## 5 Train-to-Ground Communication Architectures with Leaky Waveguide
### On-Train Leaky Lens
#### 5.2.1. Refracted Surface Equation  
As is shown in Figure 5, there are two surfaces in the leaky lens: (a) inner-surface and (b) outer-surface. Assume that the leaky lens is a single-surface refracting lens, which means that only one surface could refract the leaky wave. The design for the single-surface refracting lens could be classified into two types in terms of the shape of the refracted surface:  
1. Hyperboloid lens: As is shown in Figure 5a, the outer-surface is a plane perpendicular to \(x\)-axis, and thus there is no refraction for the wireless wave in this surface. Therefore, the inner-surface is the refracted surface, and its analytic expression could be deduced based on the optical theories. Assume that \(n=\sqrt{\varepsilon_{r}l_{r}}\) is the refractive index, where \(\varepsilon_{r}\) and \(\mu_{r}\) are relative dielectric constant and relative magnetic permeability of the lens, respectively. According to the equal optical path principle, the optical path of the leaky wave from leaky waveguide to the inner-surface can be expressed as \[\rho\left(\varphi\right)=\frac{\left(n-1\right)f}{n\cos\left(\varphi\right)}\] (3) where \[\rho\left(\varphi\right)=\sqrt{\left(f+x\right)^{2}+y^{2}}\] (4) \[\cos\left(\varphi\right)=\frac{f+x}{\sqrt{\left(f+x\right)^{2}+y^{2}}}\] (5) Combine Equations (3)-(5), the analytic expression of the inner-surface can be expressed as \[\left(n^{2}-1\right)x^{2}+2f\left(n-1\right)x-y^{2}=0\] (6) According to Equation (6), we can see that the inner-surface is a hyperboloid, and the leaky waveguide (radiation sources) needs to be located at the focus or focus plane of the hyperboloid. The hyperboloid lens has the advantages of simple design, easy processing, as well as the effective suppression for sidelobe.
2. Ellipsoid lens: As is shown in Figure 5b, the inner-surface is a sphere with the radius \(r\) of \(\rho_{1}\), and it is parallel to the equiphase surface of the leaky wave, which means that there is no refraction in this surface. Similar to the hyperboloid lens, the analytic expression of the outer-surface (refracted surface)  
Figure 5: Two types of leaky lens: (**a**) Hyperboloid lens and (**b**) ellipsoid lens.  
could be deduced based on the optical theories. According to the equal optical path principle, the optical path of the leaky wave from leaky waveguide to the outer-surface can be expressed as  
\[\rho\left(\varphi\right)=\frac{\left(n-1\right)f}{n-\cos\left(\varphi\right)} \tag{7}\]  
where  
\[\rho\left(\varphi\right)=\sqrt{\left(f+x\right)^{2}+y^{2}} \tag{8}\]  
\[\cos\left(\varphi\right)=\frac{f+x}{\sqrt{\left(f+x\right)^{2}+y^{2}}} \tag{9}\]  
Combine Equations (7)-(9), the analytic expression of the inner-surface can be expressed as  
\[\left(n^{2}-1\right)x^{2}+2n\left(n+1\right)fx+n^{2}y^{2}=0 \tag{10}\]  
According to Equation (10), we can learn that the outer-surface is an ellipsoid, and the leaky waveguide (radiation sources) needs to be located at the focus or focus plane of the ellipsoid. Compared to the hyperboloid lens, the caliber efficiency of ellipsoid lens is higher. Thus, the acceptable angle range of the incident leaky wave is larger, which implies the more even field distribution of the generated wave, the higher radiation efficiency, and then the higher system efficiency. However, there are two surfaces that need to be proceed for ellipsoid lens, which implies a large manufacturing cost.