## 2 Circular Cavity Metasurface Antenna with a Diameter of 0.25 \(\lambda_{0}\)
### Bandwidth Performance of Small Circular Cavity Antennas  
The transmission line (TL) models given above are very attractive to provide a qualitative explanation of the BW behaviors of patch antennas and metasurface antennas in cavities. As demonstrated in [8], the inductance is the blocking point, which does not permit the antennas to achieve the maximum achievable BW (according to the predictions based on [PERSON]'s bound [8; 13]). Nevertheless, this upper limit can be reached using a structure having parasitic capacitance only (instead of the patch). Moreover, it was demonstrated in [8] that the \(Q\) factor and BW predicted by the TL model are in good agreement with theoretical values for minimum \(Q\) (maximum BW), as shown in [13]. Hence, we concentrate here only on the predicted values of \(Q\) and BW using the TL model.  
These numerical results, obtained with the TL model and full-wave electromagnetic simulations using the commercial software CST Microwave Studio [14], are plotted in Figure 5 for a metasurface antenna design with a cavity diameter \(a=32\) mm (\(0.25\lambda_{0}\)). The center frequency is chosen to be 2300 MHz (\(\lambda_{0}=130\) mm).  
Figure 6 shows that, for the considered cavity diameter \(a\), the BW increases with the cavity height \(h_{c}\) up to a given value, after which it stays quasi-constant. These theoretical results have been validated by full-wave simulations of metasurfaces using the CST software [14]. The maximum achievable BW is 77 MHz for \(h_{c}=20.0\) mm. Nevertheless, for a smaller value of \(h_{c}\) (15.0 mm), the BW (75 MHz) is quite close to this maximum. It is to be noted that these simulations are carried out considering ideal materials without any losses (\(g_{t}=0\)) in order to compare the full-wave results to the ones from the TL models, and also to assess the theoretical maximum achievable BW induced by the cavity resonance. In practice, losses would result in widening the BW but decreasing the antenna efficiency.  
Figure 6: Maximum achievable bandwidth (BW) calculated with the transmission line (TL) model and simulated BW (with CST software) for three reference patches (with three different dielectric substrates) and for a metasurface within cavities of diameter \(a=32\) mm (no glue). The solid blue line displays the maximum achievable BW computed with the TL model [8; 12; 13].  
For comparison purposes, the BW of standard circular patch antennas (Figure 3) embedded in a circular cavity of the same diameter (\(a=32\) mm) has been also reported in Figure 6 for three permittivity values. These results have been obtained using CST Microwave Studio [14], and no glue is considered. Comparison between the results plotted in Figure 6 shows that the BW achieved with patch antennas is narrower than with metasurfaces: 64 MHz for reference patches printed on \(\varepsilon_{\mathrm{r}}=6.15\) (\(h_{\mathrm{c}}=12.0\) mm) or on \(\varepsilon_{\mathrm{r}}=10.2\) (\(h_{\mathrm{c}}=12.0\) mm), compared to 77 MHz (\(h_{\mathrm{c}}=20.0\) mm) or 75 MHz (\(h_{\mathrm{c}}=15.0\) mm) for the metasurface. Furthermore, it has to be noted that for the patch case, the maximum of BW is not achieved with the lowest permittivity material, and even that the BW is similar for \(\varepsilon_{\mathrm{r}}=6.15\) and \(\varepsilon_{\mathrm{r}}=10.2\) for a thickness of \(h_{\mathrm{c}}=12.0\) mm. Increasing the thickness further does not result in significantly enlarging the BW.  
To be more exhaustive in the comparison, a patch antenna in a cavity of diameter \(a=32\) mm and thickness \(h_{\mathrm{c}}=12.0\) mm was also considered on a dual-layer substrate: 10.1 mm of air (\(\varepsilon_{\mathrm{r}}=1\)) and 1.9 mm of RO3010 (\(\varepsilon_{\mathrm{r}}=10.2\)), as is the case for the metasurface. The cross-section view is given in Figure 7. After optimization, the BW was 65 MHz, demonstrating that the achievable BW with a patch is narrower than with the metasurface in a small cavity. The above conclusions are in agreement with the one derived for rectangular cavities [8; 12].