# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 3 Fabry-Perot Cavity Antennas
### Full-Wave Validation of the Theoretical Model  
This subsection simultaneously provides a rapid full-wave simulation setup for FPCAs and a validation of the theoretical leaky-wave model and of the correct antenna design. In order to carry this out, a case study is considered with an equivalent sheet impedance \(X_{s}=30\)\(\Omega\) and a dielectric slab with negligible losses (lossy cases are thoroughly discussed in [71]), thickness \(h=2.77\) mm, and relative permittivity \(\varepsilon_{r}=3\). It is worthwhile to point out that the \(X_{s}\), \(\varepsilon_{r}\), and \(h\) values can be properly designed to achieve the desired radiating features at a certain frequency \(f_{0}\), as previously discussed in Section 3.1. Once the design parameters of the proposed FPCA are fixed, one can easily evaluate the leaky-wave dispersion curves through the transverse resonance technique applied to the TEN of the device, as thoroughly discussed in Section 3.2. In this manner, the profile of the phase and attenuation constant normalized with respect to the vacuum wavenumber \(k_{0}\), viz. \(\hat{\beta}=\beta/k_{0}\) and \(\hat{\alpha}=\alpha/k_{0}\), respectively, are retrieved vs. frequency \(f\), as shown in Figure 7a. From the dispersion curves, it is clear that the proposed device is able to generate a pencil beam with a maximum radiated power at broadside at \(f_{c}=29.88\) GHz, when \(\beta^{\text{TE}}\simeq\alpha^{\text{TE}}\simeq\beta^{\text{TM}}\simeq\alpha^{ \text{TM}}\)[74]. Moreover, as previously discussed in Section 3.2, one can assert that the device is working in the reactive regime for \(f<f_{c}\)[75] and it is generating a conical beam for \(f>f_{c}\). By exploiting the evaluation of the dispersion curves, it is also possible to choose the lateral extension of the device to achieve a radiation efficiency of, at least, \(90\) % at a certain working frequency \(f_{w}=33\) GHz. In particular, by considering \(\alpha=\min\{\alpha^{\text{TE}}(f_{w}),\alpha^{\text{TM}}(f_{w})\}\), the minimum extension of the radiating device is set at \(L=38.96\) mm through (2). Figure 7: **(a)** Dispersion curves of both the TE and TM modes of the phase and attenuation constants normalized with respect to the vacuum wavenumber \(k_{0}\), viz. \(\hat{\beta}=\beta/k_{0}\) and \(\hat{\alpha}=\alpha/k_{0}\), respectively. **(b)** Radiation patterns, normalized with respect to their maximum on a linear scale, of the simulated FPCA at different frequencies. Once the minimum lateral dimension of the FPCA has been found, the device is ready to be simulated on a commercial full-wave solver to verify its radiation property. In order to perform that and to strongly reduce the computational effort, one can implement the source as an ideal dipole (its physical implementation and matching techniques are discussed in Section 3.4) and the PRS through a surface impedance boundary condition (SIBC)--the design of the real reflecting layer can be addressed through a homogenized metasurface designed with the approach shown in Section 3.3. It is worthwhile to point out that the SIBC has to be described with a frequency-dispersive behavior which respects the [PERSON]'s reactance theorem [90]. Therefore, depending on the inductive-like or capacitive-like nature of the PRS, one has to enforce \(Z_{s}=j\omega L\) or \(Z_{s}=-j/\omega C\), respectively, on the full-wave solver. Specifically, the lumped inductance \(L\) and capacitance \(C\) value are fixed so as to match the desired \(X_{s}\) value at the working frequency of the design. By applying the simulation guidelines discussed so far, a three-dimensional (3-D) model of the proposed FPCA has been implemented on CST Microwave Studio [85] through a _transparent_ SIBC on top of a cylindrical grounded dielectric slab with height \(h=2.77\) mm, aperture diameter \(L=40\) mm, and relative permittivity \(\varepsilon_{r}=3\). By exciting the device through an HED in the middle of the cavity [55], the far-field radiation patterns for \(f>f_{c}\) reported in Figure 7b were achieved.