# Oblique Wide-Angle Multi-Sector Metamaterial Absorber for Space Applications
## 2 Design
### Description of the Absorbers  
The base case absorber (Figure 2a) is composed of a resistive FSS mounted at a distance of approximately \(\lambda/4\) from a copper ground plane. They are separated by a low mass dielectric substrate (Nomex Honeycomb) and the FSS is covered with a thin dielectric layer (RO4003C), which serves both as a mechanical support and as protection. The FSS is a periodic structure (Figure 2b) comprising rectangular metallic patches connected by resistive strips made of a TCR foil of 100 \(\Omega/\Box\).  
An equivalent electric model (Figure 2c) is established for fast calculation. The dielectric layers are assimilated to transmission line sections and the FSS is described by its equivalent admittance [17] (\(G\), \(C\) parallel circuit). The conductance \(G\) is determined by the surface resistance \(R_{\mathrm{s}}\) of the TCR foil and the dimensions of the resistive strips. The shunt capacitance \(C\) is related to the periodicity \(p\), to the  
Figure 1: Angular range targeted for the present study. The platform is separated in two sectors for which the absorber can be optimized on a narrower range of incident angles.  
Figure 2: Geometry of the absorber (**a**) and of the rectangular unit cell of the frequency selective surface (FSS) (**b**). Dimensions along x and y directions are dedicated to TE and TM polarizations respectively. (**c**) Equivalent circuit valid for each polarization.  
gap \(g\), and to the dielectric permittivity of the surrounding layers. As each layer contribution is in parallel in the model (Figure 2c), it is convenient to work with admittance.  
For an oblique angle of incidence \(\theta\), the admittance of the incoming wave depends on the polarization as:  
\[Y_{0}^{TE}=Y_{0}\cdot\cos\theta \tag{1}\]  
\[Y_{0}^{TM}=Y_{0}/\cos\theta \tag{2}\]  
where \(Y_{0}\) is the free space admittance. From transmission line theory, the reflection coefficient at the interface \(\varGamma\) is given by  
\[\varGamma^{TE}=\frac{Y_{0}\cdot\cos\theta-Y_{in}}{Y_{0}\cdot\cos\theta+Y_{in}}, \tag{3}\]  
\[\varGamma^{TM}=\frac{Y_{0}/\cos\theta-Y_{in}}{Y_{0}/\cos\theta+Y_{in}}, \tag{4}\]  
where \(Y_{in}\) is the input admittance of the absorber. This leads to a seemingly contradictory problem when trying to minimize the reflection coefficient for both polarizations simultaneously. To tackle this problem, a practical solution proposed in [14] is to consider an FSS with different dimensions along \(x\) and \(y\) directions, each one dedicated to one polarization. This leads to a rectangular unit cell, as shown in Figure 2b. Note that for this approach the wave vector has to be comprised in the \(xz\) plane (i.e., \(\varphi=0\)).  
For optimization, the transmission line (TL) modeling approach [13; 14; 15; 16; 17] is used. It is however possible to derive a simple analytical model using two basic approximations. First, we assimilate the honeycomb (\(\varepsilon_{r}=1.08\)) to free space. Second, due to its small thickness (\(h_{1}=0.2\) mm), the admittance of the RO4003C dielectric sheet is regarded as a shunt capacitance for which the value for normal incidence is given [18] by \(C_{1}=Y_{0}\frac{(\varepsilon_{r}-1)\cdot h_{1}}{c}\). For oblique incidence, the effective value \(C_{1}\) also depends on incidence \(\theta\) and on polarization [18]. Considering the equivalent circuit of Figure 2c, \(Y_{in}\) is then given by  
\[Y_{in}{}^{TE}=G+\int\!\!\left[\omega C_{1}\cdot\cos\theta+\omega C\left(1- \varepsilon_{f}\cdot\sin 2\theta\right)-Y_{0}\cdot\cos\theta\cdot\cot(\beta \cdot h_{2}\cdot\cos\theta)\right]\!, \tag{5}\]  
\[Y_{in}{}^{TM}=G+j[\omega C_{1}/\cos\theta+\omega C-Y_{0}/\cos\theta\cdot\cot( \beta\cdot h_{2}\cdot\cos\theta)], \tag{6}\]  
where \(\varepsilon_{f}\) is the effective permittivity of the FSS calculated with the help of [19]. The maximum absorption is obtained when \(Y_{in}\) is matched with the admittance of the incoming wave Equations (3) and (4). This appends at the resonance [Im\((Y_{in})=0\)] when \(G\) equals \(Y_{0}^{TE,\ TM}\).