## 3 Results and Discussion  
### Numerical Examples  
We first investigate the applicability of the AFGSM method for determining the passband and stopband regions of corrugated PPW structures. The final phase is dimensional synthesis in accordance with the design objectives. Figures 3 and 4 serve as analysis examples to evaluate the efficacy of the AFGSM method in distinguishing passband and stopband regions including not only wide but also narrow stopbands inside single and glide-symmetric double-corrugated PPW structures. Parametric details of Figure 3a,b are given with \(\varepsilon_{r}=1\), \(p=6\) mm, \(h_{1}=3.5\) mm, \(h_{2}=4.33\) mm, \(a=5.1\) mm, and \(b=0.45\) mm and \(\varepsilon_{r}=1\), \(p=9\) mm, \(h_{1}=6\) mm, \(h_{2}=1\) mm, \(a=1\) mm, and \(b=4\) mm, respectively. All parameter values of Figure 4a,b are taken as \(\varepsilon_{r}=1\), \(p=6\) mm, \(h_{1}=3.5\) mm, \(h_{2}=h_{3}=15\) mm, \(a=3\) mm, and \(b=1.5\) mm and \(\varepsilon_{r}=1\), \(p=9\) mm, \(h_{1}=10\) mm, \(h_{2}=h_{3}=1\) mm, \(a=0.5\) mm, and \(b=4.25\) mm, respectively. The first and second stopbands of the periodic structure are obtained by applying the eigenvalue equation to the wide stopband example in Figure 3. These stopbands are located within the ranges of 12.14-19.58 GHz and 29.89-37.26 GHz, respectively. The AFGSM method yields the band edge frequencies of 12.136 GHz, 19.584 GHz, 29.881 GHz, and 37.263 GHz, respectively. The AFGSM method identified the band edges with a maximum difference of 6 MHz, which coincides with the results derived from the eigenvalue equation. Figure 3b presents a narrowband example, indicating the first and second stopband edges at nearly 16.33 GHz, 16.71 GHz, 32.29 GHz, and 33.43 GHz for AFGSM and EE. We achieved band edges near the eigenvalue equation in the narrow stopband scenario, with a maximum deviation of 7 MHz. Figure 2: Equivalent circuit model of (**a**) single-corrugated and (**b**) glide-symmetric double-corrugated PPW structures. Figure 4a,b demonstrates that the zero crossings of the \(X_{+}\) and \(X_{-}\) functions in the AFGSM method consistently align closely with the band edge frequencies in the dispersion diagrams derived from the eigenvalue equation solutions. Floquet mode transitions are observed in certain regions of the dispersion diagrams without stopbands in Figure 4a,b, with transitions occurring at 4.52 GHz and 13.49 GHz, respectively, and 16.57 GHz in Figure 4b. In these regions, the zero crossings of the \(X_{+}\) and \(X_{-}\) functions occur at identical frequencies. Table 1 presents the band edge frequency results associated with Figure 4a,b. Figure 4: Dispersion diagram and \(X_{+}\), \(X_{-}\) frequency behavior of glide-symmetric double-corrugated PPW for (**a**) wide and (**b**) narrow stopband region analysis. Figure 3: Dispersion diagram and \(X_{+}\), \(X_{-}\) frequency behavior of single-corrugated PPW for (**a**) wide and (**b**) narrow stopband region analysis. Based on all these analysis results, it is possible to say that Equation (5) provides an alternative method to determine the passband/stopband regions of one-dimensional corrugated PPW structures. Furthermore, the AFGSM functions clearly demonstrate the separation of the band edges, even if the stop bandwidth is narrow. The interval division root-finding routine determines the roots of the \(X_{+}\) and \(X_{-}\) functions, eliminating the need for fine frequency sweeps and reducing the computation time by a factor of six compared to the eigenvalue equation. Figures 5 and 6 illustrate the dispersion diagrams and the characteristics of auxiliary functions for single and glide-symmetric double-corrugated PPW unit cells with varying dielectric constants. Parameter values of Figure 5a,b are given as \(\varepsilon_{r}=1\), \(\varepsilon_{r}=2.25\), and \(\varepsilon_{r}=3\), \(\varepsilon_{r}=11.2\) with \(p=12\) mm, \(h_{1}=3\) mm, \(h_{2}=0.5\) mm, \(a=5\) mm, and \(b=3.5\) mm for all cases, respectively.