## 2 Characteristic Mode-Based Antenna Methodology
The eigenvector current and radiation pattern of the respective modes at 25.50 and 38 GHz are shown in Figure 2a,b, respectively. In Figure 2a, Mode 1 and 2 have in-phase vector currents (currents on the parallel edges are in the same direction) along the vertical/horizontal edges of the ground (black arrow) and radiator (redarrow). Mode 1 has an in-phase current along the vertical edges, causing a bidirectional radiation pattern with a null in the azimuthal plane at \(\phi\) = 0\({}^{\circ}\). Similarly, Mode 2 has an in-phase current at horizontal edges, providing a bidirectional radiation pattern with nulls in the azimuthal plane at \(\phi\) = 90\({}^{\circ}\). Mode 3-5 have an anti-phase eigenvector current (current in the opposite direction of the parallel edges) along the antenna edges. Thus, a quad/end-fire radiation pattern is formed, as seen in Figure 2a. Similar effects can be observed in Figure 2b at 38 GHz but with the most current in the radiator. Mode 1 and 2 have an in-phase current on one set of parallel edges and an anti-phase current on the other, resulting in bidirectional quad radiation. However, Mode 3-5 have an anti-phase current, resulting in end-fire radiation. As discussed above, the first resonance occurred at 30 GHz with good impedance matching and the second at 38 GHz with poor impedance matching. Further, in Stage 2, to decrease the first resonance to the expected frequency of 25.5 GHz and improve the impedance matching at 38 GHz, a semi-elliptical stub was added at the bottom of the radiator, as displayed in Figure 3a. From the CMA of Stage 2, the results of the MS in Figure 3b indicate that Mode 1-3 drifted from 30 to 28 GHz due to structure modification. However, at 25.5 GHz, only Mode 1 and 2 were found to be significant. On the other hand, Mode 1, 2, 3, and 5 drifted slightly higher, from 38.5 to 39 GHz. The eigenvalue of these modes converged to zero, as shown in Figure 3c, at respective frequencies, resulting in resonant modes with bandwidths of 24.1-26, 27-28, and 37.5-39 GHz. The characteristic angle shows a 180\({}^{\circ}\) phase lag between the surface current and the electric field at 25.5, 28, and 39 GHz of the resonant modes. However, the Mode 5 surface current led the electric field by 210\({}^{\circ}\) until 39 GHz, abruptly changing to a phase lag of 120\({}^{\circ}\) after 39 GHz. The Mode 4 surface current had a phase lag of 150\({}^{\circ}\) from 36 to 40 GHz. Due to these phase differences in Mode 4 and 5, the structure could not radiate complete energy at 39 GHz. As a result, an impedance mismatch occurred. The port excitation substantiated the CMA. The  
Figure 1: CMA of Mode 1–5 and reflection coefficients of the Stage 1 MS antenna. (**a**) Stage 1 antenna (dimensions in mm), (**b**) modal significance, (**c**) eigenvalues, (**d**) characteristic angle, and (**e**) reflection coefficient with port excitation. results indicate the occurrence of resonances at 28 and 38 GHz with impedance matching of 25 and 10.5 dB, as shown in Figure 3e. Figure 4 represents the eigenvector current at 25.5 and 38 GHz. From the above eigenvalue and characteristic angle, only Mode 1 and 2 are resonant, but these are insufficient to generate the resonance at 25.5 GHz. This is evident from the eigenvector current in Figure 4a, as most of the current is in the ground plane rather than the radiator. At 38 GHz, Mode 1-3 have an in-phase current generating resonance, as shown in Figure 4b. However, an anti-phase current arose due to the non-resonant Mode 4 and 5, causing impedance matching. Stage 3 with a semi-elliptical stub at the top of the radiator was introduced, as displayed in Figure 5a, to lower the resonance from 28 to 25.5 GHz and to improve the impedance matching at 38 GHz. This improved the significance of Mode 1-3 at 25.5 and 38 GHz, as shown in Figure 5b. The convergence of these modes to zero in the eigenvalue, as shown in Figure 5c, represents resonant modes (Mode 3 at 25.5 GHz is partially resonant) with bandwidths of 25-27 and 37-39 GHz.