## 3 Design of the Wide Passband SIW-SSPP Filters with Multi-Sub-Bands
### The Wide Passband SIW-SSPP Filter
#### 3.1.2 The Single Wide Bandpass Filter Loaded with Archimedean Spiral Structure  
The single-bandpass filter structure designed in the previous subsection exhibited limitations in both stopband attenuation and in-band performance. To address these issues, a new groove structure with Archimedean spirals on both the top and bottom surfaces of the dielectric substrate is introduced, as illustrated in Figure 5a. The Archimedean spiral structure, due to its equidistant rotational symmetry, is suitable for the construction of periodic metamaterials. The resonant frequencies of rings with different radii within a unit cover a wide frequency band, and the coupling of multiple rings enables the smooth phase superposition and the gentle dispersion of the equivalent refractive index. These two mechanisms work together to significantly improve the flatness of the phase response of the metasurface unit within a wide frequency band. The resonant characteristics of the Archimedean spiral are determined by its geometric parameters: the lowest resonant frequency is predominantly governed by the outer radius, while the highest resonant frequency is primarily determined by the inner radius. Between these two extremes, the spiral structure exhibits multiple intermediate resonances, whose frequencies are critically dependent on the inter-turn spacing [36]. This multi-resonant behavior demonstrates remarkable electromagnetic synergy with the SSPP unit described in the preceding section. This synergistic integration enhances the high-frequency cutoff characteristics. The location and sizes of the grooves are optimized out as follows: \(move_{x1}=8.5\) mm and \(move_{y1}=1.4\) mm  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(L_{sinv}\) & 20.5 & \(W_{sinv}\) & 4.5 & \(L_{ms}\) & 3.65 \\ \(W_{ms}\) & 1.63 & \(L_{t}\) & 2.4 & \(W_{t}\) & 1 \\ \(L_{t2}\) & 1.2 & \(W_{sub}\) & 6 & \(d\) & 0.4 \\ \(s\) & 0.38 & \(a_{1}\) & 0.5 & \(a_{2}\) & 0.4 \\ \(a_{3}\) & 0.2 & \(b_{1}\) & 1 & \(b_{2}\) & 0.8 \\ \(b_{3}\) & 0.6 & \(p\) & 2.88 & \(g_{1}\) & 0.1 \\ \hline \end{tabular}
\end{table}
Table 1: Design parameters of the designed single wide bandpass filter structure (unit: mm).  
are distance from spiral center to starting point, \(r_{Arc1}\) = 0.63 mm and \(r_{Arc2}\) = 0.21 mm outer and inner radii of the spiral structures, and \(g_{Arc}\) = 0.1 mm is the groove clearance width.  
Figure 5c presents the S-parameters results. Within the 3 dB of 24-30 GHz passband, S\({}_{11}\) remains below \(-\)12 dB, while the insertion loss is 0.5 dB. Compared to the unloaded Archimedean spiral structure, the passband exhibits improved flatness. Moreover, S\({}_{21}\) drops sharply to \(-\)70 dB with a high roll-off rate, indicating significant performance enhancement. To further visualize the effects of loading the Archimedean spiral, the surface current at different frequency points is subsequently analyzed. As observed from the surface current in Figure 5c, at 23 GHz, the current is interrupted by the SIW effect. At 31 GHz, the current is blocked by the combined action of the comb-shaped SSPP structure and the Archimedean spiral. Compared with the surface current at 31 GHz in Figure 3b, the current received at Port 2 is significantly reduced. Within the passband (24-30 GHz), the current propagates through the entire region from Port 1 to Port 2. In summary, after loading the Archimedean spiral, the SIW-SSPP-based wide bandpass filter exhibits improved performance in both the passband and high-frequency cutoff characteristics.