## 2 Principle and Design Analysis of Novel SSPP and SIW with Wide Passband Characteristics
### The Novel SSPP  
SSPP operates in the slow-wave region, exhibiting low-pass filter characteristics, and features a higher cutoff frequency when used in bandpass filters. Its dispersion characteristics are influenced by the structure. Here a novel SSPP structure is proposed utilizing elliptical grooves. Figure 2a illustrates the periodic SSPP unit cell configuration (front-side architecture) obtained. The yellow regions represent metallic layers and cyan coloration denotes the substrate. SSPP formed by substrate-exposed grooves patterns are  
Figure 1: Structural and dispersion characteristics of designed SIW: (**a**) schematic unit structure; (**b**) dispersion relation.  
shown in the dielectric substrate color. It has a unit periodicity of \(p\), and \(a_{1}\) denotes the SSPP comb spacing, \(b_{1}\) denotes the longitudinal length of SSPP, and \(g_{1}\) denotes the SSPP units groove clearance width. The dispersion properties of SSPPs are typically determined by the parameters of the periodic grooves as follows [33]:  
\[\beta=\sqrt{\varepsilon_{eff}k_{0}^{2}+\left(\frac{g_{1}}{p-4a_{1}}\right)^{2} \varepsilon_{eff}k_{0}^{2}\tan^{2}(k_{0}\sqrt{\varepsilon_{eff}}2b_{1})} \tag{3}\]  
The lengths of these elliptical grooves can be expressed with Ramanujan formulas as:  
\[C_{n}=\pi(na_{1}+b_{1})\left(1+\frac{3\Big{(}\frac{b_{1}-na_{1}}{b_{1}+na_{1}} \Big{)}^{2}}{10+\sqrt{4-3\Big{(}\frac{b_{1}-na_{1}}{b_{1}+na_{1}}\Big{)}^{2}}}\right) \tag{4}\]  
in which \(n\) is the number counted from 0 and 0, and in the middle of an ellipse, a segment of the line is \(C_{0}=2b_{1}\). The elliptical grooves with semi-axes are \((na_{1},b_{1})\).  
The corresponding cut-off frequency of SSPP is given as follows [34]:  
\[f_{2}=\frac{c}{4\Big{(}2b_{1}+\sum\limits_{i=1}^{n}C_{i}\Big{)}\sqrt{ \varepsilon_{eff}}} \tag{5}\]  
Figure 2: Structural and dispersion characteristics of designed SSPP unit: (**a**) schematic unit structure; (**b**–**d**) SSPP units dispersion diagram under different \(a_{1}\), \(b_{1}\), \(g_{1}\), respectively.  
where \(f_{2}\) denotes the SSPP cutoff frequency, and \(\epsilon_{eff}\) is the effective dielectric constant of the metallic groove structure [35]:  
\[\varepsilon_{eff}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2}\left( 1+12\frac{h}{g_{1}}\right)^{-\frac{1}{2}} \tag{6}\]  
where the groove depth (\(h\)) of SSPP is 0.035 mm; the groove width (\(g_{1}\)) of SSPP is 0.1 mm; and the relative permittivity (\(\varepsilon_{r}\)) of the bottom dielectric is 2.2.  
The dispersion characteristics or low-pass effects of the SSPP structure are obtained with the calculation based on (3)-(6) and optimized with CST. Figure 2b-d illustrate the dispersion characteristics of the SSPP unit of different structure size parameters, which are obtained to meet the cutoff frequency control requirement of n261 when the optimum of \(p=2.88\). It shows that the SSPPs operate in the slow-wave region and exhibit a high cutoff frequency, thereby generating low-pass filtering behavior. The cutoff frequency of SSPP decreases with \(a_{1}\) and \(b_{1}\), while demonstrating an increasing relationship with \(g_{1}\). Figure 2b illustrates how the dispersion relation evolves as parameter \(a_{1}\) increases from 0.45 mm to 0.55 mm, while all other parameters remain at their initial values. Figure 2c,d illustrate similar results for parameter \(b_{1}\) and \(g_{1}\) in turn. Figure 2b shows the results when \(b_{1}=1\) and \(g_{1}=0.1\). It shows that \(a_{1}=0.5\) at the lowest cutoff frequency 29.5 GHz. Figure 2c shows the results when \(a_{1}=0.5\) and \(g_{1}=0.1\). It shows that \(b_{1}=1\) at the lowest cutoff frequency 29.5 GHz. Figure 2d shows the results when \(a_{1}=0.5\) and \(b_{1}=1\). It shows that \(g_{1}=0.1\) at the lowest cutoff frequency 29.5 GHz. So, the optimum sizes are \(a_{1}=0.5\), \(b_{1}=1\), and \(g_{1}=0.1\) for the lowest cutoff frequency 29.5 GHz.