# A Dual-Beam Leaky-Wave Antenna Based on Squarely Modulated Reactance Surface
## 2 Dispersion Characteristics Analysis  
According to the Floquet theory, leaky-wave radiation can be generated by the periodically modulated structure. Therefore, for the construction of spatial distribution of surface impedance, squarely modulated reactance surface (SquMRS), instead of SinMRS, can be adopted to achieve the conversion from surface wave (electromagnetic wave that propagates along the interface between different media) to leaky-wave (electromagnetic wave that is coupled or transferred to a propagation medium outside the interface). The distributions of surface impedances along one period for SquMRS and SinMRS under different modulations, which have identical average value \(X_{s}\), modulation depth \(M\) and modulation period \(p\), are shown in Figure 1. Compared with SinMRS, the structure of SquMRS is much simpler since it needs fewer kinds of unit cells. Assume the SquMRS is placed in the _xoy_ plane, while the direction of the propagation along the surface is the _x_-axis, as shown in Figure 2. For simplicity, only transverse magnetic (TM) polarization is discussed. The surface impedance distribution in one period exhibits a form:  
\[Z_{s}(x)=\left\{ \begin{array}{cc}jX_{s}(1+M)&if\ \ 0\leq x\leq p/4\ \ \ or\ \ 3p/4\leq x\leq p\\ jX_{s}(1-M)&if\ \ \ p/4<x<3p/4\end{array} \right. \tag{1}\]  
where \(X_{s}\) is the average value, \(M\) is the modulation depth and \(p\) is the modulation period. Figure 1: The distributions of surface impedances along one period under different modulations. Due to the periodic function of Equation (1), it can be expressed as a Fourier series expansion, and the impedance profile of squared modulation can be rewritten as:  
\[Z_{s}(x)=jX_{s}+jMX_{s}\sum_{m=1}^{\infty}\frac{4}{(2m-1)\pi}e^{-j(2m-1)\frac{2 \pi}{p}x} \tag{2}\]  
Here, an equivalent transmission lines model is established to analyze the dispersion characteristics, as shown in Figure 2 and originally proposed in [19]. An infinite number of independent transmission lines along the normal direction are coupled together at the impedance surface. \(Z_{eff}(^{n})=V_{n}/I_{n}\) is interpreted as the effective impedance of mode \(n\). The mode voltage \(V_{n}\) and mode current \(I_{n}\) are used to represent the field energy above the surface:  
\[\overrightarrow{E_{t}}=(1/\sqrt{2\pi})\overrightarrow{x}\sum_{n=-\infty}^{ \infty}V_{n}e^{-jk_{m}x} \tag{3}\]  
\[\overrightarrow{H_{t}}=(1/\sqrt{2\pi})\overrightarrow{y}\sum_{n=-\infty}^{ \infty}I_{n}e^{-jk_{m}x} \tag{4}\]  
According to the Floquet theory, spatial harmonics can be generated by adjusting the modulation period \(p\):  
\[k_{x}^{(n)}=k_{x}^{(0)}+2\pi n/p=\beta_{eff}^{(n)}+2\pi n/p+j\alpha_{eff} \tag{5}\]  
in which \(k_{x}^{(n)}\) and \(\beta_{eff}^{(n)}\) are the wave number and phase constant of the \(n\)th spatial harmonic, \(\alpha_{eff}\) is the attenuation constant. It has to be noticed that the periodical modulation has few influences on the attenuation constant according to (5). Therefore, all space harmonics share an identical \(\alpha_{eff}\) and possess different \(\beta_{eff}^{(n)}\) for each mode \(n\). The effective impedance of mode \(n\)\(Z_{eff}^{(n)}\) can be rewritten as:  
\[Z_{eff}^{(n)}=j\eta_{0}\sqrt{\left(k_{x}^{(n)}/k_{0}\right)^{2}-1} \tag{6}\]  
where \(\eta_{0}\) is the free-space wave impedance and \(k_{0}\) is the free-space wavenumber. Using circuit theory, the effective impedance of mode \(n=0\) can be derived as an infinite continued fraction:  
\[Z_{eff}^{(0)}=jX_{s}+(4X_{s}M/\pi)^{2}\sum_{m=1}^{\infty}\frac{1}{(2m-1)}[ \frac{1}{jX_{s}+Z_{eff}^{(2m-1)}}+\frac{1}{jX_{s}+Z_{eff}^{-(2m-1)}}] \tag{7}\]  
Figure 2: The equivalent transmission lines model. Substituting the Equations (5) and (6) into Equation (7):  
\[