## 2 Antenna Structure and Design Principles  
Figure 2 shows the geometry of the proposed antenna element with detailed dimensions, designed over Rogers RT/Duroid 5880, with substrate height of 0.508 mm. The key design parameters of the RC are \(A^{\prime},B^{\prime}\), and the cavity opening width \(C_{V}\). The fine tuning of \(C_{V}\) guarantees the antenna's impedance matching for better, return-loss performance, and the radiation efficiency. The initial value of the \(A^{\prime},B^{\prime}\) can be calculated by [19]  
\[f_{mnl}=\frac{c}{2\pi\sqrt{\mu_{r}\epsilon_{r}}}\sqrt{\left(\frac{m\pi\tau}{A^ {\prime}}\right)^{2}+\left(\frac{n\pi\tau}{B^{\prime}}\right)^{2}+\left(\frac{ l\pi}{C^{\prime}}\right)^{2}} \tag{1}\]  
where, \(A^{\prime}\), \(B^{\prime}\), and \(C^{\prime}\) are the internal length, width, and height (substrate height) of the SIW-cavity, respectively. \(f_{mnl}\) represent the cutoff frequency of RC cavity modes. In reference to the RC cavity of Figure 2 and the values of \(A^{\prime},B^{\prime}\) in Table 1. The first three resonance modes cutoff frequency points should be the TE\({}_{110}\) at 28.02 GHz, TE\({}_{120}\) at 41.99 GHz, and TE\({}_{210}\) at 46.5 GHz. The value of \(C_{V}\) can be obtained through simulation and the optimization in Ansys HFSS-2021. As stated above the RC is merely for injecting the energy in radiator and to block the back scattering--the main radiator is the CPatch antenna, operating at its dominant mode TM\({}_{110}\). The resonance frequency \(f_{r110}\) of the CPatch can be calculated by [20]  
\[f_{r110}=\frac{1.8412c}{2\pi a\sqrt{\mu_{r}\epsilon_{r}}} \tag{2}\]  
here \(c\) is the speed of light, \(a\) is the radius of the Cpatch; \(\mu_{r}\) and \(\epsilon_{r}\) are the substrate relative permeability and permittivity, respectively. As per the dimensions given in Figure 2 the patch resonance frequency is 34.882 GHz. It has been observed that as in [11] where the TCSRS-antenna is shorted to the SIW cavity ground, if we shorted the Cpatch within RC is shorted to the cavity's upper metallic-layer (in same plane), the \"total antenna structure\" resonates on its CPatch's dominant mode \(TM_{110}\). Therefore, in Figure 2 we have shorted the CPatch from two different points, orthogonal to each other, separated by quarter-wavelength. This leads to the excitation of CP wave in the antenna structure. The parametric study suggests that by changing the \"angular-separation (\(\alpha\)) between the two shorting-strips\" or \"the angular-orientation (\(\updelta\)) of the first shorting strips in-reference to the main axis of the antenna passing through the center of the Cpatch\", the resonance points of the degenerated modes \(TM_{110}\), \(TM_{110}\) change significantly. The good starting point for the value of \(\alpha\) start with quarter-wavelength at 34.8 GHz in Rogers RT/Duroid 5880 material. The width of the slot within cavity, in  
Figure 1: Topology of the proposed CP patch antenna via two feeding points separated through the quarter wavelength. surrounding of the patch antenna, and the width of metallic-strip (MStrip) connecting the patch periphery to the RC ground are 0.5 mm and 0.2 mm, respectively. These variables precise value obtained through the optimization of the total antenna performance--in terms of impedance matching and the 3 dB AR bandwidth. Through simulation, it has been observed that the narrower the Metrip the better the antenna's AR performance; due to the manufacturing limitation its value has been limited to 0.2 mm. The dimensions of the SIW feed-line should be large enough that the TE\({}_{10}\) mode can be excited at a frequency--slightly lower than the working frequency of antenna this feed-line is feeding. The position of the CPatch, in the X-axis also plays the key role in optimizing the total antenna performance obtained through simulation. Figure 3 illustrate the surface current-density (SCD) on the antenna's metallic surface at 34 GHz, obtained through simulation in Ansys HFSS-2021.