# Two-Dimensional Scanning of Circularly Polarized Beams via Array-Fed Fabry-Perot Cavity Antennas
## 4 FPCA: Array Feeding and Azimuth Scan  
The presented FPCA results demonstrate the feasibility of frequency-based scanning of the elevation plane using a directive CP beam, enabled by the proposed PCM. However, this solution does not allow for both having a pencil beam off broadside and varying the azimuthal angle of the beam. For this purpose, a 2-D grid of sources can be used by exploiting the pattern multiplication principle of array theory. This idea has been demonstrated in [48] (an overview of array-fed 2-D LWAs is given in [49]) for LP beams using a planar array of feeders to gain further control of the beam features of an FPCA. In particular, the leaky-wave dispersion is exploited to scan by frequency the beam in elevation, whereas the phasing of the feeders allows for scanning in azimuth. This technique is here suitably modified for CP beams. Figure 4: Fictorial representation of the co-polar, far-field component \(E_{\text{RHCP}}\), in a 3-D format. From the broadside pencil beam starting condition in (**a**), by stepping up the operation frequency, the radiation pattern evolves to azimuthally invariant, conical patterns (**b–e**). Figure 5: The cross-polar far-field component \(E_{\text{LHCP}}\) is reported after being normalized with respect to the \(E_{\text{RHCP}}\) maximum at each frequency. As it manifests, being on average \(-20\) dB under the desired co-polar component within the main beam, its contribution to the far-field radiation is negligible. As is known from basic theory, the electric far-field pattern \(\mathbf{E}_{\text{AP}}\) radiated by the array is given by  
\[\mathbf{E}_{\text{AP}}=A_{\text{F}}\mathbf{E}_{\text{EP}} \tag{3}\]  
where \(\mathbf{E}_{\text{EP}}\) is the element-pattern radiated when a single HED source is placed in the cavity and \(A_{\text{F}}\) is the relevant array factor. Here, the HEDs are arranged in a uniform planar 2-D array configuration so that the array factor is given by [41]  
\[A_{\text{F}}=\frac{1}{NM}\frac{\sin(N\frac{\psi_{x}}{2})}{\sin(\frac{\psi_{x}}{ 2})}\frac{\sin(M\frac{\psi_{y}}{2})}{\sin(\frac{\psi_{y}}{2})} \tag{4}\]  
where \(N\) is the number of elements along the \(x\) axis, \(M\) is the number along the \(y\) axis, the quantities \(\psi_{x}\) and \(\psi_{y}\) depend on the spherical angles \(\theta,\phi\), and the phase shifts \(\Delta\phi_{x},\Delta\phi_{y}\) along \(x,y\) through  
\[\psi_{x} =kd_{x}\sin\theta\cos\phi+\Delta\phi_{x} \tag{5}\] \[\psi_{y} =kd_{y}\sin\theta\sin\phi+\Delta\phi_{y}\]  
where \(k\) is the wavenumber, and \(d_{x}\) and \(d_{y}\) stand for inter-element spacing along \(x\) and \(y\) axes, respectively. Here, we consider \(N=4\) and \(M=10\), for a total number of 40 elements. A common design criterion in standard antenna array theory consists of requiring the distance between adjacent elements to be lower than half a wavelength. This is to avoid grating lobes, specially if it is necessary to scan elevation angles far from the broadside, namely up to \(60^{\circ}\) and beyond. Here, the elevation angle is limited to approximately \(40^{\circ}\), so grating lobes should not appear, even if such a condition is not strictly met. Additionally, possible unwanted lobes at different pointing angles are washed away due to high elevation selectivity of the element pattern, suggesting that sparser feeding grids can be considered [42]. Consequently, an equal inter-element spacing along the two directions, viz. \(d_{x}=d_{y}\), has been chosen, setting a distance equal to twice the unit-cell periodicity, \(d_{x}=d_{y}=2l_{c}=6.66\) mm. Since the lower frequency is 27 GHz, corresponding to \(\lambda=11.11\) mm, the element spacing is slightly higher than half a wavelength, but the results in Figure 6 confirm that we still manage to suppress all of the spurious lobes. In particular, in Figure 6, the ability of the proposed FPCA to radiate a CP pencil beam in any elevation direction (between \(\theta=0^{\circ}\) and \(\theta\simeq 40^{\circ}\)), as well as any azimuthal direction is demonstrated.