# A Dual-Beam Leaky-Wave Antenna Based on Squarely Modulated Reactance Surface
## 2 Dispersion Characteristics Analysis
\begin{split} j\eta_{0}\sqrt{(\frac{k_{s}^{(0)}}{k_{0}})^{2}-1}& =jX_{s}+(4X_{s}M/\pi)^{2}[[\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(1)}}{k_ {0}})^{2}-1}}+\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(1)}}{k_{0}})^{2}-1} }]\\ &+\frac{1}{3}[\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(0)}}{k _{0}})^{2}-1}}+\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(3)}}{k_{0}})^{2}-1} }]\\ &+\frac{1}{5}[\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(5)}}{k _{0}})^{2}-1}}+\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(5)}}{k_{0}})^{2}-1} }]\\ &=jX_{s}+(4X_{s}M/\pi)^{2}[[\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{ k_{s}^{(0)}+2\eta/p}{k_{0}})^{2}-1}}+\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(0)} -2\eta/p}{k_{0}})^{2}-1}}]\\ &\qquad+\frac{1}{3}[\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{( 0)}+6\eta/p}{k_{0}})^{2}-1}}+\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(0)}- 6\eta/p}{k_{0}})^{2}-1}}]\\ &\frac{1}{5}[\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(0)}+1 \eta/p}{k_{0}})^{2}-1}}+\frac{1}{jX_{s}+\eta_{0}\sqrt{(\frac{k_{s}^{(0)}-10 \eta/p}{k_{0}})^{2}-1}}]+\bullet\bullet\bullet]\end{split}  \tag{8}\]  
Due to the presence of an additional factor of \(4X_{S}M/\pi\) in each successive term, the continued fractions converge rapidly when the modulation depth \(M\) has a small value. It is sufficient to yield the solution for \(k_{x}{}^{(0)}\) to a high degree of accuracy by using (8) with only 3 modes considered, and then the effective phase constant \(\beta_{eff}{}^{(0)}\) and attenuation constant \(\alpha_{eff}\) of SquMRS can be calculated as:  
\[\left\{ \begin{array}{l}\beta_{eff}^{(0)}=\text{Re}\Big{[}k_{x}^{(0)}\Big{]} \\ \alpha_{eff}=\text{Im}\Big{[}k_{x}^{(0)}\Big{]}\end{array} \right. \tag{9}\]  
By using the equations mentioned above, the comparison between SinMRS and SquMRS is presented with parameters \(X_{s}=430\)\(\Omega\) and \(p=30\) mm. Figure 2(a) shows the relations between the dispersion characteristics and modulation depth \(M\) at operating frequency of 8.5 GHz. It can be seen that both \(\beta_{eff}{}^{(0)}\) and \(\alpha_{eff}\) are increased as \(M\) raises from 0 to 0.4, and the changes of SquMRS are more drastic than SinMRS. For both modulation methods, the variation of \(\beta_{eff}{}^{(0)}\) is slight when the value of \(M\) is less than 0.2. Meanwhile, SquMRS allows for more design flexibility to choose any required value of \(\alpha_{eff}\), since it has a larger variation range of the effective attenuation constant. Figure 2(b) presents \(\beta_{eff}{}^{(0)}\) and \(\alpha_{eff}\) as a function of the average impedance \(X_{s}\) with \(M=0.3\). As can be seen, \(\beta_{eff}{}^{(0)}\) is mainly affected by \(X_{s}\) when \(M\) is fixed. In conclusion, the values of \(X_{s}\), \(M\) and \(p\) should be modified to assure the independent control of attenuation constant and phase constant. Figure 3: The calculated effective phase constant \(\beta_{eff}{}^{(0)}\) and attenuation constant \(\alpha_{eff}\) of SinMRS and SquMRS (**a**) as the function of modulation depth \(M\) with \(X_{s}=430\)\(\Omega\) and \(p=30\) mm, (**b**) as the function of average value \(X_{s}\) with \(M=0.3\) and \(p=30\) mm. The spatial harmonics with \(n\leq 0\) can be fast wave (\(\beta_{\textit{eff}}(n)<k_{0}\)) by choosing suitable modulation period \(p\) as \(\beta_{\textit{eff}}(0)\) is larger than \(k_{0}\). In this paper, the \(n=-1\) spatial harmonic is designed to be radiated. The radiation angle \(\theta\) can be derived as:  
\[\theta=\arcsin(\beta_{eff}^{(0)}/k_{0}-2\pi/k_{0}p)\approx\arcsin \big{[}\sqrt{\big{(}X_{s}/\eta_{0}\big{)}^{2}+1}-2\pi/k_{0}p\big{]} \tag{10}\]  
Therefore, SquMRS can generate the radiating beam with the same angle as SinMRS with a relatively simple structure. In order to construct the LWAs based on SquMRS for verification, the unit cell shown in Figure 4a is proposed and analyzed in detail. The \"H\"-shaped unit cell was printed on a dielectric substrate with a relative permittivity of 2.65 and a thickness of 1 mm.