# Leaky Wave Generation Through a Phased-Patch Array
## 2 Materials and Methods
### The Sampling Methodology  
To sample the current on the aperture, we had to choose the elementary radiating elements and their locations, i.e., the spatial sampling. In regard to the radiating element, a simple patch antenna [24; 25] was chosen, due to its similarity to the leaky-wave antenna considered.  
Figure 6 shows the patch antenna that was designed as an element of the array: even if it was not necessary, the same substrate of the LWA was used:  
When in the array, the patch antenna had to be placed so that the field was polarized in the same direction as the one radiated by the leaky-wave antenna; see Figure 7. To determine the exact number of patches in the array, a straightforward observation could be made: in order to radiate the same field, it was necessary that the two structures had the same currents on the same electrical area. The simplest way to assure this for the two antennas considered was to make them the same geometrical length.  
Placing the patches too close together would have resulted in a poor active reflection coefficient, due to the mutual coupling. On the other hand, it was not advisable to place them too far apart, since grating lobes might occur and, furthermore, a coarse sampling would have resulted in a poor representation of the field emitted by the LWA.  
For these reasons, it was decided to keep \(\lambda/2\) spacing between the patches: this was the maximum distance that allowed us to avoid grating the lobes in the visible region when scanning up to endfire. Given the geometry considered, the array that we designed consisted of 9 radiating elements (patches) placed at a constant distance.  
To finalize the array design, it was essential to define the amplitudes and phases of the patch elements. Regarding the phases, we began by setting the LWA to radiate at a \(\theta^{\text{LWA}}=45^{\circ}\) angle; then, the phases for the array were set to values equal to the ones required by a phased array to point the beam in the same direction as the LWA.  
Figure 6: Patch antenna used as element of the array.  
Figure 7: Schematic representation of a continuous current source (on the **left**) and of a discrete current source obtained by sampling the continuous source (on the **right**).  
For the amplitudes, the normalised power in the dielectric of the LWA was simulated along the longitudinal direction. Then, it was sampled at each position in which a radiating element was present in the phased array along the symmetry axis. Figure 8 illustrates the resulting power distribution along the longitudinal direction of the patch antenna.  
The phasors used for the 9 elements are represented in Table 1:  
With these assumptions, the patch array of Figure 9 was realised:  
\begin{table}
\begin{tabular}{c c c c} \hline
**Element** & **Power [W]** & **Amplitude [V]** & **Phase [deg]** \\ \hline
1 & 1 & 1.4142 & 0 \\
2 & 0.8290 & 1.2876 & 127.37 \\
3 & 0.6610 & 1.1498 & 254.74 \\
4 & 0.5157 & 1.0155 & 382.11 \\
5 & 0.3917 & 0.8851 & 509.48 \\
6 & 0.2922 & 0.7645 & 636.85 \\
7 & 0.2219 & 0.6661 & 764.22 \\
8 & 0.1684 & 0.8504 & 891.59 \\
9 & 0.1266 & 0.5032 & 1019.01 \\ \hline \end{tabular}
\end{table}
Table 1: The table shows how the 9 radiating elements are excited in terms of power (in watts), amplitude (in volts), and phase (given as accumulated phase shift in degrees).  
Figure 8: Sampling of the power inside the dielectric of the LWA.  
Figure 9: Layout of the phased array constituted by the patch antennas chosen.  
The performances of the array obtained could be rapidly assessed by evaluating the active reflection coefficient of each element (Figure 10) and by the average embedded element pattern (Figure 11):