# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 3 Fabry-Perot Cavity Antennas
### Evaluation of the Dispersion Curves
This assumption is valid for a large class of metasurfaces, even in the terahertz frequency range [82]. Typical examples, shown in Figure 3a, are 2-D patch arrays, metal strip gratings, and fishnet-like metasurfaces [72; 73; 82]. A method for the evaluation of such equivalent impedance value is shown in the following, Section 3.3. Depending on the nature of the PRS (viz., if the metasurface is mainly inductive or capacitive) and on the considered polarization, a variety of modes (i.e., surface, leaky, or plasmonic) can be generated. The generation of the different kinds of waves and their nature is discussed in depth in [83]. In this manuscript, we are just interested in leaky-wave modes for achieving an effective, compact, and directive radiating device. The application of the transverse resonance techniques [77] to the TEN implies equating to zero the sum of the input impedance or admittance looking downward and looking upward at an arbitrary cross section of the equivalent transmission line. In this case study, it is convenient to assume \(z=0\) as a reference plane where the PRS is present (see Figure 3). In this manner, on the one side (the upper), there is the free-space characteristic admittance \(Y_{0}\), and, on the other side (the lower), there is the parallel admittance of the equivalent PRS admittance \(Y_{s}=1/Z_{s}\) (which is purely imaginary if the PRS is lossless, obtaining \(Y_{s}=jB_{s}\), with \(B_{s}=-1/X_{s}\)) and of the short-circuited transmission-line section \(Y_{\text{sc}}=-jY_{d}\cot\left(k_{zd}h\right)\). Therefore, the _dispersion equation_ of the FPCA in the presence of a lossless PRS reads:  
\[Y_{0}+Y_{\text{sc}}=Y_{0}+jB_{s}-jY_{d}\cot\left(k_{zd}h\right)=0 \tag{9}\]  
It is worthwhile to point out that, in FPCAs able to produce a broadside pencil beam, by exciting the structure through an HED or an HMD both TE and TM modes are excited. It is thus important to report here the different quantities that appear in (9) for both polarizations, indicating with the hat \(\hat{\cdot}\) the normalization with respect to the free-space wavenumber \(k_{0}\) and with \(\eta_{0}=120\pi\)\(\Omega\) the vacuum characteristic impedance. \[Y_{0}^{\text{TE}}=\frac{\hat{k}_{z0}}{\eta_{0}},\ \ Y_{d}^{\text{TE}}=\frac{\hat{k} _{zd}}{\eta_{0}\mu_{r}},\ \ \hat{k}_{z0}=\sqrt{1-\hat{k}_{\rho}^{2}},\ \ \hat{k}_{zd}=\sqrt{\varepsilon_{r}\mu_{r}-\hat{k}_{\rho}^{2}} \tag{10}\]  
\[Y_{0}^{\text{TM}}=\frac{1}{\hat{k}_{z0}\eta_{0}},\ \ Y_{d}^{\text{TM}}=\frac{ \varepsilon_{r}}{\eta_{0}\hat{k}_{zd}},\ \ \hat{k}_{z0}=\sqrt{1-\hat{k}_{\rho}^{2}},\ \ \hat{k}_{zd}=\sqrt{ \varepsilon_{r}\mu_{r}-\hat{k}_{\rho}^{2}} \tag{11}\]  
By considering a nonmagnetic material (\(\mu_{r}=1\)) and by substituting Equations (10) or (11) in (9), the dispersion equation for TE,\[\sqrt{1-\hat{k}_{p}^{2}}+jB_{s}\eta_{0}-j\sqrt{\varepsilon_{r}-\hat{k}_{p}^{2}}\cot \left(k_{0}h\sqrt{\varepsilon_{r}-\hat{k}_{p}^{2}}\right)=0 \tag{12}\]  
or TM,  
\[\left(\sqrt{1-\hat{k}_{p}^{2}}\right)^{-1}+jB_{s}\eta_{0}-j\varepsilon_{r}\left( \sqrt{\varepsilon_{r}-\hat{k}_{p}^{2}}\right)^{-1}\cot\left[k_{0}h\sqrt{ \varepsilon_{r}-\hat{k}_{p}^{2}}\right]=0 \tag{13}\]  
modes are achieved, respectively. The zeros in Equations (12) and (13) can numerically be found efficiently through the _Pade_ algorithm [84]. In particular, by searching for the complex _improper_ roots (i.e., with \(\text{Im}[k_{z}]>0\)), the dispersion curves of the _leaky-wave modes_ can be found and one can predict the radiating performance of the FPCAs under analysis.