# Leaky Wave Generation Through a Phased-Patch Array
## 3 Results  
### Comparison Between the Antennas Operating at 12 GHz  
We first compared the two antennas in terms of the generated far field, because the far field was obtained via an integral transformation of the near-field radiation, resulting in a preliminary estimator of the similarity among the two structures; see Figure 13. Also, this provided important information related to the radiating properties of the two antennas. Figure 12: Design of a compact uniform LWA realised employing a phased array. Figure 13: Comparison between the far field produced by the LWA versus the one produced by the phased array. We then examined the near field, to identify the maximum vertical distance from the antenna where the array still produced a field that satisfactorily approximated the one generated by the Menzel antenna. After having verified the similarity in the far field, before focusing on the near field, it is worth making some observations to extend the comparison between these two structures to the frequency-scan behaviour. It is well known that the LWA radiation mechanism is given by the radiation losses provoked into a guiding structure. So, it is clear that the propagating mode inside the considered lossy waveguide is of chief importance as far as the radiation of the LWA is concerned. If this mode is in cut-off, there will be no radiation at all. By scanning the frequency, the mode will start propagating inside the structure (i.e., its wave number component along the direction of propagation passes from predominantly imaginary to predominantly real), and the LWA will start radiating near broadside. Again, increasing the frequency, the LWA will scan the beam up to endfire. For the reasons just stated, for a one-dimensional LWA it is impossible to radiate at broadside, since it would correspond to the radiation when the mode inside the guide is in cut-off. This issue, known as stop-band [11], can be overcome in several ways that will not be considered here. The interested reader may refer to the relevant literature, e.g., Ref. [11]. For a phased array, the situation is quite different. In this case, a propagating mode inside the structure is not present, and, to enforce a radiation angle for the beam, a specific phase relation between the radiating elements has been super-imposed here. It is then interesting to analyse what happens when the frequency varies, in particular focusing on the resulting spatial geometrical scanning of the radiated beam. In this case, the direction of scanning can be computed manipulating Equation (3), which we write here for the sake of clarity, it is:  
\[\Delta\phi=kd[sin(\theta)-sin(\theta_{0})]=kdsin(\theta)-\beta \tag{4}\]  
then it follows:  
\[\theta_{0}=arcsin\bigg{(}\frac{\beta}{kd}\bigg{)}=arcsin\bigg{(}\frac{\beta} {2\pi fd/c}\bigg{)} \tag{5}\]  
where \(\Delta\phi\) is the total phase between the adjacent antenna elements, while \(\beta=kd\sin(\theta_{0})\) represents the progressive phase shift between the elements of the array [4]. It follows that if the amplitude of \(\beta\) is chosen so that the antenna radiates at a \(\theta_{0}=45^{\circ}\) angle for a frequency \(f=12\) GHz then as the frequency increases the radiated beam tends to point to broadside. If the frequency decreases, the beam is directed towards endfire. This behaviour is the opposite with respect to an LWA, but it is very interesting to note that in this case there is also a stop-band behaviour at broadside as, from the relation above, it is clear that to achieve a \(\theta_{0}\) equal to 0, the operating frequency needs to tend to infinity. To reach endfire, instead, it is sufficient to reach the frequency, as follows:  
\[f=\frac{\beta}{2\pi\sin\theta_{0}d/c} \tag{6}\]  
To assess this behaviour, let us consider Figure 14, where the array factor of a linear equi-spaced array with a phased coefficient, so as to scan at 45\({}^{\circ}\) at 12 GHz, has been considered at different frequencies.