## 2 Flat Panel Antenna Design
### Final Design
#### 2.2.2 Feeding Structure  
To make the antenna compact, pillbox-based feeding is implemented underneath the disc, as shown in Figure 2. A detailed view of the feeding mechanism along with the used substrates is described in Figure 5. The feeding structure consists of an SIW horn feeding, parabolic reflector, and coupling slot, as shown in Figure 4. The geometric properties of the parabolic reflectors offer the transformation of a cylindrical wave originating from the paraboloid's focus into a plane wave directed along the parabola axis. Therefore, the feeding structure and the coupling slot are arranged and placed in different layers to avoid aperture blocking. The feeding is positioned in the focal plane of the parabolic reflector located on the lower substrate (D\({}_{1}\)), which is integrated with the horn and SIW structure to feed and guide the waver, respectively. Meanwhile, the coupling slot is placed on the upper substrate (D\({}_{2}\)). Connectivity between the two layers is facilitated through this coupling slot. The reflector surfaces are actualized using vertical metallic vias, establishing a connection between the top metal layers M\({}_{1}\) and M\({}_{3}\). Additionally, a coupling slot with a width of 5 mm is etched onto the middle metal layer (M\({}_{2}\)). The SIW horn, which has dimensions of \(L_{1}\) and \(L_{2}\), is fed by the 2.92 mm millimeter-wave 50 \(\Omega\) connector and generates a quasi-cylindrical wave (see Figure 6a). The wave propagates via substrate 3 (Figure 5) towards the parabolic reflector. The parabolic reflector is also SIW-based and converts the incoming wave into a plane wave. The parabolic reflector is generally designed using [34; 41]:  
\[r=\frac{2F}{1+cos\phi} \tag{1}\]  
where \(r\) is the parabolic surface radius, \(F\) refers to the focal length, and \(\phi\) indicates the angle between \(r\) and \(F\) vectors (see Figure 5b). The parabolic slot is positioned close to the parabolic surface for maximum power coupling. The coupling slot transforms that plane wave to the top layer (substrate 1), which has metallic patterns; see Figure 6b. The coupling slot, which has a width of \(s_{w}\), is placed just on the edge of the parabolic reflector. \(s_{w}\) is a critical parameter to improve the impedance-matching bandwidth. The optimized value of the \(s_{w}\) is 5 mm for the final design. The SIW horn and parabolic reflector are made of metallic vias that have a radius of 0.5 mm. The focal length and curve aperture are optimized to achieve a low side lobe level. The parabolic reflector has a curve aperture of 301 mm and focal point distance (\(F\)) of 126 mm.  
Figure 6 shows the electric field distributions of the antenna for the top layer (radiating strips) and bottom layer (pillbox feeding) to help understand its working mechanism. Pillbox feeding generates a plane wave in the \(x\)-direction, which excites the radiating strips as a leaky-wave antenna. This excitation produces a broadside radiation pattern when there is no rotation in patterns. However, when the metallic patterns are rotated by an angle \(\omega=\pm 20^{\circ}\), the beam can be steered in azimuth and elevation planes. This can be observed from the electric field distributions in Figure 6c,d. Specifically, to achieve beam scanning, the intersection of the plane wave and the radiating metallic strips is changed by rotating the disc, as shown in Figure 6c,d. The disc is rotated (\(\omega\)) clockwise and anti-clockwise to produce symmetrical beam scanning.