# Compact Full Ka-Band Waveguide Directional Coupler Based on Rectangular Aperture Array with Stairs
## 3 Coupler Design and Simulation Results  
To show the design process of the compact coupler, a 3 dB coupler in full Ka band with novel rectangular aperture arrays and stairs was designed. The well-known commercialsimulator HFSS (Ansys, Pittsburgh, Pennsylvania, U.S.) was used to simulate the coupler. Before the design process, we assumed that the structures of the two waveguides are the same as each other and placed parallel. The structure of the sample coupler is symmetric with the middle planes, which is shown in Figure 1. A perfect conductor was used to build the model and a discrete sweep type was used in the simulation. The process is described as follows. First, choose a standard rectangular waveguide (WR-28) (Guangsheng metal products company, Cangzhou City, Hebei Province, China) to meet the required operating frequency. Second, tune t, b\({}_{1}\), b\({}_{2}\), w\({}_{1}\), h\({}_{1}\), w\({}_{3}\), and h\({}_{2}\) (the size of rectangular holes and the height of floors) to make the coupling strength close to 3 dB. The coupling strength is determined by the size of rectangular coupling holes and the size of the height of floors simultaneously. If the rectangular holes become smaller, the height of floors must change to a bigger size. On the other side, if the rectangular holes become bigger, the height of floors must change to a smaller size. A compromise must be made between the size of rectangular holes and the height of floors. Then, adjust w\({}_{2}\) to make the S\({}_{11}\) magnitude better. Finally, a genetic algorithm is used to optimize t, b\({}_{1}\), b\({}_{2}\), w\({}_{1}\), h\({}_{1}\), w\({}_{3}\), h\({}_{2}\), and w\({}_{2}\), and to make the S\({}_{21}\) magnitude and S\({}_{31}\) magnitude balance in the operating frequency. The dimensions of the variables depicted in Figure 1 were chosen as follows: a = 7.12, b = 3.556, r = 0.5, t = 0.5, b\({}_{1}\) = 0.71, b\({}_{2}\) = 1.47, w\({}_{1}\) = 9.22, h\({}_{1}\) = 3.03, w\({}_{3}\) = 1.82, h\({}_{2}\) = 2.4, and w\({}_{2}\) = 3.3 (all in millimeter). The simulation curves are shown in Figure 3(a), in which the S\({}_{11}\) magnitude and S\({}_{41}\) magnitude were below \(-\)18.68 dB and \(-\)19.15 dB, respectively, over full waveguide bandwidth. The S\({}_{21}\) magnitude and S\({}_{31}\) magnitude varied between \(-\)3.62 dB and \(-\)2.65 dB from 26.5 GHz to 40 GHz.  
For measurement purposes, two 90\({}^{\circ}\) transitions were added to the structure in order to facilitate the measurements. The dimensions of the coupler with transitions, which are shown in Figure 4(a), are as follows: l\({}_{1}\) = 5, l\({}_{2}\) = 15, l\({}_{3}\) = 25, R\({}_{1}\) = 1.21, and R\({}_{2}\) = 8.33 (all in millimeter). The simulated results of the coupler with transitions are depicted in Figure 3(b), from which we can see that the S\({}_{21}\) parameter and S\({}_{31}\) parameter were almost unchanged compared with the no transitions situation, which is shown in Figure 3(a), and the simulated S\({}_{11}\) parameter and S\({}_{41}\) parameter were below \(-\)18.97 dB and \(-\)19.16 dB, respectively, in the full Ka band. Figure 4(b) shows the 3D-structure diagram of the prototype coupler. The exploded schematic of the coupler is depicted in Figure 4(c).  
Figure 4: Simulated results of sample coupler (**a**) without transitions and (**b**) with transitions.