## 2 Materials and Methods
### Advanced Filters
#### 2.4.2 Multiple Bandstop Frequencies  
A directional filter with \(N\) complementary bandstop and bandpass responses can be realised by increasing the number of coupling resonators from 1 to \(N\), as shown in Figure 5a. Here MI waveguides with coupling coefficients \(\pm\kappa\) are connected using vertical lines containing \(N\) weakly coupled resonators with coupling \(\kappa_{c}\). The resonator rows are labelled from 0 (bottom) to top (\(N-1\)) and have the currents \(K_{m}\) and \(P_{m}\) flowing in the left and right columns, respectively. The lines of coupling resonators can be considered as two identical uniform MI waveguides with a passband of \(\frac{1}{\sqrt{1+|\kappa_{c}|}}\leq\frac{\omega}{\omega_{0}}\leq\frac{1}{ \sqrt{1-|\kappa_{c}|}}\) and dispersion relation \(Z+Z_{mc}(e^{-jk_{c}a}+e^{jk_{c}a})=0\), where \(k_{c}\) is the propagation constant. The currents \(P_{m}\) and \(K_{m}\) may also be expressed in terms of reflected and transmitted waves as:  
\[P_{m}=P_{T}e^{-jk_{c}ma}+P_{R}e^{jk_{c}ma},\quad K_{m}=K_{T}e^{-jk_{c}ma}+K_{R}e ^{jk_{c}ma}. \tag{22}\]  
Figure 4: (**a**) Variation of \(\lambda\) with \(\mu\) for different unloaded resonator Q-factors with \(\kappa=0.6\). (**b**) Frequency dependence of scattering parameters for infinite rejection filter with lossy MI and resistive narrowband terminations with \(\kappa=0.6\), \(\kappa_{c}=0.3\), and \(Q=200\). Figure 5: (**a**) Multiple bandstop MI directional filter; (**b**) frequency dependence of S-parameters for multiple passband filter with lossless MI terminations (1) for \(\kappa=0.6\), \(\kappa_{c}=0.3\), \(N=2\), and \(N=5\). The solutions for currents \(I_{n}\) and \(J_{n}\) remain unchanged from Equation (8). Once again, we may write down the equations at the junctions as:  
\[ \begin{array}{c}ZI_{0}+Z_{m}(I_{-1}+I_{1})+Z_{mc}K_{0}=0,\ ZI_{1}+Z_{m}(I_{0}+I _{2})+Z_{mc}P_{0}=0\\ ZK_{0}+Z_{mc}(I_{0}+K_{1})=0,\ ZP_{0}+Z_{mc}(I_{1}+P_{1})=0\\ ZK_{N-1}+Z_{mc}(K_{N-2}+J_{0})=0,\ ZP_{N-1}+Z_{mc}(P_{N-2}+J_{1})=0\\ ZJ_{0}-Z_{m}(J_{-1}+J_{1})+Z_{mc}K_{N-1}=0,\ ZJ_{1}-Z_{m}(J_{0}+J_{2})+Z_{mc}P_{ N-1}=0.\end{array}  \tag{23}\]  
Equation (23) can again be solved for the reflection coefficient at port 1 and transmission coefficients at ports 2-4 by substituting the dispersion relations and eliminating the terms \(P_{T},P_{R},\) and \(K_{T},\ K_{R}\). When \(N=1\), Equation (24) is identical to Equation (11). \[ \begin{array}{c}\frac{R}{I}=(e^{j2k_{\epsilon}a(N+1)}-1)\Big{\{}e^{j2k_{ \epsilon}a}\Big{(}1+e^{j2k_{\epsilon}a}\Big{)}-\mu e^{j(k_{f}+k_{\epsilon})a} \Big{(}1+e^{j2k_{\epsilon}a}\Big{)}+\mu^{2}e^{j2(k_{f}+k_{\epsilon})a}\Big{\}} \ /D\\ \frac{T_{2}}{I}=\Big{(}1-e^{j2(N+1)k_{\epsilon}a}\Big{\{}2e^{j2(k_{f}+k_{ \epsilon})a}-\mu e^{j(3k_{f}+k_{\epsilon})a}\Big{(}1+e^{j2k_{\epsilon}a}\Big{)} \Big{\}}/D\\ \frac{T_{4}}{I}=\mu e^{j(N+1)k_{\epsilon}a}\Big{\{}2e^{j(k_{f}+k_{\epsilon})a }\Big{(}1-e^{j2k_{\epsilon}a}\Big{)}-\mu e^{j2k_{\epsilon}a}\Big{(}1-e^{j4k_{ \epsilon}a}\Big{)}\Big{\}}/D\\ D=\Big{(}e^{j2k_{\epsilon}a}-1\Big{)}e^{j2k_{\epsilon}a}\Big{(}1-e^{j2k_{ \epsilon}a(N+1)}\Big{)}+2\mu e^{j(k_{f}+k_{\epsilon})a}\Big{(}1-e^{j2k_{ \epsilon}a(N+2)}\Big{)}-\mu^{2}e^{j2k_{\epsilon}a}\Big{(}1-e^{j2k_{\epsilon}( N+3)a}\Big{)}\end{array}  \tag{24}\]  
The coefficients can be converted to scattering parameters as before. Figure 5b shows the frequency dependence of S-parameters for \(N=2\) and \(N=5\). The additional coupling resonators generate multiple bandpass responses in \(S_{41}\) and complementary notches in \(S_{21}\). The value of \(|x_{c}|\) determines the sharpness of the peaks in \(S_{41}\), as well as the notch frequencies, with a larger separation between peaks for higher values. The quality factors of the peaks are higher for increasing \(N\), and the effect of loss is to reduce rejection in \(S_{21}\) and increase insertion loss in \(S_{41}\). The frequencies of peaks and notches can be found by computing the maxima of \(S_{41}\) or the minima of \(S_{21}\) within the MI passband.