# A Clutter Loss Model for Satellite Communication Systems
## 4 Reflection and Diffraction Loss  
Although the concept of plane waves is largely theoretical, as the wave is associated with plane fronts that extend indefinitely orthogonally to the propagation direction, in practice, such a wave represents a satisfactory approximation of the reality when the observation point lies in the far-field region. Under these assumptions, the reflection and the diffraction (i.e., transmission) phenomena of a plane wave depend on the polarization, on the incident angle and on the electromagnetic features of the two materials separated by a planar interface.  
Referring to Figure 4, let us consider a plane wave impinging on the discontinuity between two homogeneous and isotropic media with different electromagnetic properties (electric permittivity \(\varepsilon\), magnetic permeability \(\mu\) and conductivity \(\sigma\)). The electric \(\overset{\rightarrow}{(E)}\) and magnetic \(\overset{\rightarrow}{(H)}\) fields of the incident wave are identified by the subscript \(i\), while those corresponding to the transmitted (refracted) and reflected waves are identified by subscripts \(t\) and \(r\), respectively. Finally, \(\overset{\rightarrow}{k}\) represents the propagation vectors, all lying on the same (\(x\), \(z\)) plane. The reflection and transmission mechanisms differ for the two orthogonal polarizations of the incident waves, i.e., the transverse electric (TE) component, for which \(\overset{\rightarrow}{E}_{i}\) is perpendicular to the (\(x\), \(z\)) plane, and the transverse magnetic (TM) component, for which \(\overset{\rightarrow}{E}_{i}\) lies on the (\(x\), \(z\)) plane.  
For both cases, the E-field reflection coefficient \(\Gamma\) is defined as the ratio of the reflected electric field to the corresponding incident electric field at the interface and is given by [13]  
\[\Gamma^{TE}=\frac{E_{r}^{TE}}{E_{i}^{TE}}=\frac{\eta_{2}\cos(\theta_{1})-\eta_ {1}\cos(\theta_{2})}{\eta_{2}\cos(\theta_{1})+\eta_{1}\cos(\theta_{2})}, \tag{1}\]  
\[\Gamma^{TM}=\frac{E_{r,X}^{TM}}{E_{i,X}^{TM}}=\frac{-\eta_{1}\cos(\theta_{1})+ \eta_{2}\cos(\theta_{2})}{\eta_{1}\cos(\theta_{1})+\eta_{2}\cos(\theta_{2})}, \tag{2}\]  
where  
\(\eta_{i}=\sqrt{\frac{j\omega\mu_{i}}{\sigma_{i}+J\omega\varepsilon_{i}}}\),  
\(\sin(\theta_{2})=\frac{\gamma_{i}}{\gamma_{2}}\sin(\theta_{1})\),  
\(\gamma_{i}=\sqrt{-\omega^{2}\mu_{i}\varepsilon_{i}+j\omega\mu_{i}\sigma_{i}}\),  
\(\omega=2\pi f\),  
\(E_{i,X}^{TM}\) and \(E_{r,X}^{TM}\) are, respectively, the \(x\)-components of \(E_{i}^{TM}\) and \(E_{r}^{TM}\),\(f\) is the frequency (Hz), \(\theta_{1}\) is the incidence angle, \(\theta_{2}\) is the refraction angle calculated using [PERSON]'s law, \(\varepsilon_{i}\) is the electric permittivity, \(\mu_{i}\) is the magnetic permeability, \(\sigma_{i}\) is the conductivity, \(\gamma_{i}\) is the propagation constant and \(\eta_{i}\) is the intrinsic impedance of medium \(i\).  
For multiple slabs of different materials, the overall value of \(\Gamma\) results from the combination of the multiple reflections of the wave occurring within each slab [13].  
Figure 4: Reflection and transmission of a plane wave at an interface between two isotropic and homogeneous media with different electromagnetic characteristics.