## 2 Array Design
### Numerical Results  
The analytical techniques presented in the previous paragraphs are exploited here in a proof of concept to test their effectiveness and accuracy in the array synthesis and its performance evaluation. The proof of concept is chosen to fit the constraints mentioned at the beginning of Section 2, namely, \(\mathrm{SLL}<-30\) dB, \(\theta_{0,\mathrm{max}}=60^{\circ}\), \(\Theta_{0\mathrm{h}}=3^{\circ}\) over the \(f_{\mathrm{min}}\)-\(f_{\mathrm{max}}\) range. In order to fulfill the SLL requirement, we have to accept a spreading factor of about 1.3, but only a slight modification to the condition for avoiding grating lobes (\(w_{0}\simeq 1.0057\)). We thus set \(d_{x}=d_{y}=\lambda_{0}/2\), which mostly satisfies (3). With these numbers at hand, it is found from (5) that \(N_{x}=N_{y}=40\) provides for \(\Theta_{0\mathrm{h}}\) around \(2.5^{\circ}\) that, after taking into account the spreading factor (viz., \(s_{\mathrm{f}}\simeq 1.3\)), yields \(3.3^{\circ}\). Under this condition, the \(-3\) dB fractional bandwidth at the maximum scan angle would be around 6.6%. Using (11), it is found that the targeted bandwidth performance would be reached by either limiting the maximum scan angle to \(40^{\circ}\), or broadening the broadside beamwidth to about \(4.5^{\circ}\). In the following, we consider the original setting, thus accepting a reduction in the operating bandwidth. Therefore, we consider a \(40\times 40\) planar array with \(d/\lambda=0.5\) spacing among the elements and apply the Tseng-Cheng technique to obtain an invariant SLL of \(-30\) dB. The distribution of the amplitude coefficients is reported in Figure 1. As shown, the coefficients at the four corners are almost zero, thus the corresponding elements may not be fed, with minor impact on the radiating performance. In particular, in Figure 2a-d a comparison is shown among the normalized power radiation patterns produced with an _ideal_ (i.e., keeping all the coefficients) Tseng-Cheng distribution (see Figure 2a), and those produced by setting to 0 the coefficients below 0.01 (see Figure 2b), 0.05 (see Figure 2c), and 0.1 (see Figure 2d), when the array is scanned at (\(\theta_{0}=60^{\circ}\), \(\phi_{0}=45^{\circ}\)). For a threshold of 0.01 no appreciable difference is seen with the ideal case, but in the former case one would achieve a reduced filling factor of 83%. As the threshold raises, the filling factor reduces further (viz., 62% and 43% for thresholds of 0.05 and 0.1, respectively), but the SLL increases as well (viz., peaks of \(-26\) dB and \(-21\) dB for thresholds of 0.05 and 0.01, respectively). As a result, further reductions in the number of elements should not be considered as this would require optimization techniques for sparse arrays, an aspect that goes beyond the scope of this work. The effect of the element reduction is even more appreciable looking at the 1-D normalized power patterns along the two principal planes (solid black and gray dashed lines in Figure 2e-h). It is noted that the patterns along the elevation plane are more affected than those along the orthogonal plane in all cases. Interestingly, the beamwidths are not affected by the element reduction, and their numerical evaluations agree well (a percent error less than 10%) with the formulas provided in (5)-(7). With respect to the bandwidth performance, the beam squint \(\Delta\theta\) and the gain loss are evaluated as a function of the frequency in Figure 3a,b. It is seen that the numerical evaluation of the beam squint (solid black line in Figure 3a) compares well with the theoretical prediction of (10) (dashed gray line in Figure 3a), and reveals a beam squint of about \(3^{\circ}\) within the \(-3\) dB bandwidth shown in Figure 3b. The beamwidths along the principal planes and as a function of frequency are also shown in Figure 4, where their numerical evaluation is compared against the analytical expressions provided in Equations (6) and (7), accounting for the spreading factor.