# Parametric Testing of Metasurface Stirrers for Metasurface Reverberation Chambers
## 1 Introduction
### Analysis of Simulation Results  
To evaluate whether a stirrer provides independent field conditions for the different rotation positions, the correlation coefficient is the key parameter. It is assumed that uncorrelated stirrer positions yield independent field conditions [15]. Thus, the correlation coefficient between different three-dimensional far-field patterns of the metasurface corresponding to different rotation angles, as introduced in [11], was calculated. The correlation coefficients with the variations of the rotation angle from 0\({}^{\circ}\) to 45\({}^{\circ}\) in steps of 3\({}^{\circ}\) are shown in Figure 4. In the diagram, it can be observed that the correlation coefficients of stirrers #1 and #3 decrease sharply within 10 degrees of the rotation angle. This decrease in the correlation coefficient is indicative of the different electric field patterns between the different angles (due to changing boundary conditions), giving rise to an effective improvement of field uniformity. The TSCS reveals an interesting criterion to quantify the target scattering ability. In the EMC regard, the TSCS of an RC provides an indication of the capability of the chamber to generate statistical  
Figure 4: Correlation coefficients of the metallic and metasurfaces stirrers at 1 GHz. Figure 3: Simulated far-field pattern of the metallic stirrer (**a**) and metasurfaces stirrers (**b**) #1, (**c**) #2, and (**d**) #3 under normal incidence at 1 GHz (excitation with a plane wave is considered). field uniformity, through stirring efficiency [16; 17; 18; 19; 20]. To compute the TSCS in a reverberant environment, the commercial software CST Microwave Studio(r) (CST China Ltd. Nanjing, China) was employed. Nevertheless, the theoretical foundations of TSCS are given as follows. The expression giving the mean value of the E-field \(E_{\eta}(t)\) (\(\eta=x\), \(y\), \(z\): Cartesian component) over the sources \(\alpha\) (in our case, \(\alpha=1\)), probes \(\beta\) (in our case, \(\beta=8\)) and over the various object locations \(\gamma\) (in our case, \(\gamma=1\)) at time \(t\), is as follows [18; 21]:  
\[\left\langle\left\langle E_{\eta}(t)\right\rangle_{\gamma}^{2}\right\rangle_{ \alpha,\beta}=\left\langle\left\langle E_{\eta}(t=0)\right\rangle_{\gamma}^{ 2}\right\rangle_{\alpha,\beta}e^{-t/\tau_{\rm s}}. \tag{1}\]  
The TSCS can be inferred from the scattering damping time \(\tau_{\rm s}\):  
\[TSCS=V/N\tau_{\rm s}c, \tag{2}\]  
where \(V\) is the volume of the chamber, \(N\) is the number of moving objects (in our case, \(N=1\)) and \(c\) is the speed of light in vacuum. After rewriting (1), we can define the ratio \(C(t)\) as follows:  
\[C(t)=\frac{\left\langle\left\langle E_{\eta}(t)\right\rangle_{\alpha,\beta}^{ 2}\right\rangle_{\alpha,\beta}}{\left\langle\left\langle E_{\eta}^{2}(t=0) \right\rangle\right\rangle_{\alpha,\beta,\gamma}}=e^{-t/\tau_{\rm s}}. \tag{3}\]  
Therefore, the scattering damping time \(\tau_{\rm s}\) can be linearly derived from relations (1) and (3) according to:  
\[\tau_{\rm s}=-\frac{t}{\ln[C(t)]}, \tag{4}\]  
where \(C(t)\) is inferred through the CST simulation. In Figure 5, we plotted \(\ln C(t)\) corresponding to the four stirrers (#1, #2, #3, and the metallic stirrer). After the estimation of the damping time (\(\tau_{s}\)), the TSCS was computed for the four stirrers (TSCS is inversely proportional to \(\tau_{s}\), according to Expression 2). Note that for a high Q cavity, waves outperform many bounces within the cavity before they decay. Thus, the higher the Q value, the longer the \(\tau_{s}\)[22]. In other words, small \(\tau_{s}\) is associated with significant stirring. From the results of Figure 5, it can be clearly seen that \(\ln C(t)\) of stirrers #1 and #3 reduces sharply, as compared to the metallic stirrer and stirrer #2. In other words, the stirrers #1 and #3 exhibited better TSCS and stirring efficiency.