# Nanogap Plasmon Resonator: An Analytical Model
## 2 Analytical Model
(**a**,**b**) Conformal map of two cylinders into the \(u-v\) strip; (**c**) metallic cylinder above the metal plate; the electromagnetic wave is incident from the top; (**d**) electric field between Ag plate and cylinder with a radius of \(a=50\) nm and the gap \(d\ =\ 0.5\) nm calculated at the excitation wavelength of \(\lambda=405\) nm. fields \(E^{R}=-\
abla_{w}\varphi^{R}\), \(E^{G}=-\
abla_{w}\varphi^{G}\), and \(E^{L}=-\
abla_{w}\varphi^{L}\) at the surface of the \"\(a\)\" and \"\(a_{1}\)\" cylinders:  
\[E^{R}_{v}=E^{G}_{v},\;\varepsilon_{m}E^{R}_{u}=\varepsilon_{d}E^{G}_{u},\;u=u_{d} \tag{6}\]  
\[E^{L}_{v}=E^{G}_{v},\;\varepsilon_{m_{1}}E^{L}_{u}=\varepsilon_{d}E^{G}_{u},\;u= u_{d_{1}} \tag{7}\]  
where \(\varepsilon_{m}\), \(\varepsilon_{d}\), and \(\varepsilon_{m_{1}}\) are the permittivities of the \"\(a\)\" cylinder, the outer space, and the \"\(a_{1}\)\" cylinder, respectively. We expanded the external field \(E_{0}=-\
abla\varphi_{0}=\{E_{0x},E_{0y}\}\) in a series of functions \(f^{(q)}_{RC}\), \(f^{(q)}_{Rs}\), \(f^{(q)}_{LC}\), \(f^{(q)}_{Ls}\). For the simplest case of the constant external field, the expansion:  
\[\{E_{0u},E_{0v}\}=\sum_{q}\Bigl{\{}E^{(q)}_{0u},E^{(q)}_{0v}\Bigr{\}} \tag{8}\]  
has the following form:  
\[\Bigl{\{}E^{(q)}_{0u},E^{(q)}_{0v}\Bigr{\}}=-2L(-1)^{q}q\times \begin{cases}E_{0 x}f^{(q)}_{RC}-E_{0y}f^{(q)}_{Rs},\;E_{0x}f^{(q)}_{Rs}+E_{0y}f^{(q)}_{RC},\;u>0 \\ E_{0x}f^{(q)}_{LC}+E_{0y}f^{(q)}_{Ls},\;E_{0y}f^{(q)}_{LC}-E_{0x}f^{(q)}_{Ls}, \;u<0\end{cases}  \tag{9}\]  
where \(q=1,2,\ldots\). We substituted the above equation into the boundary Equation (6) for the surface of the \"\(a\)\" cylinder and assimilated the coefficients to the same \(f^{(q)}\) functions by obtaining the following equations for the coefficients \(\{A^{(q)},B^{(q)}\}\) in the Equations (3)-(5):  
\[\varepsilon_{d}\Bigl{(}A^{(q)}_{3}+A^{(q)}_{1}g^{2q}\Bigr{)}- \varepsilon_{m}B^{(q)}_{1}=E_{0x}(\varepsilon_{m}-\varepsilon_{d}), \tag{10}\] \[\varepsilon_{d}(A^{(q)}_{4}-A^{(q)}_{2}g^{2q})-\varepsilon_{m}B^{ (q)}_{2}=E_{0y}(\varepsilon_{m}-\varepsilon_{d}),\] (11) \[A^{(q)}_{2}g^{2q}+A^{(q)}_{4}-B^{(q)}_{2}=A^{(q)}_{1}g^{2q}-A^{ (q)}_{3}+B^{(q)}_{1}=0 \tag{12}\]  
where \(g=(L+d)/(L-d)\). By matching the electric field and the displacement on the surface of the \"\(a_{1}\)\" cylinder, we obtain from the Equation (7):  
\[\varepsilon_{d}\Bigl{(}A^{(q)}_{1}+A^{(q)}_{3}g^{2q}_{1}\Bigr{)}- \varepsilon_{m_{1}}B^{(q)}_{3}=E_{0x}(\varepsilon_{m_{1}}-\varepsilon_{d}), \tag{13}\] \[\varepsilon_{d}\Bigl{(}A^{(q)}_{2}-A^{(q)}_{4}g^{2q}_{1}\Bigr{)}- \varepsilon_{m_{1}}B^{(q)}_{4}=E_{0y}(\varepsilon_{m_{1}}-\varepsilon_{d}),\] (14) \[A^{(q)}_{4}g^{2q}_{1}+A^{(q)}_{2}-B^{(q)}_{4}=A^{(q)}_{3}g^{2q}_ {1}-A^{(q)}_{1}+B^{(q)}_{3}=0 \tag{15}\]  
where \(g_{1}=(L+d_{1})/(L-d_{1})\). Setting the determinant of Equations (10)-(12) and (13)-(15) to zero gives the \"\(a\)\" resonance condition. The cylinders with the same permittivity \(\varepsilon_{m_{1}}=\varepsilon_{m}\) resonate when \((gg_{1})^{2q}(\varepsilon_{d}+\varepsilon_{m})^{2}-(\varepsilon_{m}- \varepsilon_{d})^{2}=0\). The first term dominates when the distance \(d+d_{1}\) between the cylinders increases and \(gg_{1}\to\infty\). Then, all resonant frequencies \(\omega^{(q)}_{r}\) collapse into a single value \(\omega^{(1)}_{r}\) given by the well-known equation \(\Re[\varepsilon_{d}(\omega^{(1)}_{r})+\varepsilon_{m}(\omega^{(1)}_{r})]=0\) for the plasmon resonance in a metallic cylinder. On the other hand, when the gap size vanishes \(d+d_{1}\to 0\), the factor \(gg_{1}\to 1\) from above and the resonance frequencies \(\omega^{(q)}_{r}\) spread out from \(\omega^{(1)}_{r}\) to the minimum resonance frequency estimated from the equation \(\varepsilon_{m}(\omega^{(m)}_{r})\simeq-\frac{2\varepsilon_{d}a_{1}}{L(a+ \varepsilon_{1})}\to-\infty\).