# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 3 Fabry-Perot Cavity Antennas
### Evaluation of the Equivalent Sheet Impedance of a Partially Reflecting Sheet  
The evaluation of the equivalent sheet impedance \(Z_{s}\) for a homogenized metasurface is a crucial step in the design of the realistic devices for FPCAs. A scalar sheet impedance \(Z_{s}\), which generally depends on the polarization, can represent a planar patterned metal sheet if its periodicity \(p\) is much lower than the operative wavelength \(\lambda\). This assumption, which corresponds to the _homogenization principle_, is needed to let only the fundamental \(n=0\) Floquet harmonic propagate.  
In order to evaluate the \(Z_{s}\) value, the unit cell of the homogenized metasurface is implemented on a commercial solver (such as CST Microwave Studio [85] or HFSS [86]) in a periodic environment (which means the application of _phase-shift_ walls as boundary conditions on the periodicity directions). On top of the patterned metal, an air block is considered and, at the bottom, a grounded dielectric slab, where the metasurface lies, is assumed. In Figure 5, for instance, the case of a fishnet-like metasurface printed on top of a grounded dielectric slab of thickness \(h\) is considered.  
By de-embedding the reference plane for the wave port shown in Figure 5 at the metasurface level (\(z=0\)), the reflection coefficient \(S_{11}\) at the air-PRS interface is retrieved from the full-wave solver. At this point, one can easily evaluate the input admittance \(Y_{\text{in}}\) (see Figure 5) as  
\[Y_{\text{in}}=Y_{0}\frac{1-S_{11}}{1+S_{11}} \tag{14}\]  
where \(Y_{0}\) is the vacuum wave admittance of the mode in propagation.  
By observing the equivalent circuit in Figure 5, it is clear that \(Y_{\text{in}}\), evaluated through a full-wave simulation, is given by the parallel of the homogenized-metasurface admittance  
Figure 5: Full-wave model of a fishnet-like metasurface unit cell in a periodic environment on the left. A wave port is considered on top and it impinges on the homogenized-metasurface unit cell, taking into account the substrate effect. On the bottom, a perfect-electric-conductor (PEC) condition is considered in order to represent the ground plane of the dielectric slab. On the right, the equivalent transmission-line model of the problem is shown.  
\(Y_{s}\) and the input admittance of an \(h\)-thick, short-circuited, transmission-line segment with the dielectric characteristic admittance \(Y_{d}\) and propagation constant \(k_{zd}\). Therefore, the desired \(Y_{s}\) value is straightforwardly evaluated as  
\[Y_{s}=Y_{\text{in}}-jY_{d}\cot(k_{zd}h) \tag{15}\]  
Once the equivalent admittance is known, the evaluation of the equivalent sheet impedance of the homogenized metasurface is trivial:  
\[Z_{s}=R_{s}+jX_{s}=1/Y_{s} \tag{16}\]  
with \(R_{s}\) the equivalent resistance (which takes into account ohmic and dielectric losses) and \(X_{s}\) the equivalent sheet reactance (in lossless material, one commonly has \(Z_{s}=jX_{s}\)). The reactance sign can be either positive or negative, representing an inductive- or capacitive-like metasurface, respectively.  
In order to corroborate the proposed approach, an inductive-like partially reflecting sheet with \(X_{s}\) closed-form homogenization expressions is considered. In particular, a metal strip grating printed on an air-like grounded dielectric slab with strip width \(w\) and periodicity \(p\) is assumed (see Figure 6). For this kind of unit cell, it has been demonstrated in [72] that  
\[X_{s}=\frac{\eta_{0}k_{0}p}{2\pi}\ln\left(\csc\frac{\pi w}{2p}\right) \tag{17}\]  
In Figure 6, for a fixed working frequency and periodicity, the equivalent reactance computed through full-wave simulations and theoretical formulas for different \(w/p\) filling factors are in a perfect agreement, thus corroborating the proposed approach.