# Resonant Coupling Effects in a Double-Layer THz Bandpass Filter
## 3 Results and Discussion
When the transmittance is the maximum, the circuit resonates, and the imaginary parts of the paralleled impedance \(Z_{imp1}\) and \(Z_{FSS}\) should be equal in amplitude but with opposite signs. Therefore, we add up the imaginary parts of \(Z_{imp1}\) and \(Z_{FSS}\) and plot the sum in Figure 4a for \(d=555\)\(\upmu\)m and \(d=585\)\(\upmu\)m, and Figure 4b is the enlarged view at the sum equal to zero. As it can be seen from the figure, for \(d=555\)\(\upmu\)m, at about 0.487 and 0.621 THz, the sum lines pass zero. At these two zero points, the upper and lower FSSs resonate and the anomalous transmission peaks \(f_{1}\) and \(f_{3}\) appear. For \(d=585\)\(\upmu\)m, at 0.473 and 0.607 THz, the sum lines pass zero, corresponding to the anomalous transmission peaks of \(f_{4}\) and \(f_{6}\). Therefore, according to ECM theory, the anomalous transmission peaks (\(f_{1}\),\(f_{3}\),\(f_{4}\), and \(f_{6}\)) can be attributed to the resonant coupling between double FSS layers. At 0.564 THz, that is, the transmission peaks \(f_{2}\) and \(f_{5}\) (also the position of line \(B\)), the sum is not equal to zero, because the peak corresponds to the intrinsic transmission of the single-layer FSS and is irrelevant to the double-layer coupling. In addition, the ECM also predicts peaks at about 0.25 and 0.8 THz, which were not observed in our experimental data. For the peak at 0.25 THz, the real part of \(Z_{L2}\) is only about a few ohms, mismatching with the impedance of the free space (377 ohms), indicating a weak resonance/peak. The peak at 0.8 THz is too sharp to be observed in our experiments because we have a limited frequency resolution. Furthermore, for the periodic appearance of the anomalous transmission peaks/wings, we found that \(Z_{imp1}\) has a certain periodicity. Increasing \(d\) by integer multiple of half wavelength (\(\frac{mA}{2}\)) and using Equation (6) to calculate \(Z_{\text{inp1}}\) (\(d+\frac{mA}{2}\)), we have  
\[Z_{\text{inp1}}\left(d+\frac{mA}{2}\right)=Z_{0}\frac{Z_{L1}+jZ_{0}tan(\beta_ {0}\times d+m\uptau)}{Z_{0}+jZ_{L1}tan(\beta_{0}\times d+m\uptau)}=Z_{0}\frac {Z_{L1}+jZ_{0}tan(\beta_{0}\times d)}{Z_{0}+jZ_{L1}tan(\beta_{0}\times d)}=Z_ {\text{inp}} \tag{10}\]  
Figure 4: (a) The sum of the imaginary parts of \(Z_{\text{inp1}}\) and \(Z_{\text{FSS}}\); (b) the enlarged view at the sum equal to zero. This indicates that the impedances of two points with \(\frac{m\lambda}{2}\) spacing on the transmission line are equal. We can convert the repetitive phase \(\Delta\varphi=m\pi\) (\(m\) is a positive integer) into distance. The repetition period of \(d\) is calculated as follows:  
\[\Delta d=\frac{mc}{2f} \tag{11}\]  
By using Equation (11) and substituting different \(m\) values, we can plot lines \(A_{i}\) and \(C_{i}\), which coincide with the center of the periodic wings shown in Figure 3. Therefore, by calculating the sum of the imaginary parts of \(Z_{imp1}\) and \(Z_{FSS}\), we can predict these periodically appearing anomalous transmission peaks/wings that are caused by the resonant coupling between the two FSS layers. Our interpretation is consistent with the explanation in terms of a strongly coupled Fabry-Perot resonator [21]. In addition, if we draw a horizontal line at a larger spacing \(d\) in Figure 3, we obtain more anomalous peaks. They can also be perfectly predicted and explained using our model. At THz frequencies, the thicknesses of commercially available substrates are comparable to the free-space wavelength. Substrates cause unwanted substrate resonances or Fabry-Perot resonances, which otherwise degrade the transmission characteristics of the cascaded FSS structure, smearing out the anomalous transmission features observed in this work.