# Nanogap Plasmon Resonator: An Analytical Model
## 2 Analytical Model  
We considered two parallel metallic cylinders whose respective radii were \(a\) and \(a_{1}>a\), which were separated by a distance \(d+d_{1}\), so that the spacing between the cylinder axes was \(a+d+a_{1}+d_{1}\). The \(x\)-axis connects the cylinders' centers, which have the coordinates \(\{x,y\}=\{a+d,0\}\) and \(\{-a_{1}-d_{1},0\}\), as displayed in Figure 1. In order to assess the electric field located in the nanogap lit with an incident light (see Figure 1d), alternative coordinates (\(u\) and \(v\)) were employed, facilitating the introduction of the complex variables \(z\) and \(w\) associated with the following equation:  
\[z=L\tanh\frac{w}{2} \tag{1}\]  
with \(z\) corresponding to \(x+iy\), \(w\) corresponding to \(u+iv\), and \(L\) equal to:  
\[L=\sqrt{d(2a+d)} \tag{2}\]  
where \(d\) is the length between the cylinder surface (named \(a\)) and the origin of the coordinate system. Note that \(a\), \(a_{1}\), and \(L\) fully define the system geometry, if we define \(d_{1}=\sqrt{a_{1}^{2}+L^{2}}-a_{1}<d\). The electrodynamics of two parallel cylinders is defined by their radii \(a\), \(a_{1}>a\) and the distance between the cylinders, which equals \(d+d_{1}=\sqrt{a^{2}+L^{2}}-a+\sqrt{a_{1}^{2}+L^{2}}-a_{1}\approx L^{2}(a+a_{1} )/(2 aa_{1})\). A particular value of \(d\) or \(d_{1}\) simply defines the position of the origin of the coordinates as soon as the values \(a\)\(a_{1}\), and \(L\) are given. The length \(L\) corresponds to the characteristic length of the electric field variation in the gap, as shown below. In the transition from \(z\) to \(w\), the whole plane \(\{x,y\}\) transforms to the strip \(-\infty<u<\infty\), \(-\pi<v<\pi\). The surfaces of the cylinders \"\(a\)\" and \"\(a_{1}\)\" are transformed to the vertical lines \(u_{d}=\ln[(a+d)/(a-d)]>0\) and \(u_{d_{1}}=-\ln[(a_{1}+d_{1})/(a_{1}-d_{1})]<0\) (Figure 1). The entire outer space of the cylinders shrunk into the rectangular \([u_{d_{1}}<u<u_{d},\ -\pi<v<\pi]\). The approach was similar to the map used in [35; 60]. The solution of the Laplace equation in the rectangle can be presented as a sum of functions \(f_{Rc}^{(q)}=e^{-qt}\cos qv\), \(f_{Rs}^{(q)}=e^{-qu}\sin qv\), \(f_{Lc}^{(q)}=e^{qu}\cos qv\), \(f_{Ls}^{(q)}=e^{qu}\sin qv\), \(q=1,2\ldots\):  
\[\varphi^{\rm G}(u,v)=\varphi_{0}+\sum_{q=1}^{\infty}\Bigl{[}A_{1}^{(q)}f_{Rc} ^{(q)}+A_{2}^{(q)}f_{Rs}^{(q)}+A_{3}^{(q)}f_{Lc}^{(q)}+A_{4}^{(q)}f_{Ls}^{(q) }\Bigr{]} \tag{3}\]  
where \(\varphi_{0}\) is the potential of the external electric field. We assumed that the electric field concentrates in the nanogap whose size is much smaller than the wavelength of the incident light \(d+d_{1}\ll\lambda\). The electric potentials in the \"\(a\)\" and \"\(a_{1}\)\" cylinders are given by the following expressions, respectively:  
\[\varphi^{\rm R}(u,v)=\varphi_{0}+\sum_{q=1}^{\infty}\Bigl{[}B_{1}^{(q)}f_{Rc }^{(q)}+B_{2}^{(q)}f_{Rs}^{(q)}\Bigr{]},\ \ u>u_{d} \tag{4}\]  
\[\varphi^{L}(u,v)=\varphi_{0}+\sum_{q=1}^{\infty}\Bigl{[}B_{3}^{(q)}f_{Lc}^{(q )}+B_{4}^{(q)}f_{Ls}^{(q)}\Bigr{]},\ \ u<u_{d_{1}} \tag{5}\]  
The components of the electric field in \(\{u,v\}\) space \(\{E_{u},E_{v}\}=-\
abla_{w}\varphi=-\{\partial\varphi/\partial u,\)\(\partial\varphi/\partial v\}\) are complex since the metal permittivity is complex. The electric field in \(\{x,y\}\) space is equal to \(\{E_{x},E_{y}\}=\hat{f}\{E_{u},E_{v}\}\), where the Jacobian \(\hat{f}\) was obtained from the derivative \(w_{z}=dw/dz\), namely \(J_{11}=J_{22}=\Re w_{z}\), \(J_{21}=-J_{12}=\Im w_{z}\). The coefficients \(A^{(q)}\) and \(B^{(q)}\) in the Equations (3)-(5) were found from the boundary conditions for the electric  
Figure 1: The design of cylindrical nanoparticle-on-mirror cavity.