## 2 Antenna Array and Design Method
### Operation Mechanism and Leaky-Wave Generation  
The proposed antenna was a conductor-backed alumina DIL with two different dielectric layers mounted on top of each other and two sets of metal circular discs periodically sandwiched between the layers along the transmitting direction with a periodicity of \(p\). As it is well known, travelling waves inside the proposed layered dielectric image waveguide without the periodic loadings is regarded as a slow wave. Hence, it is not possible to radiate and produce leaky waves directly from the dominant mode, even if it is a semi-open structure. However, with periodic modulations, the leaky conditions can be modified. Through the placement of periodic metallic overlapped discs with the distance p on the top of the layered DIL, an infinite number of leaky modes with a wavenumber \(\beta_{m}<k_{0}\) were excited. The propagation constant of the mth order space harmonic is defined by [18]  
\[\beta_{m}=\beta_{0}+\frac{2m\pi}{p} \tag{2}\]  
where \(\beta_{0}\) and \(\beta_{m}\) are the propagation constants for the fundamental and mth order space harmonics, correspondingly and \(m\) covers all integer ranges. As the fundamental space harmonic for open dielectric waveguides is usually a slow-wave, the negative spatial harmonics (m = \(-1\), \(-2\), \(-3\), etc.) can only be leaky and radiative. It should be noted that not all leaky modes are physically important, and more than one leaky mode may be excited at the same time. An explanation of how each leaky mode contributes to the antenna radiation performance follows from the dispersion plot in Figure 2. This dispersion diagram of a unit-cell including the propagation (\(\beta_{m}\)) and attenuation constant (\(\alpha\)) graphs was obtained using the following equations [18]  
\[ \begin{split}\beta_{m}p=\left|\text{Im}(\cosh^{-1}(\frac{1-S_{1 1}S_{22}+S_{21}S_{12}}{2S_{21}}))\right|\\ \alpha p=\left|\text{Re}(\cosh^{-1}(\frac{1-S_{11}S_{22}+S_{21}S_ {12}}{2S_{21}}))\right|\end{split}  \tag{3}\]  
where the \(S\)-parameters of the unit-cell are simply obtained by HFSS simulations. In Figure 2, we observed that the n = \(-1\) space harmonic enters the radiation region ( \(\beta_{-1}\) \(<k_{0}\)), at 157.5 GHz. When the frequency reaches 172 GHz, the leaky-wave antenna scans to the broadside without encountering an open-stop-band (OSB) problem, which corresponds to a \(\beta_{-1}\) = 0. The OSB, as a known problem of many LWAs, is mainly caused by the coupling between the forward- and the backward-travelling of the n = \(-1\) leak wave harmonic [46]. In the OSB, the reflection coefficient was increased and the gain drops considerably which avoided continuous beam scanning. In the proposed design, by printing two circular metal discs a quarter wavelength apart, OSB is suppressed. This will be explained in detail later in this section. As the frequency increased above 172 GHz, the main beam scans in the forward direction. When the frequency reached 190 GHz, the n = \(-2\) space harmonic also enters the radiation region (\(1\,\beta_{-2}\,|<k_{0}\)) and \(\beta_{-1}p=\pi\). The coupling between the forward-traveling n = \(-1\) and the backward-traveling n = \(-2\), generally encounters the LWAs with another stop band. This can be observed as a sudden increase in the reflection coefficient which restricts the scanning range in the forward direction as well, appearing as a grating lobe around the backward end fire [4]. To suppress this stop band, the radiation from the n = \(-2\) space harmonic should be avoided or weakened. One commonly used technique is to prevent the n = \(-2\) space harmonic from entering the radiation region (i.e., (\(1\,\beta_{-2}\,|>k_{0}\))). This method has been previously mentioned in the literature [22; 24; 27]. Another approach is to make the attenuation constant of the n = \(-2\) space harmonic as small as possible (near zero). We used the latter technique for designing the proposed antenna using the optimization algorithms in HFSS.