## 2 Array Design
### Analytical Techniques
#### 2.1.2 Beamwidth  
As is known, the HPBW (\(\Theta_{x(y)0}\)) at broadside along the principal planes of a uniform planar array is well approximated by the formula [18]:  
\[\Theta_{x(y)0}=\pi-2\arccos\left(\frac{0.886\lambda}{2N_{x(y)}d_{x(y)}}\right) \simeq\frac{0.886\lambda}{N_{x(y)}d_{x(y)}} \tag{5}\]  
where the last approximation is obtained in the limit of large arrays (we recall that \(\arccos(x)\simeq\pi/2-x\) for \(x\to 0\)).  
As the beam is scanned at a generic angle (\(\theta_{0}\), \(\phi_{0}\)), the beamwidth over the two principal planes changes according to the following equations [18]:  
\[\Theta_{\rm h}=\left(\sqrt{\cos^{2}\theta_{0}[\Theta_{x0}^{-2} \cos^{2}\phi_{0}+\Theta_{y0}^{-2}\sin^{2}\phi_{0},\phi_{0}]}\right)^{-1} \tag{6}\] \[\Psi_{\rm h}=\left(\sqrt{\Theta_{x0}^{-2}\sin^{2}\phi_{0}+\Theta _{y0}^{-2}\cos^{2}\phi_{0}}\right)^{-1} \tag{7}\]  
where \(\Theta_{\rm h}\) and \(\Psi_{\rm h}\) represent the HPBW on the elevation plane, and its orthogonal plane, respectively. When the array has the same number of elements and the same spacing over the \(x\) and \(y\) axes (as in the case shown discussed here), \(\Theta_{x0}=\Theta_{y0}=\Theta_{0\rm th}\), thus \(\Psi_{\rm h}=\Theta_{0\rm th}\) is constant, whereas \(\Theta_{\rm h}=\Theta_{0\rm th}\sec\theta_{0}\), giving rise to an unavoidable pattern distortion in the scanning towards endfire (viz., for \(\theta_{0}=60^{\circ}\), \(\sec\theta_{0}=2\), hence \(\Theta_{\rm h}\) is doubled). While the numerical evaluation of the beamwidth along the plane \(\phi=\phi_{0}\) is quite straightforward, a few words should be spent on the evaluation along the orthogonal plane. Indeed, the AF is represented as a function of \(\theta\) and \(\phi\), hence the HPBW over the \(\phi_{0}\)-plane is simply given by the distance between the \(-3\) dB points along \(\theta\), while the HPBW over the orthogonal plane needs an explicit definition of the angle \(\psi\). This definition can be recovered from simple geometrical considerations and leads to the following expression:  
\[\psi=\arcsin[\sin\bar{\theta}\sin(\bar{\phi}-\phi_{0})] \tag{8}\]  
where \(\bar{\theta}\) and \(\bar{\phi}\) are the zeros of the equation \(\tan\theta\cot\theta_{0}=\sec(\phi-\phi_{0})\), that is, the equation of the plane orthogonal to the unit vector \(\hat{\theta}\) in (\(\theta_{0}\), \(\phi_{0}\)), i.e., the plane defined by \(\hat{r}\cdot\hat{\theta}|_{\theta_{0},\phi_{0}}=0\), where \(\hat{r}\) is the position unit vector. We should note that \(\bar{\theta}\) and \(\bar{\phi}\) should be sought by means of a root-finding algorithm. Once all the \(\bar{\theta}\), \(\bar{\phi}\) pairs are found, the AF calculated at \((\bar{\theta},\bar{\phi})\) must be rearranged according to the ordering given by \(-\pi/2\leq\psi(\bar{\theta},\bar{\phi})\leq\pi/2\). Then, the \(-3\) dB points along \(\psi\) can finally be found.  
However, when tapering techniques for SLL reduction are taken, the HPBWs are notably affected and accounted for by the _spreading factor_\(s_{\rm f}\). In particular, if one applies a Dolph-Chebyshev technique (as that of Tseng-Cheng) to achieve an SLL of \(-30\) dB, the \(s_{\rm f}\) is given by [18]  
\[s_{\rm f}=1+0.636\left\{\frac{2}{\rm PSR}\cosh\left[\sqrt{(\arccos\rm{PSR})^ {2}-\pi^{2}}\right]\right\}^{2}. \tag{9}\]  
Therefore, \(\Theta_{x0}\) and \(\Theta_{y0}\), as they appear in (5), have to be multiplied by \(s_{\rm f}\), as given by (9).