# Nanogap Plasmon Resonator: An Analytical Model
## 2 Analytical Model
The real part of the permittivity of silver, gold, and many other metals is well described by the Drude formula in the red and infrared spectral range \(\Re\varepsilon_{m}(\omega)\propto-\;(\omega_{p}/\omega)^{2}\)[61] and the minimal resonant frequency \(\omega^{(m)}_{r}\propto\omega_{p}(d/a)^{1/4}\ll\omega_{p}\). Observing plasmon resonance between the metal cylinders is a challenging experimental problem involving positioning two nanocylinders parallel to each other at a nanometric distance. Yet, the investigation of the plasmon resonance in the metallic nanocylinder placed at a nanodistance above a flat metallic surface (i.e., \(a_{1}=\infty\)) is the focus of current experimental studies. In order to evaluate the electric field located within this nanogap, the limit case was employed (i.e., \(a_{1}\to\infty\) and \(d_{1}\to 0\)). The electric field (named \(E_{0}\)) had a fixed value at the level of the cylinder surface \(a_{1}\) (far from the cylinder \(a\)). Thus, all expressions of the electric field are expressed in terms of \(E_{0}\). For the limit case (\(a_{1}\to\infty\)), \(E_{0}\) corresponds to that attached to the surface of the metal plate (i.e., at \(x=0,\ a\ll|y|\); see Figure 1c). From there, the electric field \(E(x,y)\) was assessed at the central coordinates (i.e., \((x,y)=(0,0)\) and \((u,v)=(0,0)\)), and its value was maximal, as depicted in Figure 1c,d. Thus, the Jacobian \(\bar{J}\) simplifies to the scalar quantity \(L\), and the expression of \(E_{x,y}\) was determined by solving the Equations (10)-(12) and (13)-(14). The field equation is simplified in the center of coordinates, namely  
\[E(x=0,y=0)=\{E_{x0}\bigg{[}1+\frac{8\epsilon_{m_{1}}\Sigma}{\epsilon_{d}- \epsilon_{m}}\bigg{]},E_{y0}\bigg{[}1-\frac{8\epsilon_{d}\Sigma}{\epsilon_{d}- \epsilon_{m}}\bigg{]}\}, \tag{16}\]  
\[\Sigma=\sum_{q=1}^{\infty}\frac{(-1)^{q}q\,h}{(b+1)^{q}-h},\ h=\frac{( \epsilon_{d}-\epsilon_{m})(\epsilon_{d}-\epsilon_{m_{1}})}{(\epsilon_{d}+ \epsilon_{m})(\epsilon_{d}+\epsilon_{m_{1}})}, \tag{17}\]  
\[b=(d+L)(2a+d+L)/a^{2}\simeq\sqrt{8d/a} \tag{18}\]  
where \(\epsilon_{m}\), \(\epsilon_{m_{1}}\), and \(\epsilon_{d}\) are the permittivities of the cylinder, metal plate, and outer space, respectively. \(a\) and \(d\) are the cylinder radius and the gap size, respectively. The last approximation in the Equation (18) holds for \(d\ll a\). The system resonances if a denominator in the first equation in (17) almost vanishes, i.e., the dimensionless parameter \(\Re h>1\) and the loss factor \(\kappa=\Im h/|h|\simeq 2\epsilon_{d}\big{(}3\epsilon_{m}|\epsilon_{m}|^{-2}+3 \epsilon_{m1}|\epsilon_{m1}|^{-2}\big{)}\ll 1\) for \(|\epsilon_{m}|\), \(|\epsilon_{m_{1}}|\gg\epsilon_{d}\). Despite the simplicity of the Equations (16) and (17), the gap field has a rather rich behavior when the gap vanishes, i.e., \(b\to 0\). Suppose that the parameter \(\Re h<1\), then the denominator \(n^{(q)}=(b+1)^{q}-h\) in the sum \(\Sigma\) in the Equation (17) is not close to zero for all the values of \(q\), and hence, it can be linearized as \(n^{(q)}\simeq b_{1}q-h_{1}\), where \(h_{1}=h-1\), \(\Re h_{1}<0\), \(b_{1}=\log(1+b)\simeq b\). Then, the sum \(\Sigma\) in the Equation (17) is approximated as  
\[\Sigma\simeq S(h_{1},b_{1})=\sum_{q}^{\infty}(-1)^{q}q/(b_{1}q-h_{1}) \tag{19}\]  
which can be solved by the Laplace transformation and Abel regularization. The function \(S\) has a simple integral presentation:  
\[\varepsilon_{d}(A^{(q)}_{4}-A^{(q)}_{2}g^{2q})-\varepsilon_{m}B^{ (q)}_{2}=E_{0y}(\varepsilon_{m}-\varepsilon_{d}),\]0  
where \(\psi_{1}=\psi\Big{(}\frac{b_{1}-h_{1}}{2b_{1}}\Big{)}\), \(\psi_{2}=\psi\Big{(}-\frac{h_{1}}{2b_{1}}\Big{)}\) and \(\psi=\Gamma^{\prime}/\Gamma\) is the polygamma function. The sum is approximated as \(\Sigma\simeq-\frac{1}{4(b_{1}-h_{1})}\left(\frac{b_{1}}{2b_{1}-h_{1}}+1 \right)\to\frac{1}{a_{1}}\) for the fixed \(h_{1}\) and \(|b_{1}/h_{1}|\to 0\). In other words, the electric field tends towards a finite value when the gap between the cylinder and the plate vanishes.