## 3 CRLH Antenna with Butler Matrix
The reflected waves from ports \(1^{\prime}\) to \(4^{\prime}\) are given as \({b^{\prime}}_{e}=\left[0,\ \left(\frac{a^{\prime}_{1}}{2}\right)e^{-j\beta_{10}l},\ \left(\frac{a^{\prime}_{1}}{2}\right)e^{-j\beta_{10}l},0\right]\) with \(\beta_{10}\) as the phase constant of the \(TE_{10}\) mode. The total reflected waves from ports \(1^{\prime}\) to \(4^{\prime}\) can be found out as \({b^{\prime}}=\left[0,\ \left(\frac{a^{\prime}_{1}}{2}\right)\left(e^{-j\beta_{10}l} +e^{-j\beta_{20}l}\right),\ \left(\frac{a^{\prime}_{2}}{2}\right)\left(e^{-j\beta_{10}l} -e^{-j\beta_{20}l}\right),0\right]\). Since the slot or the coupling region considered is symmetric, the S parameters at port \(1\) are sufficient to fully characterize the region. The S parameters are given as \(S_{1^{\prime}1^{\prime}}=0\), \(S_{2^{\prime}1^{\prime}}=\cos\Bigl{(}(\beta_{10}-\beta_{20})\frac{l}{2}\Bigr{)} e^{-j(\beta_{10}-\beta_{20})\frac{l}{2}}\Bigr{)}\), \(S_{3^{\prime}1^{\prime}}=-j\sin\Bigl{(}(\beta_{10}-\beta_{20})\frac{l}{2} \Bigr{)}e^{-j(\beta_{10}-\beta_{20})\frac{l}{2}}\Bigr{)}\), and \(S_{4^{\prime}1^{\prime}}=0\). From \(S_{3^{\prime}1^{\prime}}\), it can be written as  
\[(\beta_{10}-\beta_{20})\frac{l}{2}=sin^{-1}|S_{3^{\prime}1^{\prime}}| \tag{1}\]  
The values of the phase constants \(\beta_{10}\) and \(\beta_{20}\) are given as  
\[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]  
and  
\[{\beta_{20}}^{2}=k^{2}-\[PERSON]{(}\frac{2\pi}{w}\[PERSON]{)}^{2} \tag{3}\]  
Substituting (2) and (3) in (1), the equation obtained is  
\[\left(\sqrt{\left(k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2}\right)}-\sqrt{ \left(k^{2}-\[PERSON]{(}\frac{2\pi}{w}\[PERSON]{)}^{2}\right)}\right)=sin^{-1}|S_{3^{ \prime}1^{\prime}}| \tag{4}\]  
where \(k\) is the wavenumber and \(w\) is the width of the slot or the coupling region. As previously mentioned in the even mode or \(TE_{10}\), there are reflections and these reflections can be cancelled out if the following approximate condition is satisfied [40]:  
\[1+e^{-j\beta_{10}l}=0 \tag{5}\]  
Figure 9: Short clot coupler with the ports \(1\) to \(4\) and the ports at the extremes of the coupling region are from \(1^{\prime}\) to \(4^{\prime}\). where \(l\) is the length of the slot or the coupling region. From (5), \(l\) can be found out as  
\[l=\frac{(2n+1)\pi}{2\beta_{10}} \tag{6}\]  
where \(n\) can be zero or a positive integer. Substituting (6) in (4) and substituting the value of \(\beta_{10}\), the design equations of the short slot coupler can be found out. A detailed analysis of the short slot coupler can be found in [36; 40]. The width and length of the slot region of the short slot coupler are given by  
\[w=\frac{\pi}{k}\sqrt{\frac{\left[\pi(2n+1)+4 sin^{-1}|S_{3^{\prime}1^{\prime}}| \right]\left[3\pi(2n+1)-4 sin^{-1}|S_{3^{\prime}1^{\prime}}|\right]}{8 sin^{-1}| S_{3^{\prime}1^{\prime}}|\left[\pi(2n+1)-2 sin^{-1}|S_{3^{\prime}1^{\prime}}| \right]}} \tag{7}\]  
\[l=\frac{(2n+1)\pi}{2\sqrt{k^{2}-\left(\frac{\pi}{w}\right)^{2}}} \tag{8}\]  
where \(k\) is the wavenumber, \(n\) can be zero or a positive integer, and \(S_{3^{\prime}1^{\prime}}\) is the S parameter response in the slot region between ports \(3^{\prime}\) and \(1^{\prime}\). For \(90^{\circ}\) hybrid couplers, the S parameter response \(|S_{3^{\prime}1^{\prime}}|\) is \(1/\sqrt{2}\). Substituting this in (7) and (8) gives the width and length of the slot region to the \(90^{\circ}\) hybrid coupler. They are given as [36]  
\[w_{1}=\frac{\pi}{k}\sqrt{\frac{4(3n+1)(n+1)}{4n+1}} \tag{9}\]  
\[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]0  
The S parameter responses for the case of crossover are given as \(|S_{3^{\prime}1  
The designed 90\({}^{\circ}\) hybrid coupler and crossover are used to construct the Butler matrix. The phase shifters are designed using the curved SIW lines.