# Plasmonic Filter and Demultiplexer Based on Square Ring Resonator
## 2 Materials, Structures, and Methods  
As shown in the Figure 1a, an MIM band-stop filter based on a single side-coupled SRR is introduced. The detailed structural parameters are given in the caption. The bus waveguide and SSR can be etched on the silver plate by the focused ion beam (FIB) method. The insulator in the bus waveguide and the SRR is air as \(\varepsilon_{i}=1\), while the complex relative permittivity of silver is described by the Drude model [29]:  
\[\varepsilon_{\text{m}}=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega( \omega+i\gamma)} \tag{1}\]  
Here \(\varepsilon_{\infty}\) gives the medium constant for the infinite frequency, \(\omega_{p}\) refers to the bulk frequency for plasma, \(\gamma\) denotes the damping frequency for electron oscillation, and \(\omega\) represents the incident light angular frequency. The parameters for silver are \(\varepsilon_{\infty}\) = 3.7, \(\omega_{p}\) = 1.38 \(\times 10^{16}\) Hz, and \(\gamma\) = 2.73 \(\times 10^{13}\) Hz. The transverse magnetic (TM) mode can propagate at the interface of waveguide when coupled into the MIM structure. Compared with incident wavelength, the thickness of the bus waveguide is much smaller, so only the fundamental TM mode can exist. The dispersion relation of this fundamental mode is described as follows [2]:  
\[ \begin{split}&\frac{\varepsilon_{i}p}{\varepsilon_{m}k}=\frac{1-e^ {kw}}{1+e^{kw}}\\ & k=k_{0}\sqrt{(\frac{\beta_{spp}}{k_{0}})^{2}-\varepsilon_{i}},p=k_{0}\sqrt{(\frac{\beta_{spp}}{k_{0}})^{2}-\varepsilon_{m}}\\ &\beta_{spp}=n_{eff}k_{0}=n_{eff}\frac{2\pi}{\lambda}\end{split}  \tag{2}\]  
Here, \(w\) refers to the width of the bus waveguide, \(\lambda\) is the incident light wavelength in vacuum, \(\varepsilon_{i}\) and \(\varepsilon_{m}\) are the relative dielectric and metal permittivity, \(\beta_{spp}\) and \(n_{eff}\) are the propagation constant and effective refractive index (RI) of SPPs, respectively, and \(k_{0}=2\pi/\lambda\) indicates the wave number. Here, the width of the bus waveguide is 100 nm, as is usually selected [13], and the width of SRR is chosen as 15 nm to increase the energy density in the resonator, which is beneficial for improving the drop efficiency. The Lumerical Finite-Difference Time-Domain (FDTD) solutions with a mesh grid size of 2.5 nm is utilized for simulating features of such a device with the boundary condition of stabilized perfectly matched layers (PML) to maintain convergence [4]. A mode source is set at the input port and a fundamental mode is selected. To collect the incident and transmitted power, two power monitors are put respectively at \(P_{in}\) and \(P_{out}\). The transmittance of power can be calculated by \(T=P_{out}/P_{in}\). The transmission spectrum for band-stop filter is shown in Figure 1b, which shows that the resonant wavelength is approximately 1310 nm. The transmission for the 1310 nm wavelength is almost zero, indicating that there is a strong resonance inside the SRR and the energy can be confined in the cavity very well. Therefore, this filter has a great performance in stopping a specific wavelength. The inset of Figure 1b gives the field distributions (\(|\mathbf{H}|\)) in the SRR for the resonant wavelength. In the ring resonator, the resonant wavelength \(\lambda\) can be calculated from [21]:  
\[\lambda=\frac{4L_{eff}\text{Re}(n_{eff})}{N},N=1,\;2,\;3\;\dots \tag{3}\]  
where \(n_{eff}\) refers to the effective RI in the ring resonator, which can be solved by Equation (2), and \(L_{eff}\) indicates the effective side length, which generally refers to the average of the inner and outer side lengths of the ring. \(N\) refers to the mode number, which is an integer. Figure 1c gives the transmission spectra related to different side lengths \(L\) of the SRR with a gap of 15 nm. The resonant peak shows a red-shift when increasing the side length, which agrees with Equation (3). Besides, the influence of the gap \(g\) on the resonant wavelength is investigated with the side length of 550 nm.