# Reflection Characteristics Measurements of Indoor Wireless Link in D-Band
## 2 Principle of Reflection Measurements and Theoretical Model
### Reflection Theoretical Model  
For nonmagnetic materials, the Fresnel reflection coefficient of the parallel polarized wave is shown as (4),  
\[|\Gamma_{//}|=\left|\frac{\varepsilon_{r2}\cdot\cos\theta_{i}-\sqrt{\frac{ \varepsilon_{r2}}{\varepsilon_{r1}}-\sin^{2}\!\theta_{i}}}{\frac{\varepsilon_{ r2}}{\varepsilon_{r1}}\cdot\cos\theta_{i}+\sqrt{\frac{\varepsilon_{r2}}{ \varepsilon_{r1}}-\sin^{2}\!\theta_{i}}}\right|=\left|\frac{\varepsilon_{r} \cdot\cos\theta_{i}-\sqrt{\varepsilon_{r}-\sin^{2}\!\theta_{i}}}{\varepsilon _{r}\cdot\cos\theta_{i}+\sqrt{\varepsilon_{r}-\sin^{2}\!\theta_{i}}}\right|, \tag{4}\]  
where \(\theta_{i}\) is the incident angle, \(\varepsilon_{r1}\) and \(\varepsilon_{r2}\) denote the relative permittivity of incident medium and reflecting medium respectively, and \(\varepsilon_{r}\) is the ratio of \(\varepsilon_{r2}\) to \(\varepsilon_{r1}\). Generally, the incident medium is air (\(\varepsilon_{r1}=1\)), so \(\varepsilon_{r}\) is equal to \(\varepsilon_{r2}\).The reflection coefficient in (4) is just suitable for a completely smooth surface, because the specular reflection without scattering only occurs for the ideal surface. Actually, there is no completely smooth plane at all, so when the electromagnetic wave hits a rough surface, it will scatter at the other angle besides the reflection angle, thus causing the energy reduction in specular reflection. Therefore, (4) needs to be modified.  
Figure 1: Transmission in free space. (**a**) LoS propagation measurement, (**b**) Reflection measurement.  
The first step is to characterize the roughness of the surface using [PERSON]'s rule. [PERSON] criterion defines the critical height of the surface, which is determined by the wavelength and the incident angle, as shown in (5),  
\[h_{c}=\frac{\lambda}{8\cos\theta_{i}}. \tag{5}\]  
If the maximum height drop of the surface is greater than \(h_{c}\) at a certain wavelength and an incident angle, the surface of the object is considered rough. On the contrary, the surface of the object is smooth [35]. The effects of the scattering can be modeled by plenty of simulations based on the Maxwell boundary value problem [36]. Instead, there has been a simpler analytic approximation method applied to D-band scattering issue [20]. The previous descriptions have shown that the surface roughness is usually characterized by two parameters, namely root mean square height and correlation length [37]. Here, the root mean square height is defined as the standard deviation of the surface height from the average height, which is used to characterize the surface roughness in this paper, and the larger it is, the rougher the surface will be.  
In order to mitigate the energy loss caused by the scattering in specular reflection, the Rayleigh roughness factor from Beckman-Kirchhoff theory [38] is introduced, as illustrated in (6) [35; 39],  
\[\rho_{s}=\exp\left[-8\cdot\left(\frac{\pi h_{rms}\cos\theta_{i}}{\lambda} \right)^{2}\right]\cdot J_{0}\left[8\cdot\left(\frac{\pi h_{rms}\cos\theta_{i }}{\lambda}\right)\right], \tag{6}\]  
where \(\lambda\) is the wavelength, \(\theta_{i}\) is the incident angle, \(h_{rms}\) denotes the root mean square height, namely surface roughness, and \(J_{0}(\cdot)\) is the first type of zero-order Bessel function. Gaussian height distribution of the surface is assumed, and the sharp edge and shadowing can be neglected in this model. Therefore, the modified reflection coefficient is as follows:  
\[|\Gamma_{//}|=\rho_{s}\cdot\left|\frac{\varepsilon_{r}\cdot\cos\theta_{i}- \sqrt{\varepsilon_{r}-\sin^{2}\theta_{i}}}{\varepsilon_{r}\cdot\cos\theta_{i }+\sqrt{\varepsilon_{r}-\sin^{2}\theta_{i}}}\right|. \tag{7}\]