# Design and Experiment of Wideband Filters Based on Double-Layered Square-Loop Arrays in the F-Band
## 3 Analysis and Discussion  
In order to examine the intrinsic mechanism of the MDM structure, the surface current and electric field distributions were calculated for the two resonant frequencies \(f_{1}\) and \(f_{2}\), and the obtained results are shown in Figures 2 and 3. Figure 2 shows the surface current distributions in the first and second metal layers in \(f_{1}\) and \(f_{2}\), respectively. The surface currents along the square-loop mainly distribute at the edge of internal patch and external metal loop and flow parallel to the direction of the electric field vector of the incident wave, which excites a dipole response. One can see that the same resonant mode was excited from the surface current distribution of the two resonant frequencies in the first metal layer and excites two different resonant modes in the second metal layer with surface currents flowing in opposite directions.  
Figure 1: Transmission spectra though the metal\(-\)dielectric (MD) and metal\(-\)dielectric\(-\)metal (MDM) structures for a normally incident plane wave.  
Figure 2: Numerical simulation results. Distributions of the surface current for (**a**) the first metal layer and (**b**) second metal layer in \(f_{1}\) and (**c**) the first metal layer and (**d**) second metal layer in \(f_{2}\).  
Figure 3 shows the electric field distributions in the y-z plane (\(x=\) L/2) for the two resonant frequencies \(f_{1}\) and \(f_{2}\). It is evident from Figure 3 that the incident electromagnetic wave propagates through the first metal layer and subsequently excites two different resonant modes with different electric field distributions in the second metal layer. However, the electric field intensity of the transmitted wave is approximately equal to the incident wave, indicating that the MDM structure possesses great transmission characteristics at both resonant frequencies.  
Transmittance through the proposed structure for various oblique angles of incidence is also shown in Figure 4. \(\theta\) is the angle of incidence and is defined as the angle between \(k\) and the x-axis in the x-z plane. The electric field is parallel to the y-axis, whereas the wave propagation direction changes by varying \(\theta\). As shown in Figure 4, transmittance changes are not obvious as \(\theta\) changes from \(0^{\circ}\) to \(30^{\circ}\). If \(\theta\) continues to increase the transmittance of the model decreases drastically. However, the minimum transmittance is slightly larger than 0.63 in the 98.84-132.77 GHz frequency range when the incident angle is \(30^{\circ}\), which indicates broadband transmission.  
It is generally challenging to perfectly align the two metallic layers due to limitations in the manufacturing process. Hence, it is necessary to investigate how periodic misalignment influences wave propagation. Figure 5 shows how the two resonant peaks \(f_{1}\) and \(f_{2}\) vary for different P values ranging from 0.1 to 0.9 in two different directions (diagonal and axis directions), where P is defined as a measure of misalignment. For periodic misalignment along the diagonal direction, resonant peak \(f_{2}\) (red line) decreases and eventually reaches a minimum value of 122.15 GHz when P = 0.5. It then increases in a manner symmetrical to the first half of the curve when P varies from 0.6 to 0.9. This is because the proposed structure is symmetrical and periodic in both diagonal and axis directions. The value of \(f_{1}\) (black line) changes in an opposite manner compared to \(f_{2}\). For periodic misalignment  
Figure 4: Oblique incidence response of the proposed structure.  
Figure 3: Electric field distributions in the y-z plane (\(x=\) L/2) for (**a**) \(f_{1}\) and (**b**) \(f_{2}\).  
along the axis direction, it is evident that variations in the two resonant peaks \(f_{1}\) and \(f_{2}\) are similar to those along the diagonal direction. The results shown in Figure 5 show that misalignment along both the diagonal and axis directions decreases the bandwidth of the MDM structure.