## 4 The Asymmetric Case  
The asymmetric case used a square patch for the first case (Model A). Specifically, one of the corners was trimmed by connecting two points: one located 4 mm along the horizontal (\(x\)-axis) direction from the top-left corner, and another located 4 mm downward (along the \(y\)-axis) from the same corner. A straight diagonal line was drawn between these two points to create the chamfered edge, resulting in an asymmetrical polygonal shape, as illustrated in Figure 11a. Aside from this geometric modification, all other structural parameters remain identical to those of the first case (Model A). This model is also evaluated through simulations and experiments under linearly polarized wave incidence. The purpose of evaluating the asymmetric structure is to investigate whether introducing geometric asymmetry can mitigate the severe voltage dips observed in symmetric designs, rather than to achieve full polarization independence. As shown in Figure 11a, the asymmetric model demonstrates a smoother voltage profile with reduced sharp voltage dips across the frequency range, even when the polarization angle is varied. This indicates that the asymmetrical structure helps modify the surface current distribution in a way that is less sensitive to changes in polarization, reducing the occurrence of sharp voltage dips. Although the asymmetric structure does not eliminate polarization dependence entirely it significantly mitigates voltage dips, leading to more stable energy harvesting performance. In addition, the dip angle varies with frequency because the surface current changes in a complex manner, depending on the relationship between the electric field direction, the position of the via (resistor), and the shape of the cut. Figure 11: (**a**) The asymmetric model of the unit cell and the simulated voltage results. The black dashed line represents the frequency-dependent position of the voltage dips. (**b**) Comparison of simulation results of the voltage on the asymmetric model (blue line) and symmetric model of the first case Model A (orange line). (**c**) Comparison of measurement and simulation results for the voltage on the asymmetric model at the maximum voltage frequencies of 4.75 GHz and 4.998 GHz, respectively. Simulation results are the blue line and measurement results are the orange dots. Figure 11b shows a comparison between the simulated voltage responses of the asymmetric patch model (blue line) and the symmetric patch model (Model A, orange line) as a function of polarization angle from 0\({}^{\circ}\) to 180\({}^{\circ}\). The results are shown for fixed operating frequencies corresponding to the maximum voltage observed in each design based on earlier simulations: 5.114 GHz for the symmetric case (see Figure 3a) and 4.998 GHz for the asymmetric case (see Figure 11a). Figure 11b clearly shows that the symmetric patch (Model A) exhibits a sharp voltage dip at 90\({}^{\circ}\) polarization, indicating high sensitivity to polarization angle. In contrast, the asymmetric patch maintains a more stable voltage response across the entire polarization range, demonstrating less sensitivity. However, this stability comes with a lower maximum voltage as the peak voltage for the asymmetric case reaches 12.7 dBV, while the symmetric design reaches a higher maximum of 14.7 dBV. This reduction in voltage is due to the asymmetric geometry, which interacts more favorably with circularly polarized components and less effectively with linearly polarized waves. This observation clearly illustrates a performance trade-off. The asymmetric structure sacrifices approximately 2 dB in peak voltage compared to the symmetric case, which corresponds to roughly a 37% reduction in power efficiency. However, it offers improved robustness across varying polarization angles. Such robustness is particularly valuable for practical RF energy harvesting applications where the incident polarization may vary unpredictably or randomly over time.