## 2 Characteristic Mode-Based Antenna Methodology  
The characteristic mode analysis (CMA) is a rapid development tool applied to the antenna development stages to comprehend its performance and radiation pattern. The structure was modified to achieve the desired results. CMA estimates the effective modes of an arbitrarily shaped antenna structure, where the first five modes are mostly found to be significant. However, modifying the structure and changing the feed position can generate undesired modes. In Stage 1 of the antenna design, an initial rectangular patch antenna was designed based on the design equations in [20] for a resonating frequency of 25.5 GHz. From the above calculations, the approximate patch width and length were 4.65 and 3.718 mm, respectively. However, to retain compactness and to study the modified structure behavior, the patch width and length were chosen to be 5 and 3 mm in the first stage, as shown in Figure 1a. For CMA, a 0.51\(\lambda_{0}\)\(\times\) 0.51\(\lambda_{0}\) lossless Roger 5880 substrate with a thickness of 0.25 mm and relative permittivity (\(\varepsilon_{r}\)) of 2.2 was selected. The electromagnetic simulation software CST v21 was used to analyze antenna development stages without port excitation. The antenna was designed to generate dual-band resonances at 25.5 and 38 GHz. Therefore, the CMA was performed separately for each of these frequencies. During CMA, the total surface current on the antenna structure of the patch and ground plane was estimated by considering the sum of the eigenvector (\(J_{n}\)) using Equation (1) [15; 18]. \[J=\sum_{n}\frac{V_{n}J_{n}}{1+j\lambda_{n}} \tag{1}\]  
where \(V_{n}\) is the model-excitation coefficient. The \(V_{n}\) along with model significance (MS) defines the significant modes at desired frequencies, where \(MS=\left|\frac{1}{1+j\lambda_{n}}\right|\). In Figure 1b, the modal significance graph indicates that Mode 1 and 2 are significant at 25.5 GHz, Mode 1-3 are significant at 28 GHz with a wide bandwidth, and Mode 1-5 are significant at 38.5 GHz with a narrow bandwidth. The convergence of these modes eigenvalue (\(\lambda_{n}\)) to zero indicates that the modes are resonant. However, the modes with an eigenvalue (\(\lambda_{n}\)) > or <0 are non-resonant inductive and capacitive modes. These modes store the energy in the form of electric and magnetic fields in the near-field region. The eigenvalue (\(\lambda_{n}\)) indicates that Mode 1-3 and 1-4 at 28 and 38.5 GHz are resonances, as shown in Figure 1c. The characteristics angle (CA) is another feature of CMA, which represents the resonant and non-resonant modes by analyzing the phase lag between the electric field and surface current on the antenna structure. It is represented in terms of angle (\(\alpha_{n}\)). The characteristics angle in Figure 1d depicts that Mode 1-3 from 27-28.5 GHz have zero eigenvalues, resulting in \(\alpha_{n}=180^{\circ}\), which means these modes are resonant modes with good impedance matching. Beyond this frequency, the Mode 1 and 2 surface current's phase lag from the electric field led to an inductive (non-resonant mode) mode. At 38.5 GHz, a poor impedance matching with a narrow bandwidth can be seen due to the summation of phase-lead and phase-lag of Mode 3 and 5 and Mode 2, respectively. The port excitation corroborates in the same manner, which indicates that, at 30 GHz, a good impedance matching 1S11  of 37 dB with a bandwidth of 29.5-30.5 GHz is achieved, as shown in Figure 1e. However, at 38 GHz, due to inductive and capacitive Mode 2, 3, and 5, a poor impedance matching of 12 dB can be seen. Therefore, the CMA estimation for the first stage closely agrees with the port excitation results. The sum of all modes of eigenvector currents (\(J_{n}\)) represents the total current on the antenna structure. In Figure 2, the eigenvector current (\(J_{n}\)) is concentrated over the antenna surface edges. Based on the \(J_{n}\) movement and direction, the respective modes characterize the radiation behavior.