# A Dual-Beam Leaky-Wave Antenna Based on Squarely Modulated Reactance Surface
## 3 Dual-Beam Leaky-Wave Antenna Implementation  
A dual-beam LWA pointing at \(\theta_{1}=-30^{\circ}\) (beam 1) and \(\theta_{2}=18^{\circ}\) (beam 2) was designed to operate at 8.5 GHz. In previous studies about producing multi-beams, the modulation pattern was divided into several regions, and each region provided a desired radiation beam [24]. Usually, a monopole antenna was chosen as the feed source so that it was difficult to reduce the height of the antenna. In addition, a holographic synthesizing method, which translated any requested electromagnetic modulation of \(\beta\) and \(\alpha\) along the antenna into the corresponding geometrical modulation, was used to produce multiple beams [25]. Several previous designs, based on superposing multiple objective field patterns on the SinMRS, were proposed to produce multi-beams and achieve near-field focusing [26; 27; 28]. In this work, a superposition method based on SquMRS was used to design a dual-beam LWA, due to the capability of flexible control of phase constant and attenuation constant. Unlike using \(n=-1\) and \(n=-2\) spatial harmonics to generate dual-beam (\(\beta_{\textit{eff}}^{-(-1)}<\)\(k_{0}\), \(\beta_{\textit{eff}}^{-(-2)}<\)\(k_{0}\)) [29; 30; 31], the produced two beams in this work are both excited by the \(n=-1\) space harmonic, and other space harmonics are all slow waves (only \(\beta_{\textit{eff}}^{-(-1)}<\)\(k_{0}\)). \begin{table}
\begin{tabular}{c c c c c} \hline \(M\) & \(\theta\)(SinMRS) & \(\theta\)(SquMRS) & \(\Delta\theta\)(SinMRS) & \(\Delta\theta\)(SquMRS) \\ \hline
0.2 & 22\({}^{\circ}\) & 21\({}^{\circ}\) & 6.1\({}^{\circ}\) & 6.7\({}^{\circ}\) \\
0.3 & 21\({}^{\circ}\) & 19\({}^{\circ}\) & 6.4\({}^{\circ}\) & 7.2\({}^{\circ}\) \\
0.4 & 21\({}^{\circ}\) & 20\({}^{\circ}\) & 6.9\({}^{\circ}\) & 8.6\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Radiation angles and beam width for various values of \(M\). Figure 5: The schematic of LWA based on SquMRS, in which \(w_{0}=3\) mm, \(w=20\) mm and \(g=0.1\) mm. Figure 6: The simulated radiation patterns with different modulation depths at 8.5 GHz. Assuming these two beams possess the identical \(X_{s}\) and \(M\), the radiation angles are decided by the modulation period \(p_{1}\) and \(p_{2}\), respectively. The individual modulation patterns for each beam are added up at each unit of the proposed antenna [22; 24]. According to the holographic antenna theory, once the interference pattern is recorded by the interaction between reference wave \(\Psi_{ref}\) and multi-beam radiation wave \(\Psi_{rad}\) on the modulated surface, the multiple beams can be reconstructed. The multi-beam radiation wave can be written as the sum of two individual beams: \(\Psi_{rad}=\Psi_{md}(^{1})+\Psi_{md}(^{2})\). The distribution of surface impedance is used to embody the whole surface interferogram:  
\[ \begin{array}{l}Z_{s}(x)=j[X_{s}+M\text{Re}(\Psi_{rad}\cdot\Psi_{rad}^{*})]=j [X_{s}+M\text{Re}[(\Psi_{rad}^{(1)}+\Psi_{rad}^{(2)})\cdot\Psi_{rad}^{*}]]\\ =jX_{s}[1+a_{1}Mf_{1}(x)+a_{2}Mf_{2}(x)]=a_{1}jX_{s}[1+Mf_{1}(x)]+a_{2}jX_{s}[ 1+Mf_{2}(x)]\end{array}  \tag{11}\]  
where \(f_{1}(x)\) and \(f_{2}(x)\) are periodic functions of the desired two beams with the surface impedance coefficient \(a_{1}\) and \(a_{2}\) (\(a_{1}+a_{2}=1\)), respectively. Therefore, the precondition of this superposing method is that two beams possess the identical \(X_{s}\) and \(M\). Figure 7 shows the simulated radiation patterns of the LWA with different surface impedance coefficients. In theory, two beams should obtain an identical gain when \(a_{1}=a_{2}=0.5\), but the simulated result shows that a gain difference (about 1 dB) existed. As \(a_{1}\) raises to 0.6, the gain of beam 1 is 3 dB higher than the gain of beam 2. In these two cases, the changing of surface impedance coefficients has almost no effect on the radiation angles. Finally, the gain difference between two beams is about 6.5 dB when \(a_{1}=0.7\), \(a_{2}=0.3\).