## 2 Design of the Mode Converters for the Sum and Difference Patterns  
The monopulse system will have two dedicated ports in rectangular waveguide mode, associated to the \(\Sigma\) and \(\Delta\) patterns, respectively. They operate in their fundamental \(TE_{10}{}^{rec}\) rectangular waveguide modes. Thus, conversion between modes of the rectangular and circular waveguides is required. For the conversion between the \(TE_{10}{}^{rec}\) and the \(TE_{11}{}^{cir}\) modes in the sum \(\Sigma\) pattern, a novel geometry will be introduced. This conversion only involves fundamental modes. It is therefore simpler than that required for the difference \(\Delta\) pattern. In this last case, the conversion to the \(TE_{01}{}^{cir}\) mode is required, avoiding as much as possible the generation of other non-desired modes in the circular waveguide. There are several methods to design mode converters involving the \(TE_{01}{}^{cir}\) mode [44; 45; 46; 47; 48; 49; 50], which are used in different applications. The side wall coupling used for instance in [46; 48] and the inline coupling in [49; 50; 51] are two suitable candidates. The sidewall coupling approach was utilized in this work since it leads to shorter and more compact designs, while allowing a dedicated control of the spurious excitation of non-desired higher order modes. Both mode converters between the \(TE_{10}{}^{rec}\) and the \(TE_{11}{}^{cir}\), \(TE_{01}{}^{cir}\) modes have to be integrated within the same horn feeder, making up a novel dual-mode converter (DMC) feeding network. The DMC will excite a dual-mode conical horn antenna with a 2-D azimuthal monopulse pattern (vertical polarization) in the frequency band of 37.5-39 GHz. Initial work was carried out by [52]. Here, the full design methodology is described, detailing the fabrication process and the experimental results, which will be compared with the theoretical ones. The CHA is fabricated using additive manufacturing by Direct Metal Laser Sintering (DMLS). The DMC is fabricated using subtractive manufacturing by a standard Computer Numerical Control (CNC) process. In this section, we present the design process of the two mode converters found in the system. One is associated to the sum pattern, while the other is associated to the difference pattern. In this Section 2, the designs are carried out independently. Section 3 later on explains how they are integrated to provide the DMC that is connected to the Conical Horn Antenna (CHA). Mode Converter from \(TE_{10}{}^{rec}\) Mode in the Rectangular Waveguide to the \(TE_{11}{}^{cir}\) Mode in the Circular Waveguide  
The proposed \(TE_{10}{}^{rec}\)-\(TE_{11}{}^{cir}\) mode converter is shown in Figure 1. Only the inner air channel is represented, without the metallic enclosure. Since the converter only involves fundamental modes at the input and output ports, a gradual transition is used [53; 54]. It is based on intermediate waveguides with a cross-section evolving between the rectangular and the circular waveguides. In the present work, it consists of three sections. Section 1 is the nominal WR-28 standard rectangular waveguide at \(port\) 1, which allocates the dedicatedport for the sum pattern (\(\Sigma\)). Section 2 is a waveguide with a bow-tie shape [55; 56], which converts the \(TE_{10}{}^{rec}\) mode of the input rectangular waveguide into the \(TE_{11}{}^{circ}\) mode of the output circular waveguide. Section 3 includes stepped intermediate circular waveguides to reach the required final diameter of 10 mm (1.27 \(\lambda_{0}\)) at _port_ 2. Although the design is narrow band (37.5-39 GHz) for this type of transition, the presented configuration is intended to have enough degrees of freedom to allow its integration with the mode converter for the difference pattern (\(\Delta\)) described later. Moreover, the circular waveguide at the output is over-dimensioned to allow the propagation of the \(TE_{01}{}^{circ}\) mode, which is a difference with respect to classical transitions between rectangular and circular waveguides.