# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 4 Radially Periodic Leaky-Wave Antennas
### A Simple Technique for the Bloch Analysis  
As mentioned above, a two-dimensional radially periodic LWA is constituted by a subwavelength-thick grounded dielectric slab with an annular metal strip on top (see Figure 2e) [7]. By considering an azimuthally symmetric source, such as a VED, it is possible to locally linearize the radial structure as a 1-D periodic metal strip grating, which can be conveniently described in terms of space harmonics [7]. The radial leaky-wave propagation over an annular metal strip grating can indeed be seen as a 1-D propagation over an infinite metal strip grating with phase advance normal to the printed strips thanks to the azimuthal symmetry of the device [7]. This aspect is well explained in Figure 8, where the linearization process for an azimuthally symmetric radial periodicity on a 2-D structure is reported straightforwardly. The device can thus be analyzed as a 1-D periodic LWA. In this context, the generic \(m\)-th Floquet space harmonic (with \(m=0,\ \pm 1,\ \pm 2,\dots\)) has a dispersive behavior related to the fundamental harmonic wavenumber \(k_{\rho 0}=\beta_{0}-j\alpha\) and a periodicity (along the radial direction) \(p\) as follows:\[k_{\rho m}=\beta_{0}+\frac{2\pi m}{p}-j\alpha \tag{18}\]  
The structure is usually designed to let the \(m=-1\) harmonic propagate in the working bandwidth. Thus, it is manifest that, in this case too, it is crucial to accurately determine the leaky wavenumber for the theoretical design and performance assessment of the LWA. In this context, the possibility to describe the device through its linearized counterpart [7] (see Figure 8) is crucial for the evaluation of the leaky radial wavenumber. This is because different techniques have been already proposed for retrieving the dispersion diagram of leaky modes propagating in 1-D periodic LWAs (see, e.g., [91; 92; 93; 94] and the references therein). In this paper, in order to avoid the implementation of an ad hoc method-of-moments (MoM) approach for the considered structures [91], the need for complicated numerical routines [93], or the correct selection of higher-order modes in the simulation model [92], a method originally proposed for the analysis of _frozen_ modes is exploited [95]. In order to corroborate the proposed approach, the reference case in [87] is considered since the dispersion curves are retrieved through an MoM [91]. While, in [87], the radially periodic device was considered to be a wideband Bessel-beam launcher (BBL) in the near-field region, the same structure will be considered here as an LWA in the far-field region. The wideband BBL proposed in [87] works with a central frequency \(f_{0}=23\) GHz through a GDS with relative permittivity \(\varepsilon_{r}=2.2\) and thickness \(h=3.14\) mm perturbed by a metal strip grating with periodicity \(p=10\) mm and slot width \(s=6\) mm (see Figure 9a). In order to apply the approach proposed in [95], two waveguide ports are defined here in the periodic direction with a height \(h_{\text{port}}=3.5\) h and lateral dimension \(w_{\text{port}}=\lambda_{0}/4\) to ensure the correct representation of the fundamental TM\({}_{0}\) surface wave of the grounded dielectric slab. Moreover, as shown in Figure 9a, perfect-magnetic-conductor (PMC) boundaries and an open boundary condition are assigned along \(y\) and \(z\), respectively. At this stage, the transfer matrix of a single unit cell is obtained through CST full-wave simulations [85] by multiplying the transfer matrix obtained considering \(N+1\) unit cells with the inverse of the one with \(N\) unit cells [95]. The \(T\) matrix of the single unit cell, which takes into account the mutual coupling of adjacent cells, can thus be found. Then, the leaky phase and attenuation constants can be derived through well-known formulas [88; 92; 94].