## 3 CRLH Antenna with Butler Matrix
The structure of the Butler matrix and a photograph of the fabricated Butler matrix are shown in \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]1 The simulated and measured S parameter response of the Butler matrix is shown in \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]2  
\[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]3 Crossover (**a**) structure with dimensions of \(w_{2}\) = 7.13 mm and \(l_{2}\) = 6 mm; (**b**) simulated S parameter response of the crossover for the input at port 1. \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]4 A 90\({}^{\circ}\) hybrid coupler (**a**) structure with dimensions \(w_{1}\) = 7.51 mm and \(l_{1}\) = 2.4 mm; (**b**) simulated S parameter responses of the hybrid coupler for input at port 1 and a phase difference between \(S_{31}\) and \(S_{21}\). The simulated phases from ports 5 to 8 for the input at port 1 are approximately \(20.82^{\circ}\), \(-9.53^{\circ}\), \(-38.58^{\circ}\), and \(-92.60^{\circ}\) respectively at 30 GHz. The measured phases from ports 5 to 8 for input at port 1 are approximately \(-26.06^{\circ}\), \(-15.07^{\circ}\), \(-44.48^{\circ}\), and \(-170.00^{\circ}\), respectively, at 30 GHz. The final fabricated structure of the Butler matrix with CRLH antenna is shown in \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]5 The same Rogers RO4003c substrate that was used in the fabrication of the CRLH antenna is also used to fabricate the Butler matrix with the CRLH antenna. \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]6 The \(4\times 4\) SIW Butler matrix (**a**) structure; (**b**) photograph of the fabricated structure. \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]7 Measured and simulated S parameter responses of the \(4\times 4\) SIW Butler matrix for port 1 excitation. The CRLH antenna described in Section 2 is added as the radiating element for a \(4\times 4\) Butler matrix at the four output ports. The CRLH antenna is designed without via holes to obtain the radiation in the broad side direction. The distance between the antennas is 0.75 \(\lambda_{g}\). The simulated electric field of the Butler matrix during port 1 excitation is shown in \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]8 From the electric field distribution, it can be found that the waves from port 1 reach the four output ports at different phases to tilt the radiated beam in the desired direction. The radiation pattern of the antenna in the XZ (\(\varphi=0^{{}^{\circ}}\)) plane for excitations in the four input ports is shown in \[{\beta_{10}}^{2}=k^{2}-\[PERSON]{(}\frac{\pi}{w}\[PERSON]{)}^{2} \tag{2}\]9 The simulated total realized gain is above 7 dBi in all input port excitations, and the simulated scanning angles are \(12^{\circ}\) for port 1 excitation, \(-68^{\circ}\) for port 2 excitation, \(64^{\circ}\) for port 3 excitation, and \(-11^{\circ}\) for port 4 excitation. \[{\beta_{20}}^{2}=k^{2}-\[PERSON]{(}\frac{2\pi}{w}\[PERSON]{)}^{2} \tag{3}\]0 The Butler matrix with a CRLH antenna (**a**) structure; (**b**) photograph of the fabricated structure. \[{\beta_{20}}^{2}=k^{2}-\[PERSON]{(}\frac{2\pi}{w}\[PERSON]{)}^{2} \tag{3}\]1 Simulated electric distribution of the Butler matrix with an antenna in SIW.