## 3 CRLH Antenna with Butler Matrix  
The detailed description of the design of the substrate integrated waveguide (SIW) Butler matrix at 30 GHz is given in [36; 37]. Figure 8 shows the block diagram of a \(4\times 4\) Butler matrix with four 90\({}^{\circ}\) hybrid couplers, two crossovers, two 45\({}^{\circ}\) phase shifters, and two 0\({}^{\circ}\) phase shifters that are required. Short-slot couplers are used to implement hybrid couplers and crossovers [36]. The short slot coupler is formed by combining two SIWs sideways and removing the central via hole wall to form the slot region. The short slot coupler along with the coupling region are shown in Figure 9. By adjusting the dimensions of the coupling region of the short slot coupler, the ratio of power transmitted to the output ports can be varied. The even and odd-mode analyses are used to analyze the short slot coupler. The ports of the coupler in Figure 9 are denoted as 1 to 4. The ports at the extremes of the coupling regions are denoted by 1\({}^{\prime}\) to 4\({}^{\prime}\) and the reference planes are denoted by dotted lines in Figure 9. Consider an input at port 1. The incident waves for this condition are given as \(a=[a_{1},\,0,\,0,\,0]\). For even- and odd-mode cases, the incident waves are represented as \(a_{e}=[a_{1}/2,\,0,\,0,\,a_{1}/2]\) and \(a_{o}=[a_{1}/2,\,0,\,0,\,-a_{1}/2]\) respectively and odd and even mode cases are added to obtain the total incident wave, which is \(a=a_{e}+a_{o}\). The even- and odd-mode excitations introduce modes \(TE_{10}\) and \(TE_{20}\), respectively, in the coupling region. The incident waves from ports 1\({}^{\prime}\) to 4\({}^{\prime}\) are given as \(a^{\prime}_{e}=[a^{\prime}_{1}/2,\,0,\,0,\,a^{\prime}_{1}/2]\) and \(a^{\prime}_{o}=[a^{\prime}_{1}/2,\,0,\,0,\,-a^{\prime}_{1}/2]\) for the even- and odd-mode excitations. Since ports 1\({}^{\prime}\) to 4\({}^{\prime}\) have a few electrical lengths from ports 1 to 4, the waves \(a^{\prime}_{e}\) and \(a^{\prime}_{o}\) are obtained by phase shifting the waves \(a_{e}\) and \(a_{o}\), respectively. In the odd-mode case, there is an electric wall at  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **Frequency of** & **Antenna Footprint** & **Radiation Pattern** & **Gain** & **Bandwidth \%** \\ \hline This work & 30 GHz & 2.46 mm \(\times\) 0.8 mm & Patch like & 5.35 dBi & 4.2\% \\ \hline
[6] & 3.38 GHz (CRLH) & \(\frac{\lambda_{0}}{6}\times\frac{\lambda_{0}}{6}\times\frac{\lambda_{0}}{35}\) & Monopole like & 0.87 dBi (CRLH) & \\  & 3.37 GHz (induced) & \(\frac{\lambda_{0}}{6}\times\frac{\lambda_{0}}{6}\times\frac{\lambda_{0}}{35}\) & Monopole like & 0.7 dBi (indicator loaded) & Not mentioned \\ \hline
[7] & 4.88 GHz & 75\% reduction to patch & Patch like & Not mentioned & Not mentioned \\ \hline
[8] & 1.93 GHz and & & & \(-3.21\) dBi (lower band), 7.45 dBi & Not mentioned \\  & 4.16 GHz & Not mentioned & Patch like & (higher band) & Not mentioned \\ \hline
[9] & 5.83 GHz & Not mentioned & Patch like & 5.06 dBi & Not mentioned \\ \hline
[10] & 5.76 GHz & \ \begin{tabular}{c} 0.28 \(\lambda_{0}\times\) 0.191 \(\lambda_{0}\) \\ and ground plane \\ size 0.861 \(\lambda_{0}\times\) 0.766 \(\lambda_{0}\) \\ \end{tabular}  &
\begin{tabular}{c} 0.  
the center parallel to the electric walls at the sides in the coupling region. So, the reflected waves from ports \(1^{\prime}\) to \(4^{\prime}\) are given as \({b^{\prime}}_{o}=\left[0,\ \left(\frac{a^{\prime}_{1}}{2}\right)e^{-j\beta_{20}l},- \left(\frac{a^{\prime}_{1}}{2}\right)e^{-j\beta_{20}l},0\right],\) where \(\beta_{20}\) is the phase constant of the \(TE_{20}\) mode. In contrast to the odd-mode case, for the even-mode excitation, there is a magnetic wall at the center of the coupling region parallel to electric walls at the sides. Because of this, there are small reflections of the incident waves that are cancelled by imposing conditions on the length of the coupling or slot region.