# Optimization of Multilayer Absorbers Using the Bald Eagle Optimization Algorithm
## 2 Materials and Methods  
### Physical Model and Chew Approach of Multilayer Microwave Absorbers  
With the development of stealth technology, absorber materials have been used to make them invisible. Absorbers are mounted on the surface of objects to reduce the scattered electromagnetic waves. Single or multilayer absorbers are used for this purpose. In this study, an absorber of \(N\) planar layers backed with a perfect electric conductor (PEC) surface is considered (Figure 1). All layers of the MMA are supposed to be infinite and each layer is characterized by a different thickness and frequency-dependent magnetic/electrical properties. For oblique incidences, the first layer at the interface of the multilayer material structure receives the incoming electromagnetic wave at a certain angle with the normal incidence. As the wave travels through successive layers of the multilayer material structure, some of the wave's energy is absorbed and multiple reflections occur between layers. The PEC (layer \(N+1\)), which offers a perfect reflection, is the last point of reflection for the wave.  
The generalized RC between the successive two layers of such a multilayer material structure is given by the recursive formula in Equation (1) given by Transmission Line Theory [10; 15; 16]. The multilayer material structure's overall reflection loss is determined by calculating Equation (1) from the last layer to the first, iteratively.  
\[R_{i,i+1}=\frac{r_{i,i+1}+R_{i+1,i+2}e^{-2\beta k_{i+1}d_{i+1}}}{1+r_{i+1}R_{i +1,i+2}e^{-2\beta k_{i+1}d_{i+1}}} \tag{1}\]  
Figure 1: Illustration of the microwave absorber structure.  
In Equation (1), for TM or Parallel polarization mode where the magnetic field is transverse to the direction of propagation of the wave, while the electric field is normal to this direction of propagation:  
\[r_{i,i+1}=\frac{\varepsilon_{i+1}k_{i}-\varepsilon_{i}k_{i+1}}{\varepsilon_{i+1} k_{i}+\varepsilon_{i}k_{i+1}}\quad,\;\;i<N \tag{2}\]  
When the incidence wave is in TE or Vertical polarization mode where the electric field is transverse to the wave propagation direction, the magnetic field is normal to this propagation direction:  
\[r_{i,i+1}=\frac{\mu_{i+1}k_{i}-\mu_{i}k_{i+1}}{\mu_{i+1}k_{i}+\mu_{i}k_{i+1}} \quad,\;\;i<N \tag{3}\]  
Here, \(\varepsilon_{i}\) and \(\mu_{i}\) are the frequency-dependent electrical permittivity and magnetic permeability of the \(i\)th layer of the multilayer material structure, respectively. \(k_{i}\) is the wave number of the \(i\)th layer, whose formulation is given by Snell's Law (Equation (4)).  
\[k_{i}=\omega\sqrt{\varepsilon_{i}\mu_{i}-\varepsilon_{0}\mu_{0}\text{sin}^{2 }\theta} \tag{4}\]  
The frequency of the incident wave and the values of the free space magnetic permeability (\(\mu_{0}=4\pi 10^{-7}\) [H/m]) and electrical permittivity (\(\varepsilon_{0}=10^{-9}/36\pi\) [H/m]) are used in Equation (4). The \(\theta\) in Equation (4) is for the angle of incidence which is illustrated in Figure 1.  
The RC of the final interface between the last layer of the multilayer material structure and the PEC (\(R_{N,N+1}\)) should be set as \(-1\) for TE polarization and \(+1\) for TM polarization. Calculating the initial interface's RC is important in determining the multilayer material structure's overall reflection loss. This is performed by calculating Equation (1) from the last layer to the first, recursively.