## 2 Materials and Methods  
The ANSYS High Frequency Simulation Software (HFSS) (Ansys, Canonsburg, PA, USA) was employed for glass antenna design and modelling. The structure of the proposed microstrip patch antenna is depicted in Figure 1, and it is a conventional sandwich-likestructure. The antenna is constructed using silicon-boron glass dielectric substrate, with its dielectric constant (\(\epsilon_{r}\)) and dielectric loss tangent (tangent \(\delta\)) equal to 6 and 0.002, respectively, at 10 GHz and 23 \({}^{\circ}\)C. The glass substrate has a size of Sub_L \(\times\) Sub_W \(\times\) Sub_h = 30 mm \(\times\) 30 mm \(\times\) 0.7 mm, and it is sandwiched between the upper radiator and lower ground plane. The upper layer radiator consists of three parts: the rectangular patch, the quarter wavelength converter and the feeding line. First determine the width W of the rectangular patch, according to the following empirical formula:  
\[\text{W}=\frac{c}{2f_{0}}(\frac{\epsilon_{r}+1}{2})^{-1/2} \tag{1}\]  
where \(c\) is the speed of light,\(f_{0}\) is the operating frequency and \(\epsilon_{r}\) is the dielectric constant of the dielectric substrate. The radiation patch length \(L\) is a half wavelength. Moreover, since the electromagnetic wave propagates within the medium, the guided wave wavelength \(\lambda_{e}\) is defined as follows:  
\[\lambda_{e}=\frac{c}{f_{0}\sqrt{\epsilon_{e}}} \tag{2}\]  
where \(\epsilon_{e}\) is the effective dielectric constant, which is calculated as follows. \[\epsilon_{e}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}(1+\frac{12h}{ W})^{-1/2} \tag{3}\]  
Moreover, consider the radiation slit length \(\Delta L\); thus, the equivalent slit length is calculated as follows. \[\Delta L=0.412h\frac{\epsilon_{e}+0.3}{\epsilon_{e}-0.258}\frac{\frac{w}{h}+0. 264}{\frac{w}{h}+0.8} \tag{4}\]  
Therefore, the radiation patch length \(L\) is described as follows. \[L=\frac{\lambda_{e}}{2}-2\Delta L \tag{5}\]  
After the radiation patch design is completed, the line width of the feed line needs to be considered since the microstrip feed is chosen as the feed method. The equation for the characteristic impedance \(Z_{0}\) of the microstrip line is described as follows:  
\[Z_{0}=\frac{377}{\sqrt{\epsilon_{r}}}\left\{\frac{w^{\prime}}{h}+0.883+0.165 \frac{\epsilon_{r}-1}{\epsilon_{r}{}^{2}}+\frac{\epsilon_{r}+1}{\pi\epsilon _{r}}\bigg{[}\ln\left(\frac{w^{\prime}}{h}+1.88\right)+0.758\bigg{]}\right\}^ {-1} \tag{6}\]  
where \(w^{\prime}\) is the line width of the microstrip line, and it can be observed that the impedance \(Z_{0}\) is related to line width \(w^{\prime}\), dielectric substrate thickness h and dielectric constant \(\epsilon_{r}\). The  
Figure 1: Design and dimension of micro-strip patch antenna. The image on the left is 3-dimensional structure. The image on the right is top view. above equation is written as a MATLAB program, and the corresponding data are brought into the calculation to obtain the approximate size of the desired microstrip antenna. Set the parameters: For example, the operating frequency is 8 GHz, the thickness of the glass dielectric substrate is Sub_h is 0.7 mm and the dielectric constant is 6. The theoretical data are obtained as follows: radiation patch length Ptc_LI is 6.4 mm, radiation patch width Ptc_W is 10 mm, feed line width Fdl_W is 1 mm and quarter wavelength Mtc_L is 4.4 mm. Since the results are calculated by empirical equations, these parameters are inputted into HFSS for modeling and optimization in order to obtain the final simulation parameters. The rectangular patch size is calculated and optimized to Ptc_L \(\times\) Ptc_W = 7.1 mm \(\times\) 11.5 mm. The structure in the red circle in Figure 1 is a quarter-wavelength converter for impedance matching, with a size of Mtc_L \(\times\) Mtc_W = 3.9 mm \(\times\) 0.1 mm. In order to realize characteristic impedance, the size of the feeding line is selected at Fdl_L \(\times\) Fdl_W = 7.55 mm \(\times\) 1 mm.