# Influence of Dielectric Plate Parameters on the Reflection Coefficient of a Planar Aperture Antenna
## 2 Analysis of the Parameters of the Dielectric Plate on the Antenna Aperture
### The Thickness and Width \(\times\) Length of the Antenna Aperture Dielectric Plate
#### 2.1.1 Propagation and Reflection of a Plane Wave at the Boundary of Two Mediums  
A simplified analysis of this problem uses a plane wave. In the case of a plane wave, the wave that arrives at the boundary of the two mediums consists of the electric and magnetic field strength \(E\) and \(H\). These two fields are perpendicular to each other and, at the same time, perpendicular to the propagation plane of the total EM wave. The wave that arrives at the boundary is called an incident wave (\(E_{\mathrm{i}}\) and \(H_{\mathrm{i}}\)), whereas the wave that passes the boundary under certain conditions is called the transmitted wave (\(E_{\mathrm{t}}\) and \(H_{\mathrm{t}}\)). At the same time, part of the wave is reflected at the boundary and is called a reflected wave (\(E_{\mathrm{r}}\) and \(H_{\mathrm{r}}\)). A simplified situation is shown in Figure 1.  
In such a case, the reflection coefficient S\({}_{11}\) is calculated according to the following relation:  
\[\mathrm{S}_{11}=\frac{E_{\mathrm{r}}}{E_{\mathrm{i}}}=\frac{Z_{\mathrm{c2}}-Z _{\mathrm{c1}}}{Z_{\mathrm{c2}}+Z_{\mathrm{c1}}}=\frac{Z_{\mathrm{c2}}-Z_{ \mathrm{0}}}{Z_{\mathrm{c2}}+Z_{\mathrm{0}}}, \tag{1}\]  
where:  
E\({}_{\mathrm{r}}\)--the electric field strength of the reflected wave in V/m;  
E\({}_{\mathrm{i}}\)--the electric field strength of the incident wave in V/m;\(Z_{\rm c1}=Z_{\rm c3}\)--characteristic impedance of the first and the third medium--the impedance of air in the waveguide is equal to the impedance of free space \(Z_{0}\) in \(\Omega\);  
\(Z_{\rm c2}\)--characteristic impedance of the second medium--dielectric plate, also in (Figure 1). The characteristic impedance of the medium through which the EM wave is propagated can be determined as follows:  
\[Z_{\rm ci}=\sqrt{\frac{\mu_{\rm i}}{\epsilon_{\rm i}}},\,{\rm i}=1,2, \tag{2}\]  
where:  
\(\mu_{\rm i}\)--the i-th medium permeability;  
\(\epsilon_{\rm i}\)--the i-th medium permittivity. If it is assumed that mediums 1 and 2 (dielectrics) have the same relative permeability value, then the reflection coefficient at the first boundary can be calculated according to the equation:  
\[|\Gamma_{1}|=|S_{11}|=\frac{\sqrt{\epsilon_{1}}-\sqrt{\epsilon_{2}}}{\sqrt{ \epsilon_{1}}+\sqrt{\epsilon_{2}}};\,\,{\rm and}\,\,{\rm for}\,\,\epsilon_{1} =\,\,\epsilon_{0},|\Gamma_{1}|=|S_{11}|=\frac{1-\sqrt{\epsilon_{\rm r}}}{1+ \sqrt{\epsilon_{\rm r}}}. \tag{3}\]  
For the purpose of this research, there are two boundaries, the first is air-dielectric plate, and the second is dielectric plate-air (Figure 2a). The one-boundary approach is applicable to the case of multiple boundaries and to each of the boundaries individually. For example, for the case of two boundaries: (a) for the first boundary \(\epsilon_{1}=\epsilon_{0}\) (the permittivity of the air in the waveguide), and \(\epsilon_{2}\) is the dielectric constant of the dielectric plate; (b) for the second boundary, conversely, \(\epsilon_{1}\) is the permittivity of the dielectric plate, and \(\epsilon_{2}=\epsilon_{0}\). In the geometry of the two boundaries (the space inside the waveguide--dielectric plate and the dielectric plate-free space), the electric field strength \(E\) (incident, transmitted, and reflected) in each of the mentioned mediums is presented in detail in Figure 2a. Figure 1: Incident and reflected waves in medium 1 and the transmitted wave in medium 2. Figure 2a shows that part of the incident wave \(\mathrm{E_{I}}\) passes the boundary between two mediums (\(\mathrm{E_{t}}\)), whereas part of the wave is reflected (\(\mathrm{E_{r}}\)). In the problem of the EM wave arriving at two boundaries, the reflected wave in the first medium will consist of the reflected wave at the first boundary and all the waves that are transmitted into the first medium after multiple reflections between the two boundaries (within the dielectric plate).