# Compact Full Ka-Band Waveguide Directional Coupler Based on Rectangular Aperture Array with Stairs
## 2 Coupler Configuration  
The configuration and port nominations of the proposed coupler, which includes three pairs of rectangular aperture arrays and the stairs structures placed in the coupling region of two rectangular waveguides, are shown in Figure 1. The main waveguide as well as the branch one are placed in parallel, and they are coupled to each other by three pairs of large aperture arrays in their common broad wall. Stairs structures are placed in the coupling region to enhance the coupling strength and to match the dimensions of standard rectangular waveguides (a = 7.12 mm, b = 3.56 mm).  
The design results from the different numbers of rectangular apertures are listed in Table 2, with the bandwidth all assumed to be the same as the full WR-28 waveguide bandwidth, which is from 26.5 GHz to 40 GHz. The performance of the presented coupler is improved as the number of apertures and stairs increases. However, as the number of apertures increases, the length of the coupling region becomes longer. As a compromise, three pairs of rectangular aperture arrays are used in our prototype coupler.  
TE\({}_{10}\) mode is used in the proposed coupler. The coupling is less frequency dependent. The simulated surface current density is shown at 33 GHz in Figure 2. It can be observed that the signal, fed at port 1, is properly coupled at port 3, and port 4 is isolated. The surface current can flow into another waveguide across the large aperture array (Figure 1a),  
Figure 1: The proposed coupler: (**a**) rectangular aperture array, (**b**) side view, (**c**) top view, and (**d**) 3D-structure diagram.  
which is placed at the edge of the rectangular waveguide broadside. Further, the proposed structure reduces the size more significantly than the conventional designs.  
Figure 3a-f illustrates the simulated S\({}_{21}\) and S\({}_{31}\) magnitudes of the proposed coupler with one parameter varied, where the other dimensions are fixed. Figure 3a,b shows the relationship between the heights of the aperture array and S\({}_{21}\), S\({}_{31}\) magnitudes. As h\({}_{1}\), h\({}_{2}\) become bigger, the S\({}_{21}\) magnitude becomes smaller and the S\({}_{31}\) magnitude becomes bigger. The higher frequency S parameter changes faster than the lower frequency S parameter. As is shown in Figure 3c,d, as w\({}_{1}\) and w\({}_{3}\) become bigger, the S\({}_{21}\) magnitude becomes smaller and the S\({}_{31}\) magnitude becomes bigger. The lower frequency is more sensitive to the width of coupling holes. In Figure 3e, we can see that the S\({}_{31}\) magnitude is sensitive to b\({}_{1}\) around 35 GHz. The higher frequency is more sensitive to b\({}_{2}\) than the lower frequency, which is depicted in Figure 3f. It is found that there are two ways to change the coupling coefficient: one is to adjust the height of stairs (b\({}_{1}\), b\({}_{2}\)) and the thickness of the common wall (t), while the other way is to change the parameters of the rectangular apertures (w\({}_{1}\), h\({}_{1}\), w\({}_{3}\), and h\({}_{2}\)). The isolation of the coupler is influenced by the dimension of w\({}_{4}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Apertures** & **Coupling Imbalance (dB)** & **Through Flatness (dB)** & **Return Loss (dB)** & **Coupler Model** \\ \hline \(2\times 1\) & \(3.17\pm 0.92\) & \(3.4\pm 1.2\) & 15.19 \\ \hline \(2\times 2\) & \(3.11\pm 0.7\) & \(3.23\pm 0.89\) & 17.65 \\ \hline \(2\times 3\) & \(3.14\pm 0.46\) & \(3.13\pm 0.48\) & 18.68 \\ \hline \end{tabular}
\end{table}
Table 2: Performance of 3 dB directional coupler with different number of large apertures.  
Figure 2: Surface current density at 33 GHz: (**a**) bottom view, (**b**) top view, and (**c**) side view.