## 2 Transition Design  
The transition configuration consists of a standard WR75 waveguide connected perpendicularly to a SIW through a H-shaped coupling slot etched in the SIW bottom ground plane, as shown in Figure 1. The SIW top ground plane is hidden for better clarity. Figure 1: _Cont._  
The characteristic impedance and the propagation constant of this SIW were equal to a dielectric-filled RWG with an equivalent width _ae_ and the same height, as explained in [22; 23]. The other edge of the SIW ends with a short circuit at a distance \(s\), which was kept fixed for reasons to be discussed in the next section. Also, the slot dimensions remained unchanged and were optimized in order to maximize the transmission coefficient to the SIW with no matching elements. The H-shaped slot could be seen as a short double-ridge waveguide section and its cut-off frequency, equal to 9.15 GHz, was low enough to let the electromagnetic field propagate between the two transmission lines. A 2:1 hole pitch-to-diameter ratio was chosen for a reliable manufacturing. To minimize the insertion loss, the adopted substrate was RT5880 by Rogers Corporation, with a dielectric constant \(\varepsilon_{r}=2.2\) and tan\(\delta=0.001\) measured at 10 GHz. The substrate height was \(b=1.575\) mm and the metal thickness was \(t=35\)\(\upmu\)m, though it was increased to 45\(\upmu\)m in the design to include the effects of the metal hole deposition in a following manufacturing process. The design center frequency was around 12.5 GHz and the simulation bandwidth was between 10 to 15 GHz, which is the typical operating frequency range for a WR75 waveguide. The transition geometrical parameters and simulated scattering parameters are presented in Table 1 and Figure 2. The RWG and the SIW are respectively indicated with port indices 1 and 2. It is evident that, with no matching elements included, such transition exhibits poor return loss performance and cannot be utilized in hybrid feeding networks for large planar array antennas. It is worth mentioning that similar performance would be obtained with any other slot shape, given the fact that the equivalent load connected to the RWG presents a very low impedance, tending towards a short circuit when the substrate thickness becomes very small. Nevertheless, the thinner the slot width _sw_, the higher is the transmission coefficient. In addition to that, by adopting an H-shaped slot, the return loss at the RWG port can be further improved. In Figure 2 the scattering parameters obtained with a standard rectangular slot (_sh = sw_, dashed lines) and the optimized H-shaped slot (solid lines)  
Figure 1: RWG-to-SIW transition geometry with no matching elements. (**a**) Top view; (**b**) Bottom view; (**c**) 3D view. are compared. An improvement of about 1 dB in the return loss can be observed. The transition with the H-shaped slot therefore represents a good starting point for the design of a distributed impedance matching network. The latter can be realized by a number of symmetric irises carved in the RWG E-plane, as depicted in Figure 3. Each iris is specified by its height \(h\) and its width \(w\). Without loss of generality, a constant iris width equal to \(w=1\) mm was adopted. A blending radius \(br=1.25\) mm was considered to ease a fabrication process in the case of CNC (computer numerical control) milling machine manufacturing. Wideband matching can be achieved with multi-section matching networks, which can be quickly synthesized with the aid of a circuit simulation software. On the other hand, a full-wave simulation software can accurately predict the performance of an electromagnetic structure, at the expense of