specify the incident wave. The radius \(a\) of the nanocylinder was fixed, and the size of the nanogap \(d\) was a variable parameter. The electric field in the gap and the S-parameters were obtained by solving [PERSON]'s equations via the finite-element method. Comparing the results of these numerical simulations with the analytical model, a relatively good agreement was observed for the enhancement of the electric field within the nanogap (see Figure 2a-c). By considering a metal-semiconductor nanogap, significant values of the electric field within the latter can be achieved (see Figure 2d). The concentration of electromagnetic energy in the nanogaps can be noticed by a decrease in reflectance (see Figure 3b,d). To compare our analytical model with the computer simulations described above, we set the electric field \(E_{s}\) at the point \(x=d+2a,y=0\) on the surface of a cylinder and calculated the field \(E(x,y)\) throughout space using our analytical theory. To simplify the consideration, the direct interaction between cylinders was neglected since the electric field was localized in the nanogaps of size \(d\ll a\). Then, we averaged the field over the layer \(0<x<d+2a\) and introduced the effective permittivities \(\epsilon_{xx}^{(c)}=\left\langle\epsilon(x,y)E_{x}(x,y)\right\rangle/\left\langle E _{x}(x,y)\right\rangle\) and \(e_{yy}^{(c)}=\left\langle\epsilon(x,y)E_{y}(x,y)\right\rangle/\left\langle E _{y}(x,y)\right\rangle\). An electromagnetic wave of amplitude \(E_{in}\) and the \(p\)-polarization arrives on the layer placed above the metal plate. [PERSON]'s equations were solved in terms of amplitude \(E_{in}\) in order to find an electromagnetic field in the layer of thickness \(d+2a\) and permittivity \(\hat{\varepsilon}^{(e)}\). In particular, we found the electric field \(E_{0}=\{E_{x0},E_{y0}\}\) at the interface between the layer and the metal plate (see Figure 1c). In the dilute case \(a\ll D\ll\lambda\), the field \(E_{0}\) was equal to the field that was on the metal surface without a cylinder, namely  
\[E_{0}=E_{in}\{\frac{\varepsilon_{m_{1}}\sin 2\theta}{\sqrt{\varepsilon_{d} \varepsilon_{m_{1}}-\varepsilon_{d}^{2}\sin^{2}\theta+\varepsilon_{m_{1}} \cos\theta}},\frac{2\cos\theta\sqrt{\varepsilon_{d}\varepsilon_{m_{1}}- \varepsilon_{d}^{2}\sin^{2}\theta}}{\sqrt{\varepsilon_{d}\varepsilon_{m_{1}}- \varepsilon_{d}^{2}\sin^{2}\theta+\varepsilon_{m_{1}}\cos\theta}}\} \tag{25}\]  
where \(\theta\) is the angle of incidence. This field was introduced into Equation (16) in order to obtain the analytical estimate of the electric field in the nanogap. The theoretical results were consistent with full-scale computer simulations for small cylinders (\(ka\ll 1\)).