## 2 Triangular vs. Square Lattice Planar Arrays
In this case the grating lobes are avoided if the following equations are satisfied:  
\[ \begin{array}{c}\left(\frac{\lambda_{HF}}{2d_{x}}-u_{0}^{*}\right)^{2}+ \left(\frac{\lambda_{HF}}{d_{y}}-v_{0}^{*}\right)^{2}\geq R^{2}\\ \frac{\lambda_{HF}}{d_{y}}-\sin(\theta_{\max})\geq R\\ \frac{\lambda_{HF}}{d_{x}}-\sin(\theta_{\max})\geq R\\ u_{0}^{*}=\sin(\theta_{\max})\cos(\gamma)\\ v_{0}^{*}=\sin(\theta_{\max})\sin(\gamma)\end{array}  \tag{5}\]  
where \(u_{0}^{*}\) and \(v_{0}^{*}\) represent the \(u\)-\(v\) plane coordinates where the grating lobes can appear. It is worthwhile to note that the previous grating lobes absence equations are only valid for equilateral or isosceles triangular lattices. Scalene lattices have not been considered since they provide a lower value for \(d_{0}\). Equation (5) was used to understand the effect of \(\gamma\) on the element spacing in a circular scan sector. The elements' spacing (\(d_{x}\), \(d_{y}\)), along with the minimum distance between them, were evaluated for \(\gamma\) within the interval [20\({}^{\circ}\),80\({}^{\circ}\)] in the event of \(R=1.1\). The plot summarizing the results is shown in Figure 4.  
Figure 4: Elements’ spacing (\(d_{x}\), \(d_{y}\)), the antenna elements’ distance (\(D\)), and its minimum (\(d_{0}\)) as a function of the \(\gamma\) value for a triangular lattice array in the event of \(R=1.1\). Figure 3: Planar antenna array with a triangular lattice. It is apparent that the \(\gamma\) value has a considerable effect on \(d_{0}\). Specifically, \(\gamma=30^{\circ}\) and \(\gamma=60^{\circ}\) provide the largest minimum distance between elements, which is equal to 5.97 mm (0.587 \(\lambda_{\rm HF}\)) in the addressed scenario. Moreover, Figure 4 emphasizes how the triangular lattice provides the designer more degrees of freedom due to different possible combinations of element spacing (\(d_{x}\), \(d_{y}\)). For example, let us consider the \(RP\) of an 8 \(\times\) 8 array of isotropic elements (i.e., \(E_{nm}(\theta,\phi)=1\)) when the main beam is steered along one of the \(\phi\) directions where there are grating lobes and when \(\theta=60^{\circ}\). Figure 5 illustrates the outcomes for two values of \(\gamma\), namely, \(30^{\circ}\) and \(60^{\circ}\). It is apparent that the different antenna element spacing (Figure 4) determines a different pattern shape, although the minimum distance remains the same. In particular, in the event of \(\gamma=30^{\circ}\) (Figure 5a), the antenna array presents a superior spatial resolution along the \(v\) plane, whereas the equilateral triangular lattice (\(\gamma=60^{\circ}\)) provides an almost circular footprint in the \(u\)-\(v\) plane. Moreover, in both cases the grating lobe is evidently outside the visible region (red circle), located at a distance of \(R=1.1\) from the center (\(u=0\), \(v=0\)), as expected. Therefore, according to the desired spatial resolution in the \(u\)-\(v\) plane, it is possible to select the proper \(\gamma\) value. In order to find the most suitable triangular lattice grid--and hence, the \(\gamma\) value--the angular average gain (\(\eta_{Gain}\)), namely, the mean value of the gain attained during the beam steering inside the all-circular sector, in the case of a planar array composed of 8 \(\times\) 8 elements, was evaluated as a function of the \(\gamma\) value within the addressed bandwidth (24.25-29.5 GHz), and by considering the beam scan within the circular sector cell, as shown in Figure 2. An element pattern shape equal to \(E(\theta,\phi)=cos(\theta)\), with a half-power beamwidth (HPBW) of \(90^{\circ}\), was assumed as the element factor during the average gain evaluation. From the color map of Figure 6 it can be concluded that \(\eta_{Gain}\) always increases with frequency, although it exhibits higher value fluctuations with respect to the \(\gamma\) value, especially in the lower band.