# Resonant Coupling Effects in a Double-Layer THz Bandpass Filter
## 3 Results and Discussion  
Figure 2a,b shows the FDTD (blue), ECM (green), and experimental (red) results when the interlayer spacing \(d\) equals 555 and 585 \(\upmu\)m, respectively. The ECM results are credible only when the calculated wavelength is larger than the period \(p\) = 350 \(\upmu\)m [24]. The transition point at \(\lambda\approx p\) = 350 \(\upmu\)m (\(f\) = \(\sigma\)/\(\lambda\approx\) 0.857 THz) was traditionally referred to as [PERSON]'s anomaly [32]. Thus, the simulation THz range is set from 0.1 to 0.85 THz. In the simulated range, the results obtained by ECM agree with those obtained by FDTD and experiments. We see three peaks in each experimental spectrum, which are labeled as \(f_{1}\),\(f_{2}\),\(f_{3}\) for \(d\) = 555 \(\upmu\)m and \(f_{4}\),\(f_{5}\),\(f_{6}\) for \(d\) = 585 \(\upmu\)m. The ECM spectrum shows narrower peaks, because both experiments and FDTD simulations have a limited time window. In the time-domain THz spectroscopy measurements, the time window is confined by the internal reflection pulse of a 1 mm ZnTe detection crystal, appearing at about 21.6 ps after the main THz pulse. We scanned the interlayer spacing \(d\) from 50 to 1050 micron to explore the evolution of the transmission peaks at different interlayer spacing using ECM. We obtained the transmittance contour map for a range from 0.1 to 0.85 THz, as shown in Figure 3a, which coincides with the results of FDTD simulation (Figure 3b). The transmission peak, which does not vary with the spacing \(d\), is highlighted with a white line \(B\), and two transmission lines, periodically appearing like wings on both sides of line \(B\) with varying \(d\), are highlighted as line \(A_{i}\) and \(C_{i}\) (\(i\) denotes the series of wings). Figure 3: (**a**) The transmittance contour map in a THz range from 0.1 to 0.85 THz for spacing \(d\) changing from 50 to 1050 \(\upmu\)m, calculated by ECM. Line \(B\) highlights the central transmission peak, and line \(A_{i}\) and line \(C_{i}\) highlight the left and right wings, respectively. \(i\) denotes the series of wings. (**b**) The transmittance contour map calculated by FDTD. Figure 2: Comparison between the experimental, equivalent circuit model (ECM), and Finite-Difference Time-Domain (FDTD) results for spacing \(d\) = 555 \(\upmu\)m (**a**) and 585 \(\upmu\)m (**b**). It can be seen from Figures 2 and 3 that the positions of \(f_{2}\) and \(f_{5}\) coincide with that of line \(B\), at about 0.557 THz. Obviously, the peak highlighted by line \(B\) is the intrinsic transmission of the single-layer FSS. In order to analyze the effect of double-layer coupling, we need to look into \(Z_{L2}\). Equation (6) shows that \(Z_{L2}\) is composed of \(Z_{imp1}\) and \(Z_{FSS}\) connected in parallel. The air interlayer is represented by \(Z_{0}\) and \(\beta_{0}\). \(Z_{imp1}\) represents the impedance of the first layer FSS plus interlayer air, and \(Z_{FSS}\) represents the equivalent impedance of the second FSS layer. The single-layer slot-type FSS is equivalent to the parallel connection of inductance and capacitance. When resonance occurs, the inductance reactance and capacitance reactance of the parallel circuit are equal, that is, the inductance current and the capacitance current are equal. Because the inductance current lags behind the voltage by \(90^{\circ}\) in phase and the capacitance current leads the voltage by \(90^{\circ}\) in phase, the phase difference between the inductance current and the capacitance current is exactly \(180^{\circ}\). At this time, there is no current passing through the circuit (no loss) and the transmittance of the FSS is the maximum. Similarly, for the double-layer FSS, the total input impedance \(Z_{L2}\) is equivalent to the parallel connection of \(Z_{\text{inp1}}\) and \(Z_{FSS}\). As we know, the real part of the impedance represents resistance, while the imaginary part represents reactance. A positive imaginary part represents inductance, while a negative imaginary part represents capacitance.