# The Leaky-Wave Perspective for Array-Fed Fabry-Perot Cavity and Bull's-Eye Antennas
## 2 Fabry-Perot Cavity Antennas
### Fabry-Perot Cavities and Far-Field Twisted Beams  
Recently, FPCAs have been employed in a number of cases to launch waves carrying orbital angular momentum (OAM) [28]. While polarization control is a widespread topic, OAM engineering has recently gained interest due to potential applications to telecommunications, radar, imaging, and sensing. Indeed, electromagnetic waves carrying different OAM states are orthogonal; thus, the bandwidth of a channel theoretically increases linearly with the number of OAM states [29; 30]. OAM waves have been introduced in [31] for optical frequencies, rapidly finding interest in other bands, from radio frequencies up to X- and gamma-rays. Field twists are due to the phase distribution depending on the azimuth \(\phi\) coordinate with the form \(\exp{(j\ell\phi)}\), where \(\ell\) is known as _topological charge_; its sign determines the handedness of the vortex and its absolute value the relevant number of twists in a period. The interest in OAM waves mainly depends on the fact that, unlike circular polarization that is only able to provide right-handed or left-handed rotation, a potentially unlimited number of waves with different values of \(\ell\) can be generated. In a real-world scenario, OAM applications have limitations, mainly due to the phenomena of strong attenuation and fading [32; 33]. One distinct issue consists in finding a suitable method of successful OAM excitation. It has been demonstrated that the most versatile method consists in circular arrays, which can be properly phased to obtain a radiated field transporting a given momentum. By switching from one inter-element phase difference value to another, the order of the momentum is varied. An issue for the latter scheme in the free-space configuration is the unavoidable dependence between the pointing angle and \(\ell\), as can be inferred from the expression of the radiation pattern below (2):  
\[P^{\text{FS}}(\theta)=-2\pi j^{\ell}P_{0}\sin\theta J_{\ell}(k_{0}a\sin\theta) \tag{2}\]  
where \(P_{0}\) is the complex amplitude coefficient impressed to feeders, \(a\) is the radius of the array, and \(J_{\ell}\) represents the first-kind Bessel function of \(\ell\)-th order. When \(\ell\) varies, the maximum of the pattern moves to different angles, for fixed values of array radius \(a\) and free-space propagation constant \(k_{0}\). This issue has been overcome by considering concentric circular arrays, one for each OAM order [34], increasing the overall number of feeders and the total complexity of the system. An alternative solution in order to reduce the number of sources was addressed in [28], where the effect of source embedding in Fabry-Perot cavities was considered. In a resonant environment, a simple planar circular array of coaxial feeders, approximated by vertical electrical dipoles (VEDs), is inserted through the ground plane of the cavity (see Figure 2). This choice is largely motivated by the fact that, according to (1), the main-beam features are mainly set by the radial wavenumber \(k_{\rho}\) of the operating leaky mode (3):  
\[P^{\text{FP}}(\theta)=\frac{2\cos\theta\csc k_{zz}hP^{\text{FS}}(\theta)e^{jk_ {0}h\cos\theta}}{j\bar{k}_{zx}(1+\cos\theta\overline{Y}_{\text{PRS}})+\varepsilon _{r}\cos\theta\cot k_{zx}h} \tag{3}\]  
in which \(k_{zx}\) corresponds to the complex transverse propagation constant [1], \(\bar{k}_{zx}=k_{zx}/k_{0}\) is the relevant quantity normalized by the free-space wavenumber, \(\overline{Y}_{\text{PRS}}=Y_{\text{PRS}}\eta_{0}\) is the normalized sheet admittance of the PRS, \(\eta_{0}\) is the free-space impedance, \(\varepsilon_{r}\) the value of dielectric constant of the material filling the cavity, and \(h\) the relevant height. Observing the numerator in (3), the \(P^{\text{FS}}\) term is still present, but the nulls of the denominator, which appear due to the resonant nature of the cavity, mitigate the shifting effect due to OAM order selection, at least for low values of \(\ell\).