## 2 Design of the Mode Converters for the Sum and Difference Patterns
The mode chart of the standard WR-28 rectangular waveguide is illustrated in Figure 2a. This waveguide supports a monomode propagation bandwidth between 21.07 GHz (cutoff frequency of the fundamental \(TE_{10}{}^{rec}\) mode) and 42.14 GHz (cutoff frequency of the first degenerated \(TE_{01}{}^{rec}\) and \(TE_{20}{}^{rec}\) higher order modes). Figure 2b presents the mode chart of the circular waveguide of diameter 10 mm. The cutoff frequency of the fundamental \(TE_{11}{}^{circ}\) mode is 17.58 GHz. In addition, this waveguide will have several potential propagating modes in the desired operating bandwidth (37.5-39 GHz). One of them is the \(TE_{01}{}^{circ}\) higher order mode. The others are undesired modes whose levels will be either not excited by the symmetry of the structure, or with a level small enough to not degrade the system performance. The reason to work with the \(TE_{11}{}^{circ}\) fundamental mode and at the same time with the \(TE_{01}{}^{circ}\) mode, will be explained in detail in the next sections. Each mode will be associated to the sum (\(\Sigma\)) and difference (\(\Delta\)) patterns, respectively. Figure 1: Schematic view of the mode converter from the \(TE_{10}{}^{rec}\) mode in the rectangular waveguide (which will be later the sum port) to the \(TE_{11}{}^{circ}\) mode in the circular waveguide (connected later to the conical horn). It should be noted that the \(TE_{mn}{}^{cir}\) and \(TM_{mn}{}^{cir}\) modes with index \(m\
eq 0\) represent two independent modes, with \(\sin(m\phi)\) and \(\cos(m\phi)\) angular variation, respectively, in its longitudinal field component [57, 58]. Thus, they have different E-field vector polarization. In the case of \(TE_{11}{}^{cir}\), for example, one mode is aligned along the \(\hat{x}\)-axes (\(TE_{11}{}^{cir}\)), while the other is oriented along the \(\hat{y}\)-axes (\(TE_{11}{}^{cir}\)). As shown in Figure 3, the fundamental mode of the rectangular waveguide \(TE_{10}{}^{rec}\) at \(port\) 1 with polarization along the \(\hat{y}\) direction, will excite the fundamental mode of the circular waveguide \(TE_{11}{}^{cir}\) at \(port\) 2, polarized along the same \(\hat{y}\) direction. The S-parameters for this mode converter are depicted in Figure 4. CST Studio Suite [59] was used for the simulations. \(S_{11}(TE_{10}{}^{rec})\) represents the return loss of the fundamental mode (\(TE_{10}{}^{rec}\)) within the standard rectangular waveguide. The result is better than 30 dB from 36 to 40 GHz, in a 10.5% bandwidth. Figure 3: The electric field of the fundamental modes related to rectangular waveguide (\(TE_{10}{}^{rec}\), \(port\) 1) and circular waveguide (\(TE_{11}{}^{cir}\), \(port\) 2) with polarization along the \(\hat{y}\) direction. Figure 2: Cutoff frequency of the modes at the ports of the mode converters. (**a**) Nominal WR28 standard rectangular waveguide used for the dedicated ports associated to the sum (\(\Sigma\)) and difference (\(\Delta\)) patterns. (**b**) Common circular waveguide used as input to the conical horn antenna (diameter of 10 mm). The structure avoids the generation of the modes without perfect magnetic wall boundary conditions at the symmetry plane [57; 58]. Since symmetry can be degraded due to the tolerances in the fabrication, a full analysis is conducted as shown in Figure 4b. It includes all the modes within the band of interest according to Figure 2b, regardless of their symmetry, to be sure about their effect on the operating bandwidth. The cutoff frequency of the \(TE_{10}{}^{circ}\) mode is around 49 GHz which is out of the operating frequency bandwidth. Thus, Figure 4b shows the generation of the first eight higher order modes in the circular waveguide, under the excitation of the mode converter by the \(TE_{10}{}^{rec}\). It is clear that the higher order modes have been adequately attenuated, with the exception of the \(TM_{11y}{}^{circ}\). This mode exhibits a level worse than \(-30\) dB, since the E-field is partially oriented in the same direction as the \(TE_{11y}{}^{circ}\).