## 2 Materials and Methods  
### Silicon Lens for Eliminating Thick Substrate Modes  
THz antenna on thick dielectric suffers power loss because of substrate modes, shown in Figure 1a. More than 90% of the radiated power is trapped in the dielectric [25]. If the substrate thickness can be made infinite, the loss in the substrate mode can be eliminated. Additionally, the infinite dielectric can be synthesized by a lens with the same dielectric constant attached to the antenna substrate [26]. The substrate lens can be hemispherical, hyper-hemispherical and elliptical, seen in Figure 1b-d. As hyper-hemispherical lens [27; 28] is widely used and easier to design, we focus on the hyper-hemispherical lens in our study. If a dielectric hemisphere is attached to the back of the substrate with the same relative permittivity as the interface between them, there will be no refraction or reflection. All radiated rays will directly strike the circular surface of the hemisphere without any disturbance, and there will be only loss due to the reflection loss [29]. This type of lens is called hemispherical lens, shown in Figure 1b. By introducing a matching layer of quarter wavelength, this loss can be eliminated or minimized, the layer is called anti-reflection (AR) coating. If the hemisphere lens is extended by adding a cylinder with the same diameter and attached with the antenna then it is called the hyper-hemispherical lens, shown in Figure 1c. The thickness of this cylinder is called the extension length of the lens. As the superconducting YBCO film was deposited on a bicrystal magnesium oxide (MgO) substrate with the dielectric constant of 9.6, the high-resistivity, float-zone (HRFZ) silicon (Si), whose dielectric constant of 11.9 is close to that of MgO, is chosen to prepare the hyper-hemispherical lens. Furthermore, the loss tangent of HRFZ Si can be ignored in THz band. \[\tan\delta=\frac{1}{\omega\cdot\varepsilon_{\mathrm{v}}\cdot\varepsilon_{ \mathrm{r}}\cdot R} \tag{1}\]  
Loss tangent is calculated according to the Formula (1) [30], where \(\omega\) is the circular frequency, \(\varepsilon_{\mathrm{v}}\) is the dielectric constant of vacuum (8.85 \(\times\) 10\({}^{-12}\) F/m), \(\varepsilon_{\mathrm{r}}\) is the relative permittivity of Si (11.9), and \(R\) is the specific resistance. For example, loss tangent of HRFZ Si with a resistivity of 10 k\(\Omega\) cm at 1 THz is 1.5 \(\times\) 10\({}^{-5}\). \[L_{\mathrm{h}}=R_{\mathrm{L}}/n_{\mathrm{r}} \tag{2}\]  
The hyper-hemispherical lens can be designed by Equation (2) [31], where \(L_{\mathrm{h}}\) is the length of the extension, \(R_{\mathrm{L}}\) is the radius of the hemisphere and \(n_{\mathrm{r}}\) is the refractive index of the lens, which is equal to \(\sqrt{\varepsilon_{\mathrm{r}}}\). In the literature [26], by changing the diameter of the hyper-hemispherical lens from 4 mm to 9 mm, the directivity increases with the increase in diameter in the low-frequency band of THz. Thus, we make the radius of the hyper-hemispherical lens of 4.5 mm. According to Equation (2), the extension length of the lens should be 1.3 mm. However, due to the thickness of the integrated chip being 0.5 mm or 0.6 mm, the extended length of the customized hyper-hemispherical lens is 0.7 mm. \[n_{1}=\sqrt{n_{2}\cdot n_{0}} \tag{3}\]  
AR coating of the substrate lens should have a thickness of a quarter wavelength (\(\lambda_{\mathrm{air}}/4\)), and the refractive index of the coating material meets the Formula (3) [26], where \(n_{1}\), \(n_{2}\) and \(n_{0}\) are the refractive indexes of AR coating, Si and the air, respectively. Thus, the thickness of the AR coating is 357 \(\mu\)m at 0.21 THz, and \(n_{1}\) equals to 1.86.