## 2 Antenna Design and Optimization
### Evolution of Antenna Geometry  
The designed antenna gained the desired design requirements in three main steps. The dual-frequency antenna in this article was obtained via the evolution of the planar inverted-F antenna (PIFA), and the initial dimensions of the PIFA were obtained via the following equation [29]:  
\[f_{r}=\frac{c}{4(w+l)} \tag{1}\]  
where \(c\) represents the velocity of light in free space, \(w\) and \(l\) represent the width and length of the radiation component, and \(f_{r}\) represents the operating frequency.  
The calculated center frequency of the antenna is around 5.1 GHz when the radiation patch is without slots, and the antenna does not gain dual-band characteristics. As shown in Table 3, the design is divided into three main steps. Step 1: By cutting two F-shaped slots into the left part of the radiation patch, the resonant frequency is reduced to around 2.4 GHz due to the increased current path, as shown in Figure 3a. Step 2: After cutting three I-slots into the right of the radiation patch, the antenna has a new resonance point near 1.4 GHz but it cannot cover the 2.4 GHz band well; see the green line in Figure 3a. Additionally, the AR in the 1.4 GHz band is poor as shown in Figure 3b (green line). Then, we used the characteristic mode method to analyze its polarization characteristics. The simulated model significance (MS), determined using CST, is shown in Figure 4a. The MS values of modes 1 and 2 are less than 0.8 in the 1.4 GHz band, and the simulated current vectors of modes 1 and 2 are not orthogonal, as shown in Figure 5a. So, its AR is not good enough in this case. The MS values of modes 3 and 4 are less than 0.2, so we do not consider these two modes. To gain CP characteristics in the 1.4 GHz band, step 3 is led out, and we adjust the feed appropriately and cut a cross slot on the GND. The simulated MS for the antenna and the current vector distribution at the patch surface at this time are displayed in Figures 4b and 5b, respectively. The MS values of mode 1 and mode 2 are nearly equal to 1 in this case. Moreover, the simulated current vectors of the two modes are orthogonal to each other, allowing the antenna to operate in the desired band and achieve CP characteristics (Table 4).  
Figure 3: Simulation results of S11 and AR in three cases. **(a)** S11 and **(b)** AR.  
Figure 4: The simulated MS of different cases. (**a**) Case 2 and (**b**) case 3.  
Figure 5: The simulated current vector distributions on the radiation patch of different cases. (**a**) Case 2 and (**b**) case 3. (The darker the arrow color, the greater the current intensity).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **(1)** & **(2)** & **(3)** \\ \hline Top View & & \\ Bottom View & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation of the evolution process of the antenna’s geometry.  
From this, we can draw the following conclusion:  
1. Because of the slots, the dimensions of the antenna are reduced for equal operating frequencies.
2. Two operating bands can be produced by cutting I-shaped grooves into the right side of the rectangular radiation patch.
3. The cross grooves cut on the GND allow for better tuning of the resonant frequency and CP performance.