## 3 Evolution Process and Parametric Study
### Evolution Process  
The design process of the proposed quad-band antenna begins with the design of the MMw antenna element. Thus, a full-mode SIW (FMSIW) cavity structure is designed to have resonant frequency of its fundamental mode (TE\({}_{101}\)) at 28 GHz. The general geometry of the FMSIW is shown in Figure 3a, in which all design parameters are derived using Equations (1)-(3) [24; 27]. \[\frac{d}{s}\geq 0.5 \tag{1}\]  
\[W_{equ}=\frac{C}{f_{TE101}\sqrt{2e_{r}}} \tag{2}\]  
\[W_{siw}=W_{equ}+\frac{d^{2}}{0.95\;s} \tag{3}\]  
where, \(d\) represents vias diameter, s represents the separation period between vias, \(f_{TE101}\) is the resonant frequency of the fundamental mode inside the FMSIW cavity, \(W_{equ}\) represents the equivalent width of the traditional RWG, and \(W_{siw}\) represents the width of the SIW cavity wall (transverse spacing between the centers of the vias). Based on the calculated design parameters, a 3D model of the FMSIW cavity antenna element is created using CST Microwave Studio. As shown in Figure 3a, a standard 50 Ohm microstrip excites the FMSIW cavity. The resulting electric field distribution at the resonant frequency (28 GHz) of the dominant mode (TE101) is shown in Figure 4a. A quarter of the entire FMSIW cavity is highlighted in both Figures 3 and 4; it has a quarter cycle of the full mode at 28 GHz, as shown in the left QMSIW in Figure 4b. This QMSIW can be excited instead of the FMSIW, and at a much smaller size (miniaturized design). In addition to the dominant mode electric field distribution at 28 GHz, the electric field distributions at two higher order modes are presented in Figure 4 (TE102 and TE106 at 38 GHz and 0.13 THz). Their field distribution will help in determining the best locations for slot radiators at their frequencies. This is because they have weak electric field magnitudes at the open-edge walls of the sub-cavity (QMSIW) and are unable to efficiently radiate at  
Figure 3: Top view of the proposed SIW cavity. (**a**) Full-mode SIW, (**b**) quarter-mode SIW (REF1). 38 GHz and 0.13 THz. The REF1 antenna (shown in Figure 5) operates with a single mode of radiation at 28 GHz. This fact is clearly evident from the simulation results of the REF1 antenna, as depicted in Figure 6a. Figure 4: Intended E-field distributions of **(a)** FMSIW at 28 GHz and higher modes at 38 GHz and 0.13 THz. **(b)** Electric field distributions on the quarter cut of the FMSIW cavity. Figure 5: Antenna structure evolution steps. Based on the electric field distribution at 38 GHz (see Figure 4), the REF2 antenna is developed by carving an inverted-L slot (Slot 1) on the top of the QMSIW, as depicted in Figure 5. The location and size of the slot are optimized to efficiently radiate the incident energy of the TE\({}_{102}\) QMSIW mode at 38 GHz. A new radiated mode is presented at 38 GHz, as shown in Figure 6a. In order to have tri-band operation, another inverted-L slot (Slot 2) is created as shown in REF3 in Figure 5, which releases the electromagnetic energy of the TE\({}_{106}\) QMSIW mode into the free space at 0.13 THz. Figure 6b presents the simulated results of the reflection parameters at 0.13 THz. Accordingly, the proposed single-element dual-slot QMSIW antenna supports three frequency bands: 28 GHz, 38 GHz, and 0.13 THz, respectively. In order to design a four-element MIMO antenna for the three MMw bands, four QMSIW antennas (REF3) are arranged in a way that permits both polarization and space diversities. This can be simply seen in the REF4 antenna in Figure 5. Two pairs of QMSIWs are connected through a new feeding port (Port 5). Port 5 is used to activate structure-reuse by exciting a half-wavelength dipole-like Mw antenna, which is made from the four QMSIW antenna elements.