# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 3 Fabry-Perot Cavity Antennas
### Evaluation of the Dispersion Curves  
The standard procedure for determining the eigenvalues in waveguiding structures requires one to solve [PERSON]'s equations in any homogeneous region of the domain of interest and then apply the appropriate boundary conditions [76]. A nontrivial solution is then obtained by discretizing the problem, thus reducing it to a linear algebraic system, and enforcing the determinant of the coefficient matrix to be equal to zero. The complex roots of the resulting determinantal equation--which is commonly referred to as a _dispersion equation_--are the desired eigenvalues. This approach enables the expression of any solution to the problem in terms of its eigenvalue (the waveguide mode) and eigenfunction (the modal field), up to a multiplicative constant that depends on the excitation. This procedure, however, could be lengthy for devices with different media and complicated structures. Since, in the leaky-wave perspective, we are usually interested only in the evaluation of the eigenvalues, a faster approach is given by the _transverse resonance technique_[77]. It is indeed well known [78] that the eigenvalues of a waveguide problem correspond to the pole singularities of an appropriate characteristic Green's function in the \(k_{z}\) complex plane, which represents the voltage (or current) in a transmission line along one of the transverse directions of the waveguide [5,6]. In this network formalism, the pole singularities correspond to the resonances of the transverse equivalent network (TEN) model, which can be effectively calculated using both analytical and numerical methods. Since FPCAs are constituted by a grounded dielectric slab with a PRS on top, their TEN is formed by a transmission-line segment--with characteristic admittance \(Y_{d}\) and transverse number \(k_{zd}=\sqrt{k_{0}^{2}e_{r}-k_{P}^{2}}\)--terminated, on one side, on the equivalent sheet admittance of the PRS \(Y_{s}\), and, on the other side, on a short circuit representing the ground plane (see Figure 3b). The term PRS generally denotes any type of screen that provides partial shielding of the radiation emitted by the primary dipole-like source located within the cavity [16]. For this reason, PRSs may take various forms that are conveniently categorized into three different classes [16] (see Figure 4):  
Figure 4: Schematic representation of Fabry–Perot cavity antennas with different physical implementation of their partially reflecting screen: (**a**) single-layer superstrate, (**b**) dielectric multilayer, and (**c**) homogenized metasurface. Different gray-scale colors represent different dielectric relative permittivities, while the black color represents metal areas. Single-layer dielectric covers which are typically implemented through a quarter-wavelength-thick high-permittivity slab [79; 80];
* Dielectric multilayers consisting of alternating quarter-wavelength-thick slabs of high and low permittivity to realize a distributed Bragg reflector [81];
* Homogenized metasurfaces which are realized through subwavelength (with period \(p\ll\lambda_{0}\)) periodic planar arrangements of metal scatterers [82]. The last class is the most common at microwave frequencies and beyond to reduce dielectric losses [82]. This kind of PRS is rigorously represented by an equivalent impedance \(\underline{\mathbf{Z}}_{s}\) with a tensor-like form and a spatially and frequency-dispersive model. However, since PRSs with simple geometries are commonly implemented in FPCAs that are typically required to operate over a narrow FBW and/or at a fixed beam angle (or over a small angular variation) [82], the model of the PRS is usually constituted by a single, scalar, generally complex, sheet impedance \(Z_{s}=R_{s}+jX_{s}\), with \(R_{s}\) and \(X_{s}\) being the equivalent sheet resistance and reactance of the PRS, respectively. For lossless isotropic metasurfaces, this contribution is given by a scalar, purely imaginary, sheet impedance \(Z_{s}=jX_{s}\).