# The Leaky-Wave Perspective for Array-Fed Fabry-Perot Cavity and Bull's-Eye Antennas
## 3 Bull's-Eye Antennas  
The second class of planar structures considered in this paper consists of _bull's eye_ antennas (BEAs). First proposed in [48], and sketched in Figure 17, BEAs constitute an evolution of the classic grounded dielectric slab, loaded by a certain number (ideally infinite) of concentric,  
Figure 16: The feeding network realized through a circular array of \(N=6\) coaxial cables. Two concentric via-hole rings are inserted to enhance the broadband impedance matching. Reproduced from [47]. Figure 15: FPCA fed by a circular array with \(N\) elements: Simulated maximum directivity for \(N=6\) (green dotted line) and main sidelobe level at 20 GHz for with \(N=6,8,10\) (solid lines). Reproduced from [47]. metallic rings. As a modification of the slab, the typical operative regime consists in exciting the fundamental TM\({}_{0}\) mode which, differently from the case of the simple dielectric slab, may exhibit a radiative fast-wave (i.e., leaky) nature. As is known, propagation modes (also known as Bloch modes) in periodic structures are constituted by an infinite number of space harmonics, each one being the solution of the relevant dispersion equation [1]. It is known that, even if the corresponding mode in the unloaded structure is a slow mode, the perturbation due to the periodicity may push, in certain frequency ranges, one or more space harmonics into a fast-wave regime, thus producing radiation leakage. Should one wish to draw a comparison, a BEA is a _radially evolved_ classical metal strip grating on a grounded dielectric slab (MSG-CDS) [49; 50; 51]. Capable of being excited by simple dipole-like sources as in an FPCA resonant scenario, due to high similarity to the MSG-GDS, the spectral analysis of a BEA may be assumed to be congruent to that performed for the latter. Further details on spectral analysis and design rules can be found in [48]. Among many possible applications, BEAs can be employed in designing near-field focusing devices. As shown in [52; 53], a diffraction-limited beam requires an inward cylindrical-wave distribution. Taking advantage of the symmetry properties of Hankel functions recalled by equations in (5), an inward cylindrical-wave distribution can be obtained from an outward cylindrical-wave distribution once a backward leaky wave is excited. \[H_{0}^{(2)}(ze^{-j\pi})=-H_{0}^{(1)}(z)\quad H_{0}^{(1)}(ze^{j\pi})=-H_{0}^{(1 )}(z) \tag{5}\]  
Therefore, if a zeroth-order TM cylindrical wave is launched in a BEA through a coaxial feeder, the annular strip grating can be designed to have a space harmonic with a fast and negative phase constant in the desired frequency range. Under such conditions, the BEA is capable of radiating a beam that retains its limited-diffraction properties up to a finite distance known as the nondiffractive range \(z_{\text{NDR}}\)[54; 55]. The microwave and millimeter-wave generation of limited-diffraction beams through BEAs has gained interest for the creation of cutting-edge antenna systems with additional innovative features. For instance, nondiffractive beams, often referred to as Bessel beams (BBs), may carry an OAM, similarly to what happens in the far-field region. While different methods have been proposed to produce such beams, e.g., employing metasurfaces, lenses, and radial line slot arrays [56; 57; 58; 59], here we consider array-based solutions. In particular, a novel method to generate OAM-carrying near-field focused beams has been explored considering the insertion of multiple VEDs in the form of a planar, circular coaxial probe array to excite a BEA [60], in order to impress a phase dependence on the azimuth angle of the type \(e^{j\phi}\). Referring specifically to the structure analyzed in [60] and sketched in Figure 18, the dielectric slab is loaded with a number of \(N=13\) metallic rings with a certain width,  
Figure 17: Pictorial representation of a _bull’s eye_ antenna, constituted by an arrangement of concentric microstrip rings on top of a grounded dielectric slab.