## 2 Design of Proposed Array  
### A Stand-Alone Antenna  
Figure 1 shows the geometry of a compact stand-alone SIW cavity-backed wideband patch antenna using an indirect slot feeding for a high directivity. The proposed antenna consists of three dielectric layers: a radiator layer (Layer 1), an ASIWC layer (Layer 2), and an indirect feeding network layer (Layer 3). Figure 1: Geometry of the proposed antenna. On Layer 1, a rectangular patch radiator, having a width \(w_{1}\) and a length \(l_{1}\), is printed on a TLY-5 (\(\epsilon_{r}=2.2\), tan\(\delta=0.0009\)) dielectric substrate (\(w_{2}\times l_{2}\)) to operate in Ka-band. The patch radiator is driven by the indirect slot feeding network to provide a high-directive radiation pattern with a narrow HPBW. On Layer 2, the ASIWC using the printed patch and vias is designed to substitute a conventional cavity wall [4]. The substrate is perforated in a rectangular shape with dimensions of width \(\times\) a length \(\times\) a thickness (\(w_{3}\times l_{3}\times h_{2}\)) to reduce the dielectric loss. In addition, the perforated structure is surrounded by the multiple cylindrical vias with a diameter of \(d\) and an interval of \(s\). Then, the vias are electrically connected by a ring-shaped patch printed on the substrate to create the ASIWC-backed structure. The width and length of the ASIWC are \(w_{4}\) and \(l_{4}\), and these parameters, including the substrate height \(h_{2}\), can achieve the high directivity with the narrow HPBW. On the top of Layer 3, a thin rectangular slot (\(w_{5}\times l_{5}\)) is etched on the ground to indirectly excite the field to the radiator, and the microstrip line (\(w_{6}\times l_{6}\)) is printed on the bottom of Layer 3 to have a coupled feed for the slot aperture. This indirect feeding network can obtain a wide impedance-matching characteristic in Ka-band compared to the conventional direct feeding technique using a feed pin [24]. It is because two separated resonances of the patch and slot are provided by the proposed feeding network, and the resonances are close to each other for enhancing the matching bandwidth when optimizing the slot and the ASIWC parameters. In addition, the indirect feeding network is essential in Ka-band due to the difficulty in implementing the feed pin, which has an extremely small radius to directly connect the patch radiator and also has a vulnerable structure to an external shock. To optimize the antenna performances, we have conducted parametric studies for the critical geometry parameters, such as \(h_{2}\), \(l_{4}\), and \(w_{5}\). The optimized design parameters are listed in Table 1.  
In addition, the low thickness of the ASIWC can concentrate strong EM fields in the patch antenna to minimize the fringing field, which can generate the narrow HPBW. In the increment of the ASIWC length, while maintaining the aspect ratio (\(w_{4}/l_{4}=1.35\)), the boresight gain continuously increases due to the reduction in the dielectric loss, and this case allows the HPBW to become narrower because of the high-directive properties of the radiation pattern, as shown in Figure 2b. Figure 2c presents the simulated bore-sight gain and the reflection coefficient according to the slot width of the proposed antenna. The gain and the reflection coefficient have an optimal point in the range of \(w_{5}\) from 2.6 mm to 5 mm. The length of the slot is an important parameter used for the impedance matching of the proposed antenna, which determines the EM coupling strengths between the patch radiator and the ground as well as the radiation characteristic [25].