# Oblique Wide-Angle Multi-Sector Metamaterial Absorber for Space Applications
## 2 Design
### Optimization of Sector A  
A first optimization is conducted for one single absorber sector using the TL model and the genetic algorithm of Matlab. The optimization results are presented in Figure 3 and the corresponding values of the geometric parameters are provided in Table 1. The Matlab results are compared against full wave simulations (CST Microwave Studio, F solver with Floquet boundaries). There is a very good agreement, which validates the model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **p (mm)** & **g (mm)** & **w (mm)** & **Z (\(\Omega\))** \\ \hline TE (x) & 5.1 & 1.6 & 0.32 & 500 \\ TM (y) & 5 & 1 & 0.4 & 250 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical dimensions of sector A for each polarization. The optimization results presented in Figure 3 show that the desired absorption level (\(-13\) dB) is achieved only for a limited range of angle of incidence corresponding to \([35^{\circ},55^{\circ}]\) for both TE and TM polarizations. This limitation is due to two phenomena typical from oblique incidence:  
* The resonant frequency, for which the reflection is minimum, shifts toward high frequency for increasing angles. For TE (Figure 3a), we observed a shift from \(1.9\) GHz to \(2.5\) GHz for \(35^{\circ}\) and \(55^{\circ}\) respectively. * The absorption level varies with \(\theta\). For TE, we observe that the absorption ranges from \(-30\) dB to \(-15\) dB for \(35^{\circ}\) and \(55^{\circ}\), respectively. These two phenomena can be understood by inspecting the evolution of \(Y_{in}\) with respect to \(\theta\) at the fixed frequency of \(2.15\) GHz (Figure 4). As represented in Figure 4a, the imaginary part of \(Y_{in}\) varies with \(\theta\). At \(2.15\) GHz, we see that the dimensions of sector A are chosen such that the imaginary part of \(Y_{in}\) cancels out at \(45^{\circ}\) for both TE and TM polarizations. We know, from Equations (5) and (6), that \(\text{Im}(Y_{in})\) depends both on \(\theta\) and on frequency, which explains the scan behavior observed in Figure 3a. The different shapes of the curves observed for TE and TM polarizations in Figure 4a are mainly due to the last term of Equations (5) and (6), which is directly related to the stub (honeycomb + ground plane). For TE, the variation of the  
Figure 4: Imaginary (**a**) and real (**b**) part of \(Y_{in}/Y_{0}\) with respect to the angle of incidence at \(2.15\) GHz for sector A and both polarizations. Figure 3: Comparison between TL model and EM simulation of sector A for TE polarization (**a**) and TM polarization (**b**). The rectangle corresponds to an objective of \(-13\) dB in the band \([2\) GHz, \(2.3\) GHz]. admittance (\(Y_{0}\cdot\cos\theta\)) compensates the cotangent term such that \(\text{Im}(Y_{in})\) remains bounded for grazing incidence. This is not the case for TM, for which the variation rate increases with \(\theta\) (\(\Delta Y_{in}{}^{TM}=0.3Y_{0}\) for \([0^{\circ},\,35^{\circ}]\), \(0.7Y_{0}\) for \([35^{\circ},\,55^{\circ}]\), and \(0.9Y_{0}\) for \([55^{\circ},\,65^{\circ}]\)). This evolution illustrates the difficulty of designing wide-angle absorbers for high oblique incidence. The real part of \(Y_{in}\) (Figure 4b) is relatively constant with respect to \(\theta\) but needs to be matched with the admittance of the incoming wave, which varies according to Equations (1) and (2). This explains the second point, as when \(\theta\) varies, this directly affects the mismatch according to Equations (3) and (4), resulting in a lower absorption level. Interestingly, the cancelling out condition \(G=Y_{0}^{TE,\,TM}\) is not chosen at resonance (\(40^{\circ}\) for TE and \(48^{\circ}\) for TM instead of being at exactly \(45^{\circ}\)), which allows the widening of the frequency bandwidth at the expense of the minimum reflection level [20]. Similarly to what is observed for the imaginary part, the variation rate of \(Y_{0}^{TE}\) and \(Y_{0}^{TM}\) increases with \(\theta\).