# Metallic, 3D-Printed, K-Band-Stepped, Double-Ridged Square Horn Antennas
## 2 Design and Fabrication of the Antennas  
The corrugated horn is used either as a single radiator or as the feed for a parabolic reflector for its symmetric pattern, low side lobe level and low cross-polarization. The corrugations are designed to excite high order modes such that the field distribution on the aperture demonstrates a hybrid HE\({}_{11}\) mode. To achieve acceptable matching across the whole K-band of the horn, ridges are designed to extend the impedance bandwidth toward the lower side band. From the perspective of electromagnetics, the ridges form a parallel-plate structure that supports the propagation of the transverse electro-magnetic (TEM) mode, which has a zero cut-off frequency. As a result, the impedance bandwidth of the horn is extended downwards. To model the ridges in an equivalent circuit, they function as a shunt capacitor that is generally a low-pass device. Hence, the impedance bandwidth is also extended toward the low frequency. Figure 1 shows the geometries of the K-band, corrugated, ridged, square horn antenna. It is composed of a WR-42 rectangular waveguide (10.7 mm \(\times\) 4.3 mm) interface, a standard UBR 220 type flange, a rectangular-to-square waveguide taper and the stepped double ridged horn. In Figure 1a, the horn is laid referring to the coordinate origins on the flange. The H-plane of the horn is on _x-o-z_ as in Figure 1b, while the E-plane is on _y-o-z_ in Figure 1c. Since the horn is designed as a linearly polarized horn, as well as for the ease of fabrication, only two ridges were applied on the E-plane to support the propagation of the TEM mode. Table 1 lists the geometries of the horn. To verify the design, the horn was printed with the selective laser melting (SLM) technology using aluminum alloy and copper, respectively. They were printed along the \(z\)-direction in a whole piece in one run, which eliminates the necessity for post electro-plating and assembly compared with the dielectric 3D printed horns. The outer surface of the horns were polished by post processes. The inner surface of the horn, which was indeed the functional surface, was left untouched to maintain the sound structure of the steps and ridges. Outperforming the 125 um surface roughness of the reported powder bed fusion (PBF) technology in [13], the SLM technology used in this paper turned out to be 6 um and 3 um for the aluminum alloy and copper horns respectively on the inner surface. The flange interface is milled for a tight connection with the vector network analyzer (VNA) extender. Figure 2a,b show the horns from the outlook; the post-polishing endows them with a shiny appearance. Figure 2c,d present the horns from the aperture, in which the corrugations and ridges are seen. However, for the unprocessed inner surface, the roughness is much worse than the outer surface. The milled flanges are shown in Figure 2e,f. The roughness on the waveguide opening in both cases gives alternative views on the process tolerance. The roughness on the waveguide interface may give rise to increased reflection, unwanted passive intermodulation (PIM) and conductor loss. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & **(mm)** & **Parameters** & **(mm)** & **Parameters** & **(mm)** & **Parameters** & **(mm)** \\ \hline _d1_ & 9 & _d2_ & 17 & _l1_ & 4.83 & _l2_ & 13.73 \\ _l3_ & 18.87 & _ld1_ & 4 & _ld2_ & 2 & _rl1_ & 4 \\ _rl2_ & 2 & _rl3_ & 2 & _rl4_ & 2 & _rl5_ & 2 \\ _rl6_ & 2 & _rl7_ & 2 & _rl8_ & 2 & _rl9_ & 2 \\ _rl1_ & 0.5 & _rt2_ & 0.5 & _rt3_ & 0.5 & _rt4_ & 0.5 \\ _rl5_ & 0.5 & _rt6_ & 1.5 & _rl7_ & 0.5 & _rl8_ & 0.5 \\ _rt9_ & 0.5 & _sl1_ & 2 & _sl2_ & 2 & _sl3_ & 2 \\ _sl4_ & 2 & _sl5_ & 2 & _sl6_ & 2 & _sl7_ & 2 \\ _st1_ & 0.5 & _st2_ & 0.5 & _st3_ & 0.5 & _st4_ & 0.5 \\ _st5_ & 0.5 & _st6_ & 0.5 & _st7_ & 0.5 & \(t\) & 2 \\ _wr_ & 1 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometries of the horn.