# Building Penetration Losses at 3.5 GHz: Dependence on Polarization and Incidence Angle
## 2 Theoretical Background
### Uniform Plane Wave Incident on A Semi-Infinite Medium  
Consider a uniform plane wave propagating in free space that impinges on a planar obstacle, as seen in Figure 1.  
Figure 1: Plane wave incident on a planar boundary: (**a**) with electric field perpendicular to the incidence plane; (**b**) with electric field parallel to the incidence plane. The surface of this obstacle represents a planar boundary of an infinite extent. As a consequence of this discontinuity on the propagating medium, reflected and transmitted waves emerge. The electric and magnetic field of the three waves at the boundary should satisfy the boundary conditions derived from [PERSON]'s equations. From these boundary conditions, the ratio of the amplitudes of the reflected and transmitted electric fields, relative to the amplitude of the incident electric field, known as reflection (\(\Gamma\)) and transmission (T) coefficients, respectively, can be calculated. The expressions for these coefficients are also known as [PERSON]'s equations. Both reflection and transmission coefficients depend on the polarization of the incident field relative to the incidence plane, which is the plane that contains the direction of propagation of the electromagnetic wave and a vector normal to the discontinuity surface (\(\overrightarrow{n}\)). In Figure 1, the incidence plane is the plane of the paper sheet. We assume that the first medium corresponds to free space, while the second one is a good dielectric, with a relative dielectric constant \(\varepsilon_{\mathrm{r}}\) and a conductivity \(\sigma\). This assumption means that:  
\[\left(\frac{\sigma}{2\pi\,\mathrm{f}\,\varepsilon_{\mathrm{r}}\,\varepsilon_{ 0}}\right)^{2}\ll\ 1 \tag{3}\]  
where f is the frequency and \(\varepsilon_{0}\) the permittivity of the vacuum. If the electric field is polarized perpendicular to the incidence plane, as in Figure 1a, [PERSON]'s coefficients are [21]:  
\[\Gamma_{\mathrm{perpendicular}}\ =\ \frac{\cos(\theta_{\mathrm{incident}})-\sqrt{ \varepsilon_{\mathrm{r}}-\sin^{2}(\theta_{\mathrm{incident}})}}{\cos( \theta_{\mathrm{incident}})+\sqrt{\varepsilon_{\mathrm{r}}-\sin^{2}( \theta_{\mathrm{incident}})}} \tag{4}\]  
\[\Gamma_{\mathrm{perpendicular}}\ =\ \frac{2\ \cos(\theta_{\mathrm{incident}})}{\cos( \theta_{\mathrm{incident}})\ +\sqrt{\varepsilon_{\mathrm{r}}-\sin^{2}( \theta_{\mathrm{incident}})}} \tag{5}\]  
while if it is polarised parallel to the incidence plane, as in Figure 1b, the coefficients are:  
\[\Gamma_{\mathrm{parallel}}\ =\ \frac{-\varepsilon_{\mathrm{r}}\cos(\theta_{ \mathrm{incident}})+\sqrt{\varepsilon_{\mathrm{r}}-\sin^{2}(\theta_{ \mathrm{incident}})}}{\varepsilon_{\mathrm{r}}\cos(\theta_{\mathrm{incident}})+ \sqrt{\varepsilon_{\mathrm{r}}-\sin^{2}(\theta_{\mathrm{incident}})}} \tag{6}\]  
\[\Gamma_{\mathrm{parallel}}\ =\ \frac{2\sqrt{\varepsilon_{\mathrm{r}}}\cos(\theta_{ \mathrm{incident}})}{\varepsilon_{\mathrm{r}}\cos(\theta_{\mathrm{incident}})\ +\sqrt{\varepsilon_{\mathrm{r}}-\sin^{2}(\theta_{\mathrm{incident}})}} \tag{7}\]  
where \(\theta_{\mathrm{incident}}\) is the incidence angle, as depicted in Figure 1.  
The attenuation, in dB, may be calculated as:  
\[\mathrm{A}=-10\cdot\log_{10}(1-|\Gamma|^{2}) \tag{8}\]  
In Figure 2, we plotted the attenuation as a function of the incidence angle for obstacles made of concrete (\(\varepsilon_{\mathrm{r}}\)= 5.31, \(\sigma=0.089\) S/m), brick (\(\varepsilon_{\mathrm{r}}\)= 3.75, \(\sigma=0.038\) S/m) and glass (\(\varepsilon_{\mathrm{r}}\)= 6.27, \(\sigma=0.02\) S/m) as an illustrative example to observe the clear theoretical differences due to the polarization and incidence angle. The electromagnetic properties of these materials were calculated from [7] for a frequency of 3.5 GHz. As can be seen in Figure 2, the values of the attenuation vary with the polarization of the electric field as well as with the incidence angle of the electromagnetic wave, even when the electric field is small.