# 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
The total reflection coefficient \(\mathrm{S_{11}}\) for two mediums is (Figure 2b) as follows:  
\[\mathrm{S_{11}}=\frac{\mathrm{E_{r}}}{\mathrm{E_{i}}}=\frac{\Gamma_{1}\left(1- \mathrm{e}^{\mathrm{j}\mathrm{2k_{2}}\mathrm{d}}\right)}{1-\Gamma_{1}^{2} \mathrm{e}^{\mathrm{j}\mathrm{2k_{2}}\mathrm{d}}}. \tag{4}\]  
The wave number of the second medium can be written as:  
\[\mathrm{k_{2}}=\sqrt{\epsilon_{r}}\mathrm{k_{1}}, \tag{5}\]  
where:  
\(\Gamma_{1}\)--the first medium reflection coefficient;  
\(\mathrm{S_{11}}\)--the total reflection coefficient;  
\(\mathrm{d--}\)dielectric plate thickness;  
\(\mathrm{k_{1}}\)--the wave number of the first medium. In the labels referring to electric and magnetic field strengths in Figure 2a, the numerical value of the index indicates the medium in which the fields propagate, whereas the numerical value of the exponent of these fields indicates the order of the boundary at which reflection or transmission occurs. Letters \(i\), \(r\), and \(t\) in the index indicate the incident, reflected, and transmitted wave, respectively. A minus sign represents a reflected wave propagating from boundary 2 to boundary 1, whereas a plus sign represents the opposite direction. The total reflected wave is obtained by superposition of all waves coming from the direction of the first boundary into the first medium. These are the first reflected wave at the first boundary and all transmitted waves from the second medium (dielectric plate) to the first medium (air) (Figure 2a). The total transmitted wave is obtained by superposition of all waves passing the second boundary from medium 2 to medium 3 (Figure 2a). Figure 2: Propagation of an EM wave through a dielectric plate of thickness d (one dielectric layer): (**a**) existing electric field strengths (incident, transmitted, and reflected); (**b**) the absolute value of the reflection and transmission coefficient \(\mathrm{S_{11}}\). 1.2 Propagation and Reflection of an EM Wave Generated in a Circular Cross-Section Waveguide at the Boundary of Two Mediums  
Due to the simplicity of the construction, the microwave antenna with a circular opening is very widely used (Figure 3). A circular aperture antenna is most often excited by a circular cross-section waveguide. Waveguide modes are generated in a circular waveguide excited by a quarter-wave monopole antenna (probe). The mode with the lowest cut frequency is the dominant mode. In the case of a circular waveguide, it is mode TE\({}_{11}\). Figure 4 shows the \(E\) and \(H\) field lines of the dominant mode of a circular cross-section waveguide The straight line of the arrow indicates the strength of the electric field \(E\). The dashed arrow indicates the strength of the magnetic field \(H\) (the \(H\)-field is perpendicular to the \(E\)-field). The signs \(\times\) and dot indicate the direction of the \(H\)-field--inwards and outwards respectively. Figure 4: Lines \(E\) and \(H\) of the dominant mode (TE\({}_{11}\)) of waveguides with a circular cross-section. Figure 3: Dominant TE\({}_{11}\) mode distribution aperture on the grounded plane. The tangential component of the TE\({}_{11}\) mode electric field inside the waveguide can be expressed by the equation:  
\[\mathrm{S}_{11}=\frac{E_{\mathrm{r}}}{E_{\mathrm{i}}}=\frac{Z_{\mathrm{c2}}-Z _{\mathrm{c1}}}{Z_{\mathrm{c2}}+Z_{\mathrm{c1}}}=\frac{Z_{\mathrm{c2}}-Z_{ \mathrm{0}}}{Z_{\mathrm{c2}}+Z_{\mathrm{0}}}, \tag{1}\]0  
where \(\hat{a}_{\rho}\) and \(\hat{a}_{\varphi}\). are unit direction vectors, and electric field components \(\mathrm{E}_{\rho}\) and \(\mathrm{E}_{\varphi}\).