# The Leaky-Wave Perspective for Array-Fed Fabry-Perot Cavity and Bull's-Eye Antennas
## 2 Fabry-Perot Cavity Antennas
### Fabry-Perot Cavities for Beam Scanning in Elevation and Azimuth
Full-wave results obtained through reciprocity theorem were verified to be in good agreement with analytical leaky-wave patterns. Further comparison with the truncated version of the cylindrical wavefront model  
Figure 8: FPCA 3-D model and a planar feeding grid realized with coaxial cables. A schematic of the employed PRS is also reported. Reproduced from [46]. Figure 9: Dispersion of radial phase (blue lines) and attenuation constants (red lines) for multiple values of distance \(s\) between metallic patches composing the PRS. Higher values of \(s\) induce higher attenuation, while weakly affecting the phase constant. Reproduced from [46]. was carried out taking into account the finite dimension of the cavity and relevant effects on the pattern [46], as shown in Figure 10.  
Once element-pattern radiation capabilities have been assessed and validated by different approaches, multiple feeders are considered to extend the beam-scanning effect in the azimuthal dimension. A highly reduced number of \(N\times N=9\) VEDs is considered in a planar square grating, breaking the azimuthal invariance of the conically shaped element pattern with the multiplication by the scalar array factor, (4):  
\[\mathbf{AP(\theta,\phi)=EP(\theta,\phi)AF(\theta,\phi)} \tag{4}\]  
where \(\mathbf{AP(\theta,\phi)}\) corresponds to the array pattern, \(\mathbf{EP(\theta,\phi)}\) to the element pattern, and \(\mathrm{AF(\theta,\phi)}\) to the scalar array factor. An expression for uniform arrays may be found in [41] for different planar geometries. Beam-shaping effects are shown, for multiple frequencies, in Figure 11, through directivity patterns. Figure 11: _Cont._  
Figure 10: Cylindrical leaky-wave model, truncated cylindrical leaky-wave model, and reciprocity-obtained elevation pattern (TEN) at 16 GHz. (**a**) \(s=50\) μm, (**b**) \(s=10\) μm, (**c**) as in (**b**) but radiating 99% input power. Reproduced from [46]. When multiple inputs are considered, it is important to assess the cross-coupling among elements. This latter aspect is properly addressed by considering the input scattering matrix, reported in Figure 12 for three independent elements (the others may be deduced by symmetry) [46]. Cross-coupling coefficients have been proven to be lower than \(-10\) dB for adjacent elements, under \(-18\) dB for elements on the same side of the square lattice, and lower than \(-25\) dB for elements at opposite corners. The resulting effect is a bandwidth enhancement, which in turn widens the elevation scan from about \(21^{\circ}\) to \(68^{\circ}\). A full electronic steering effect is achieved with high level of beam directivity using simple and inexpensive materials in compact and easily resizable geometries. The fruitful results obtained in [46] turned into a later study presented in [47], with a novel multilayer resonant cavity, where the usual quasi-TEM mode was confined to a slow-wave regime inserting multiple, commercial, low-cost dielectric layers realizing a low-permittivity dielectric constant, as sketched in Figure 13.  
Figure 7:0 Input reflection coefficients for a \(3\times 3\) square planar VEDs array inside an FPCA. The horizontal dashed black line corresponds to the \(-10\) dB threshold. Reproduced from [46]. Exploiting both the leaky-wave theory and the phased-array theory to separately obtain a scan on elevation and azimuth planes, an increased matching and a shift to higher frequencies of the input reflection coefficient are achieved by concentrically inserting two rings of vias contacting the ground plane of the cavity and the lower dielectric laminate (see Figure 14). A comparison between square and circular planar arrays was also performed, finding out that the azimuthal symmetry of the latter induces a saturation of the maximum gain obtained by the overall antenna when the number \(N\) of coaxial probes is larger than or equal to 6.