## 2 Design
### HMSIW CP Slot Antenna  
Currently, substrate-integrated waveguide (SIW) technology is promising for millimeter-wave applications by virtue of its low loss and ease of fabrication with RF circuits. CP slot antennas based on the SIW technology have been presented [22; 23; 24]. HMSIW structures have been used for size reduction, which was achieved by bisecting a conventional SIW along a fictitious magnetic wall. A slot antenna based on the HMSIW technology and leaky waves has been presented for linear and circular polarizations (LP and CP, respectively) [25]. In HMSIW, the quasi-TE(m-0.5),0 modes could propagate, and the dominant quasi-TE0.5,0 mode was addressed (m = 1) for slot antennas [26]. A 45\({}^{\circ}\)-inclined HMSIW slot analysis model is illustrated in Figure 3. Both series impedance \(Z_{1}\) and parallel impedance \(Z_{2}\) existed because the slot was off the centerline (bisection line for HMSIW) of the SIW [27]. From the equivalent circuit model, the calculation formulas for \(Z_{1}\) and \(Z_{2}\) are expressed as follows [27]:  
\[\frac{Z_{1}}{Z_{0}} =\frac{1+S_{11}-S_{21}}{1-S_{11}}, \tag{1}\] \[\frac{Z_{2}}{Z_{0}} =\frac{S_{21}}{1-S_{11}-S_{21}}. \tag{2}\]  
The radiation power rate can be calculated as follows:  
\[\frac{P_{r}}{P_{in}}=1-|S_{11}|^{2}-|S_{21}|^{2}, \tag{3}\]  
where \(P_{r}\) and \(P_{in}\) are the radiation and input powers, respectively.  
In this study, an HMSIW slot antenna providing \(-45^{\circ}\) and \(+45^{\circ}\) LPs, left-handed CP (LHCP), and right-handed CP (RHCP) with input ports on the same side for easier integration with other components was introduced. The proposed HMSIW slot antenna is shown in Figure 4. The antenna was constructed using \(-45^{\circ}\) and \(+45^{\circ}\) LP antennas, each of which had 16 radiating slots and a short-circited termination. If the slot spacing of a slot array is not just half of the guided wavelength and a matched termination is placed at the end, a leaky-wave array may be obtained [25]. The proposed slot array antenna, whose slot spacing was not just half of the guided wavelength, operated as a leaky-wave array, even though it used a short-circited termination because the array with radiating slots was long enough to allow the power at the termination to become a small fraction of the radiated power. The solution using short-circited termination with the residual power at the termination being less than 2% of the radiated power was virtually equivalent to the solution using matched termination [22]. The HMSIWs of both antennas faced each other by sharing inner via for the metallic walls of the HMSIW. An additional row of shunt vias (edge via) near the open edge was responsible for the improvement in the return loss [25] and the reduction in antenna coupling between the two adjacent arrays. Each slot of the \(-45^{\circ}\) LP antenna was placed at the middle of each adjacent slot of the \(+45^{\circ}\) LP antenna along the \(z\)-axis for a negligible effect on the radiation patterns. For speculation of the CP, the HMSIW slot antenna was realized using a Branch-line coupler for a circular polarizer and a line with a phase angle of \(180^{\circ}\) for port 2 (Figure 4). The CP mechanism was confirmed by vector current distribution in simulation [28; 29]. The surface current  
Figure 3: Equivalent circuit of the slot.