## 2 Characteristic Theory-Based Antenna Methodology  
The characteristic mode analysis (CMA) has been applied to arbitrary complex antenna structures where the mathematical proofing of such structures becomes difficult. It determines the naturally resonant modes and position of excitation on the antenna structure. Initially, [PERSON] [18; 19] proposed this method, which was later improved by [PERSON] and [PERSON] [20], who applied integral equations (IEs) to derive surface currents. An IE is solved for independent surface current modes that result closely to the impedance matrix obtained using the method of moments (MoM) [21]. For our case, analysis was performed by considering the impinging incident wave on a perfectly conducting surface and lossless substrate material. An eigenvalue (\(\lambda_{n}\)) with a modal excitation coefficient (\(V_{n}^{i}\)) relates the electric field (\(E^{i}\)) and surface current (\(J_{n}\)) on the structure. This coefficient provides adequate information on the feed position, magnitude, and phase effects of individual modes resulting in total surface current [22]. Therefore, the total surface current on the conductor surface due to the incident wave was computed using Equation (1):  
\[J=\sum_{n}a_{n}J_{n}=\sum_{n}\frac{V_{n}^{i}J_{n}}{1+j\lambda_{n}}=\sum_{n}\frac{ \langle J_{n},\ E^{i}\rangle J_{n}}{1+j\lambda_{n}} \tag{1}\]  
where \(a_{n}\) is the excitation coefficient of the \(n\)-th naturally resonant mode. When the eigenvalue (\(\lambda_{n}\)) approaches zero, this indicates that the resulting individual surface current/mode is significant, which is estimated using Equation (2):  
\[MS_{n}=\left|\frac{1}{1+j\lambda_{n}}\right| \tag{2}\]  
However, structural properties and their behavior can be further studied through relative phase differences between these modes with the aid of the characteristic angle (CA). The incident electric field (\(E^{i}\)) on the conductor induces the surface current (\(J_{n}\)), which is relayed by Equation (3) [22]:  
\[E^{i}=\iint\limits_{V_{n}^{i}}J_{n}\cdot\overline{G}dv^{i} \tag{3}\]  
where \(\overline{G}\) is the dyadic Green's function for the current source \(J_{n}\). Based on the material property and construction of the structure, phase lag may occur between the oscillating electric field and induced surface current due to the structure's capacitance or inductance effect. The resulting phase difference is calculated from the eigenvalue (\(\lambda_{n}\)) using Equation (4):  
\[\varphi=180^{\circ}-tan^{-1}\lambda_{n} \tag{4}\]  
Here, the phase difference between \(90^{\circ}\) and \(180^{\circ}\) indicates that the structure has an inductance effect that generates opposing electromotive force (EMF) in response to the incident electric field, causing phase delay in the oscillating mode surface current (\(J_{n}\)). This phase lag consequently stores the energy in magnetic form, causing no radiation. On the other hand, the phase difference of \(180^{\circ}\) to \(270^{\circ}\) indicates the structure stores electric charges in an incident electric field, leading to a capacitance effect. When \(\varphi=180^{\circ}\), the structure has no capacitance or inductance effect for that respective mode, and thus, the mode is naturally resonant and radiating completely. In [23], a series of H-shaped and metallic strip metasurface characteristics were studied through CMA. The distance and dimension of the H-shaped structure was optimized by analyzing its resultant eigenvalue (\(\lambda_{n}\)). In another article, ref. [24], the CP behavior of a slotted U-shaped antenna was analyzed by observing the direction of surface current using CMA. In our case, the final antenna structure evolved with modification of the circular ring shown in Figure 1a. It was designed on a Rogers 5880 substrate with a thickness of 0.254 mm. The CMA of the circular ring of Figure 1a sorted it at 28 GHz with five modes. CMA indicated that Mode 1 was significant from 20 to 34 GHz, considering a modal significance at 0.8. However, Modes 2 and 3 were significant from 20 to 44 GHz.