# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 3 Fabry-Perot Cavity Antennas
### Theoretical Background
In _lossless_ FPCAs, the radiation efficiency is given by [1; 18]  
\[\eta_{L}=1-e^{-\alpha L} \tag{1}\]  
A simple design equation for \(L\) is thus found for a desired value of the aperture efficiency (a typical minimum lower bound is set at \(\eta_{L}=90\%\)[16]):  
\[L=-\frac{\ln\left(1-\eta_{L}\right)}{\alpha} \tag{2}\]  
Thus, the importance of accurately evaluating the leaky attenuation constant \(\alpha\) for the correct design of the device is clear. Such a parameter not only affects the aperture efficiency, but also the beamwidth, the directivity, and the fractional bandwidth (FBW). On the other hand, the leaky phase constant \(\beta\) mainly determines the pointing angle of the device at a given frequency. (provided the attenuation constant is small, as commented next.) For this reason, in general, the correct evaluation of the leaky wavenumber \(k_{\rho}=\beta-ja\)--with \(\beta\) and \(\alpha\) being the leaky phase and attenuation constants, respectively--plays a fundamental role. For this reason, simple yet effective methods for the evaluation of \(k_{\rho}\) are reported both in the case of FPCAs and radially periodic LWAs in this work. While a Bloch analysis is needed for the latter [7], this procedure is addressed through the _transverse resonance technique_ in the former case, as discussed in detail in Section 3.2. This method requires the representation of the FPCA through a transverse equivalent network (see Figure 3b): the GDS is considered to be a transmission-line section whose dimension and characteristic impedance depend on the cavity height, the considered polarization, and the dielectric permittivity of the material (assuming \(\mu_{r}=1\) for simplicity); the PRS, if lossless (the lossy case is a natural evolution of this analysis and it is discussed in [71]), is commonly well represented through a purely imaginary impedance sheet \(Z_{s}=jX_{s}\)[72; 73] (see, e.g., the inset on the left in Figure 3a,b). In the leaky-wave perspective, the radiation features of the FPCAs are known once the leaky-wave wavenumber is correctly evaluated. By assuming that the feeder launches an omnidirectional cylindrical leaky wave with small attenuation constant \(\alpha\ll k_{0}\) (where \(k_{0}\) is the vacuum wavenumber), the latter provides the main contribution to the field on the antenna aperture plane at \(z=0\) (see Figure 3a) [5; 6]. These cylindrical leaky wavesare known in closed formulas (reported for the first time in [53]) and they can be used to evaluate the far-field radiation pattern through a simple Fourier transform, thus obtaining the following (assuming an infinite aperture):  
\[\text{SF}(\theta)=\frac{4|k_{\rho}|^{2}}{|k_{\rho}^{2}-k_{0}^{2}\sin^{2}\theta|^ {2}} \tag{3}\]  
where SF stands for space factor. From the far-field radiation pattern in (3), one can easily retrieve important design relations [53]. For instance, the maximum value of SF for different \(\theta\) angles is obtained by minimizing the denominator in (3). It is thus clear that the pointing angle for the FPCA is given by  
\[\theta_{0}=\arcsin\left[\sqrt{\left(\frac{\beta}{k_{0}}\right)^{2}-\left( \frac{\alpha}{k_{0}}\right)^{2}}\right] \tag{4}\]  
Since \(\alpha\ll k_{0}\) in order to have a dominant leaky-wave contribution to the leaky-wave aperture [5; 6], Equation (4) reveals that \(\beta\) mainly determines the pointing angle \(\theta_{0}\). Starting from Equation (3), one can also easily derive the half-power beamwidth (HPBW) \(\Delta\theta\) as [16]  
\[\Delta\theta=2\frac{\alpha}{k_{0}}\sec\theta_{0} \tag{5}\]  
Also, Equation (5) reveals an important result. The beamwidth, which is strictly related to the directivity, mainly depends on the attenuation constant \(\alpha\). The lower \(\alpha\) is, the narrower the beamwidth is and, in turn, the pattern peaks in the pointing-angle direction. Therefore, the more reflective the PRS is and, in turn, the lower \(\alpha\) is (since the PRS reactance controls the amount of power which leaks out), the more directive the antenna is.