## 2 Double-Ridged Waveguide Rotman Lens Design in the K/K\({}_{\text{a}}\)-Band
### Rotman Lens Design  
A 10 \(\times\) 10 Rotman lens was selected as a proof-of-concept, which required a trade-off of the main design parameters [24]. In particular, a compromise was required between frequency bandwidth and scanning range. Larger values of \(\alpha\) led to degraded performance in the upper range of the targeted frequency band as a consequence of the larger incidence angle at the feeding ports. It was found that \(\alpha=20^{\circ}\) was acceptable for a K/K\({}_{\text{a}}\)-band design considering the size of the inner lens contour, \(\Sigma_{1}\). Similarly, an optimal focal arc design leading to low theoretical phase aberrations typically requires \(g>1\). However, this increases the curvature of the focal arc and degrades the performance when secondary effects, such as feeding port scattering, residual side wall reflections, etc., are taken into account. An acceptable trade-off was found for \(g=0.97\). Considering the nominal feeding port aperture width reported above, the off-axis focal length was set to \(F=110\,\)mm or \(F=11\lambda_{0}\), where \(\lambda_{0}\) is the wavelength in free space at the highest operating frequency, \(30\,\)GHz. An internal Matlab tool was used to generate a step file combining the design parameters of the lens (Figure 1) with those of the feeding ports (Figure 2). The step file was then imported into Ansys HFSS to create the full-wave model. The shape of the side walls in the PPW cavity was adjusted to reduce residual reflections while implementing adequate absorbing boundary conditions [39]. The final full-wave model is illustrated in Figure 3.  
The scattering parameters of the full-wave model are reported in Figure 4, with port 1 corresponding to one of the outer ports (\(\theta=\alpha=20^{\circ}\)), while port 5 is one of the inner ports (\(\theta=4.44^{\circ}\)), following the port numbering in Figure 3. These results are representative of the quasi-optical beamformer, with port 1 displaying some degradation in the transmission coefficients at the highest frequencies, while port 5 shows some degradation of the reflection and port-to-port coupling at the lowest frequencies. The results remained, nevertheless, acceptable for the targeted application, with a reflection and port-to-port coupling below \(-15\,\)dB over the frequency bands of interest (light grey area in Figure 4), while the degradation on the transmission only affected port 11, corresponding to the extreme inner lens contour port on the same side as port 1. Figure 3: Full-wave model (Ansys HFSS) of the double-ridged 10 \(\times\) 10 Rotman lens inner waveguide cavity, including transmission lines at the output ports. The phase difference between adjacent ports also remained fairly constant at any frequency point across the band analyzed, providing the desired uniform phase progression, setting the beam pointing direction while having a phase slop characteristic of true-time delay beamformers. As expected, the phase difference between adjacent output ports increased with the distance of the input port considered to the center of the focal arc, with port 1 having the largest phase difference, while port 5 had a phase difference closer to zero producing a beam pointing near the boresight direction (\(\theta_{0}=0^{\circ}\)). The phase error, defined as the deviation from the average phase difference between adjacent output ports, is quantified as a function of frequency and the results are reported in Figure 5. The root mean square (RMS) error, corresponding to the standard deviation, remained below \(5^{\circ}\) across the operating bands, while the peak-to-peak error remained below \(18^{\circ}\). The transmission line lengths, \(W_{i}\) in Figure 1, were included in the model as double-ridged waveguides, which were more dispersive closer to the cut-off frequency.