# Plasmonic Filter and Demultiplexer Based on Square Ring Resonator
## 2 Materials, Structures, and Methods
As shown in Figure 1d, the resonant wavelength performs a blue-shift with the increase of the gap. Therefore, the resonant wavelength could be controlled by varying the side length or the gap. However, when the SRR comes to the traditional add-drop structure, the drop performance is terrible, as shown in Figure 2a. Therefore, we introduce a unique side-coupling method as presented in Figure 2b. The drop waveguide is side-coupled in such a way for two reasons. First, the position of the drop waveguide is selected according to the local mode distribution in the SRR. Besides, since the  
Figure 1: (**a**) Two-dimensional (2D) structure of the band-stop filter with a single square ring resonator (SRR). \(L=550\) nm, \(D=20\) nm, \(w=100\) nm, \(g=15\) nm. (**b**) The transmission spectra of the band-stop filter. The inset is the corresponding field distribution. (**c**) The transmission spectra pertinent to different side lengths \(L\) of the SRR. (**d**) The transmission spectra pertinent to different gaps \(g\). resonant wave propagates anticlockwise in the SRR, the drop wave will have a shorter optical path by this method than the traditional add-drop coupling way, consequently suffering less from ohmic loss. The influence of the gap \(g\) between the drop waveguide and the SRR on the drop efficiency is studied with \(s=320\) nm. As given in Figure 2d, the increase of \(g\) will cause the decrease of the transmission peak and bandwidth, but the FWHM is lower. Moreover, there is a blue-shift of the transmitted peak wavelength. The effect of the coupling length \(s\) on the transmission is also discussed with \(g=15\) nm. It is shown in Figure 2c that there is an optimum coupling length, \(s=320\) nm, to achieve the best drop efficiency. According to the inset of Figure 2e, the optimum coupling length is dependent on the mode distribution of the resonant wavelength at the SRR edge adjacent to the drop waveguide. After the optimization of structural parameters, the final result from FDTD is shown in Figure 2e. The drop efficiency at 1310 nm, defined as the ratio of drop to input power, is approximately 52%, and the corresponding FWHM is 45 nm. The physical mechanism of such a channel drop filter can be theoretically analyzed by CMT [30]. The normalized field amplitude \(a\) of the SRR mode can be described as:  
\[\frac{da}{dt}=j\omega_{0}a-(\frac{2}{\tau_{b}}+\frac{1}{\tau_{b}}+\frac{1}{\tau _{0}})a+e^{j\theta_{1}}\sqrt{\frac{2}{\tau_{b}}}S_{+1}+e^{j\theta_{2}}\sqrt{ \frac{2}{\tau_{b}}}S_{+2}+e^{j\theta_{3}}\sqrt{\frac{2}{\tau_{d}}}S_{+3} \tag{4}\]  
Here, \(\omega_{0}\) refers to the resonant frequency, and \(1/\tau_{b}\) and \(1/\tau_{d}\) represent the decay rates inside the bus and drop waveguide, respectively. \(1/\tau_{0}\) is the decay rate due to the ohmic loss of the plasmonic system. \(S_{+i}\) and \(S_{-i}\) (\(i=1\), 2, 3) represent the normalized amplitudes of input and output SPPs, respectively, as demonstrated in Figure 2b. \(\theta_{k}\) (\(k=1\), 2, 3) are the phases of the corresponding  
Figure 2: (**a**) The drop spectrum for the traditional add-drop SRR. (**b**) The schematic diagram of the band-pass filter. (**c**) The drop spectrum pertinent to different coupling lengths. (**d**) The drop spectrum pertinent to different gaps. (**e**) The transmission spectrum for a channel drop filter with \(L_{1}=546\) nm, \(w=100\) nm, \(g=15\) nm, \(s=320\) nm from Finite-Difference Time-Domain (FDTD) and coupled mode theory (CMT). The inset is the field distribution of the channel drop filter. coupling coefficients. If the reference plane is set at the center of the SRR, there will be \(\theta_{1}=\theta_{2}\). According to the power conservation and the time reversal symmetry, we can obtain:  
\[S_{-1} =S_{+2}-e^{-j\theta_{1}}\sqrt{\frac{2}{\tau_{b}}}a \tag{5}\] \[S_{-2} =S_{+1}-e^{-j\theta_{1}}\sqrt{\frac{2}{\tau_{b}}}a\] \[S_{-3} =-S_{+3}+e^{-j\theta_{3}}\sqrt{\frac{2}{\tau_{d}}}a\]  
In this case, \(S_{+2}=S_{+3}=0\).