# Leaky Wave Generation Through a Phased-Patch Array
## 3 Results
### Comparison Between the Antennas Operating at 12 GHz
Here, the squinting behaviour is exaggerated to show how, given a set of coefficients to scan at a given angle, a stop-band behaviour is experienced at broadside. In Figure 14, only the array factor is shown, for three main reasons: firstly, it would not be feasible to realise a radiating element that could cover such a bandwidth. This is even more true for patches that typically cover fractional bandwidths of 1-5% when in standard configuration [29; 30]. Secondly, this phenomenon is due to the relative amplitude and phase of the electromagnetic field for the elements in a given lattice. All these characteristics, according to the elementary theory of array, belong to the array factor and not to the element factor that is, or at least should be, common to all radiating elements [16]. After having compared the far fields, we then analysed the differences in the near field, and, given the difference in behaviour in the case of the frequency scan, we concentrated on the situation in which both antennas pointed at a 45\({}^{\circ}\) angle, as in this case the two antennas operated at the same frequency, by design choice. To make this evaluation, referring to the reference system shown in Figure 15, we firstly compared the amplitude of the electric field \(|E_{x}(0,y_{i},z)|\) as a function of \(z\), sampling it at a different position, \(y_{i}\), where the subscript \(i\) denoted a distinct sampling point along the \(y\) axis:  
Our results are shown in Figure 16.  
After this evaluation, we compared the field on planes for different constant values of the \(z\) coordinate, i.e., we plotted \(|E_{x}(x,y,z_{i})|\), choosing different values, \(z_{i}\), along the \(z\) axis. Our results are visible in Figure 17.  
The calculation of the amplitude of the field normalised by its maximum value was a useful estimator for evaluating the behaviour of the two structures without taking into account the different power radiated, i.e., without considering the efficiency of the antennas. Figure 14: Array factor (AF) of a linear equi-spaced phased array with fixed phase-relation at different frequencies. Figure 15: Reference system used for the comparison. Another interesting parameter that allowed us to obtain a deep understanding of the variation of the field was the derivative of such a field, as shown in Figure 12:0  
Figure 12:1 Comparison between the amplitude \(|E_{x}|\) of the electric field normalised by its maximum for the LWA and the phased array along the \(z\) axis when \(x=0\) and for different constant values of the \(y\) coordinate: (**a**) \((x,y)=(0,30)\) mm; (**b**) \((x,y)=(0,40)\) mm; (**c**) \((x,y)=(0,50)\) mm; (**d**) \((x,y)=(0,60)\) mm; (**e**) \((x,y)=(0,70)\) mm; (**f**) \((x,y)=(0,80)\) mm. Figure 12:2 \(|E_{x}|\) field component sampled at distinct planes \(z=z_{i}\) for the LWA, on the left, and the phased array (PA), on the right: (**a**) \(|E_{x}^{LWA}|\) for \(z=2\) mm; (**b**) \(|E_{x}^{PA}|\) for \(z=2\) mm; (**c**) \(|E_{x}^{LWA}|\) for \(z=5\) mm; (**d**) \(|E_{x}^{PA}|\) for \(z=5\) mm; (**e**) \(|E_{x}^{LWA}|\) for \(z=10\) mm; (**f**) \(|E_{x}^{PA}|\) for \(z=10\) mm; (**g**) \(|E_{x}^{LWA}|\) for \(z=20\) mm; (**h**) \(|E_{x}^{PA}|\) for \(z=20\) mm. Figure 12:3 Comparison of the derivative of the \(|E_{x}|\) component with respect to y along \(z\) for different constant values chosen along the \(y\) axis. Values: (**a**) \((x,y)=(0,30)\) mm; (**b**) \((x,y)=(0,40)\) mm; (**c**) \((x,y)=(0,50)\) mm; (**d**) \((x,y)=(0,60)\) mm; (**e**) \((x,y)=(0,70)\) mm; (**f**) \((x,y)=(0,80)\) mm. To make a quantitative evaluation of the similarity, we computed the root mean square (RMS) value of the error, visible in Figure 12:4 For every point in the \(z\) direction (i.e., the longitudinal direction for the antenna), we evaluated the RMS error on the field values over the (x,y) plane. The applied formula was  
Figure 12:5  
where \(N\), \(M\) were the numbers of samples on, respectively, the \(x\) and \(y\) axis.