## 2 Method of Multi-Ray-Tracing in Tunnels  
The propagation of millimeter waves along a tunnel can be described using the ray-tracing approach [18]. Figure 1 illustrates the ray paths and the resulting reflections along the tunnel. To find the link performances, it is necessary to consider the line-of-sight (LOS) and all the reflected rays. For convenience, the received power is presented in a normalized form:  
\[\frac{P_{r}(Tunnel)}{P_{r}(LOS)}=\left|\sum_{m=-\infty}^{\infty}\sum_{n=- \infty}^{\infty}\frac{R_{0}}{R(m,n)}\cdot\Gamma^{|m|}\cdot\Gamma^{|n|}\cdot \frac{G(\theta,\phi)}{G_{LOS}}\cdot e^{-j2\pi f\Delta\tau(m,n)}\right|^{2} \tag{1}\]  
where \(f\) is the wave  
where a and b are defined as half of the tunnel width and height, respectively. The reflection coefficients of the incident electric transverse electric (TE) and transverse magnetic (TM) waves are found using [PERSON]'s equations [19]:  
\[ \begin{split}\Gamma_{TM}=\frac{\epsilon_{r}\sin(\theta_{g})-\sqrt{ \epsilon_{r}-\cos^{2}(\theta_{g})}}{\epsilon_{r}\sin(\theta_{g})+\sqrt{ \epsilon_{r}-\cos^{2}(\theta_{g})}}\\ \Gamma_{TE}=\frac{\sin(\theta_{g})-\sqrt{\epsilon_{r}-\cos^{2}( \theta_{g})}}{\sin(\theta_{g})+\sqrt{\epsilon_{r}-\cos^{2}(\theta_{g})}}\end{split}  \tag{5}\]  
where \(\epsilon_{r}\) is the relative dielectric permittivity of the surface and \(\theta_{g}\) is defined as the grazing angle between the incident ray to the surface (see Figure 1b):  
\[\theta_{g}(m,n)= \begin{cases}\arcsin\left(\frac{|x_{t}(m)-x_{r}|}{R(m,n)} \right),&\text{vertical surfaces}\\ \arcsin\left(\frac{|y_{t}(n)-y_{r}|}{R(m,n)}\right),&\text{horizontal surfaces}\end{cases}  \tag{6}\]  
Table 1 summarizes the reflection coefficient type according to the antenna polarization [20]. To examine the sensitivity of the model to variations in the reflectivity of the surfaces, it is important to evaluate the Fresnel coefficients for the dielectric permittivity of typical construction materials. Table 2 presents different construction materials and their relative dielectric constant \(\epsilon_{r}\) at 28 GHz [21; 22; 23]. Calculations were made to find reflection coefficients for transverse electric (TE) and transverse magnetic (TM) waves and for different materials. Figure 2 shows that the differences are minor and mainly expressed for high incident angles \(\theta_{g}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Antenna Polarization** & **Walls’ Polarization** & **Floor and Ceiling Polarization** \\ \hline Vertical & \(\Gamma_{TE}\) & \(\Gamma_{TM}\) \\ Horizontal & \(\Gamma_{TM}\) & \(\Gamma_{TE}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reflection coefficients for fundamental polarization. Figure 1: (**a**) Transmitter and receiver transverse coordinates inside the tunnel. (**b**) Longitudinal tunnel side view. Here, \(2a\) and \(2b\) are the width and height of the tunnel, respectively. The ray traveling distances \(R(m,n)\) can be expressed as follows:  
\[R(m,n)=R_{0}\sqrt{\left(\frac{a}{R_{0}}\right)^{2}[2m-1+(-1)^{m}]^{2}+\left(\frac {b}{R_{0}}\right)^{2}[2n-1+(-1)^{n}]^{2}+1} \tag{7}\]  
We define the following parameters:  
\[A_{m}=[2m-1+(-1)^{m}]^{2} \tag{8}\] \[B_{n}=[2n-1+(-1)^{n}]^{2}\]  
The above expression Equation (7) can be re-written using the above parameter definitions:  
\[R(m,n)=R_{0}\sqrt{\left(\frac{a}{R_{0}}\right)^{2}A_{m}+\left(\frac{b}{R_{0}} \right)^{2}B_{n}+1} \tag{9}\]  
Substitute Equation (9) into Equation (2), where \(\tau_{0}=R_{0}/c\) results in the incremental delays \(\Delta\tau(m,n)\) of the multi-path rays, which are given by the following:  
\[\frac{\Delta\tau(m,n)}{\tau_{0}}=\left[\sqrt{\left(\frac{a}{R_{0}}\right)^{2}A _{m}+\left(\frac{b}{R_{0}}\right)^{2}B_{n}+1}-1\right] \tag{10}\]  
Figure 3 illustrates graphs of the parameters \(A_{m}\), \(B_{n}\) and the resultant time delay of the multi-path rays as a function of their m\({}^{\prime}\)th and \(n^{\prime}\)th orders.