# The Leaky-Wave Perspective for Array-Fed Fabry-Perot Cavity and Bull's-Eye Antennas
## 1 Introduction  
Two-dimensional (2-D) leaky-wave antennas (LWAs) belong to the class of traveling-wave antennas. Based on planar geometries, either metallic or metal-dielectric, their working principle depends on the excitation on the antenna aperture of _cylindrical leaky waves_, excited by a simple nondirective source and propagating with a complex leaky radial wavenumber \(k_{\rho}=\beta-j\alpha\) (with respect to the cylindrical coordinates reported in Figure 1), with \(\beta\) and \(\alpha\) being the radial leaky phase and attenuation constants, respectively [1, 2].  
Leaky waves in uniform open waveguides made using ordinary materials are _improper_ solutions of the relevant dispersion equations, i.e., they do not satisfy the Sommerfeld radiation condition at infinity [3, 4, 5], while they may be either proper or improper in periodic open waveguides [6], depending on the operating frequency. In any case, their physical validity is restricted to certain angular ranges within which they are exponentially damped and hence negligible at infinity; nevertheless, if they are effectively excited in frequency ranges where they are both fast (\(|\beta|<k_{0}\)) and characterized by an attenuation constant that is small with respect to the free-space wavenumber \(k_{0}\) (\(\alpha\ll k_{0}\)), they may dominate the antenna aperture field: leaky waves can therefore provide an asymptotic representation of the relevant waveguide continuous spectrum and thus, via Fourier transformation, of the antenna radiation pattern [7, 8].  
The condition \(\alpha\ll k_{0}\), crucial for the leaky wave to dominate the aperture field, implies that the complex wave provides a large aperture illumination, in terms of the free-space wavelength \(\lambda_{0}\). An immediate consequence is that, considering the properties of the Fourier transform, the radiation pattern is characterized by high directivity in elevation.  
A cylindrical leaky wave with \(\beta>\alpha\) radiates a conical scanned beam with pointing angle \(\theta_{0}>0\), whereas, when \(\beta\leq\alpha\), a broadside beam (\(\theta_{0}=0\)) is produced [7]. Two important formulas for the pointing angle \(\theta_{0}\) and the \(-3\) dB elevation beamwidth \(\Delta\theta\), valid for the case of a scanned beam, are (1)  
\[\theta_{0}\simeq\arcsin\,\hat{\beta}\hskip 28.452756 pt\Delta\theta\simeq 2 \hat{a}\sec\theta_{0} \tag{1}\]  
where the hat \(\hat{\cdot}\) indicates the normalization of radial phase and attenuation constants \(\beta\) and \(\alpha\), respectively, by the free-space wavenumber \(k_{0}\) (additional general pattern-related formulas may be found in [7; 9; 10], for a great variety of cases encountered in applications). The normalized radial wavenumber \(\hat{k}_{p}\) typically shows a frequency-dispersive behavior. Therefore, as a consequence of (1), to perform an elevation scan (and also potentially change the elevation beamwidth), it is sufficient to vary the operating frequency [11; 12; 13].  
The aim of this work is to introduce 2-D LWAs with multiple feeders, i.e., array-fed 2-D LWAs. In such designs, the attractive features of LWAs in terms of elevation directivity, pattern reconfigurability, and feed simplicity are combined with additional degrees of freedom. In particular, there is the possibility to independently control multiple sources in amplitude and phase, thus shaping the associated array factor. Although various other 2-D LWAs exist (such as holographic [14; 15] and modulated surface antennas [16; 17; 18; 19]), here we specifically address two categories, namely Fabry-Perot cavity antennas (FPCAs) and _bull's-eye_ antennas (BEAs). In Section 2, we present array-fed implementations of the former class, whereas the latter configuration with multiple feeders is discussed in Section 3. Finally, conclusions are drawn in Section 4.