## 2 Materials and Methods
### Improvement of Cross-Polarization  
Figure 4 presents two different SIW-fed dipole antennas. Antenna A is similar to the traditional printed dipole antenna fed by SIW. Antenna B is the proposed folded dipole fed by SIW. The positions of the connecting metallic vias and the length of the arms are optimized to realize a good impedance matching. In order to reduce the parasitic effect of the SIW-to-CPW transition, both of the antennas are fed by wave ports for simulation. The simulated \(1\,\mathrm{S_{11}}\,\mathrm{l}\) of the two investigated dipole antennas are shown in Figure 5a. The antenna works in the traditional half-wavelength dipole mode. Thus, it has only one resonance at 42 GHz and has a narrow impedance bandwidth. According to the above broadband principle, the designed SIW-fed folded dipole has two resonances, which are the traditional half-wavelength dipole mode at a low frequency and the high-order dipole mode at a high frequency. Thus, the proposed folded dipole realizes a wide impedance bandwidth of 58% covering 30 to 55 GHz. The simulated maximum gain and cross-polarization of the two antennas are shown in Figure 5b. The simulated results display that both antennas have a comparable gain that varies between 5 to 7 dBi from 30 to 55 GHz. However, the cross-polarization performance of Antenna B is greatly improved compared with that of Antenna A due to the introduction of the folded structure. Figure 4: Two different SIW-fed printed dipole antennas: (**a**) routine half-wavelength dipole; (**b**) the proposed wideband folded dipole. Figure 3: The evolution of wideband printed long dipole with dual resonant mode. Figure 6 shows the surface currents for the conventional SIW-fed dipole (Antenna A) and the presented SIW-fed folded dipole (Antenna B) at 45 GHz. The aperture of the SIW terminal is equivalent to a magnetic dipole [19] aligning with the y-axis, and it mainly contributes to the vertical polarization electric field while the printed electric dipole mainly contributes to the horizontal polarization electric field, as shown in Figure 6a. Therefore, the conventional SIW-fed dipole has a poor cross-polarization higher than \(-13\) dB. However, for the SIW-fed folded dipole antenna, it can be observed from Figure 4b that the surface of the connecting metallic vias have a current in vertical direction and the currents have the same phase and amplitude. Therefore, the vertical polarization electric field of the SIW-fed folded dipole comes from two parts: one is the magnetic dipole and the other is the surface currents on the surface of the connecting metallic vias. If the two vertical polarization electric fields are adjusted with the out-of-phase, the cross-polarization of the SIW-fed folded dipole will be improved. The relationship between the two electric fields generated by the magnetic dipole and the connecting metallic vias is studied to investigate the cross-polarization improvement. The conceptual configuration consisting of a magnetic dipole and two short electric dipoles is shown in Figure 7. The aperture serves as a virtual magnetic dipole, which is located  
Figure 5: The simulated \(\ket{S_{11}}\), gain and cross-polarization of Antenna A and Antenna B: (**a**) \(\ket{S_{11}}\); (**b**) gain and cross-polarization. Figure 6: The surface electric current distribution for the two antennas at 45 GHz: (**a**) Antenna A, (**b**) Antenna B.  
on the y-axis and operates at one-half-wavelength mode. Thus, the electric field of the aperture at an arbitrary observation point P in the far field is found to be [20]  
\[\overset{\rightarrow}{E}_{aperture}=-\
abla\times\overset{\rightarrow}{F}_{y}= jkK_{1}\big{(}-\theta\cos\varphi+\phi\cos\theta\sin\varphi\big{)}\frac{e^{-jkr}}{r} \tag{1}\]  
where \(k=2\pi/\lambda\) is the wave number, and \(K_{1}=\mu_{0}M_{0}L_{1}/4\pi\) is the total amplitude of the vector potential function.