## 1 Introduction
To avoid the metallic vias, a few papers designed the resonant antenna based on coplanar waveguide (CPW) [13; 14], but CPW configuration increased the back lobes. An artificial magnetic conductor (AMC) screen has been used to minimize the back lobes [14], and it also acts as a separation layer between the human body and the antenna. In the mmWave frequency range, a small change in the dimensions will affect the performance to a greater extent. Therefore, for an antenna at mmWave, it is desirable to design it without metallic vias. A broad-band balun [15] and filter [16] where the CRLH is designed without metallic-via were reported. Only a few works have reported patch-like radiation at zero-order resonance, and that too at the lower frequency range. In this paper, ZOR with patch-like radiation at 30 GHz is designed. The Butler matrix [17] is one of the methods to realize passive-beam-forming circuits. Since the operating frequency is 30 GHz, the substrate integrated waveguide (SIW) configuration is chosen to design the Butler matrix. The analysis of properties of SIW like dispersion characteristics, propagation constant, modeling, cutoff frequency, and relation between the width of the rectangular waveguide and SIW was reported in [18; 19; 20]. A review of different microwave circuits implemented on SIW was reported in [21]. The SIW discontinuities were analyzed in [22]. The taper was used in the integration of the microstrip and SIW [23; 24] on a single substrate. The different transitions between SIW and other transmission lines were discussed in [25]. There are many beam-forming networks reported, like the Rotman lens [26], Blass matrix [27], Nolan matrix [28], and Butler matrix. The Blass matrix is a lossy network since it requires a direction coupler with a coupling coefficient of one to achieve a lossless Blass matrix [27]. The Nolan matrix is also a type of Blass matrix with losses suppressed, and it also has directional couplers and phase shifters [28]. Among the beam-forming networks, the Butler matrix is chosen because of its simple and modular design. A wide-band Butler matrix in SIW was proposed without crossover in [29]. The comparison between losses in microstrip bends and SIW bends at 77 GHz was also discussed [29]. A compact two-layer SIW Butler matrix was reported in [30] with reduced errors in magnitude and phase. Another two-layer \(4\times 8\) Butler matrix in SIW at 38 GHz was proposed in [31] with an antenna array in the third layer, and the crossover was avoided in this reported work also. A \(4\times 8\) Butler matrix with a reduced number of crossovers was presented in [32] with amplitude tapering. The number of crossovers was reduced in the two-layer SIW Butler matrix reported in [33] by changing the \(-3\) dB hybrid couplers near the input ports to E-plane hybrid couplers and by reversing the \(-3\) dB hybrid couplers near the output ports. A \(7\times 8\) Butler matrix with an antenna array was implanted in three SIW layers in [34] where the boresight beam was created by merging two inputs at the center. The Butler matrix with a modified hybrid coupler was reported in [35]. A mmWave SIW Butler matrix designed at 60 GHz was systematically explained in [36]. The detailed design of the 30 GHz Butler matrix used in this paper is explained in [37]. Metamaterial properties were used to enhance the performance of the antenna implemented in SIW [38]. A power splitter was designed to support an array with two dipole antennas. An epsilon-negative lattice with \(3\times 2\) unit cells was used to increase the broadside gain and reduce the cross-section radiation in [38]. Here, the newly designed ZOR antenna is added to the Butler matrix implemented in SIW. In this paper, a new CRLH resonant antenna at 30 GHz is designed without metallic vias to obtain patch-like radiation. The open-circuited CRLH transmission line is used to design the resonant structure, which is different from the balanced structures reported in [15; 16].