## 2 Antenna Array and Design Method  
### Radiating Element  
A cross-sectional view of the proposed radiating element stacked over a conductor-backed alumina DIL dielectric image line (DIL) is depicted in Figure 1a. The radiation element, composed of different layers mounted on top of each other, consists of a pair of overlapped copper discs. The discs are placed with \(\lambda\)g/4 center-to-center spacing to avoid the stop-band phenomenon at the broadside, where \(\lambda\)g is the wavelength seen by these metal discs at 172 GHz. A 0.254 mm thick RO5880 dielectric substrate with a low permittivity of 2.4 was used to prevent a higher-order mode excitation. The diameter of the discs was optimized by means of high-frequency simulation software (HFSS) to achieve the maximum directivity in the normal direction, where its initial value was \(\lambda\)g/2. The thickness and permittivity of the dielectric layers were presumed to be fixed based on the accessibility of these materials. Figure 1b shows a 15-element array view of the proposed LWA with an equal inter-element spacing of one wavelength (\(\lambda\)g) at 172 GHz along the DIL direction.  
The DIL operating as the feed line of the radiating elements was a rectangular dielectric slab with a relative dielectric constant of \(\varepsilon_{\mathrm{r}}\), backed by a copper ground plane as presented in Figure 1a. The relative permittivity (\(\varepsilon_{\mathrm{r}}\)), width (2a), and thickness (b) of the substrate are the key design parameters of the DIL. The substrate material used in the antenna structure should have a high permittivity (\(\sim\)10 or above in practice) to achieve a high field confinement around the unshielded dielectric and a compact antenna with a large scanning angle. Additionally, to realize a high-gain and high-efficiency antenna, the considered substrate should have low dielectric losses in the sub-THz regimes. Accordingly, in this work, an alumina substrate of 0.254 mm thickness with a dielectric constant of 9.7 and a loss tangent of 0.0005 was considered for the DIL [27]. After choosing the substrate, the width (2a) and thickness (b) of the DIL was designed. The DIL thickness is the most crucial parameter to control the operating frequency range. However, the DIL width has less  
Figure 1: Layout of the proposed LWA. (**a**) Cross-sectional view of the array and (**b**) a top view of the structure along with the feeding network; 2a = 0.3, p = 0.8, t\({}_{\mathrm{y}}\) = 0.7, S = 0.12, g = 0.1, r = 0.1 mm, b = 0.254, h\({}_{2}\) = 0.254, h\({}_{1}\) = 0.127, (Unit: mm).  
influence on the working frequency band and strongly affects the second DIL mode cutoff frequency. The widest practical bandwidth is achieved with a nearly equal height and half-width (a unity aspect ratio b = a), using the following formula [45]  
\[\frac{a}{\lambda_{0}}\approx\frac{0.32}{\sqrt{\varepsilon_{r}-1}} \tag{1}\]  
where \(\lambda_{0}\) is the free space wavelength at the operational frequency. The estimated value of \(a\) is equal to the substrate thickness \(b\) which equals 0.15 mm using (1) at the center frequency of 172 GHz. Therefore, the initial value of the DIL width was chosen as 2a = 0.3 mm. Since the commercially available alumina substrate does not have the thickness b = 0.15 mm, the closest available thickness b = 0.25 mm was considered in our design.