# A Dual-Beam Leaky-Wave Antenna Based on Squarely Modulated Reactance Surface
## 2 Dispersion Characteristics Analysis
The advantage of this unit cell is that its surface impedance can be easily modulated by changing the width of the central bar \(b\). The length of this unit cell \(a\) was chosen to be 3 mm (0.085 \(\lambda_{0}\) at 8.5 GHz) to guarantee the sub-wavelength structure. Other dimensions of unit cell were \(d=1.1\) mm, \(l=10\) mm. The dispersion curves of the unit cell with different \(b\) were obtained by using the commercial software CST Microwave Studio (Computer Simulation Technology TM, Darmstadt, Germany) [21], as shown in Figure 4b. The phase difference across the unit cell \(\beta a\) was simulated by eigen mode solver under periodic boundary conditions [22]. When \(b\) decreases from 8 mm to 2 mm, the dispersion curves gradually bend away from the air line, giving rise to a much slower travelling wave. The results show that the phase constant \(\beta\) of the dominant mode (\(n=0\)) becomes higher as \(b\) decreases. Then, the surface impedance of the unit cell can be calculated as \(Z_{s}=j\eta_{0}\sqrt{(\beta a/k_{0}a)^{2}-1}\). The relation between the surface impedance and \(b\) is demonstrated in Figure 4b. The surface impedance ranges from the minimum value of \(137.5\,j\Omega\) to the maximum value of \(752.3\,j\Omega\) at 8.5 GHz as \(b\) decreases from 9 mm to 1 mm. LWAs, based on these two modulation methods, were designed with a similar structure. The schematic of the LWA based on SquMRS is shown in Figure 5, which consists of coplanar waveguides (CPWs), matching transitions and modulation pattern. The proposed LWAs were printed on dielectric substrates with a relative permittivity of 2.65 and a thickness of 1 mm. The dimensions of the CPWs shown in Figure 5 are \(w_{0}=3\) mm, \(w=20\) mm and \(g=0.1\) mm. In order to feed and receive energy, both ends of CPWs are designed to achieve 50 \(\Omega\) impedance. The matching transition, which consists of a flaring ground and gradient grooves, provides a good matching of impedance and momentum between CPWs and modulation pattern [23]. Therefore, a high-efficiency transmission is ensured by the matching transitions. The function of the modulation pattern is to generate the desired leaky-wave radiation by using the periodically modulated reactance surface, which is composed of 10 periods, and each period contains 10 unit cells. The distribution of surface impedance within one period is modulated as described in Figure 1 with \(X_{s}=430\)\(\Omega\) and \(p=30\) mm (10 unit cells per period). The  
Figure 4: (**a**) Structure of the proposed unit cell. (**b**) The dispersion curves with different \(b\) and surface impedance of the unit cell with different \(b\) at 8.5 GHz. desired radiation angle is set to be \(\theta=20^{\circ}\) by using Equation (8). The simulated radiation patterns of different modulation methods at 8.5 GHz are shown in Figure 6. Table 1 reports the values of radiation angle \(\theta\) and beam width \(\triangle\theta\) with different \(M\). For both modulation methods, the radiation angle changes a little when the modulation depth \(M\) varies from 0.2 to 0.4 and the beam width increases with \(M\) rising so that the feasibility of LWAs based on squared modulation is proved. It is worth noting that the broader beam is just a characteristic rather than an advantage of SquMRS.